Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing

In the asymptotic setting, the optimal test for hypotheses testing of the maximally entangled state is derived under several locality conditions for measurements. The optimal test is obtained in several cases with the asymptotic framework as well as the finite-sample framework. In addition, the experimental scheme for the optimal test is presented.


Introduction
Recently, various methods for quantum information processing have been proposed. Many of them require the use of maximally entangled states [1]- [3]. Hence, it is often desirable to be able to generate maximally entangled states experimentally. If this is done, statistical techniques are necessary to decide whether the state generated experimentally is really the required maximally entangled state.
Currently, the method 'entanglement witness' is often used as the standard method [4]- [8]. However, from a statistical viewpoint, this is not necessarily the optimal method. In mathematical statistics, the decision problem for the truth of a given hypothesis is called statistical hypothesis testing, and it has been systematically studied. Hence, we wish to investigate the problem of deciding whether a given quantum state is the required maximally entangled state in this framework of statistical hypotheses testing. In statistical hypotheses testing, we suppose given two hypotheses (the null and alternative hypotheses) to be tested, we assume that one of these is true. Based on observed data, we decide which hypothesis is true. Most earlier studies utilizing quantum hypothesis testing have used only simple hypothesis testing; that is, they have assumed that both the null and the alternative hypothesis each consist of a single quantum state. For example, the quantum Neyman-Pearson lemma [9,10], the quantum version of Stein's lemma [11]- [14], the quantum Chernoff bound [15,16] and the quantum Hoeffding bound [17]- [19] all deal with simple hypotheses.
However, from a practical standpoint, it is unnatural to specify either hypothesis using a single quantum state. Hence, we cannot directly apply the quantum Neyman-Pearson theorem and the quantum Stein's lemma; we have to utilize composite hypotheses, i.e. the case where both hypotheses consist of plural quantum states. It is also necessary to restrict our measurements for testing among measurements based on local operations and classical communications (LOCC) because the state to be tested is a maximally entangled state.
Recently, based on quantum statistical inference [10,20,21], Hayashi et al [22] discussed this testing problem in the context of statistical hypothesis testing with a locality condition. They treated a testing problem where the null hypothesis consists only of the required maximally entangled state. Their analysis has been extended to a more experimental setting [23], and its effectiveness has been experimentally demonstrated [24]. Modifying this setting, Owari and Hayashi [25] clarified the difference in performance between the one-way LOCC restriction and the two-way LOCC restriction in a specific case. In particular, Hayashi et al [22] studied the optimal test and the existence of a uniformly optimal test (whose definition will be presented later) when one or two samples of the state to be tested are provided. Their analysis mainly concentrated on the two-dimensional case.
In this paper, we treat null hypothesis testing of quantum states whose fidelity for the desired maximally entangled state is not greater than , and discuss this testing problem in three settings, for several given samples of the tested state in the range of our measurements. (Note that our previous paper [22] treats the case of = 0.) We remark that, for this problem, there are two kinds of locality restriction: L1, one is the locality of the two distinct parties; L2, the second concerns the locality of the samples. The three settings are as follows: M1, all measurements are allowed; M2, there is restriction on the locality L1, but not on the locality L2; M3, there is restriction on the locality L2 as well as on L1. The restriction M3 for measurement was first discussed by Virmani and Plenio [28]. Hayashi et al [22] treated the settings M2 and M3 more systematically. This paper mainly deals with the case of sufficiently many samples, i.e. the first-order asymptotic theory. In this context, we find that there is no difference in performance between settings M1 and M2 in an asymptotic framework, i.e. in the case when the fidelity between the true state and the target maximally entangled state is parameterized as 1 − (δ/n), where n is the number of samples of the prepared unknown state (theorem 3), whereas Hayashi et al [22] and Virmani and Plenio [28] did not treat such an asymptotic framework. In particular, the test achieving asymptotically optimal performance can be realized by quantum measurement with quantum correlations between only two samples in respective parties, as is illustrated in figure 1. That is, even if we use higher quantum correlations among samples in respective parties, no further improvement is available within the first-order asymptotic framework. In the two-dimensional case, the required measurement with quantum correlations in respective parties is the four-valued Bell measurement between the two samples in both the parties. In the setting M3, we treat the null hypothesis consisting only of the maximally entangled state. Then, it can be proved that even if we use classical correlation between samples in respective parties, there is no further improvement (theorem 5). That is, the optimal protocol can be realized by repeating the optimal measurement in the one sample case in the setting M3.
Concerning the non-asymptotic setting, we derive the optimal test with an arbitrary finite number of samples under a suitable group symmetry without any locality condition (theorem 1). This result can be trivially extended to hypothesis testing of arbitrary pure states. Moreover, we derive the optimal test with two samples under several conditions, and calculate its optimal performance (theorems 4 and 6).
We also consider the case when each sample system consists of two or three different quantum systems whose state is a tensor product state of different states. In this case, even if we have just one sample, every party consists of multiple systems. In this situation, we obtain the optimal test for the one-sample case in both settings M2 and M3. It is proved that repeating the optimal measurement for one sample yields a test achieving asymptotically optimal performance (theorem 8). Moreover, when each sample system consists of two different 5 systems, we show that the optimal measurement for the one-sample case can be realized by a four-valued Bell measurement on the respective parties (theorem 7). Repeating this measurement yields the optimal performance in the first-order asymptotic framework. (In fact, it is difficult to perform the quantum measurement with quantum correlation between two samples because we need to prepare two samples from the same source at the same time. However, in this formulation, it is sufficient to prepare two states from different sources.) When each sample system consists of three different systems, the optimal measurement can be described by the Greenberger-Horne-Zeilinger (GHZ) state (1/ √ d) i |i |i |i , where d is the dimension of the system (theorem 9). This fact seems to indicate the importance of the GHZ state for the three systems.
Concerning locality restrictions on our measurements, it is natural to treat two-way LOCC, but we treat one-way LOCC and separable measurements, because the separability condition is easier to treat than two-way LOCC. We generally assume separability in this paper, as it is a useful mathematical condition. In contrast, Virmani and Plenio [28] and Hayashi et al [22] used the positive partial transpose (PPT) condition as well as the separability condition.
This paper is organized as follows. The mathematical formulation of statistical hypothesis testing is presented in section 2 and, the group theoretical symmetry is explained in section 3.2. In section 3.3, we explain the restrictions of our measurement for our testing, for example, one-way LOCC, two-way LOCC, separability, etc. In section 4, we review the fundamentals of statistical hypothesis testing for probability distributions. In section 5 (sections 6 and 7), the settings M1 (M2 and M3) are discussed, respectively. Further results for the two-dimensional case are presented in section 8. Finally, in sections 9 and 10, we discuss the cases of two and three different quantum states, respectively.

