Can a quantum state over time resemble a quantum state at a single time? Chris Heunen 1 / 20
History Barbados, 2011 J. Barrett Information processing in generalized probabilistic theories , Physical Review A 75(3):032304, 2007 2 / 20
History Barbados, 2011 J. Barrett Information processing in generalized probabilistic theories , Physical Review A 75(3):032304, 2007 Oxford, 2012 C. Heunen and C. Horsman Matrix multiplication is determined by orthogonality and trace , Linear Algebra and its Applications 439(12):4130–4134, 2013. 2 / 20
Prehistory Barbados, 2011 J. Barrett Information processing in generalized probabilistic theories , Physical Review A 75(3):032304, 2007 Oxford, 2012 C. Heunen and C. Horsman Matrix multiplication is determined by orthogonality and trace , Linear Algebra and its Applications 439(12):4130–4134, 2013. London, 2013 M. Pusey Is quantum steering spooky? PhD, Imperial College London, 2013. 2 / 20
Prehistory Barbados, 2011 J. Barrett Information processing in generalized probabilistic theories , Physical Review A 75(3):032304, 2007 Oxford, 2012 C. Heunen and C. Horsman Matrix multiplication is determined by orthogonality and trace , Linear Algebra and its Applications 439(12):4130–4134, 2013. London, 2013 M. Pusey Is quantum steering spooky? PhD, Imperial College London, 2013. Kitchener, 2014 M. Leifer and R. Spekkens Towards a formulation of quantum theory as a causality neutral theory of Bayesian inference , Physical Review A 88(5):052130, 2013 2 / 20
Prehistory Barbados, 2011 Oxford, 2013 J. Barrett Information processing in generalized probabilistic theories , Physical Review A 75(3):032304, 2007 Oxford, 2012 Durham, 2015 C. Heunen and C. D. Horsman Matrix multiplication is determined by orthogonality and trace , Linear Algebra and its Applications 439(12):4130–4134, 2013. London, 2013 Kitchener, 2014 M. Pusey Is quantum steering spooky? PhD, Imperial College London, 2013. Kitchener, 2014 M. Leifer and R. Spekkens Towards a formulation of quantum theory as a causality neutral theory of Bayesian inference , Physical Review A 88(5):052130, 2013 2 / 20
Idea ◮ ‘problem of time’ in quantum gravity asymmetry between space and time in quantum theory ◮ one solution: remove asymmetry quantum states across time as well as space ◮ many approaches to states over time path integrals, consistent histories, multi-time states ◮ but: all depend on spatio-temporal relationships classical probability theory does not ◮ can it be done? 3 / 20
Outline ◮ Three proposals ◮ Four axioms ◮ The theorem ◮ Its proof ◮ What does it mean? 4 / 20
States across space first region second region A B H A H B ρ A ρ B
States across space first region composite system second region A B H A H A ⊗ H B H B ρ A ρ B ρ A ⊗ ρ B 5 / 20
States across time second time B evolution first time A
States across time H B second time B evolution H A first time A
States across time ρ B H B second time B E AB evolution ρ A H A first time A 6 / 20
States across time ρ B H B second time B E AB evolution ρ A H A first time A Composite system AB , operator ρ AB on H AB = H A ⊗ H B No restrictions on ρ AB , but fully defined by ρ A , ρ B , and E AB ρ AB = f ( ρ B , E AB , ρ A ) 6 / 20
States across time ρ B H B second time B E AB evolution ρ A H A first time A Composite system AB , operator ρ AB on H AB = H A ⊗ H B No restrictions on ρ AB , but fully defined by ρ A and E AB ρ AB = f ( E AB , ρ A ) What form could f take? 6 / 20
States across time ρ B H B second time B E AB evolution ρ A H A first time A Composite system AB , operator ρ AB on H AB = H A ⊗ H B No restrictions on ρ AB , but fully defined by ρ A and E AB ρ AB = E AB ⋆ ρ A What form could star product ⋆ : B ( H AB ) × B ( H AB ) → B ( H AB ) take? channel state E AB = � ij E AB | i �� j | A ⊗ | j �� i | B satisfying ρ B = Tr A ( E AB ρ A ) 6 / 20
Leifer-Spekkens “Quantum theory is like Bayesian inference” Towards a formulation of quantum theory as a causality neutral theory of Bayesian inference Physical Review A 88(5):052130, 2013 channel state conditional probabilities E AB P ( B | A ) Tr B ( E AB ) = 1 � B P ( B | A ) = 1 � ρ B = Tr A ( E AB ρ A ) A P ( B | A ) P ( A ) ? P ( AB ) = P ( B | A ) P ( A ) 7 / 20
