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Quantum Theology
        Conditional Density Operators
           Conditional Independence
               Quantum State Pooling
                         Conclusions




   The Church of the Smaller Hilbert Space
(a.k.a. An Approach to Quantum State Pooling
  from Quantum Conditional Independence)

                              M. S. Leifer

                   Institute for Quantum Computing
                         University of Waterloo

                            Perimeter Institute


       March 11th 2008 / APS March Meeting


                         M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
               Conditional Density Operators
                  Conditional Independence
                      Quantum State Pooling
                                Conclusions


Outline


      Quantum Theology
  1


      Conditional Density Operators
  2


      Conditional Independence
  3


      Quantum State Pooling
  4


      Conclusions
  5




                                M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators
                                             The Church of the Larger Hilbert Space
                Conditional Independence
                                             The Church of the Smaller Hilbert Space
                    Quantum State Pooling
                              Conclusions


Quantum Theology


    The Two Churches of Quantum Theory
        The Church of the Larger Hilbert Space
        The Church of the Smaller Hilbert Space

    Each church consists of:
        A moral code, i.e. a set of proof techniques.
        A set of core beliefs, i.e. interpretation of quantum theory.


    Secular theorists are free to draw their moral code from
    both churches.


                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators
                                             The Church of the Larger Hilbert Space
                Conditional Independence
                                             The Church of the Smaller Hilbert Space
                    Quantum State Pooling
                              Conclusions


Quantum Theology


    The Two Churches of Quantum Theory
        The Church of the Larger Hilbert Space
        The Church of the Smaller Hilbert Space

    Each church consists of:
        A moral code, i.e. a set of proof techniques.
        A set of core beliefs, i.e. interpretation of quantum theory.


    Secular theorists are free to draw their moral code from
    both churches.


                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators
                                             The Church of the Larger Hilbert Space
                Conditional Independence
                                             The Church of the Smaller Hilbert Space
                    Quantum State Pooling
                              Conclusions


Quantum Theology


    The Two Churches of Quantum Theory
        The Church of the Larger Hilbert Space
        The Church of the Smaller Hilbert Space

    Each church consists of:
        A moral code, i.e. a set of proof techniques.
        A set of core beliefs, i.e. interpretation of quantum theory.


    Secular theorists are free to draw their moral code from
    both churches.


                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators
                                             The Church of the Larger Hilbert Space
                Conditional Independence
                                             The Church of the Smaller Hilbert Space
                    Quantum State Pooling
                              Conclusions


Quantum Theology


    The Two Churches of Quantum Theory
        The Church of the Larger Hilbert Space
        The Church of the Smaller Hilbert Space

    Each church consists of:
        A moral code, i.e. a set of proof techniques.
        A set of core beliefs, i.e. interpretation of quantum theory.


    Secular theorists are free to draw their moral code from
    both churches.


                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Larger Hilbert Space
Moral Code

        Thou shalt purify mixed states.

                                   ρA = TrE (|ψ ψ|AE )


        Thou shalt Steinspring dilate TPCP maps.
                                                    †
               E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR


        Thou shalt Naimark extend POVMs.
                      (j)                        (j)
              Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R

                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Larger Hilbert Space
Moral Code

        Thou shalt purify mixed states.

                                   ρA = TrE (|ψ ψ|AE )


        Thou shalt Steinspring dilate TPCP maps.
                                                    †
               E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR


        Thou shalt Naimark extend POVMs.
                      (j)                        (j)
              Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R

                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Larger Hilbert Space
Moral Code

        Thou shalt purify mixed states.

                                   ρA = TrE (|ψ ψ|AE )


        Thou shalt Steinspring dilate TPCP maps.
                                                    †
               E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR


        Thou shalt Naimark extend POVMs.
                      (j)                        (j)
              Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R

                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Larger Hilbert Space
Core Beliefs



         The entire universe is described by a massively entangled
         pure state, |Ψ U , defined on an enormous number of
         subsystems.

         Quantum mechanics is a well-defined dynamical theory.
         |Ψ U evolves unitarily according to the Schrödinger
         equation and that’s all there is to it!

         Taken seriously this leads to Everett/many worlds.


                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Larger Hilbert Space
Core Beliefs



         The entire universe is described by a massively entangled
         pure state, |Ψ U , defined on an enormous number of
         subsystems.

         Quantum mechanics is a well-defined dynamical theory.
         |Ψ U evolves unitarily according to the Schrödinger
         equation and that’s all there is to it!

         Taken seriously this leads to Everett/many worlds.


                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Larger Hilbert Space
Core Beliefs



         The entire universe is described by a massively entangled
         pure state, |Ψ U , defined on an enormous number of
         subsystems.

         Quantum mechanics is a well-defined dynamical theory.
         |Ψ U evolves unitarily according to the Schrödinger
         equation and that’s all there is to it!

         Taken seriously this leads to Everett/many worlds.


                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Smaller Hilbert Space
Moral Code




        Thou shalt not adorn your church with unnecessary
        ornaments.
             Thou shalt not purify mixed states.
             Thou shalt not Steinspring dilate TPCP maps.
             Thou shalt not Naimark extend POVMs.


        This talk is about what thou shouldst do instead.




                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                 Conditional Density Operators
                                                 The Church of the Larger Hilbert Space
                    Conditional Independence
                                                 The Church of the Smaller Hilbert Space
                        Quantum State Pooling
                                  Conclusions


The Church of The Smaller Hilbert Space
Moral Code




        Thou shalt not adorn your church with unnecessary
        ornaments.
             Thou shalt not purify mixed states.
             Thou shalt not Steinspring dilate TPCP maps.
             Thou shalt not Naimark extend POVMs.


        This talk is about what thou shouldst do instead.




                                  M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                    Conditional Density Operators
                                                    The Church of the Larger Hilbert Space
                       Conditional Independence
                                                    The Church of the Smaller Hilbert Space
                           Quantum State Pooling
                                     Conclusions


The Church of The Smaller Hilbert Space
Core Beliefs



         Quantum theory is best thought of as a noncommutative
         generalization of classical probability theory.

         Classical probability distributions do not have purifications.
               We will lose sight of useful analogies if we purify.


         Taken seriously this leads to quantum logic, quantum
         Bayesianism, ..., any interpretation in which the structure
         of observables is taken as primary.



                                     M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                    Conditional Density Operators
                                                    The Church of the Larger Hilbert Space
                       Conditional Independence
                                                    The Church of the Smaller Hilbert Space
                           Quantum State Pooling
                                     Conclusions


The Church of The Smaller Hilbert Space
Core Beliefs



         Quantum theory is best thought of as a noncommutative
         generalization of classical probability theory.