Mathematical formulation of quantum hypothesis testing
Let H be a finite-dimensional Hilbert space corresponding to the physical system of interest. Then, the state is described by a density matrix on H. In quantum hypothesis testing, it is assumed that the current state ρ of the system is unknown, but that the system is known to belong to a subset S 0 or S 1 of the set of densities. Hence, our task is testing based on an appropriate measurement of H. That is, we are required to decide which hypothesis is true. We call H 0 the null hypothesis, and we call H 1 the alternative hypothesis.
A test for the hypothesis (1) is given by a positive operator valued measure (POVM) on H composed of two elements, {T 0 , T 1 }, where T 0 + T 1 = I . For simplicity, the test {T 0 , T 1 } is described by the operator T = T 0 . Our decision should be made based on this test as follows: we accept H 0 (=we reject H 1 ) if we observe T 0 , and we accept H 1 (=we reject H 0 ) if we observe T 1 . In order to treat performance, we focus on the following two kinds of errors: a type 1 error is an event for which we accept H 1 though H 0 is true. A type 2 error is an event for which we accept H 0 though H 1 is true. Hence, we treat the following two kinds of error probabilities: the type 1 error probability α(T, ρ) and the type 2 error probabilities β(T, ρ) are given by α(T, ρ) = Tr (ρT 1 ) = 1 − Tr (ρT )(ρ ∈ S 0 ), β(T, ρ) = Tr (ρT 0 ) = Tr (ρT )(ρ ∈ S 1 ).
The quantity 1 − β(T, ρ) is called power. A test T is said to be at level-α if α(T, ρ) α for any ρ ∈ S 0 .

