The Theory of Quantum Statistical Comparison and Some Applications in Quantum Information Sciences Francesco Buscemi (Nagoya U) The 39th Quantum Information Technologies Symposium (QIT) . RCAST, The University of Tokyo, 27 November 2018
The Original Formulation
Statistical Models and Decision Problems experiment decision Θ − → X − → U � � � θ − → x − → u w ( x | θ ) d ( u | x ) Definition • The statistical model is given by: the parameter set Θ , the sample set X , and the PDs w ( x | θ ) . • The statistical decision problem: is given by the parameter set Θ , the action set U , and the payoff function ℓ : Θ × U → R . 1/25
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 statistical model w = � Θ , X , w ( x | θ ) � w.r.t. a decision problem ℓ = � Θ , U , ℓ ( θ, u ) � is given by ℓ ( θ, u ) d ( u | x ) w ( x | θ ) | Θ | − 1 . � def E ℓ [ w ] = max d ( u | x ) u,x,θ 2/25
Comparing Statistical Models 1/2 Second model: w ′ = � Θ , Y , w ′ ( y | θ ) � First model: w = � Θ , X , w ( x | θ ) � experiment decision experiment decision Θ − → X − → U Θ − → Y − → U � � � � � � − → − → − → − → θ x u θ y u w ( x | θ ) d ( u | x ) w ′ ( y | θ ) d ′ ( u | y ) Given a statistical decision problem ℓ = � Θ , U , ℓ ( θ, u ) � , if E ℓ [ w ] ≥ E ℓ [ w ′ ] , then one says that model w is more informative than model w ′ with respect to problem ℓ . 3/25
Comparing Statistical Models 2/2 Definition (Information Preorder) If model w = � Θ , X , w ( x | θ ) � is more informative than model w ′ = � Θ , Y , w ′ ( y | θ ) � for all decision problems ℓ = � Θ , U , ℓ ( θ, u ) � , then we say that w is (always) more informative than w ′ , and write w � w ′ . Problem. The information preorder is operational, but not really “concrete”. Can we visualize this better? 4/25
A Fundamental Result Blackwell-Sherman-Stein (1948-1953) Given two models with the same parameter space, w = � Θ , X , w ( x | θ ) � and w ′ = � Θ , Y , w ′ ( y | θ ) � , the condition w � w ′ holds iff w is sufficent for w ′ , that is, iff there exists a conditional PD 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 ) 5/25
The Precursor: Majorization
Lorenz Curves and Majorization Lorenz curve for probability distribution • two probability distributions, p and q , of the p = ( p 1 , · · · , p n ) : same dimension n • truncated sums P ( k ) = � k i =1 p ↓ i and i =1 q ↓ 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) Hardy, Littlewood, and P´ olya (1934) ( x k , y k ) = ( k/n, P ( k )) , 1 ≤ k ≤ n p � q ⇐ ⇒ q = M p , for some bistochastic matrix M . 6/25
Generalization: Relative Majorization • two pairs of probability distributions, ( p 1 , p 2 ) and ( q 1 , q 2 ) , of dimension m and n , respectively 2 and q j 1 /q j • relabel entries such that ratios p i 1 /p i 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 Blackwell (Theorem for Dichotomies), 1953 Relative Lorenz curves: ( x k , y k ) = ( P 2 ( k ) , P 1 ( k )) ( p 1 , p 2 ) � ( q 1 , q 2 ) ⇐ ⇒ q i = M p i , for some stochastic matrix M . 7/25
Formulation in Terms of Channels
Statistics vs Information Theory Statistical theory Communication theory Nature does not bother with coding a sender, instead, does code encoding decoding experiment decision channel Θ − → X − → U M − → Θ − → X − → U � � � � � � � − → − → m − → θ − → x − → u θ x u e ( θ | m ) w ( x | θ ) d ( u | x ) w ( x | θ ) d ( u | x ) 8/25
Statistics vs Information Theory Statistical theory Communication theory Nature does not bother with coding a sender, instead, does code encoding decoding experiment decision channel Θ − → X − → U M − → X − → Y − → M � � � � � � � m ′ − → − → m − → x − → y − → θ x u e ( x | m ) w ( y | x ) d ( m ′ | y ) w ( x | θ ) d ( u | x ) 9/25