Leifer-Spekkens “Quantum theory is like Bayesian inference” Towards a formulation of quantum theory as a causality neutral theory of Bayesian inference Physical Review A 88(5):052130, 2013 channel state conditional probabilities E AB P ( B | A ) Tr B ( E AB ) = 1 � B P ( B | A ) = 1 � ρ B = Tr A ( E AB ρ A ) A P ( B | A ) P ( A ) ? P ( AB ) = P ( B | A ) P ( A ) = E AB ⋆ LS ρ A = √ ρ A E AB √ ρ A ? = ρ ( LS ) AB 7 / 20
Fitzsimons-Jones-Vedral “Use pseudo-density matrices” Quantum correlations which imply causation Scientific Reports 5:18281, 2015 For (multi-)qubit systems A , B : 1. Measure σ i ∈ { 1 , σ x , σ y , σ z } on A 2. Evolve according to channel E AB 3. Measure σ j on B 8 / 20
Fitzsimons-Jones-Vedral “Use pseudo-density matrices” Quantum correlations which imply causation Scientific Reports 5:18281, 2015 For (multi-)qubit systems A , B : 1. Measure σ i ∈ { 1 , σ x , σ y , σ z } on A 2. Evolve according to channel E AB 3. Measure σ j on B � 3 � = 1 ρ ( FJV ) � � σ i ⊗ σ j � σ i ⊗ σ j AB 4 i = 1 8 / 20
Fitzsimons-Jones-Vedral “Use pseudo-density matrices” Quantum correlations which imply causation Scientific Reports 5:18281, 2015 For (multi-)qubit systems A , B : 1. Measure σ i ∈ { 1 , σ x , σ y , σ z } on A 2. Evolve according to channel E AB 3. Measure σ j on B � 3 � = 1 = E AB ⋆ FJV ρ A = 1 ρ ( FJV ) � � σ i ⊗ σ j � σ i ⊗ σ j 2 ( ρ A E AB + E AB ρ A ) AB 4 i = 1 8 / 20
Discrete Wigner functions “Use quasi-probabilities on discrete representation” On the quantum correction for thermodynamic equilibrium Physical Review 40:749-759, 1932 1. Pick phase-point operator basis { K A i } for H A Tr ( K A i K A i K A j ) = δ ij dim ( H A ) and � i = dim ( H A ) 2. Write ρ A as quasi-probability function r A : { i } → [ − 1 , 1 ] 3. Write E AB as conditional quasi-probability r B | A : { ( i , j ) } → [ − 1 , 1 ] 4. Define ρ AB by r AB ( ij ) = r B | A ( j | i ) r A ( i ) 9 / 20
Discrete Wigner functions “Use quasi-probabilities on discrete representation” On the quantum correction for thermodynamic equilibrium Physical Review 40:749-759, 1932 1. Pick phase-point operator basis { K A i } for H A Tr ( K A i K A i K A j ) = δ ij dim ( H A ) and � i = dim ( H A ) 2. Write ρ A as quasi-probability function r A : { i } → [ − 1 , 1 ] 3. Write E AB as conditional quasi-probability r B | A : { ( i , j ) } → [ − 1 , 1 ] 4. Define ρ AB by r AB ( ij ) = r B | A ( j | i ) r A ( i ) ρ ( W ) � r B | A ( j | i ) r A ( i ) K A i ⊗ K B AB = j ij 9 / 20
Discrete Wigner functions “Use quasi-probabilities on discrete representation” On the quantum correction for thermodynamic equilibrium Physical Review 40:749-759, 1932 1. Pick phase-point operator basis { K A i } for H A Tr ( K A i K A i K A j ) = δ ij dim ( H A ) and � i = dim ( H A ) 2. Write ρ A as quasi-probability function r A : { i } → [ − 1 , 1 ] 3. Write E AB as conditional quasi-probability r B | A : { ( i , j ) } → [ − 1 , 1 ] 4. Define ρ AB by r AB ( ij ) = r B | A ( j | i ) r A ( i ) ρ ( W ) � r B | A ( j | i ) r A ( i ) K A i ⊗ K B AB = j ij � Tr AB ( E AB K A i ⊗ K B j ) Tr A ( ρ A K A i ) K A i ⊗ K B = E AB ⋆ W ρ A = j ij 9 / 20
Axiom 1: preservation of probabilistic mixtures If A conditioned on fair classical coin ρ A , x = h = | 0 �� 0 | , ρ A , x = t = | 1 �� 1 | E AB = | φ + �� φ + | T B and channel is identity then should have composite state be mixture � 1 2 ρ A , x = h + 1 = 1 + 1 � � � E AB ⋆ 2 ρ A , x = t E AB ⋆ ρ A , x = h 2 ( E AB ⋆ ρ A , x = t ) 2 10 / 20
Axiom 1: preservation of probabilistic mixtures If A conditioned on fair classical coin ρ A , x = h = | 0 �� 0 | , ρ A , x = t = | 1 �� 1 | E AB = | φ + �� φ + | T B and channel is identity then should have composite state be mixture � 1 2 ρ A , x = h + 1 = 1 + 1 � � � E AB ⋆ 2 ρ A , x = t E AB ⋆ ρ A , x = h 2 ( E AB ⋆ ρ A , x = t ) 2 Axiom 1: convex-bilinearity � � px + ( 1 − p ) y ⋆ z = p ( x ⋆ z ) + ( 1 − p )( y ⋆ z ) � x ⋆ py + ( 1 − p ) z ) = p ( x ⋆ y ) + ( 1 − p )( x ⋆ z ) for all operators x , y , z and probabilities p ∈ [ 0 , 1 ] 10 / 20
Axiom 2: preservation of classical limit E AB = � i � i | ρ | i � � If channel completely dephasing j p ( j | i ) | j �� j | and input state diagonal ρ A = � i p ( i ) | i �� i | then should reproduce joint classical probabilities � � E AB ⋆ ρ A = p ( i ) | i �� i | ⊗ p ( j | i ) | j �� j | = E AB ρ A i i 11 / 20
Recommend
More recommend