6
In hypothesis testing, we restrict our attention to tests whose first error probability is smaller than a given constant α for any element ρ ∈ S 0 . That is, since a type 1 error is considered to be more serious than a type 2 error in hypothesis testing, it is necessary to ensure that the type 1 error probability is less than a constant, which is called the level of significance or simply level. Hence, a test T is said to be of level-α if α(T, ρ) α for any ρ ∈ S 0 .
Then, under this condition, the performance of the test is given by 1 − β(T, ρ) for ρ ∈ S 1 , which is called the power of the test. Therefore, we often optimize the type 2 error probability as follows: Moreover, a test T ∈ T α,S 0 is called a uniformly most powerful (UMP) test if T is MP for any level-α test ρ ∈ S 1 , that is, However, in certain instances, it is natural to restrict our tests to those satisfying one or two conditions (C 1 or C 1 and C 2 ). In such a case, we focus on the following quantity instead of β(T, ρ): If a test T ∈ T α,S 0 satisfies conditions C 1 and C 2 , and β(T, ρ) = β C 2 α,C 1 (S 0 ρ), ∀ρ ∈ S 1 , it is called a uniformly most powerful C 1 , C 2 (UMP C 1 , C 2 ) test.

Hypothesis
Our aim is to test whether the generated state is sufficiently close to the maximal entangled state for n unknown densities σ 1 , . . . , σ n . We also assume that these densities σ 1 , . . . , σ n are all equal to a particular density σ . In this case, the state ρ is called n-independent and identically distributed (n-i.i.d.). In the following, we consider two settings for our hypotheses: When the null hypothesis is 'σ ∈ S ', the set of level α-tests is given in the n-fold i.i.d. case by Similarly, when the null hypothesis is 'σ ∈ S ', the set of level α-tests is given in the n-fold i.i.d. case by In this paper, we treat only the null hypothesis S . However, a large part of the results that we obtain can be trivially extended to the case of the null hypothesis S .

Restriction I: group action
In this paper, we treat these two cases under invariance conditions for the following group actions, which preserve the two hypotheses H 0 and H 1 . The naturalness of these conditions will be discussed later.

U(1)-action:
where U θ is defined by We can easily check that this action preserves our hypotheses. The U (1)-action is so small that it is not suitable to adopt this invariance as our restriction. However, since this invariance yields easier treatment, it is often adopted only for technical reasons. 2. SU(d)-action: we consider the unitary action on the tensor product space and g is the complex conjugate of g with respect to the standard basis |0 B , |1 B , . . . , |d − 1 B on the system B. Indeed, this action preserves the maximally entangled state |φ 0 AB . Hence, this action preserves our hypotheses. Furthermore, this action preserves the entanglement property. Similarly to the U (1)-invariance, the SU (d)-action is so small that it will be adopted only for technical reasons. 3. SU(d)×U(1)-action: since the SU (d) action and the U (1)-action preserve the entanglement property, the following action of the direct sum product group SU (d) × U (1) of SU (d) and U (1) also preserves this property: Thus, this condition is most suitable to take as our restriction. 4. U (d 2 − 1)-action: as a stronger form of invariance, we can consider invariance of the U (d 2 − 1)-action, i.e. the following unitary action on the orthogonal space of |φ 0 This group action includes the U (1)-action and the SU (d)-action. Hence, invariance with respect to the U (d 2 − 1)-action is stronger than invariance with respect to the preceding three actions. However, this action does not preserve the entanglement property. Thus, based on this definition, we cannot say that this condition is natural for our setting, whereas it is natural if we do not care about entanglement.
Furthermore, in the n-fold i.i.d. setting, it is appropriate to assume the invariance of the n-fold tensor product action of the above actions, i.e. U ⊗n θ , U (g) ⊗n , U (g, θ) ⊗n , V (g) ⊗n , etc.