         Classical probability distributions do not have purifications.
               We will lose sight of useful analogies if we purify.


         Taken seriously this leads to quantum logic, quantum
         Bayesianism, ..., any interpretation in which the structure
         of observables is taken as primary.



                                     M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                    Conditional Density Operators
                                                    The Church of the Larger Hilbert Space
                       Conditional Independence
                                                    The Church of the Smaller Hilbert Space
                           Quantum State Pooling
                                     Conclusions


The Church of The Smaller Hilbert Space
Core Beliefs



         Quantum theory is best thought of as a noncommutative
         generalization of classical probability theory.

         Classical probability distributions do not have purifications.
               We will lose sight of useful analogies if we purify.


         Taken seriously this leads to quantum logic, quantum
         Bayesianism, ..., any interpretation in which the structure
         of observables is taken as primary.



                                     M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators     Quantum Analog of Conditional Probability
                 Conditional Independence       Dynamical Conditional Density Operators
                     Quantum State Pooling      Hybrid Quantum-Classical Systems
                               Conclusions


Quantum Analog of Conditional Probability?

      Classical Probability                   Quantum Theory
      Sample Space:                           Hilbert Space:
        ΩX = {1, 2, . . . , n}                            HA
      Probability distribution:               Density operator:
               P(X )                                       ρA
      Cartesian product:                      Tensor product:
             ΩX × ΩY                                   HA ⊗ H B
      Joint probability:                      Bipartite density operator:
              P(X , Y )                                   ρAB
      Conditional probability:
         P(Y |X ) = P(X ,Y )                                      ?
                      P(Y )


                               M. S. Leifer     The Church of the Smaller Hilbert Space
Quantum Theology
                  Conditional Density Operators       Quantum Analog of Conditional Probability
                     Conditional Independence         Dynamical Conditional Density Operators
                         Quantum State Pooling        Hybrid Quantum-Classical Systems
                                   Conclusions


Conditional Density Operators

  Definition
  A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
  positive operator that satisfies TrB ρB|A = IA , where IA is the
  identity operator on HA .

                 P(Y |X ) = 1
      c.f.   Y
      Note: A density operator determines a CDO via
                 −1          −1
      ρB|A = ρA 2 ρAB ρA 2 .
                                       1          1
      Notation: M ∗ N = N 2 MN 2
      ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
                    A
      c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).

                                   M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
                  Conditional Density Operators       Quantum Analog of Conditional Probability
                     Conditional Independence         Dynamical Conditional Density Operators
                         Quantum State Pooling        Hybrid Quantum-Classical Systems
                                   Conclusions


Conditional Density Operators

  Definition
  A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
  positive operator that satisfies TrB ρB|A = IA , where IA is the
  identity operator on HA .

                 P(Y |X ) = 1
      c.f.   Y
      Note: A density operator determines a CDO via
                 −1          −1
      ρB|A = ρA 2 ρAB ρA 2 .
                                       1          1
      Notation: M ∗ N = N 2 MN 2
      ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
                    A
      c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).

                                   M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
                  Conditional Density Operators       Quantum Analog of Conditional Probability
                     Conditional Independence         Dynamical Conditional Density Operators
                         Quantum State Pooling        Hybrid Quantum-Classical Systems
                                   Conclusions


Conditional Density Operators

  Definition
  A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
  positive operator that satisfies TrB ρB|A = IA , where IA is the
  identity operator on HA .

                 P(Y |X ) = 1
      c.f.   Y
      Note: A density operator determines a CDO via
                 −1          −1
      ρB|A = ρA 2 ρAB ρA 2 .
                                       1          1
      Notation: M ∗ N = N 2 MN 2
      ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
                    A
      c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).

                                   M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
                  Conditional Density Operators       Quantum Analog of Conditional Probability
                     Conditional Independence         Dynamical Conditional Density Operators
                         Quantum State Pooling        Hybrid Quantum-Classical Systems
                                   Conclusions


Conditional Density Operators

  Definition
  A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
  positive operator that satisfies TrB ρB|A = IA , where IA is the
  identity operator on HA .

                 P(Y |X ) = 1
      c.f.   Y
      Note: A density operator determines a CDO via
                 −1          −1
      ρB|A = ρA 2 ρAB ρA 2 .
                                       1          1
      Notation: M ∗ N = N 2 MN 2
      ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
                    A
      c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).

                                   M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
                  Conditional Density Operators       Quantum Analog of Conditional Probability
                     Conditional Independence         Dynamical Conditional Density Operators
                         Quantum State Pooling        Hybrid Quantum-Classical Systems
                                   Conclusions


Conditional Density Operators

  Definition
  A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
  positive operator that satisfies TrB ρB|A = IA , where IA is the
  identity operator on HA .

                 P(Y |X ) = 1
      c.f.   Y
      Note: A density operator determines a CDO via
                 −1          −1
      ρB|A = ρA 2 ρAB ρA 2 .
                                       1          1
      Notation: M ∗ N = N 2 MN 2
      ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
                    A
      c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).

                                   M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
                  Conditional Density Operators       Quantum Analog of Conditional Probability
                     Conditional Independence         Dynamical Conditional Density Operators
                         Quantum State Pooling        Hybrid Quantum-Classical Systems
                                   Conclusions


Conditional Density Operators

  Definition
  A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
  positive operator that satisfies TrB ρB|A = IA , where IA is the
  identity operator on HA .

                 P(Y |X ) = 1
      c.f.   Y
      Note: A density operator determines a CDO via
                 −1          −1
      ρB|A = ρA 2 ρAB ρA 2 .
                                       1          1
      Notation: M ∗ N = N 2 MN 2
      ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
                    A
      c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).

                                   M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
            Conditional Density Operators       Quantum Analog of Conditional Probability
               Conditional Independence         Dynamical Conditional Density Operators
                   Quantum State Pooling        Hybrid Quantum-Classical Systems
                             Conclusions


Example


    Let ρAB = |Ψ Ψ|AB be a pure state with Schmidt
    decomposition

                     |Ψ                                       ⊗ ψj
                                                p j φj
                                =                                           .
                          AB                              A             B
                                      j

    Then, ρB|A = |Ψ Ψ|B|A , where

                        |Ψ                                ⊗ ψj
                                    =            φj                     .
                              B|A                     A             B
                                            j




                             M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators      Quantum Analog of Conditional Probability
                 Conditional Independence        Dynamical Conditional Density Operators
                     Quantum State Pooling       Hybrid Quantum-Classical Systems
                               Conclusions


Ta Da!

