Strong Converse and Stein s Lemma in the Quantum Hypothesis Testing

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1 Strong Converse and Stein s Lemma in the Quantum Hypothesis Testing arxiv:uant-ph/ v 24 Jun 999 Tomohiro Ogawa and Hiroshi Nagaoka Abstract The hypothesis testing problem of two uantum states is treated. We show a new ineuality between the error of the first kind and the second kind, which complements the result of Hiai and Petz to establish the uantum version of Stein s lemma. The ineuality is also used to show a bound on the first kind error when the power exponent for the second kind error exceeds the uantum relative entropy, and the bound yields the strong converse in the uantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi in the classical hypothesis testing. Keywords Quantum hypothesis testing, Stein s lemma, strong converse, uantum relative entropy. Introduction Let H be a Hilbert space which represents a physical system in interest. We suppose dim H < for mathematical simplicity. Let B(H) be the set of linear operators on H and put S(H) = {ρ B(H) ρ = ρ 0, Tr ρ = }, which is the set of density operators on H. We treat the problem of hypothesis testing a null hypothesis ρ S(H) versus an alternative hypothesis σ S(H). Here, we assume Im ρ Im σ. To consider an asymptotic situation, suppose that either ρ n S(H n ) or σ n S(H n ) is given. The problem is to decide which hypothesis is true, and the decision is given by a two-valued uantum measurement {A n, A n } (A n B(H n ), 0 A n ), where A n corresponds to the acceptance of ρ n and A n corresponds to the acceptance of σ n. We call A n B(H n ) (0 A n ) a test in the seuel. The authors are with the Graduate School of Information Systems, University of Electro- Communications, 5 Chofugaoka, Chofu, Tokyo , Japan. ( ogawa@hn.is.uec.ac.jp, nagaoka@is.uec.ac.jp)

2 For a test A n, ine the error probability of the first kind and the second kind by α n (A n ) β n (A n ) = Tr ρ n ( A n ), = Tr σ n A n, respectively. We see that α n (A n ) is the error probability of the acceptance of σ n when ρ n is true and β n (A n ) is the error probability of the converse situation. Since we can not have α n (A n ) and β n (A n ) arbitrarily small simultaneously, we will make β n (A n ) as small as possible under the constraint α n (A n ) ε. In other words, the problem is to examine the asymptotic behavior of the following uantity: β n(ε) = min{β n (A n ) A n B(H n ), 0 A n I, α n (A n ) ε}. Concerning β n(ε), Hiai and Petz [] showed and where lim sup n log β n(ε) D(ρ σ), () D(ρ σ) lim inf ε n log β n(ε), (2) D(ρ σ) = Tr ρ(log ρ log σ), is the uantum relative entropy. As for (2), they used the monotonicity of the uantum relative entropy [2, 3] as follows: D(ρ n σ n ) α n (A n ) α n (A n ) log β n (A n ) + ( α n(a n )) log α n(a n ) β n (A n ) = h(α n (A n )) α n (A n ) log( β n (A n )) ( α n (A n )) log β n (A n ) log 2 ( α n (A n )) log β n (A n ), where h(x) is the binary entropy. Thus it holds that ( α n (A n )) n log β n(a n ) log 2 n D(ρ σ), (3) which immediately yields (2). Note that (3) also leads the weak converse property, which means that if β n (A n ) e nr (r > D(ρ σ)) then α n (A n ) does not go to zero as n. In this paper, we will show a fundamental ineuality, which complements () by Hiai and Petz to show the uantum version of Stein s lemma (see e.g. [4], p.5). We will also show a bound on α n (A n ) under the exponential-type constraint β n (A n ) e nr. The bound leads to the strong converse property [5, 6] in the uantum hypothesis testing, i.e., if β n (A n ) e nr (r > D(ρ σ)) then α n (A n ) goes to one as n. Finally, we discuss the relation with the result of Han and Kobayashi [6] in the classical hypothesis testing.