Restriction II: locality
When the system consists of two distinct parties A and B, it is natural to restrict our testing to LOCC measurements between A and B. Hence, we can consider several restrictions on the locality condition. In section 4, as a first step, in order to discuss hypothesis testing with the null hypothesis S , we will treat the following optimization: . However, since our quantum system consists of two distant systems, we cannot necessarily use all measurements. Hence, it is natural to restrict our test to a class of tests. In this paper, we focus on the following seven classes.
∅: no condition S (A, B): the test is separable between the two systems H ⊗n A and H ⊗n B , i.e. the test T has the following form: where a i 0 and the matrix  Based on the above conditions, we define the following quantity as the optimal second error probability: As can be easily checked, any LOCC operation is separable. Hence, the condition L(A B) is stronger than the condition S (A, B). Also, the condition L(A 1 , . . . , A n → B 1 , . . . , B n ) is stronger than the condition S(A 1 , . . . , A n → B 1 , . . . , B n ). The relations among these conditions can be illustrated as follows: Next, we focus on the trivial relations of the optimal second error probability. If a group G 1 is greater than G 2 , the inequality holds. Moreover, if a condition C 1 is stronger than another condition C 2 , a similar inequality holds. Similarly, we define β C α,n,G ( σ ) by replacing by on the RHS. When = 0, β C α,n,G ( σ ) is abbreviated to β C α,n,G (0 σ ). Indeed, if a condition is invariant under the action of G, it is very natural to restrict our test among G-invariant tests, as is indicated by the following lemma.

Lemma 1. Assume that a set of tests satisfying the condition C is invariant under the action of G. Then
where ν G is the invariant measure and f denotes the action of G.
In the following, we sometimes abbreviate the invariant measure ν G by ν. For a proof see appendix A. This lemma is a special version of the quantum Hunt-Stein lemma [20]. Hence, lemma 1 cannot be applied to a pair of these conditions and the actions U (1), SU (d) × U (1) and U (d 2 − 1). The following lemma is useful in such a case.

Lemma 2.
Assume that the group G 1 includes another group G 2 as a subgroup and the subgroup G 2 satisfies the condition of lemma 1. If The proof is given in appendix A.

Testing for binomial distributions
In this paper, we use several facts about testing for binomial distributions, in order to test for a maximally entangled state. Hence, we review them here.

One-sample setting
As a preliminary, we treat testing for the coin flipping probability p with a single trial. That is, we assume that the event 1 happens with probability p and the event 0 happens with probability 1 − p, and focus on the null hypothesis p ∈ [0, ]. In this case, our test can be described by a mapT from {0, 1} to [0, 1], which means that when the data k is observed, we accept the null hypothesis with probabilityT (k). Then, the minimum second error probability among level-α tests is given by When we define the testT 1 ,α bỹ Moreover, if p , Hence the testT 1 ,α is of level-α. Furthermore, we can easily check that the minimum of q(T ) under the condition (4) forT can be attained byT =T 1 ,α if q > . Hence,

n-sample setting
In the n-trial case, the data k = 0, 1, . . . , n obey the distribution P n p (k) def = n k (1 − p) n−k p k with unknown parameter p. Hence, we discuss testing of the null hypothesis P n def = {P n p (k)| p } and the alternative hypothesis (P n ) c . In this case, our testT can be described by a function 12 from the data set {0, 1, . . . , n} to the interval [0, 1]. In this case, when the data element k is observed, we accept the null hypothesis P n with probability T (k). Then, the minimum second error probability among level-α tests is given by We define the testT n ,α as follows: where the integer l n ,α and the real number γ n ,α > 0 are defined by

Proposition 1. The testT n
,α is a level-α UMP test with null hypothesis P n . Hence, For a proof, see appendix C.

Asymptotic setting
In asymptotic theory, there are at least two settings. One is a large deviation setting, in which the parameter is fixed, hence we focus on the exponential component of the error probability. The other is a small deviation setting, in which the parameter is close to a given fixed point in proportion to the number of samples and the error probability converges to a fixed number. That is, the parameter is fixed in the former, whereas the error probability is fixed in the later.

Small deviation theory.
It is useful to treat the neighbourhood around p = 0 as the small deviation theory of this problem for a discussion of asymptotic testing for a maximally entangled state. Hence, we focus on the case that p = t n , since the probability P n t/n (k) = n k (1 − (t/n)) n−k (t/n) k converges to the Poisson distribution P t (k) def = e −t (t k /k!). Hence, our testing problem with null hypothesis P δ/n and alternative hypothesis t /n is asymptotically equivalent to a test of the Poisson distribution P t (k) with null hypothesis t ∈ [0, δ] and alternative hypothesis t . That is, by defining the following proposition holds.
The proof is presented in appendix D. Similar to the testT n ,α , we define the testT δ,α as where the integer l δ,α and the real number γ δ,α > 0 are defined by The following proposition, similar to proposition 1, holds.