     Classical Probability                    Quantum Theory
     Sample Space:                            Hilbert Space:
       ΩX = {1, 2, . . . , n}                              HA
     Probability distribution:                Density operator:
              P(X )                                         ρA
     Cartesian product:                       Tensor product:
            ΩX × ΩY                                      HA ⊗ H B
     Joint probability:                       Bipartite density operator:
             P(X , Y )                                     ρAB
     Conditional probability:                 Conditional density operator:
        P(Y |X ) = P(X ,Y )                          ρB|A = ρAB ∗ ρ−1
                                                                   A
                     P(Y )


                               M. S. Leifer      The Church of the Smaller Hilbert Space
Quantum Theology
                Conditional Density Operators   Quantum Analog of Conditional Probability
                   Conditional Independence     Dynamical Conditional Density Operators
                       Quantum State Pooling    Hybrid Quantum-Classical Systems
                                 Conclusions


A problem with the analogy

     ρAB usually represents the state of two subsystems at a
     given time.

     P(X , Y ) is more flexible.
         X and Y might refer to different subsystems.
         Y might represent the value of the same quantity as X , but
         at a later time.
         Y might represent the result of a measurement of the value
         of X .
         ....


                                 M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                Conditional Density Operators   Quantum Analog of Conditional Probability
                   Conditional Independence     Dynamical Conditional Density Operators
                       Quantum State Pooling    Hybrid Quantum-Classical Systems
                                 Conclusions


A problem with the analogy

     ρAB usually represents the state of two subsystems at a
     given time.

     P(X , Y ) is more flexible.
         X and Y might refer to different subsystems.
         Y might represent the value of the same quantity as X , but
         at a later time.
         Y might represent the result of a measurement of the value
         of X .
         ....


                                 M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                Conditional Density Operators   Quantum Analog of Conditional Probability
                   Conditional Independence     Dynamical Conditional Density Operators
                       Quantum State Pooling    Hybrid Quantum-Classical Systems
                                 Conclusions


A problem with the analogy

     ρAB usually represents the state of two subsystems at a
     given time.

     P(X , Y ) is more flexible.
         X and Y might refer to different subsystems.
         Y might represent the value of the same quantity as X , but
         at a later time.
         Y might represent the result of a measurement of the value
         of X .
         ....


                                 M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                Conditional Density Operators   Quantum Analog of Conditional Probability
                   Conditional Independence     Dynamical Conditional Density Operators
                       Quantum State Pooling    Hybrid Quantum-Classical Systems
                                 Conclusions


A problem with the analogy

     ρAB usually represents the state of two subsystems at a
     given time.

     P(X , Y ) is more flexible.
         X and Y might refer to different subsystems.
         Y might represent the value of the same quantity as X , but
         at a later time.
         Y might represent the result of a measurement of the value
         of X .
         ....


                                 M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                  Conditional Density Operators   Quantum Analog of Conditional Probability
                     Conditional Independence     Dynamical Conditional Density Operators
                         Quantum State Pooling    Hybrid Quantum-Classical Systems
                                   Conclusions


Subsystems

  time

          Two classical subsystems                 Two quantum subsystems

          X                              Y           A                                      B



                                                           ρAB = ρB|A ∗ ρA
         P (X, Y ) = P (Y |X)P (X)




                                   M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                    Conditional Density Operators   Quantum Analog of Conditional Probability
                       Conditional Independence     Dynamical Conditional Density Operators
                           Quantum State Pooling    Hybrid Quantum-Classical Systems
                                     Conclusions


Dynamical CDOs

  time       Y                                                 B

                                                                   Trace-preserving
                  Classical
                  stochastic                                       completely-positive
                  dynamics                                         dynamics

             X                                                 A

                                                                           = EB|A (ρA )
         P (Y )       = ΓY |X (P (X))                        ρB
                  =           P (Y |X)P (X)                           = TrA ρB|A ∗ ρA
                          X
                      =           P (X, Y )                                = TrA (ρAB )
                              X




                                     M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
                    Conditional Density Operators   Quantum Analog of Conditional Probability
                       Conditional Independence     Dynamical Conditional Density Operators
                           Quantum State Pooling    Hybrid Quantum-Classical Systems
                                     Conclusions


Dynamical CDOs

  time       Y                                                 B

                                                                   Trace-preserving
                  Classical
                  stochastic                                       completely-positive
                  dynamics                                         dynamics

             X                                                 A

         P (Y )       = ΓY |X (P (X))
                                                                           = EB|A (ρA )
                                                             ρB
                  =           P (Y |X)P (X)
                          X
                                                                      = TrA ρTA ∗ ρA
                      =           P (X, Y )                                  B|A
                              X




                                     M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators      Quantum Analog of Conditional Probability
                 Conditional Independence        Dynamical Conditional Density Operators
                     Quantum State Pooling       Hybrid Quantum-Classical Systems
                               Conclusions


Hybrid Quantum-Classical Systems
                       X                                          A

                                                                       (j)
                                    P(X = j) |j           j|X ⊗ ρA
                   ρXA =
                                j

                                     (j)
                                                  j|X ρX |j
                   P(X = j)ρA                                         = P(X = j)
       ρA =                                                      X
               j

                                                                 (j)
                        j|X ρXA |j             = P(X = j)ρA
                                           X


                                                          (j)
                               j|X ρA|X |j           = ρA
                                                 X




                               M. S. Leifer      The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators      Quantum Analog of Conditional Probability
                 Conditional Independence        Dynamical Conditional Density Operators
                     Quantum State Pooling       Hybrid Quantum-Classical Systems
                               Conclusions


Hybrid Quantum-Classical Systems
                       X                                          A

                                                                       (j)
                                    P(X = j) |j           j|X ⊗ ρA
                   ρXA =
                                j

                                     (j)
                                                  j|X ρX |j
                   P(X = j)ρA                                         = P(X = j)
       ρA =                                                      X
               j

                                                                 (j)
                        j|X ρXA |j             = P(X = j)ρA
                                           X


                                                          (j)
                               j|X ρA|X |j           = ρA
                                                 X




                               M. S. Leifer      The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators      Quantum Analog of Conditional Probability
                 Conditional Independence        Dynamical Conditional Density Operators
                     Quantum State Pooling       Hybrid Quantum-Classical Systems
                               Conclusions


Hybrid Quantum-Classical Systems
                       X                                          A

                                                                       (j)
                                    P(X = j) |j           j|X ⊗ ρA
                   ρXA =
                                j

                                     (j)
                                                  j|X ρX |j
                   P(X = j)ρA                                         = P(X = j)
       ρA =                                                      X
               j

                                                                 (j)
                        j|X ρXA |j             = P(X = j)ρA
                                           X


                                                          (j)
                               j|X ρA|X |j           = ρA
                                                 X