Sufficiency vs Degradability Degradability relation Sufficiency relation for statistical experiments for noisy channels w ′ ( z | x ) = � w ′ ( y | θ ) = � y ϕ ( z | y ) w ( y | x ) x ϕ ( y | x ) w ( x | θ ) Only the labeling convention changes, but the two conditions are absolutely equivalent. 10/25
Decoding Problems and Codes Fidelities When dealing with communication channels, it is natural to restrict to particular decision problems that we name “decoding problems”. E = { e ( x | m ) } , N = { w ( y | x ) } , D = { d ( m ′ | y ) } Code Fidelity Given a noisy channel N : X → Y , its code fidelity , for any set M and any coding channel E : M → X , is defined as 1 � def e ( x | m ) w ( y | x ) d ( m ′ | y ) δ m,m ′ E �E� [ N ] = max |M| D : Y→M 11/25 m,x,y,m ′
Comparison of Noisy Channels Theorem (Coding Problems Are Complete) Given two noisy channels N : X → Y and N ′ : X → Y ′ , N is degradable into N ′ if and only if E �E� [ N ] ≥ E �E� [ N ′ ] , for all codes E : M → X , with M ∼ = Y ′ . 12/25
Extensions to the Quantum Case
Extending Decoding Problems decoding problems ւ ց quantum decoding problems quantum “realignment” problems 13/25
Quantum Decoding Problems � d M | Φ + 1 M ′ M � = i =1 | i � M | i � M √ d M Quantum Code Fidelity Given a quantum channel (i.e., CPTP linear map) N : A → B , its quantum code fidelity , for any Hilbert space H M ∼ = H M ′ and any quantum coding channel E : M → A , is defined as E q def D : B → M � Φ + M ′ M | ( id M ′ ⊗ D ◦ N ◦ E )(Φ + M ′ M ) | Φ + M ′ M � = d − 1 M 2 − H min ( M ′ | B ) �E� [ N ] = max 14/25
Quantum Realignment Problems Quantum Realignment Fidelity Given a quantum channel N : A → B , for any Hilbert space H C ∼ = H C ′ and any “misaligning” channel F : A ′ → C ′ , its quantum realignment fidelity is defined as F q def D : B → C � Φ + C ′ C | ( F A ′ ⊗ D ◦ N )(Φ + A ′ A ) | Φ + C ′ C � = d − 1 C 2 − H min ( C ′ | B ) �F� [ N ] = max 15/25
Comparison of Quantum Channels Theorem (Quantum Coding and Realignment Problems Are Complete) Given two quantum channels N : A → B and N ′ : A → B ′ , the following are equivalent: 1. N is degradable into N ′ ; 2. for any quantum coding channel E : M → A , with H M ∼ = H B ′ , one has E q E [ N ] ≥ E q E [ N ′ ] , or, equivalently, H min ( M ′ | B ) ≤ H min ( M ′ | B ′ ) ; 3. for any quantum misaligning channel F : A ′ → C ′ , with H C ′ ∼ = H B ′ , one has F q F [ N ] ≥ F q F [ N ′ ] , or, equivalently, H min ( C ′ | B ) ≤ H min ( C ′ | B ′ ) . 16/25
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 mapping � � N ( i ) 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 17/25
A “Zoo of Quantum Markovianities” From: Li Li, Michael J. W. Hall, Howard M. Wiseman. Concepts of quantum non-Markovianity: a hierarchy . (arXiv:1712.08879 [quant-ph]) 18/25
A “Zoo of Quantum Markovianities” From: Li Li, Michael J. W. Hall, Howard M. Wiseman. Concepts of quantum non-Markovianity: a hierarchy . (arXiv:1712.08879 [quant-ph]) 19/25
Divisibility as “Information Flow” Theorem � � N ( i ) Given a mapping i ≥ 1 , the following are equivalent to divisibility Q 0 → Q i 1. for any quantum code, its fidelity is monotonically non-increasing in time 2. for any misaligning channel, its quantum realignment fidelity is monotonically non-increasing in time 20/25
Application to Quantum Thermodynamics
3.5 years ago I presented some ideas (arXiv:1505.00535) that eventually led to (arXiv:1708.04302) 21/25
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