Large deviation theory.
Next, we proceed to large deviation theory. Using knowledge of mathematical statistics, we can calculate the exponents of the 2nd error probabilities β n α ( p) and β n α ( p) for any α > 0 as where the binary relative entropy d( p) is defined as In the case of α = 0, we have

Global tests
First, we treat hypothesis testing with a given group invariance condition with no locality restriction.

One sample setting
When only one sample is prepared, the test |φ 0 Hence, applying the discussion in section 4.1, the test

n-sample setting
In the n-sample setting, we construct a test for the null hypothesis S as follows. First, we perform the two-valued measurement {|φ 0 |} for respective n systems. Then, if the number of counts I − |φ 0 A,B φ 0 A,B | is described by k, the data k obey the binomial distribution P n p (k). In this case, our problem can be reduced to hypothesis testing with null hypothesis P n , which has been discussed in section 4.2.
For given α and , the test based on this measurement and the classical testT n ,α is described by the operator T n Note that the above sum contains all tensor products of k times of S and n − k times of T .
Since the operators |φ 0 On the other hand, as is shown in appendix E, Since (8) and (9) yield the following theorem.
. Moreover, we can derive the same results for the hypothesis S .

Asymptotic setting
Next, we proceed to the asymptotic setting. In the small deviation theory, we treat hypothesis testing with the null hypothesis S δ/n . In this setting, proposition 2 and theorem 1 guarantee that the limit of the optimal second error probability of the alternative hypothesis σ n is given by . In the large deviation setting, we can obtain the same results as in section 4.3, i.e.
. Moreover, we can derive similar results for the null hypothesis S .
In this section, we deal with optimization problems with several conditions on the locality between A and B.

Construction of one-way LOCC test
First, we introduce two important one-way LOCC tests, which will be used later. Consider a POVM with the following form on H A : where such a POVM is called rank-one. Based on a rank-one POVM M, a suitable test T (M) Next, we focus on the covariant POVM M 1 cov : where ν( dϕ) is an invariant measure in the set of pure states with full measure 1. Then, the test has the following form: where the last equation will be derived in appendix H. Note that the POVM M 1 cov can be realized as follows: Then, the realized POVM is M 1 cov .

One sample setting
First, we focus on the simplest case, i.e. the case of = 0 and α = 0. In this case, the test T (M) Tr Hence, it is a level-0 test with the null hypothesis |φ 0 In particular, in the one-way LOCC setting, our tests can be restricted to tests of this kind, in the following sense.

Lemma 3. Let T be a one-way LOCC
i

.e. the test T (M) is better than the test T .
Moreover, concerning the separable condition, the following lemma holds. Hence, corollary 1 indicates that it seems natural to restrict our test to tests of the form (13), even if we do adopt the separable condition.

Lemma 4. Assume that a separable test T satisfies
When the test T has the form The proof is given in appendix G. Note that we can easily obtain the same statement if we replace the summation i by the integral in (20). Since any separable test T has the form (20), the following corollary holds concerning the completely mixed state I /d 2 .

Corollary 1. If a separable test T satisfies the conditions
then the test T has the form (13).
Next, we apply the discussion in section 4.1 to the probability distribution {dp/(d + 1), 1 − dp On the other hand, for SU (d)-invariant and separable tests, the equation holds, which is shown in appendix I. The equation in the case of α = 0 and = 0 was obtained by Hayashi et al [22]. Virmani and Plenio [28] showed that this POVM is the extremal point under the PPT condition as well as the separable condition, the two-way LOCC condition and the one-way LOCC condition. Since U (d 2 − 1) is a larger group action than SU (d) and the condition L(A → B) is stricter than the condition S(A, B), the trivial inequalities hold. Therefore, relations (21) and (22) yield the following theorem.
is the UMP G-invariant C test at level α for the null hypothesis S .
Furthermore, similar results for the null hypothesis S can also be obtained.