                               M. S. Leifer      The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators      Quantum Analog of Conditional Probability
                 Conditional Independence        Dynamical Conditional Density Operators
                     Quantum State Pooling       Hybrid Quantum-Classical Systems
                               Conclusions


Hybrid Quantum-Classical Systems
                       X                                          A

                                                                       (j)
                                    P(X = j) |j           j|X ⊗ ρA
                   ρXA =
                                j

                                     (j)
                                                  j|X ρX |j
                   P(X = j)ρA                                         = P(X = j)
       ρA =                                                      X
               j

                                                                 (j)
                        j|X ρXA |j             = P(X = j)ρA
                                           X


                                                          (j)
                               j|X ρA|X |j           = ρA
                                                 X




                               M. S. Leifer      The Church of the Smaller Hilbert Space
Quantum Theology
               Conditional Density Operators      Quantum Analog of Conditional Probability
                  Conditional Independence        Dynamical Conditional Density Operators
                      Quantum State Pooling       Hybrid Quantum-Classical Systems
                                Conclusions


Hybrid Quantum-Classical Systems


                         X                                         A

      (j)
     EA = j|X ρX |A |j             is a POVM on HA
                               X


                             (j)
     Conversely, if EA is a POVM on HA then
                                      (j)
                    |j   j|X ⊗ EA is a valid CDO.
     ρX |A =    j




                                   M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
               Conditional Density Operators      Quantum Analog of Conditional Probability
                  Conditional Independence        Dynamical Conditional Density Operators
                      Quantum State Pooling       Hybrid Quantum-Classical Systems
                                Conclusions


Hybrid Quantum-Classical Systems


                         X                                         A

      (j)
     EA = j|X ρX |A |j             is a POVM on HA
                               X


                             (j)
     Conversely, if EA is a POVM on HA then
                                      (j)
                    |j   j|X ⊗ EA is a valid CDO.
     ρX |A =    j




                                   M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators   Quantum Analog of Conditional Probability
                Conditional Independence     Dynamical Conditional Density Operators
                    Quantum State Pooling    Hybrid Quantum-Classical Systems
                              Conclusions


Preparations and Measurements

         A                                                     X


                                                                   Measurement
             Preparation


         X                                                     A


       ρA = TrX ρA|X ∗ ρX                                 ρX = TrA ρX|A ∗ ρA




                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators       Classical Conditional Independence
                 Conditional Independence         Quantum Conditional Independence
                     Quantum State Pooling        Hybrid Conditional Independence
                               Conclusions


Classical Conditional Independence

             X                                Z                            Y


     H(X : Y |Z ) = H(X , Z ) + H(Y , Z ) − H(X , Y , Z ) − H(Z ) = 0

     P(X |Y , Z ) = P(X |Z )

     P(Y |X , Z ) = P(Y |Z )

     P(X , Y |Z ) = P(X |Z )P(Y |Z )



                               M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators       Classical Conditional Independence
                Conditional Independence         Quantum Conditional Independence
                    Quantum State Pooling        Hybrid Conditional Independence
                              Conclusions


Quantum Conditional Independence

            A                                C                            B


     S(A : B|C) = S(A, C) + S(B, C) − S(A, B, C) − S(C) = 0

     ρA|BC = ρA|C

     ρB|AC = ρB|C

     ⇒ ρAB|C = ρA|C ρB|C



                              M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
               Conditional Density Operators       Classical Conditional Independence
                  Conditional Independence         Quantum Conditional Independence
                      Quantum State Pooling        Hybrid Conditional Independence
                                Conclusions


Hybrid Conditional Independence

              X                                C                            Y


     S(X : Y |C) = S(X , C) + S(Y , C) − S(X , Y , C) − S(C) = 0

     ρX |YC = ρX |C

     ρY |XC = ρY |C

     ρXY |C = ρX |C ρY |C



                                M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                              Classical Pooling
                 Conditional Independence
                                              Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


The Pooling Problem


     Classical: Alice describes a system by P(Z ), Bob by Q(Z ).
     If they get together, what distribution should they agree
     upon?

     Quantum: Alice describes a system by ρC , Bob by σC . If
     they get together, what distribution should they agree
     upon?

     Introduce an arbiter, Penelope the pooler, who’s task it is to
     make the decision.


                               M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                              Classical Pooling
                 Conditional Independence
                                              Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


The Pooling Problem


     Classical: Alice describes a system by P(Z ), Bob by Q(Z ).
     If they get together, what distribution should they agree
     upon?

     Quantum: Alice describes a system by ρC , Bob by σC . If
     they get together, what distribution should they agree
     upon?

     Introduce an arbiter, Penelope the pooler, who’s task it is to
     make the decision.


                               M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                              Classical Pooling
                 Conditional Independence
                                              Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


The Pooling Problem


     Classical: Alice describes a system by P(Z ), Bob by Q(Z ).
     If they get together, what distribution should they agree
     upon?

     Quantum: Alice describes a system by ρC , Bob by σC . If
     they get together, what distribution should they agree
     upon?

     Introduce an arbiter, Penelope the pooler, who’s task it is to
     make the decision.


                               M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
            Conditional Density Operators
                                            Classical Pooling
               Conditional Independence
                                            Quantum Pooling via Indirect Measurements
                   Quantum State Pooling
                             Conclusions


Diplomatic Pooling




                 Alice                                      Bob




                                      Penelope

                             M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
            Conditional Density Operators
                                            Classical Pooling
               Conditional Independence
                                            Quantum Pooling via Indirect Measurements
                   Quantum State Pooling
                             Conclusions


Scientific Pooling




                 Alice                                      Bob




                                      Penelope

                             M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators
                                                 Classical Pooling
                Conditional Independence
                                                 Quantum Pooling via Indirect Measurements
                    Quantum State Pooling
                              Conclusions


Classical Pooling


                                                           Y
                            X

     P (Z)                                                                                 P (Z)
              Alice                                             Bob
   P (X|Z)                                                                            P (Y |Z)
                                             Z




                                                                        P (Z)
                                       Penelope
                                                                   P (Z|X)
                                                                    P (Z|Y )
                              M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                                  Classical Pooling
                 Conditional Independence
                                                  Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


Simon, the supra-Bayesian

     Simon, the fictitious know-it-all is going to update via
                                    ,Y |Z )P(Z
     Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) .

     Does Penelope have enough information to do what Simon
     says?

