A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks

Transcription

1 A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks Marco Tomamichel, Masahito Hayashi arxiv: Also discussing results of: Second Order Asymptotics for Quantum Hypothesis Testing Ke Li, arxiv: CQT, National University of Singapore Graduate School of Mathematics, Nagoya University Cambridge, January 2013

2 Outline 1 Hypothesis Testing and β ε 2 Main Result: Asymptotic Expansion of β ε and Smooth Entropies 3 Two Applications Randomness Extraction against Quantum Side Information Data Compression with Quantum Side Information 4 Finite Block Length Results 5 Main Proof Ideas 6 Conclusion and Discussion

3 Quantum Hypothesis Testing Given one out of two quantum states, either ρ or σ, we must decide which one we received. For a given test POVM, {Q, 1 Q}, 0 Q 1, the error of the first and second kind are α ρ (Q) = tr(ρ(1 Q)) and β σ (Q) = tr(σq), respectively. We are interested in the minimal β that can be achieved if α is required to smaller than a given constant ε, i.e. the SDP β ε ρ,σ := min β σ (Q) = 0 Q 1 αρ(q) ε min tr(σq). 0 Q 1 tr(ρq) 1 ε Alternatively, one may consider the divergence ( β Dh ε ε ) (ρ σ) := log ρ,σ, 0 < ε < 1. 1 ε (The additive normalization log(1 ε) ensures that ρ = σ D ε h(ρ σ) = 0.) 3

4 I.i.d. Asymptotic Expansion of D ε h Given n copies of either σ and ρ, a quantum generalization of Stein s Lemma (Hiai&Petz 91) and its strong converse (Ogawa&Nagaoka 00) imply D ε h( ρ n σ n) = nd(ρ σ) + o(n) This was recently improved (Audenaert,Mosonyi&Verstraete 12) Dh( ε ρ n σ n) nd(ρ σ) + O ( n ) and ( ρ n σ n) nd(ρ σ) O ( n ) D ε h by giving explicit upper and lower bounds. However, the terms proportional to n in the upper and lower bounds are different. (They do not have the same sign!) Our goal is to investigate the second order term, O( n). 4

5 Main Result Theorem For two states ρ, σ with supp{σ} supp{ρ}, and 0 < ε < 1, we find Dh ε (ρ n σ n ) nd(ρ σ) + nv (ρ σ)φ 1 (ε) + 2 log n + O(1), and Dh ε (ρ n σ n ) nd(ρ σ) + nv (ρ σ)φ 1 (ε) O(1). D and V are the mean and variance of log ρ log σ under ρ, i.e. V (ρ σ) := tr ( ρ ( log ρ log σ D(ρ σ) ) 2). Φ is the cumulative normal distribution function, and Φ 1 (ε) is The bounds given here are due to Li. A similar result was independently derived by T&H. 5.

6 Main Result Theorem For two states ρ, σ with supp{σ} supp{ρ}, and 0 < ε < 1, we find Dh ε (ρ n σ n ) nd(ρ σ) + nv (ρ σ)φ 1 (ε) + 2 log n + O(1), and Dh ε (ρ n σ n ) nd(ρ σ) + nv (ρ σ)φ 1 (ε) O(1). We also have bounds on the constant terms, enabling us to calculate upper and lower bounds on D ε h (ρ n σ n ) for finite n. Classically, the above holds with logarithmic terms in upper and lower bound equal to 1 2 log n (e.g. Strassen 62,Polyanskiy,Poor&Verdú 10). One ingredient of both proof is the Berry-Essèen theorem, which quantizes the convergence of the distribution of a sum of i.i.d. random variables to a normal distribution. Intuitively, our results can be seen as quantum, entropic formulation of the central limit theorem. The bounds given here are due to Li. A similar result was independently derived by T&H. 6

7 Smooth Entropies We also investigate the smooth min-entropy (Renner 05), where it was known (T,Colbeck&Renner 09,T 12) that, for ε (0, 1), H ε min(a n B n ) ρ n nh(a B) ρ + O ( n ), H ε min(a n B n ) ρ n nh(a B) ρ O ( n ). and We derive the following expansion Hmin(A ε n B n ) ρ n nh(a B) ρ + nv (A B) ρ Φ 1 (ε 2 ) + O(log n), Hmin(A ε n B n ) ρ n nh(a B) ρ + nv (A B) ρ Φ 1 (ε 2 ) O(log n), where H(A B) ρ = D(ρ AB 1 A ρ B ) and V (A B) ρ = V (ρ AB 1 A ρ B ). Both hypothesis testing and smooth entropies have various applications in information theory, some of which we explore next. 7