Two-sample case
In this section, we construct an SU (d) × U (1)-invariant test, which is realized by LOCC between A and B, and which attains the asymptotically optimal bound (11). For this purpose, we focus on the covariant POVM M 2 cov : where the vector u is maximally entangled and ν is the invariant measure on SU (d). Then, the operator T 2,A→B inv def = T (M 2 cov ) has the form: which is shown in appendix J. This equation implies that the testing T (M 2 cov ) does not depend on the choice of the maximally entangled state u. Now, we choose the maximally entangled state |φ 0 1,2 . Then, So, the other local POVM: . This test satisfies the equation because the form (24) guarantees the U (d 2 − 1)-invariance of the test T 2,A→B inv . Since the test T 2,A→B inv is a level-0 test with null hypothesis S 0 , the inequality holds. Next, we apply the discussion of section 4.1. Then, the test can be performed by randomized operation with T 2,A→B inv and I − T 2,A→B inv , we obtain Furthermore, as a generalization of (25), we obtain the following lemma, which, from an applied viewpoint, is more useful in the asymptotic setting. and where As will be discussed in section 6.5, the test T (M 2 Bell ) can be used as an alternative test of T 2,A→B inv in an asymptotic sense.

n-sample setting
Next, we construct a U (d 2 − 1)-invariant test when 2n samples of the unknown state σ are prepared. It follows from a discussion similar to section 5.2 that the test T 2n Since the test T 2n ,α can be realized by one-way LOCC A → B, the inequality

Asymptotic setting
We proceed to the asymptotic setting. First, we show that even if our test satisfies the A-B LOCC condition, the bound (10) can be attained in the asymptotic small deviation setting. Indeed, since P n can be proven similarly to theorem 2. Hence, equations (2) and (3) yield the following theorem.
, the bound β α ( δ t ) can be attained in the following asymptotic sense. The test T 2n (δ/2n),α,Bell may not be at level-α with the null hypothesis S δ/2n , but it is asymptotically at level-α, i.e.
holds. These relations (31) and (32) follow from lemma 5. Hence, there is no advantage in using entanglement between H A and H B for this test in the asymptotic small deviation setting. Similar results for the null hypothesis S δ/n can be obtained. The asymptotic optimal testing scheme is illustrated in figure 1.
Next, we proceed to the large deviation setting. The inequality (30) yields Hence, the relations (3) and (12) guarantee that if 1 − p 1/d, S(A, B). Hence, we can conclude that if 1 − p 1/d, there is no advantage in using entanglement between H A and H B for this test, even in this kind of asymptotic large deviation setting.

A-B locality and sample locality
In this section, we discuss locality among A 1 , B 1 , . . . , A n , B n . Since the case n = 1 of this setting is the same as that of the setting in section 6, we begin with the case n = 2.

Two-sample setting
We construct a level-0 SU (d)-invariant test for the null hypothesis The existence of {u i (ϕ)} i is shown in appendix K. 3. Alice randomly chooses g ∈ U (d − 1), which acts on the space orthogonal to ϕ and performs the projection-valued measure {|gu i (ϕ) gu i (ϕ)|} i on the second system H A 2 .
Since Bob's measurement of the test T (M 1→2 cov ) can also be realized by one-way LOCC on Bob's space, this test is an L (A 1 , A 2 → B 1 , B 2 ) test. Its POVM is given by where we choose u 1 and u 2 satisfying | u 1 |u Moreover, as is shown in appendix L, the test T A 1 →A 2 →B ⊗2 inv is U (1)-invariant. Hence, the inequality holds. On the other hand, the equation holds, as is shown in appendix M. Hayashi et al [22] have obtained a similar result for the two-dimensional case. Thus, we obtain the following theorem.

n-sample setting
Next, we proceed to the n-sample setting. Since the test T n which will be shown in appendix N.
. The condition | u|u | = 1 guarantees that t 0 = 1. The definition of T u,u guarantees that Tr T u,u d, which implies t 1 1/(d + 1). Hence, Combining (38) in the case of δ = 0, we obtain the following theorem.  B). So, in summary, there is an advantage in using quantum correlation among samples.