     Not generally, but if X and Y are conditionally independent:
                                          P(X |Z )P(Y |Z )P(Z )
                 P(Z |X , Y )                 =   P(X ,Y )
                                        P(X )P(Y ) P(Z |X )P(Z |Y )
                                       = P(X ,Y )        P(Z )
                                                P(Z |X )P(Z |Y )
                                        = NXY        P(Z )

                               M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                                  Classical Pooling
                 Conditional Independence
                                                  Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


Simon, the supra-Bayesian

     Simon, the fictitious know-it-all is going to update via
                                    ,Y |Z )P(Z
     Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) .

     Does Penelope have enough information to do what Simon
     says?

     Not generally, but if X and Y are conditionally independent:
                                          P(X |Z )P(Y |Z )P(Z )
                 P(Z |X , Y )                 =   P(X ,Y )
                                        P(X )P(Y ) P(Z |X )P(Z |Y )
                                       = P(X ,Y )        P(Z )
                                                P(Z |X )P(Z |Y )
                                        = NXY        P(Z )

                               M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                                  Classical Pooling
                 Conditional Independence
                                                  Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


Simon, the supra-Bayesian

     Simon, the fictitious know-it-all is going to update via
                                    ,Y |Z )P(Z
     Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) .

     Does Penelope have enough information to do what Simon
     says?

     Not generally, but if X and Y are conditionally independent:
                                          P(X |Z )P(Y |Z )P(Z )
                 P(Z |X , Y )                 =   P(X ,Y )
                                        P(X )P(Y ) P(Z |X )P(Z |Y )
                                       = P(X ,Y )        P(Z )
                                                P(Z |X )P(Z |Y )
                                        = NXY        P(Z )

                               M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
            Conditional Density Operators
                                                Classical Pooling
               Conditional Independence
                                                Quantum Pooling via Indirect Measurements
                   Quantum State Pooling
                             Conclusions


Quantum Pooling via indirect measurements


              X                                           B            Y
                           A
      ρC                                                                                  ρC
     ρA|C                                                                            ρB|C
             Alice                                             Bob
     ρX|A                                                                            ρY |B
                                            C




                                                                      ρC
                                      Penelope
                                                                  ρC|X
                                                                   ρC|Y

                             M. S. Leifer       The Church of the Smaller Hilbert Space
Quantum Theology
               Conditional Density Operators
                                               Classical Pooling
                  Conditional Independence
                                               Quantum Pooling via Indirect Measurements
                      Quantum State Pooling
                                Conclusions


Quantum supra-Bayesian Pooling


     If ρXY |C = ρX |C ρY |C then

                                      = ρXY |C ∗ ρC ρ−1
                  ρC|XY                              XY

                                    = ρ−1 ρX |C ρY |C ∗ ρC
                                       XY

                                = ρ−1 ρX ρY ρC|X ρ−1 ρC|Y
                                   XY             C

                                    = NXY ρC|X ρ−1 ρC|Y
                                                C




                                M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                              Classical Pooling
                 Conditional Independence
                                              Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


Quantum supra-Bayesian Pooling


     For which ρABC is pooling always possible regardless of
     ρX |A , ρY |B ?

         It is sufficient if ρAB|C = ρA|C ρB|C

                                               ρX |A ρY |B ∗ ρAB|C
                                    = TrAB
                 ρXY |C
                              = TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C
                                              = ρX |C ρY |C .




                               M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators
                                              Classical Pooling
                 Conditional Independence
                                              Quantum Pooling via Indirect Measurements
                     Quantum State Pooling
                               Conclusions


Quantum supra-Bayesian Pooling


     For which ρABC is pooling always possible regardless of
     ρX |A , ρY |B ?

         It is sufficient if ρAB|C = ρA|C ρB|C

                                               ρX |A ρY |B ∗ ρAB|C
                                    = TrAB
                 ρXY |C
                              = TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C
                                              = ρX |C ρY |C .




                               M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators   Moral
                Conditional Independence     Acknowledgments
                    Quantum State Pooling    References
                              Conclusions


The Moral of the Story

     There is a bunch of other stuff that makes more sense in
     the Church of the Smaller Hilbert Space
         The “pretty good” measurement
         “Pretty good” error correction
         Results on steering entangled states
         Entanglement in time
         Quantum sufficient statistics
         Causality


     ...but the Church of the Larger Hilbert Space has some
     pretty nifty proofs too.

     So which one is right?

                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators   Moral
                Conditional Independence     Acknowledgments
                    Quantum State Pooling    References
                              Conclusions


The Moral of the Story

     There is a bunch of other stuff that makes more sense in
     the Church of the Smaller Hilbert Space
         The “pretty good” measurement
         “Pretty good” error correction
         Results on steering entangled states
         Entanglement in time
         Quantum sufficient statistics
         Causality


     ...but the Church of the Larger Hilbert Space has some
     pretty nifty proofs too.

     So which one is right?

                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators   Moral
                 Conditional Independence     Acknowledgments
                     Quantum State Pooling    References
                               Conclusions


Blind Men and the Elephant by J. G. Saxe



     It was six men of Indostan
     To learning much inclined,
     Who went to see the Elephant
     (Though all of them were blind),
     That each by observation
     Might satisfy his mind




                               M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators   Moral
                Conditional Independence     Acknowledgments
                    Quantum State Pooling    References
                              Conclusions


Blind Men and the Elephant by J. G. Saxe
     The First approached the Elephant,
     And happening to fall
     Against his broad and sturdy side,
     At once began to bawl:
     quot;God bless me! but the Elephant
     Is very like a wall!quot;

     The Second, feeling of the tusk,
     Cried, quot;Ho! what have we here
     So very round and smooth and sharp?
     To me ’tis mighty clear
     This wonder of an Elephant
     Is very like a spear!quot;

                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
             Conditional Density Operators   Moral
                Conditional Independence     Acknowledgments
                    Quantum State Pooling    References
                              Conclusions


Blind Men and the Elephant by J. G. Saxe
     And so these men of Indostan
     Disputed loud and long,
     Each in his own opinion
     Exceeding stiff and strong,
     Though each was partly in the right,
     And all were in the wrong!

     Moral:
     So oft in theologic wars,
     The disputants, I ween,
     Rail on in utter ignorance
     Of what each other mean,
     And prate about an Elephant
     Not one of them has seen!
                              M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
            Conditional Density Operators   Moral
               Conditional Independence     Acknowledgments
                   Quantum State Pooling    References
                             Conclusions


Acknowledgments




    This work is supported by:
        The Foundational Questions Institute (http://www.fqxi.org)
        MITACS (http://www.mitacs.math.ca)
        NSERC (http://nserc.ca/)
        The Province of Ontario: ORDCF/MRI




                             M. S. Leifer   The Church of the Smaller Hilbert Space
Quantum Theology
              Conditional Density Operators   Moral
                 Conditional Independence     Acknowledgments
                     Quantum State Pooling    References
                               Conclusions


References

     Conditional Density Operators:
         M. S. Leifer, Phys. Rev. A 74, 042310 (2006).
         arXiv:quant-ph/0606022.
         M. S. Leifer (2006) arXiv:quant-ph/0611233.
     Conditional Independence:
         M. S. Leifer and D. Poulin, Ann. Phys., in press.
         arXiv:0708.1337
     Quantum State Pooling:
         M. S. Leifer and R. W. Spekkens, in preparation.
         R. W. Spekkens and H. M. Wiseman, Phys. Rev. A 75,
         042104 (2007). arXiv:quant-ph/0612190.
     Quantum Theology:
         The book with this title is unrelated to this talk.