8 Randomness Extraction against Side Information Consider a CQ random source that outputs states ρ XE = x p x x x ρ x E. Investigate the amount of randomness that can be extracted from X such that it is independent of E and the seeded randomness S. A protocol P : XS ZS extracts a random number Z from X, producing a state τ ZES when applied to ρ XE ρ S. For any 0 ε < 1 and ρ XE a CQ state, we define l ε (X E) := max { l N P, σe : Z = 2 l τ ZES ε 2 l 1 Z σ E τ S }. This quantity can be characterized in terms of the smooth min-entropy (Renner 05). We tighten this and show Theorem Consider an i.i.d. source ρ X n E n = ρ n XE and 0 < ε < 1. Then, l ε (X n E n ) nh(x E) + nv (X E)Φ 1 (ε 2 ) ± O(log n). 8

9 Data Compression with Side Information Consider a CQ random source that outputs states ρ XB = x p x x x ρ x B. Find the minimum encoding length for data reconciliation of X if quantum side information B is available. A protocol P encodes X into M and then produces an estimate X of X from B and M. For any 0 ε < 1 and ρ XB a CQ state, we define m ε (X B) ρ := min { m N P : M = 2 m P[X X ] ε }. This quantity can be characterized using hypothesis testing (H&Nagaoka 04). We tighten this and show Theorem Consider an i.i.d. source ρ X n B n = ρ n XB and 0 < ε < 1. Then, m ε (X n B n ) nh(x B) nv (X B)Φ 1 (ε) ± O(log n). 9

10 Example of Second Order Asymptotics Consider transmission of 0, 1 through a Pauli channel to B (phase and bit flip independent) with environment E. This yields the states ρ XB = 1 2 x x ( (1 p) x x + p 1 x 1 x ), ρ XE = 1 2 x x φ x φ x, φ x = p 0 + ( 1) x 1 p Plot of first and second order asymptotic approximation of 1 n lε (X E) and 1 n mε (X B) for p = 0.05 and ε =

11 Example of Finite Block Length Bounds r 0.7 1st order 2nd order converse bound direct bound n 11

12 Different Layers of Approximation Class Role Quantities Class 1 Optimal performance of protocol. m ε (X B) ρ Calculation is very difficult. l ε (X B) ρ, Class 2 One-shot bound for general source. Hh ε(a B) ρ, SDP tractable for small systems. Hmin ε (A B) ρ Class 3 Quantum information spectrum. Ds ε (ρ σ) Class 4 Classical information spectrum. Ds ε (P 0,ρ,σ P 1,ρ,σ ) Approximately possible for i.i.d. Class 5 Second order asymptotics. nh(x B)+ Calculation is easy for large n. n s(x B)Φ 1 (ε) Classes Difference Method 1 2 O(log n) Random coding and data-processing inequalities. 2 4 O(log n) Relations between entropies. 4 5 O(1) Berry-Essèen Theorem. Let ε (0, 1) and consider n i.i.d. repetitions of tasks for large n. 12

13 Step 1: One-Shot bounds I Theorem (One-Shot Randomness Extraction) Let ρ XB be a CQ state and 0 < η ε < 1. Then, H ε η min (X B) ρ log 1 η 4 3 lε (X B) ρ H ε min(x B) ρ. Theorem (One-Shot Data Compression) Let ρ XB be a CQ state and 0 < η ε < 1. Then, Hh ε (X B) ρ m ε (X B) ρ H ε η h (X B) ρ + log ε η Converse bounds keep ε intact. Achievability up to η, where η can be chosen arbitrarily small. The idea is to choose η 1/ n for the i.i.d. analysis. 13

14 Step 1: One-Shot bounds II Theorem Let ρ XB be a CQ state and 0 < η ε < 1. Then, H ε η min (X B) ρ log 1 η 4 3 lε (X B) ρ H ε min(x B) ρ. In this sense, we have l ε (X B) ρ H ε min (X B) ρ, i.e. the smooth min-entropy characterizes randomness extraction. The smooth min-entropy can be calculated efficiently (using an SDP) for small systems. The quantity l ε (X B) ρ has some natural properties, i.e. For any function: l ε (f (X ) B) ρ l ε (X B) ρ. For any quantum channel: l ε (X E(B)) ρ l ε (X B) ρ. These properties are mimicked by the smooth min-entropy. The converse follows solely from the first of these monotonicity properties. Achievability uses two-universal hashing. 14