3 2 A Fundamental Bound on the Error Probabilities In this section, we show a fundamental ineuality between the error probabilities of the first kind and the second kind. Let λ be a real number and ρ n e nλ σ n = j µ n,j E n,j, (4) be the spectral decomposition. Define a test X n,λ by X n,λ = j D n E n,j, where D n = {j µ n,j 0}. Then, the following lemma holds, which corresponds to the uantum version of the Neyman-Pearson lemma (see [7], p.08). Lemma For any test A n, we have Proof: Tr (ρ n e nλ σ n )X n,λ Tr (ρ n e nλ σ n )A n. (5) Tr (ρ n e nλ σ n )A n = µ n,j Tr E n,j A n j µ n,j Tr E n,j A n j D n µ n,j Tr E n,j j D n = Tr (ρ n e nλ σ n )X n,λ. Theorem For any test A n and any λ R, we have where α n (A n ) e nϕ(λ) + e nλ β n (A n ), (6) ϕ(λ) ψ(s) = max {λs ψ(s)}, (7) 0 s = log Tr ρ +s σ s. (8) Here, note that ϕ(λ) is the Legendre transformation of a convex function ψ(s) (see Fig. and 2). Putting A = log ρ log σ ψ (s), the convexity of ψ(s) is verified as ψ (s) = e ψ(s) Tr ρ +s σ s (log ρ log σ), ψ (s) = e ψ(s) Tr ρ +s Aσ s A = e ψ(s) Tr ( ) ( ) ρ +s 2 Aσ s +s 2 ρ 2 Aσ s 2 > 0.

4 Observing that ψ(0) = 0 and ψ (0) = D(ρ σ), we can see that if λ > D(ρ σ) then ϕ(λ) > 0. It is important to note that for any λ satisfying D(ρ σ) λ ψ (), we have s = argmax{λs ψ(s)} ψ (s ) = λ. (9) 0 s Proof of Theorem : Define probability distributions p n = {p n,j } and n = { n,j } by p n,j = Tr ρ n E n,j, n,j = Tr σ n E n,j. From (4), we have µ n,j Tr E n,j = p n,j e nλ n,j, and hence, Thus, we obtain D n = {j 0 s, e nλs p s n,j s n,j }. Tr ρ n X n,λ = j D n Tr ρ n E n,j = p n,j j D n j D n p n,j e nλs p s n,j s n,j e nλs j p +s n,j s n,j e nλs Tr (ρ n ) +s (σ n ) s, where we used the monotonicity of the uantum f-divergence [8] for an operator convex function f(u) = u s (0 s ) (see e.g. [9], p.23). Therefore, we have and hence, Tr ρ n X n,λ exp [ n{λs ψ(s)}], Tr ρ n X n,λ e nϕ(λ), by taking the maximum. Now, from (5), the theorem is proved as follows: α n (A n ) = Tr ρ n A n Tr (ρ n e nλ σ n )X n,λ + e nλ Tr σ n A n Tr ρ n X n,λ + e nλ Tr σ n A n e nϕ(λ) + e nλ β n (A n ). 3 The Quantum Stein s Lemma Theorem 2 For any 0 ε <, it holds that lim n log β n(ε) = D(ρ σ). (0)

5 Proof: From () by Hiai and Petz, we have only to show that lim inf n log β n (ε) D(ρ σ). () Let A n be an arbitrary test which satisfies α n (A n ) ε. From (6), we have and hence, By taking the minimum, we obtain ε α n (A n ) e nϕ(λ) + e nλ β n (A n ), β n (A n ) e nλ ( ε e nϕ(λ) ). β n(ε) e nλ ( ε e nϕ(λ) ). (2) Now, let λ = D(ρ σ) + δ (δ > 0), then ϕ(λ) > 0 and ε e nϕ(λ) > 0 for sufficiently large n. Thus, (2) yields and hence, n log β n (ε) λ + n log( ε e nϕ(λ) ), lim inf n log β n (ε) D(ρ σ) δ. Since δ > 0 is arbitrary, the theorem has been proved. 4 Strong Converse Theorem 3 For any test A n, if then lim sup lim sup n log β n(a n ) r, (3) n log( α n(a n )) ϕ(λ ), (4) where λ is a real number which satisfies ϕ(λ ) = r λ. Moreover, ϕ(λ ) is represented as { s ϕ(λ ) = max 0 s + s r } + s ψ(s). (5) Proof: For all δ > 0, there exists n 0 such that β n (A n ) e n(r δ), n n 0,