                               M. S. Leifer   The Church of the Smaller Hilbert Space

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The Church of the Smaller Hilbert Space

  • 1. Quantum Theology Conditional Density Operators Conditional Independence Quantum State Pooling Conclusions The Church of the Smaller Hilbert Space (a.k.a. An Approach to Quantum State Pooling from Quantum Conditional Independence) M. S. Leifer Institute for Quantum Computing University of Waterloo Perimeter Institute March 11th 2008 / APS March Meeting M. S. Leifer The Church of the Smaller Hilbert Space
  • 2. Quantum Theology Conditional Density Operators Conditional Independence Quantum State Pooling Conclusions Outline Quantum Theology 1 Conditional Density Operators 2 Conditional Independence 3 Quantum State Pooling 4 Conclusions 5 M. S. Leifer The Church of the Smaller Hilbert Space
  • 3. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
  • 4. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
  • 5. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
  • 6. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
  • 7. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Moral Code Thou shalt purify mixed states. ρA = TrE (|ψ ψ|AE ) Thou shalt Steinspring dilate TPCP maps. † E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR Thou shalt Naimark extend POVMs. (j) (j) Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R M. S. Leifer The Church of the Smaller Hilbert Space
  • 8. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Moral Code Thou shalt purify mixed states. ρA = TrE (|ψ ψ|AE ) Thou shalt Steinspring dilate TPCP maps. † E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR Thou shalt Naimark extend POVMs. (j) (j) Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R M. S. Leifer The Church of the Smaller Hilbert Space
  • 9. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Moral Code Thou shalt purify mixed states. ρA = TrE (|ψ ψ|AE ) Thou shalt Steinspring dilate TPCP maps. † E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR Thou shalt Naimark extend POVMs. (j) (j) Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R M. S. Leifer The Church of the Smaller Hilbert Space
  • 10. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Core Beliefs The entire universe is described by a massively entangled pure state, |Ψ U , defined on an enormous number of subsystems. Quantum mechanics is a well-defined dynamical theory. |Ψ U evolves unitarily according to the Schrödinger equation and that’s all there is to it! Taken seriously this leads to Everett/many worlds. M. S. Leifer The Church of the Smaller Hilbert Space
  • 11. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Core Beliefs The entire universe is described by a massively entangled pure state, |Ψ U , defined on an enormous number of subsystems. Quantum mechanics is a well-defined dynamical theory. |Ψ U evolves unitarily according to the Schrödinger equation and that’s all there is to it! Taken seriously this leads to Everett/many worlds. M. S. Leifer The Church of the Smaller Hilbert Space
  • 12. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Core Beliefs The entire universe is described by a massively entangled pure state, |Ψ U , defined on an enormous number of subsystems. Quantum mechanics is a well-defined dynamical theory. |Ψ U evolves unitarily according to the Schrödinger equation and that’s all there is to it! Taken seriously this leads to Everett/many worlds. M. S. Leifer The Church of the Smaller Hilbert Space
  • 13. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Moral Code Thou shalt not adorn your church with unnecessary ornaments. Thou shalt not purify mixed states. Thou shalt not Steinspring dilate TPCP maps. Thou shalt not Naimark extend POVMs. This talk is about what thou shouldst do instead. M. S. Leifer The Church of the Smaller Hilbert Space
  • 14. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Moral Code Thou shalt not adorn your church with unnecessary ornaments. Thou shalt not purify mixed states. Thou shalt not Steinspring dilate TPCP maps. Thou shalt not Naimark extend POVMs. This talk is about what thou shouldst do instead. M. S. Leifer The Church of the Smaller Hilbert Space
  • 15. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Core Beliefs Quantum theory is best thought of as a noncommutative generalization of classical probability theory. Classical probability distributions do not have purifications. We will lose sight of useful analogies if we purify. Taken seriously this leads to quantum logic, quantum Bayesianism, ..., any interpretation in which the structure of observables is taken as primary. M. S. Leifer The Church of the Smaller Hilbert Space
  • 16. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Core Beliefs Quantum theory is best thought of as a noncommutative generalization of classical probability theory. Classical probability distributions do not have purifications. We will lose sight of useful analogies if we purify. Taken seriously this leads to quantum logic, quantum Bayesianism, ..., any interpretation in which the structure of observables is taken as primary. M. S. Leifer The Church of the Smaller Hilbert Space
  • 17. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Core Beliefs Quantum theory is best thought of as a noncommutative generalization of classical probability theory. Classical probability distributions do not have purifications. We will lose sight of useful analogies if we purify. Taken seriously this leads to quantum logic, quantum Bayesianism, ..., any interpretation in which the structure of observables is taken as primary. M. S. Leifer The Church of the Smaller Hilbert Space
  • 18. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Quantum Analog of Conditional Probability? Classical Probability Quantum Theory Sample Space: Hilbert Space: ΩX = {1, 2, . . . , n} HA Probability distribution: Density operator: P(X ) ρA Cartesian product: Tensor product: ΩX × ΩY HA ⊗ H B Joint probability: Bipartite density operator: P(X , Y ) ρAB Conditional probability: P(Y |X ) = P(X ,Y ) ? P(Y ) M. S. Leifer The Church of the Smaller Hilbert Space
  • 19. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Conditional Density Operators Definition A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a positive operator that satisfies TrB ρB|A = IA , where IA is the identity operator on HA . P(Y |X ) = 1 c.f. Y Note: A density operator determines a CDO via −1 −1 ρB|A = ρA 2 ρAB ρA 2 . 1 1 Notation: M ∗ N = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
  • 20. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Conditional Density Operators Definition A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a positive operator that satisfies TrB ρB|A = IA , where IA is the identity operator on HA . P(Y |X ) = 1 c.f. Y Note: A density operator determines a CDO via −1 −1 ρB|A = ρA 2 ρAB ρA 2 . 1 1 Notation: M ∗ N = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
