From Statistical Decision Theory to Bell Nonlocality Francesco - PowerPoint PPT Presentation

From Statistical Decision Theory to Bell Nonlocality Francesco Buscemi * QECDT, University of Bristol, 26 July 2018 (videoconference) ∗ Dept. of Mathematical Informatics, Nagoya University, buscemi@i.nagoya-u.ac.jp

Introduction

Statistical Decision Problems experiment decision Θ − → X − → U � � � θ − → x − → u w ( x | θ ) d ( u | x ) ⇓ ℓ ( θ, u ) Definition (Statistical Models and Decisions Problems) A statistical experiment (i.e., statistical model) is a triple � Θ , X , w � , a statistical decision problem (i.e., statistical game) is a triple � Θ , U , ℓ � . 1/23

How Much Is an Experiment Worth? experiment decision Θ − → X − → U • the experiment is given , i.e., � � � it is the “resource” θ − → x − → u w ( x | θ ) d ( u | x ) • the decision instead can be ⇓ optimized ℓ ( θ, u ) Definition (Expected Payoff) The expected payoff of a statistical experiment w = � Θ , X , w � w.r.t. a decision problem � Θ , U , ℓ � is given by ℓ ( θ, u ) d ( u | x ) w ( x | θ ) | Θ | − 1 . � E � Θ , U ,ℓ � [ w ] � max d ( u | x ) 2/23 u,x,θ

Comparing Experiments 1/2 experiment w ′ = � Θ , Y , w ′ ( y | θ ) � experiment w = � Θ , X , w ( x | θ ) � experiment experiment decision decision Θ − → X − → U Θ − → Y − → U � � � � � � − → − → − → − → θ x u θ y u w ′ ( y | θ ) d ′ ( u | y ) w ( x | θ ) d ( u | x ) ⇓ ⇓ ℓ ( θ, u ) ℓ ( θ, u ) If E � Θ , U ,ℓ � [ w ] ≥ E � Θ , U ,ℓ � [ w ′ ] , then experiment � Θ , X , w � is better than experiment � Θ , Y , w ′ � for problem � Θ , U , ℓ � . 3/23

Comparing Experiments 2/2 Definition (Information Preorder) If the experiment � Θ , X , w � is better than experiment � Θ , Y , w ′ � for all decision problems � Θ , U , ℓ � , then we say that � Θ , X , w � is more informative than � Θ , Y , w ′ � , and write � Θ , X , w � � � Θ , Y , w ′ � . Problem. The information preorder is operational, but not really “concrete”. Can we visualize this better? 4/23

Blackwell’s Theorem (1948-1953) Blackwell-Sherman-Stein Theorem Given two experiments with the same parameter space, � Θ , X , w � and � Θ , Y , w ′ � , the condition � Θ , X , w � � � Θ , Y , w ′ � holds iff there exists a conditional probability ϕ ( y | x ) such that w ′ ( y | θ ) = � x ϕ ( y | x ) w ( x | θ ) . noise Θ − → Y Θ − → X − → Y � � = � � � θ − → y θ − → x − → y David H. Blackwell (1919-2010) w ′ ( y | θ ) w ( x | θ ) ϕ ( y | x ) 5/23

An Important Special Case: Majorization

Lorenz Curves and Majorization Preorder Lorenz curve for probability • two probability distributions, p and q , of distribution p = ( p 1 , · · · , p n ) : the same dimension n • truncated sums P ( k ) = � k i =1 p ↓ i and Q ( k ) = � k i =1 q ↓ i , for all k = 1 , . . . , n • p majorizes q , i.e., p � q , whenever P ( k ) ≥ Q ( k ) , for all k • minimal element: uniform distribution e = n − 1 (1 , 1 , · · · , 1) ( x k , y k ) = ( k/n, P ( k )) , 1 ≤ k ≤ n • Hardy, Littlewood, and P´ olya (1929) : p � q ⇐ ⇒ q = M p , for some bistochastic matrix M 6/23

Dichotomies and Tests • a dichotomy is a statistical experiment with a two-point parameter space: �{ 1 , 2 } , X , ( w 1 , w 2 ) � • a testing problem (or “test”) is a decision problem with a two-point action space U = { 1 , 2 } Definition (Testing Preorder) Given two dichotomies �X , ( w 1 , w 2 ) � and �Y , ( w ′ 1 , w ′ 2 ) � , we write �X , ( w 1 , w 2 ) � � 2 �Y , ( w ′ 1 , w ′ 2 ) � , whenever E �{ 1 , 2 } , { 1 , 2 } ,ℓ � [ �X , ( w 1 , w 2 ) � ] ≥ E �{ 1 , 2 } , { 1 , 2 } ,ℓ � [ �Y , ( w ′ 1 , w ′ 2 ) � ] for all testing problems. 7/23

Connection with Majorization Preorder Blackwell’s Theorem for Dichotomies (1953) Given two dichotomies �X , ( w 1 , w 2 ) � and �Y , ( w ′ 1 , w ′ 2 ) � , the relation �X , ( w 1 , w 2 ) � � 2 �Y , ( w ′ 1 , w ′ 2 ) � holds iff there exists a stochastic matrix M such that M w i = w ′ i . • majorization : p � q ⇐ ⇒ �X , ( p , e ) � � 2 �X , ( q , e ) � • thermomajorization : as above, but replace uniform e with thermal distribution γ T Hence, the information preorder is a multivariate version of the majorization preorder, and Blackwell’s theorem is a powerful generalization of that by Hardy, Littlewood, and P´ olya. 8/23