6 from (3). Putting λ = λ in (6), we have and hence, α n (A n ) e nϕ(λ ) + e n(r λ δ), n n 0, lim sup n log( α n(a n )) ϕ(λ ) + δ. Since δ > 0 is arbitrary, (4) has been proved. To show (5), suppose that ψ (0) r 2ψ () ψ() firstly (see Fig. 2), and s attains the maximum in the euation u(r) = ϕ(λ ) = max 0 s {sλ ψ(s)} = r λ. Then, taking (9) into account, u(r) is represented parametrically as u(r) = s ψ (s ) ψ(s ), (6) where, r = (s + )ψ (s ) ψ(s ). (7) By using (7) to eliminate ψ (s ) from (6), we have On the other hand, let then we have g (s) = u(r) = s s + r s + ψ(s ). g(s) = s s + r s + ψ(s), (s + ) 2 {r + ψ(s) ( + s)ψ (s)}. To examine the sign of g (s), put h(s) = r + ψ(s) ( + s)ψ (s), and we have h (s) = ( + s)ψ (s) 0, which indicates that the sign of g (s) changes at most once. Therefore g(s) takes its maximum at s = ŝ if and only if r + ψ(ŝ) ( + ŝ)ψ (ŝ) = 0. This is nothing but the condition (7), and hence, we obtain u(r) = max 0 s g(s). In the other cases, it is clear that and ϕ(λ ) = 2 r ψ() = g() = max 2 g(s), if r 0 s 2ψ () ψ(), ϕ(λ ) = 0 = g(0) = max 0 s g(s), if r ψ (0).

7 It should be noted that (5) corresponds to the representation which Blahut [5] derived, in the classical hypothesis testing (i.e., when ρ and σ commute), concerning the power exponent for α n (A n ) when r < D(ρ σ). We can easily see that if r > D(ρ σ) then ϕ(λ ) > 0 (see Fig. 2). Hence, the following corollary holds. Corollary For any test A n, if lim sup then α n (A n ) goes to one exponentially. n log β n(a n ) < D(ρ σ), (8) 5 Relation with the Classical Hypothesis Testing In this section, we discuss the relation between ϕ(λ ) and the power exponent derived by Han and Kobayashi [6] in the classical hypothesis testing. Let p and be probability distributions on a finite set X, a null hypothesis and an alternative hypothesis respectively. And ine α n (A n ) = p n (A c n) and β n (A n ) = n (A n ), where p n and n are the i.i.d. extensions of p and, and A n X n is an acceptance region of p n. Blahut [5] proved that if β n (A n ) e nr (r > D(p )) then α n (A n ) tends to one as n for all A n X n. Although Blahut showed that α n (A n ) converges at one in the polynomial order, Han and Kobayashi [6] proved a stronger result. Putting α n(r) = min{α n (A n ) A n X n, β n (A n ) e nr }, they derived the power exponent for α n(r), namely, they proved where lim inf n log( α n (r)) = ũ(r), (9) ũ(r) = min {D(ˆp p) + r D(ˆp )}. (20) ˆp:D(ˆp ) r As remarked in Han and Kobayashi [6], when r > D(p ) is not so large, the minimum of (20) is attained with euality, which we suppose here. Applying the method used in Appendix, (20) is rewritten as ũ(r) = max s 0 s + s r + s log j X p +s j Moreover, when r > D(p ) is sufficiently small, (2) yields which corresponds to (5). s ũ(r) = max 0 s + s r + s log j X p +s j j s. (2) j s,

8 It is interesting to observe that in the uantum case we have where we put min ˆρ:D(ˆρ σ) r {D(ˆρ ρ) + r D(ˆρ σ)} = max s 0 ψ(s) = log Tr e (+s) log ρ s log σ. { s + s r + s ψ(s) }, By the Golden-Thompson ineuality (see e.g. [9], p.26), we can see that ψ(s) ψ(s) and the euality holds if and only if ρ and σ commute. Thus, we have 6 Concluding Remarks min {D(ˆρ ρ) + r D(ˆρ σ)} ˆρ:D(ˆρ σ) r ϕ(λ ). (22) So far we have shown the fundamental ineuality, and seen that the uantum Stein s lemma and the strong converse in the uantum hypothesis testing are obtained as applications of the ineuality. In the classical hypothesis testing, ũ(r) is shown to be the optimal exponent. However, whether ϕ(λ ) in the uantum hypothesis testing is optimal or not is left open. Appendix We show (2) for readers convenience. Here, we will derive (2) by the information geometrical method (see e.g., [0]), although (2) can be shown by using the Lagrange multiplier method as given in [5]. Suppose that r > D(p ) is not so large that there exists a probability distribution of the form: where, p(s) j = e ψ(s) p +s ψ(s) = log j X j s j, (j X, s > 0), p +s j j s, such that D(p(s) ) = r. Note that [ ψ (s) = E p(s) log p ] = η(s), ψ (s) = E p(s) (log p ) 2 η(s) > 0. Firstly, we will show that for all probability distribution ˆp it holds that D(ˆp ) = D(p(s) ) = D(ˆp p) D(p(s) p). (23) To this end, suppose that there exists a probability distribution ˆp which satisfies D(ˆp ) = D(p(s) ) and [ E p(s) log p ] [ < Eˆp log p ].