  • 21. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Conditional Density Operators Definition A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a positive operator that satisfies TrB ρB|A = IA , where IA is the identity operator on HA . P(Y |X ) = 1 c.f. Y Note: A density operator determines a CDO via −1 −1 ρB|A = ρA 2 ρAB ρA 2 . 1 1 Notation: M ∗ N = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
  • 22. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Conditional Density Operators Definition A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a positive operator that satisfies TrB ρB|A = IA , where IA is the identity operator on HA . P(Y |X ) = 1 c.f. Y Note: A density operator determines a CDO via −1 −1 ρB|A = ρA 2 ρAB ρA 2 . 1 1 Notation: M ∗ N = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
  • 23. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Conditional Density Operators Definition A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a positive operator that satisfies TrB ρB|A = IA , where IA is the identity operator on HA . P(Y |X ) = 1 c.f. Y Note: A density operator determines a CDO via −1 −1 ρB|A = ρA 2 ρAB ρA 2 . 1 1 Notation: M ∗ N = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
  • 24. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Conditional Density Operators Definition A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a positive operator that satisfies TrB ρB|A = IA , where IA is the identity operator on HA . P(Y |X ) = 1 c.f. Y Note: A density operator determines a CDO via −1 −1 ρB|A = ρA 2 ρAB ρA 2 . 1 1 Notation: M ∗ N = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
  • 25. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Example Let ρAB = |Ψ Ψ|AB be a pure state with Schmidt decomposition |Ψ ⊗ ψj p j φj = . AB A B j Then, ρB|A = |Ψ Ψ|B|A , where |Ψ ⊗ ψj = φj . B|A A B j M. S. Leifer The Church of the Smaller Hilbert Space
  • 26. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Ta Da! Classical Probability Quantum Theory Sample Space: Hilbert Space: ΩX = {1, 2, . . . , n} HA Probability distribution: Density operator: P(X ) ρA Cartesian product: Tensor product: ΩX × ΩY HA ⊗ H B Joint probability: Bipartite density operator: P(X , Y ) ρAB Conditional probability: Conditional density operator: P(Y |X ) = P(X ,Y ) ρB|A = ρAB ∗ ρ−1 A P(Y ) M. S. Leifer The Church of the Smaller Hilbert Space
  • 27. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more flexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
  • 28. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more flexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
  • 29. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more flexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
  • 30. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more flexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
  • 31. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Subsystems time Two classical subsystems Two quantum subsystems X Y A B ρAB = ρB|A ∗ ρA P (X, Y ) = P (Y |X)P (X) M. S. Leifer The Church of the Smaller Hilbert Space
  • 32. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Dynamical CDOs time Y B Trace-preserving Classical stochastic completely-positive dynamics dynamics X A = EB|A (ρA ) P (Y ) = ΓY |X (P (X)) ρB = P (Y |X)P (X) = TrA ρB|A ∗ ρA X = P (X, Y ) = TrA (ρAB ) X M. S. Leifer The Church of the Smaller Hilbert Space
  • 33. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Dynamical CDOs time Y B Trace-preserving Classical stochastic completely-positive dynamics dynamics X A P (Y ) = ΓY |X (P (X)) = EB|A (ρA ) ρB = P (Y |X)P (X) X = TrA ρTA ∗ ρA = P (X, Y ) B|A X M. S. Leifer The Church of the Smaller Hilbert Space
  • 34. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
  • 35. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
  • 36. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
  • 37. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
  • 38. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) EA = j|X ρX |A |j is a POVM on HA X (j) Conversely, if EA is a POVM on HA then (j) |j j|X ⊗ EA is a valid CDO. ρX |A = j M. S. Leifer The Church of the Smaller Hilbert Space
  • 39. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) EA = j|X ρX |A |j is a POVM on HA X (j) Conversely, if EA is a POVM on HA then (j) |j j|X ⊗ EA is a valid CDO. ρX |A = j M. S. Leifer The Church of the Smaller Hilbert Space
  • 40. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Preparations and Measurements A X Measurement Preparation X A ρA = TrX ρA|X ∗ ρX ρX = TrA ρX|A ∗ ρA M. S. Leifer The Church of the Smaller Hilbert Space
  • 41. Quantum Theology Conditional Density Operators Classical Conditional Independence Conditional Independence Quantum Conditional Independence Quantum State Pooling Hybrid Conditional Independence Conclusions Classical Conditional Independence X Z Y H(X : Y |Z ) = H(X , Z ) + H(Y , Z ) − H(X , Y , Z ) − H(Z ) = 0 P(X |Y , Z ) = P(X |Z ) P(Y |X , Z ) = P(Y |Z ) P(X , Y |Z ) = P(X |Z )P(Y |Z ) M. S. Leifer The Church of the Smaller Hilbert Space
  • 42. Quantum Theology Conditional Density Operators Classical Conditional Independence Conditional Independence Quantum Conditional Independence Quantum State Pooling Hybrid Conditional Independence Conclusions Quantum Conditional Independence A C B S(A : B|C) = S(A, C) + S(B, C) − S(A, B, C) − S(C) = 0 ρA|BC = ρA|C ρB|AC = ρB|C ⇒ ρAB|C = ρA|C ρB|C M. S. Leifer The Church of the Smaller Hilbert Space
  • 43. Quantum Theology Conditional Density Operators Classical Conditional Independence Conditional Independence Quantum Conditional Independence Quantum State Pooling Hybrid Conditional Independence Conclusions Hybrid Conditional Independence X C Y S(X : Y |C) = S(X , C) + S(Y , C) − S(X , Y , C) − S(C) = 0 ρX |YC = ρX |C ρY |XC = ρY |C ρXY |C = ρX |C ρY |C M. S. Leifer The Church of the Smaller Hilbert Space
  • 44. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions The Pooling Problem Classical: Alice describes a system by P(Z ), Bob by Q(Z ). If they get together, what distribution should they agree upon? Quantum: Alice describes a system by ρC , Bob by σC . If they get together, what distribution should they agree upon? Introduce an arbiter, Penelope the pooler, who’s task it is to make the decision. M. S. Leifer The Church of the Smaller Hilbert Space
  • 45. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions The Pooling Problem Classical: Alice describes a system by P(Z ), Bob by Q(Z ). If they get together, what distribution should they agree upon? Quantum: Alice describes a system by ρC , Bob by σC . If they get together, what distribution should they agree upon? Introduce an arbiter, Penelope the pooler, who’s task it is to make the decision. M. S. Leifer The Church of the Smaller Hilbert Space