Visualization: Relative Lorenz Curves • two pairs of probability distributions, ( p 1 , p 2 ) and ( q 1 , q 2 ) , of dimension m and n , respectively • relabel their entries such that the ratios p i 1 /p i 2 and q j 1 /q j 2 are nonincreasing in i and j • with such labeling, construct the truncated sums P 1 , 2 ( k ) = � k 1 , 2 and Q 1 , 2 ( k ) = � k i =1 p i j =1 q i 1 , 2 Relative Lorenz curves: • ( p 1 , p 2 ) � 2 ( q 1 , q 2 ) , if and only if the relative ( x k , y k ) = ( P 2 ( k ) , P 1 ( k )) Lorenz curve of the former is never below that of the latter 9/23

The Quantum Case

Quantum Decision Theory (Holevo, 1973) classical case quantum case • decision problems � Θ , U , ℓ � • decision problems � Θ , U , ℓ � � Θ , H S , { ρ θ � • experiments w = � Θ , X , { w ( x | θ ) }� • quantum experiments E = S } • decisions d ( u | x ) • POVMs { P u S : u ∈ U} x d ( u | x ) w ( x | θ ) | Θ | − 1 | Θ | − 1 • p c ( u, θ ) = � • p q ( u, θ ) = Tr � ρ θ S P u � S � � • E � Θ , U ,ℓ � [ w ] = max • E � Θ , U ,ℓ � [ E ] = max ℓ ( θ, u ) p c ( u, θ ) ℓ ( θ, u ) p q ( u, θ ) { P u d ( u | x ) S } Hence, it is possible, for example, to compare quantum experiments with classical experiments, and introduce the information preorder as done before.

Example: Semiquantum Blackwell Theorem Theorem (FB, 2012) � Θ , H S , { ρ θ � Given a quantum experiment E = S } and a classical experiment w = � Θ , X , { w ( x | θ ) }� , the condition E � w holds iff there exists a POVM { P x � P x S ρ θ � S } such that w ( x | θ ) = Tr . S Equivalent reformulation � Θ , H S , { ρ θ � Consider two quantum experiments E = S } and E ′ = � Θ , H S ′ , { σ θ � S ′ } , and assume that the σ ’s all commute. Then, E � E ′ holds iff there exists a quantum channel (CPTP map) Φ : L ( H S ) → L ( H S ′ ) such that Φ( ρ θ S ) = σ θ S ′ , for all θ ∈ Θ . 11/23

The Theory of Quantum Statistical Comparison • fully quantum information preorder • quantum relative majorization • statistical comparison of quantum measurements (compatibility preorder) • statistical comparison of quantum channels (input-degradability preorder, output-degradability preorder, simulability preorder, etc) • applications: quantum information theory, quantum thermodynamics, open quantum systems dynamics, quantum resource theories, quantum foundations, . . . • approximate case 12/23

Application to Quantum Foundations: Distributed Decision Problems, i.e., Nonlocal Games

Nonlocal Games • nonlocal games (Bell tests) can be seen as bipartite decision problems �X , Y ; A , B ; ℓ � played “in parallel” by non-communicating players • with a classical source, p c ( a, b | x, y ) = � λ π ( λ ) d A ( a | x, λ ) d B ( b | y, λ ) • with a quantum source, � � ρ AB ( P a | x ⊗ Q b | y p q ( a, b | x, y ) = Tr B ) A � ℓ ( x, y ; a, b ) p c/q ( a, b | x, y ) |X| − 1 |Y| − 1 E �X , Y ; A , B ; ℓ � [ ∗ ] � max x,y,a,b 13/23

Semiquantum Nonlocal Games • semiquantum nonlocal games replace classical inputs with quantum inputs: �{ τ x } , { ω y } ; A , B ; ℓ � • with a classical source, p c ( a, b | x, y ) = � � Y ) ( P a | λ ⊗ Q b | λ X ⊗ ω y ( τ x � λ π ( λ ) Tr Y ) X • with a quantum source, p q ( a, b | x, y ) = X ⊗ ρ AB ⊗ ω y � ( τ x Y ) ( P a XA ⊗ Q b � Tr BY ) � ℓ ( x, y ; a, b ) p c/q ( a, b | x, y ) |X| − 1 |Y| − 1 E �{ τ x } , { ω y } ; A , B ; ℓ � [ ∗ ] � max x,y,a,b 14/23

Blackwell Theorem for Bipartite States Theorem (FB, 2012) Given two bipartite states ρ AB and σ A ′ B ′ , the condition (i.e., “nonlocality preorder”) E �{ τ x } , { ω y } ; A , B ; ℓ � [ ρ AB ] ≥ E �{ τ x } , { ω y } ; A , B ; ℓ � [ σ A ′ B ′ ] holds for all semiquantum nonlocal games, iff there exist CPTP maps Φ λ A → A ′ , Ψ λ B → B ′ , and distribution π ( λ ) such that � π ( λ )(Φ λ A → A ′ ⊗ Ψ λ σ A ′ B ′ = B → B ′ )( ρ AB ) . λ 15/23

Corollaries • For any separable state ρ AB , E �{ τ x } , { ω y } ; A , B ; ℓ � [ ρ AB ] = E �{ τ x } , { ω y } ; A , B ; ℓ � [ ρ A ⊗ ρ B ] = E sep �{ τ x } , { ω y } ; A , B ; ℓ � , for all semiquantum nonlocal games. • For any entangled state ρ AB , there exists a semiquantum nonlocal game �{ τ x } , { ω y } ; A , B ; ℓ � such that E �{ τ x } , { ω y } ; A , B ; ℓ � [ ρ AB ] > E sep �{ τ x } , { ω y } ; A , B ; ℓ � . 16/23