9 Then, we have D(ˆp ) = D(ˆp p(s)) + D(p(s) ) + j X(ˆp j p(s) j )(log p(s) j log j ) = D(ˆp p(s)) + D(p(s) ) + j X(ˆp ( j p(s) j ) ( + s) log p ) j ψ(s) j = D(ˆp p(s)) + D(p(s) ) + ( + s) > D(p(s) ), (Eˆp [ log p ] E p(s) [ log p which contradicts with the assumption. Therefore, for all probability distribution with D(ˆp ) = D(p(s) ), we have [ E p(s) log p ] [ Eˆp log p ], and hence, there exists t s such that [ E p(t) log p ] [ = Eˆp log p ], (24) since η(s) is continuous and monotone increasing. Now, from (24) and the Pythagorean relation for the Kullback-Leibler divergence we have D(ˆp ) = D(ˆp p(t)) + D(p(t) ) ]) = D(p(s) ). (25) Using the Pythagorean relation one more times and from (25), (23) is proved as follows: D(ˆp p) D(p(s) p) = D(ˆp p(t)) + D(p(t) p) D(p(s) p) = D(p(s) ) D(p(t) ) + D(p(t) p) D(p(s) p) = {D(p(s) ) D(p(s) p)} {D(p(t) ) D(p(t) p)} = η(s) η(t) 0. Now, taking (23) into account, (20) is represented as ũ(r) = min {D(p(s) p) + r D(p(s) )} s:d(p(s) ) r = min {r η(s)}. (26) s:d(p(s) ) r Here, we can see that d(s) = D(p(s) ) is a monotone increasing function of s, which is verified by d (s) = ( + s) ψ (s) > 0. Thus, the minimum of (26) is attained with euality, and ũ(r) is represented parametrically as where, ũ(r) = D(p(s) p) = s η(s) ψ(s), r = D(p(s) ) = ( + s)η(s) ψ(s). This representation corresponds to (6) (7) and we obtain (2) by following the same procedure as the proof of Theorem 3.

10 Acknowledgment The authors wish to thank Prof. Fumio Hiai for his helpful comments. They are also grateful to Prof. Te Sun Han for his suggestion for the strong converse in the uantum hypothesis testing. References [] F. Hiai and D. Petz, The proper formula for relative entropy and its asymptotics in uantum probability, Commun. Math. Phys., vol. 43, pp. 99 4, 99. [2] G. Lindblad, Completely positive maps and entropy ineualities, Commun. Math. Phys., vol. 40, pp. 47 5, 975. [3] A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys., vol. 54, pp. 2 32, 997. [4] R. E. Blahut, Principles and Practice of Information Theory, Addison-Wesley, Massachusetts, 99. [5] R. E. Blahut, Hypothesis testing and information theory, IEEE Trans. Inform. Theory, vol. IT-20, pp , 974. [6] T. S. Han and K. Kobayashi, The strong converse theorem for hypothesis testing, IEEE Trans. Inform. Theory, vol. IT-35, pp , 989. [7] C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 976. [8] D. Petz, Quasi-entropies for finite uantum systems, Rep. Math. Phys., vol. 23, pp , 986. [9] R. Bhatia, Matrix Analysis, Springer, New York, 997. [0] S. Amari, Differential-Geometrical Methods in Statistics, Springer, New York, 985.

11 ψ(s) λs ϕ(λ) D(ρ σ) s 0 s Figure : The graph of ψ(s)

12 2ψ () ψ() r ϕ(λ) D(ρ σ) r λ ϕ(λ ) = r λ 0 D(ρ σ) λ ψ () 2ψ () ψ() λ Figure 2: The graph of ϕ(λ)

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