  • 46. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions The Pooling Problem Classical: Alice describes a system by P(Z ), Bob by Q(Z ). If they get together, what distribution should they agree upon? Quantum: Alice describes a system by ρC , Bob by σC . If they get together, what distribution should they agree upon? Introduce an arbiter, Penelope the pooler, who’s task it is to make the decision. M. S. Leifer The Church of the Smaller Hilbert Space
  • 47. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Diplomatic Pooling Alice Bob Penelope M. S. Leifer The Church of the Smaller Hilbert Space
  • 48. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Scientific Pooling Alice Bob Penelope M. S. Leifer The Church of the Smaller Hilbert Space
  • 49. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Classical Pooling Y X P (Z) P (Z) Alice Bob P (X|Z) P (Y |Z) Z P (Z) Penelope P (Z|X) P (Z|Y ) M. S. Leifer The Church of the Smaller Hilbert Space
  • 50. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Simon, the supra-Bayesian Simon, the fictitious know-it-all is going to update via ,Y |Z )P(Z Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) . Does Penelope have enough information to do what Simon says? Not generally, but if X and Y are conditionally independent: P(X |Z )P(Y |Z )P(Z ) P(Z |X , Y ) = P(X ,Y ) P(X )P(Y ) P(Z |X )P(Z |Y ) = P(X ,Y ) P(Z ) P(Z |X )P(Z |Y ) = NXY P(Z ) M. S. Leifer The Church of the Smaller Hilbert Space
  • 51. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Simon, the supra-Bayesian Simon, the fictitious know-it-all is going to update via ,Y |Z )P(Z Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) . Does Penelope have enough information to do what Simon says? Not generally, but if X and Y are conditionally independent: P(X |Z )P(Y |Z )P(Z ) P(Z |X , Y ) = P(X ,Y ) P(X )P(Y ) P(Z |X )P(Z |Y ) = P(X ,Y ) P(Z ) P(Z |X )P(Z |Y ) = NXY P(Z ) M. S. Leifer The Church of the Smaller Hilbert Space
  • 52. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Simon, the supra-Bayesian Simon, the fictitious know-it-all is going to update via ,Y |Z )P(Z Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) . Does Penelope have enough information to do what Simon says? Not generally, but if X and Y are conditionally independent: P(X |Z )P(Y |Z )P(Z ) P(Z |X , Y ) = P(X ,Y ) P(X )P(Y ) P(Z |X )P(Z |Y ) = P(X ,Y ) P(Z ) P(Z |X )P(Z |Y ) = NXY P(Z ) M. S. Leifer The Church of the Smaller Hilbert Space
  • 53. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum Pooling via indirect measurements X B Y A ρC ρC ρA|C ρB|C Alice Bob ρX|A ρY |B C ρC Penelope ρC|X ρC|Y M. S. Leifer The Church of the Smaller Hilbert Space
  • 54. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum supra-Bayesian Pooling If ρXY |C = ρX |C ρY |C then = ρXY |C ∗ ρC ρ−1 ρC|XY XY = ρ−1 ρX |C ρY |C ∗ ρC XY = ρ−1 ρX ρY ρC|X ρ−1 ρC|Y XY C = NXY ρC|X ρ−1 ρC|Y C M. S. Leifer The Church of the Smaller Hilbert Space
  • 55. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum supra-Bayesian Pooling For which ρABC is pooling always possible regardless of ρX |A , ρY |B ? It is sufficient if ρAB|C = ρA|C ρB|C ρX |A ρY |B ∗ ρAB|C = TrAB ρXY |C = TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C = ρX |C ρY |C . M. S. Leifer The Church of the Smaller Hilbert Space
  • 56. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum supra-Bayesian Pooling For which ρABC is pooling always possible regardless of ρX |A , ρY |B ? It is sufficient if ρAB|C = ρA|C ρB|C ρX |A ρY |B ∗ ρAB|C = TrAB ρXY |C = TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C = ρX |C ρY |C . M. S. Leifer The Church of the Smaller Hilbert Space
  • 57. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions The Moral of the Story There is a bunch of other stuff that makes more sense in the Church of the Smaller Hilbert Space The “pretty good” measurement “Pretty good” error correction Results on steering entangled states Entanglement in time Quantum sufficient statistics Causality ...but the Church of the Larger Hilbert Space has some pretty nifty proofs too. So which one is right? M. S. Leifer The Church of the Smaller Hilbert Space
  • 58. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions The Moral of the Story There is a bunch of other stuff that makes more sense in the Church of the Smaller Hilbert Space The “pretty good” measurement “Pretty good” error correction Results on steering entangled states Entanglement in time Quantum sufficient statistics Causality ...but the Church of the Larger Hilbert Space has some pretty nifty proofs too. So which one is right? M. S. Leifer The Church of the Smaller Hilbert Space
  • 59. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions Blind Men and the Elephant by J. G. Saxe It was six men of Indostan To learning much inclined, Who went to see the Elephant (Though all of them were blind), That each by observation Might satisfy his mind M. S. Leifer The Church of the Smaller Hilbert Space
  • 60. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions Blind Men and the Elephant by J. G. Saxe The First approached the Elephant, And happening to fall Against his broad and sturdy side, At once began to bawl: quot;God bless me! but the Elephant Is very like a wall!quot; The Second, feeling of the tusk, Cried, quot;Ho! what have we here So very round and smooth and sharp? To me ’tis mighty clear This wonder of an Elephant Is very like a spear!quot; M. S. Leifer The Church of the Smaller Hilbert Space
  • 61. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions Blind Men and the Elephant by J. G. Saxe And so these men of Indostan Disputed loud and long, Each in his own opinion Exceeding stiff and strong, Though each was partly in the right, And all were in the wrong! Moral: So oft in theologic wars, The disputants, I ween, Rail on in utter ignorance Of what each other mean, And prate about an Elephant Not one of them has seen! M. S. Leifer The Church of the Smaller Hilbert Space
  • 62. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions Acknowledgments This work is supported by: The Foundational Questions Institute (http://www.fqxi.org) MITACS (http://www.mitacs.math.ca) NSERC (http://nserc.ca/) The Province of Ontario: ORDCF/MRI M. S. Leifer The Church of the Smaller Hilbert Space
  • 63. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions References Conditional Density Operators: M. S. Leifer, Phys. Rev. A 74, 042310 (2006). arXiv:quant-ph/0606022. M. S. Leifer (2006) arXiv:quant-ph/0611233. Conditional Independence: M. S. Leifer and D. Poulin, Ann. Phys., in press. arXiv:0708.1337 Quantum State Pooling: M. S. Leifer and R. W. Spekkens, in preparation. R. W. Spekkens and H. M. Wiseman, Phys. Rev. A 75, 042104 (2007). arXiv:quant-ph/0612190. Quantum Theology: The book with this title is unrelated to this talk. M. S. Leifer The Church of the Smaller Hilbert Space