The Theory of Statistical Comparison in Quantum Information and - PowerPoint PPT Presentation

The Theory of Statistical Comparison in Quantum Information and Foundations Francesco Buscemi * Modern Topics in Quantum Information: Quantum Foundations and Quantum Information . International Institute of Physics (Natal, Brazil), 31 July 2018 ∗ Dept. of Mathematical Informatics, Nagoya University, buscemi@i.nagoya-u.ac.jp

The Birth in Mathematical Statistics

Statistical Decision Problems experiment decision − → X − → U Θ � � � − → − → θ x u w ( x | θ ) d ( u | x ) payoff is ℓ ( θ, u ) ∈ R 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 � . 1/28

How Much Is an Experiment Worth? experiment decision Θ − → X − → U • the experiment is given , i.e., it is the “resource” � � � • the decision instead can be optimized θ − → x − → u w ( x | θ ) d ( u | x ) 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 . � def E � Θ , U ,ℓ � [ w ] = max d ( u | x ) u,x,θ 2/28

Comparing Experiments 1/2 First experiment: Second experiment: w ′ = � Θ , Y , w ′ ( y | θ ) � w = � Θ , X , w ( x | θ ) � experiment experiment decision decision Θ − → X − → U Θ − → Y − → U � � � � � � − → − → − → − → θ x u θ x u w ′ ( y | θ ) d ′ ( u | y ) w ( x | θ ) d ( u | x ) 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 � . 3/28

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? 4/28

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 = � � � � � David H. Blackwell (1919-2010) θ − → y θ − → x − → y w ′ ( y | θ ) w ( x | θ ) ϕ ( y | x ) 5/28

The Precursor: Majorization

Lorenz Curves and Majorization • two probability distributions, p and q , of the same dimension n Lorenz curve for probability distribution p = ( p 1 , · · · , p n ) : • truncated sums P ( k ) = � k i =1 p ↓ i i =1 q ↓ and Q ( k ) = � k 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 p � q ⇐ ⇒ q = M p , for some bistochastic matrix M . 6/28

Generalization: Relative Lorenz Curves • two pairs of probability distributions, ( p 1 , p 2 ) and ( q 1 , q 2 ) , of dimension m and n , respectively • relabel entries such that ratios p i 1 /p i 2 and q j 1 /q j 2 are nonincreasing • construct the truncated sums P 1 , 2 ( k ) = � k i =1 p i 1 , 2 and Q 1 , 2 ( k ) • ( p 1 , p 2 ) � ( q 1 , q 2 ) iff the relative Lorenz curve of the former is never below that of the latter Relative Lorenz curves: ( x k , y k ) = ( P 2 ( k ) , P 1 ( k )) Blackwell (Theorem for Dichotomies) ( p 1 , p 2 ) � ( q 1 , q 2 ) ⇐ ⇒ q i = M p i , for some stochastic matrix M . 7/28

Extension to the Quantum Case

Quantum Decision Theory A.S. Holevo, Statistical Decision Theory for Quantum Systems , 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 } • POVMs { P u • decisions d ( u | x ) S : u ∈ U} x d ( u | x ) w ( x | θ ) | Θ | − 1 � ρ θ S P u � | Θ | − 1 • p c ( u, θ ) = � • p q ( u, θ ) = Tr S � � • E � Θ , U ,ℓ � [ w ] = max ℓ ( θ, u ) p c ( u, θ ) • E � Θ , U ,ℓ � [ E ] = max 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 � 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 . 9/28

Developments • 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, coding preorder, etc) • applications: quantum information theory , quantum thermodynamics, open quantum systems dynamics, quantum resource theories, quantum foundations , . . . 10/28

The Viewpoint of Communication Theory

Statistics vs Information Theory Statistical theory : Nature does not bother with coding experiment decision Θ − → X − → U � � � θ − → x − → u w ( x | θ ) d ( u | x ) Communication theory : a sender, instead, does code encoding decoding channel M − → − → X − → U Θ � � � � − → − → − → m θ x u e ( θ | m ) w ( x | θ ) d ( u | x ) 11/28

From Decision Problems to Decoding Problems Definition (Decoding Problems) Given a channel �X , Y , w ( y | x ) � , a decoding problem is defined by an encoding �M , X , e ( x | m ) � and the payoff function is the optimum guessing probability: def � d ( m | y ) w ( y | x ) e ( x | m ) |M| − 1 E �M , X ,e ( x | m ) � [ �X , Y , w ( y | x ) � ] = max d ( m | y ) m,x,y = 2 − H min ( M | Y ) encoding channel decoding M − → X − → Y − → M � � � � − → − → − → m x y m ˆ e ( x | m ) w ( y | x ) m | y ) d ( ˆ

Comparison of Classical Noisy Channels encoding encoding channel channel M − → X − → Y M − → X − → Z � � � � � � − → − → − → − → m x y m x z w ′ ( z | x ) e ( x | m ) w ( y | x ) e ( x | m ) Theorem (FB, 2016) The following are equivalent: 1. there exists ϕ ( z | y ) : w ′ ( x | z ) = � y ϕ ( z | y ) w ( y | x ) (stochastic degradability); 2. for all codes �M , X , e ( x | m ) � , H min ( M | Y ) ≤ H min ( M | Z ) (ambiguity preorder). The above strictly imply H ( M | Y ) ≤ H ( M | Z ) (K¨ orner’s and Marton’s noisiness preorder).

Decoding Quantum Codes Definition (Quantum Decoding Problems) Given a quantum channel N : A → B , a quantum decoding problem is defined by a bipartite state ω RA and the payoff function is the optimum singlet fraction: def D � Φ + R ◦ N A → B )( ω RA ) | Φ + E ω [ N ] = max R | ( id R ⊗ D B → ¯ R � R ¯ R ¯

Comparison of Quantum Noisy Channels Theorem Given two quantum channels N : A → B and N ′ : A → B ′ , the following are equivalent: 1. there exists CPTP map C : N ′ = C ◦ N (degradability preorder); 2. for any bipartite state ω RA , E ω [ N ] ≥ E ω [ N ′ ] (coherence preorder); 3. for any bipartite state ω RA , H min ( R | B ) ( id ⊗N )( ω ) ≤ H min ( R | B ′ ) ( id ⊗N ′ )( ω ) . � by adding symmetry constraints, we have applications in quantum thermodynamics 15/28

Application to Open Quantum Systems Dynamics

Discrete-Time Stochastic Processes • Let x i , for i = 0 , 1 , . . . , index the state of a system at time t = t i • if the system can be initialized at time t = t 0 , the process is fully described by the conditional distribution p ( x N , . . . , x 1 | x 0 ) • if the system evolving is quantum, we only have a quantum dynamical � � N ( i ) mapping Q 0 → Q i i =1 ,...,N • the process is divisible if there exist channels D ( i ) such that N ( i +1) = D ( i ) ◦ N ( i ) for all i 16/28

Divisibility as “Entanglement Flow” Theorem (2016-2018) Given an initial open quantum system Q 0 , a quantum � � N ( i ) dynamical mapping i ≥ 1 is divisibile if and only if, Q 0 → Q i for any initial state ω RQ 0 , H min ( R | Q 1 ) ≤ H min ( R | Q 2 ) ≤ · · · ≤ H min ( R | Q N ) . 17/28

Application to Quantum Foundations: Probing Quantum Correlations in Space-Time