15 Step 1: One-Shot bounds III Theorem Let ρ XB be a CQ state and 0 < η ε < 1. Then, Hh ε (X B) ρ m ε (X B) ρ H ε η h (X B) ρ + log ε η In this sense, we have m ε (X B) ρ H ε h (X B) ρ, i.e. source coding is characterized by a conditional hypothesis testing entropy. The hypothesis testing entropy can be calculated efficiently (using an SDP) for small systems. The quantity m ε (X B) ρ has some natural properties, i.e. For any message: m ε (X BM) ρ m ε (X B) ρ log M. For any quantum channel: m ε (X E(B)) ρ m ε (X B) ρ. These properties are mimicked by H h. The converse follows solely from these properties. Achievability uses two-universal hashing (corresponds to random coding) and pretty good measurements. 15

16 Step 2: Relation to Information Spectrum I While the one-shot entropies can be computed for small systems, they are generally intractable for large systems. Hence, we want to approximate them further. This is done as follows. We write everything in terms of relative entropies. H ε h (A B) ρ = max σ D ε h (ρ AB 1 A σ B ) and H ε min (A B) ρ = max σ D ε max(ρ AB 1 A σ B ). Use relations between relative entropies: D ε h (ρ σ) D 1 ε max (ρ σ) D ε s (ρ σ) D ε s (P 0 P 1 ) This holds up to terms log Θ, where Θ is at most the number of distinct eigenvalues of σ, or 2 log(λ max (σ)/λ min (σ)). In particular, Θ = O(n) in the i.i.d. case. 16

17 Step 2: Relation to Information Spectrum II We focus on the last quantity, the classical information spectrum. Decompose ρ = x r x v x v x and σ = y s y u y u y. Nussbaum&Szkoła 09 introduced the distributions P 0 (x, y) := r x v x u y 2 and P 1 (x, y) := s y v x u y 2 They satisfy D(P 0 P 1 ) = D(ρ σ) and V (P 0 P 1 ) = V (ρ σ), i.e. the first two moments agree with the quantum analogue. This reduces the problem to analyzing the classical quantity { Ds ε [ (P 0 P 1 ) := max R R Pr log P0 log P 1 R ] } ε. P 0 So far we have not used any i.i.d. assumption except that we assumed that the eigenvalues of σ behave nicely. 17

18 Step 3: Asymptotic Expansion Assume i.i.d. states ρ n XB, are i.i.d. as well. P n 1. The corresponding distributions, P n 0 and We write P0 n (x n, y n ) = i P[i] 0 (x i, y i ), Z i = log P [i] 0 D ε s (P n 0 P n log P[i] 1, and 1 {R ) = max [ R Pr log P n 0 log P1 n R ] } ε P 0 { [ 1 ] } = n max R R Pr Z i R ε P 0 n where the Z i are i.i.d. distributed. The central limit theorem states that the distribution of 1 n i Z i converges to a Gaussian distribution with mean E[Z] = D(P 0 P 1 ) and variance E [ (Z E[Z]) 2] = V (P 0 P 1 ) 2. This yields the expansion D ε s (P n 0 P n 1 ) = nd(p 0 P 1 ) + nv (P 0 P 1 )Φ 1 (ε) + O(1). i 18

19 Relation to Asymptotic Information Spectrum Our results also strengthens the relation of one-shot entropies to the sup/inf-information spectrum (Datta&Renner 09). These are generally defined as (Nagaoka&Hayashi 07) D(ε ρ σ) := sup { R R lim sup trρ n {ρ n 2 nr σ n } ε }, n D(ε ρ σ) := inf { R R lim inf trρ n{ρ n 2 nr σ n } ε } n = sup { R R lim inf n trρ n{ρ n 2 nr σ n } < ε }. We find the following relations, which are valid for all ε [0, 1]. { 1 D(ε ρ σ) = sup lim inf ɛ n { 1 D(ε ρ σ) = sup lim inf ɛ n n D 1 ɛ n max n D 1 ɛ n max (ρ n σ n ) lim sup ɛ n ε }, n (ρ n σ n ) lim inf n ɛ n < ε }, The original relations by Datta&Renner only consider ε {0, 1}. 19

20 Conclusion, Open Questions and Discussion We find the second order asymptotics for quantum hypothesis testing and give bounds on β ε for finite n. We use one-shot entropies and techniques developed for hypothesis testing and the information spectrum method to find a second order expansion and finite block length bounds for operational quantities. There is a difference of 2 log n between the current upper and lower bounds on Dh ε (ρ σ). Is this fundamental, i.e. do there exist ρ and σ for which these bounds are tight? Or can this be further improved? Note that classically, the upper and lower bounds only differ in the constant term. The bounds depend on the parameter ε of the operational quantity, either l ε or m ε. This implies, in particular, that the ε in the respective one-shot entropies has clear operational meaning. Provocative question to those of us using one-shot entropies in information theory: What does it mean to characterize a quantity? When is a result considered tight? 20

21 Thank you for your attention. 21