Two-sample two-dimensional setting
Next, we proceed to the special case n = 2 and d = 2. For analysis of this case, we define the where the vector u op is defined as Then, the following theorem holds.

holds, where C = L(A → B), L(A B), S(A, B). That is, the test T (M op ) is the UMP SU (2) × U (1)-invariant C test.
This theorem is shown in appendix O. Since the quantity Tr I 3 V 2 − (Tr I 3 V ) 2 is greater than 0, the term 3 5 indicates the advantage of this optimal test compared with the test T 2,A→B inv introduced in section 6.3 (see (25)). Note that this advantage disappears if and only if the real symmetric matrix V is constant.
On the other hand, as is shown in appendix P, the RHS of (35) can be evaluated as That is, the quantity 1 5 Tr I 3 V 2 − (Tr I 3 V ) 2 indicates the effect of using classical communication between A 1 and A 2 . This is because the second type error probability is (β L(A 1 ,A 2 →B 1 ,B 2 ) 0,1,SU (2) (0 σ )) 2 = (1 − 2 3 p) 2 if we use the optimal test for one-sample case twice.

Two different systems
In section 6, we showed that if we can prepare the two identical states simultaneously and perform Bell measurement on this joint system, the asymptotically optimal test can be realized. However, it is a bit difficult to prepare two identical states from the same source simultaneously. However, as is discussed in this section, if we can prepare two quantum states independently from different sources, this Bell measurement is asymptotically optimal.

Formulation
Since the state on H ⊗2 A,B can be described as σ 1 ⊗ σ 2 , our hypotheses are given as For any group action G introduced in section 3.2, these hypotheses are invariant with respect to the G × G-action defined by

One sample setting
In this section, we treat the one sample and the = 0 case. In the first step, we focus on the case of C = ∅. For this case, the relations

27
Moreover, the optimal second error can also be calculated as . The proof is given in appendix Q. Hence, we obtain the following theorem.

Theorem 7. Assume that the quantities p
The test T 2,A→B inv is the C-UMP G-invariant test.
Using the PPT condition, Hayashi et al [22] derived this optimal test in the case of Finally, we proceed to the case of In this case, as is proven in appendix R, the optimal second error is calculated as is the C-UMP G-invariant test. Hayashi et al [22] derived this optimal test for the case of σ 1 = σ 2 and d = 2.

Asymptotic setting
In the small deviation asymptotic setting with n samples, we focus on the case = δ/n and In this setting, as is shown in appendix S, . Next, we consider the case of C = L (A → B). When we perform the test T 2,A→B inv for all systems H A 1 ⊗ H B 1 , . . . , H A n ⊗ H B n whose state is σ 1,n ⊗ σ 2,n , the number k for detecting T 2,A→B inv almost obeys the Poisson distribution e −(t 1 +t 2 ) [(t 1 + t 2 ) k /k!]. This is because Considering hypothesis testing for this Poisson distribution, we can show that the lim n→∞ β(T n,2 δ/n,α , σ 1,n ⊗ σ 2,n ) = β α ( δ t 1 + t 2 ).
Hence, combining with (46), we obtain the following theorem. S(A, B) and

Theorem 8. When t i
. Thus, the test T n,2 ,α is the C-UMP G-invariant test in the asymptotic small deviation setting. The asymptotic optimal testing scheme is illustrated in figure 1. Moreover, if we use a test based on the Bell measurement instead of the test T 2,A→B inv , the bound β α ( δ t 1 + t 2 ) can be attained for a reason similar to that which applies in lemma 5.

Three different systems
Finally, we treat the case of three quantum states that are prepared independently. Similarly to section 9.1, we consider two hypotheses where the given state is assumed to be σ 1 ⊗ σ 2 ⊗ σ 3 . Similarly we define the quantities β C α,3,G×G×G ( Similarly to section 9.2, we focus on the case of C = L(A → B), L(A B), S(A, B) with one sample. For this case, as has been mentioned, the GHZ state |G H Z 3 is irreducible, the following forms a POVM: As is proved in appendix T, the test T 3,A→B inv def = T (M 3 cov ) has the form where Hence, when we use the test T 3,A→B inv , the second error is Moreover, the optimal second error can also be calculated as The proof is given in appendix T. Hence, we obtain the following theorem.

The test T 3,A→B inv is the C-UMP G-invariant test.
On the other hand, in the case of C = L( A 2 , A 3 , B 1 , B 2 , B 3 ), we can show the optimality of the test T 1,A→B inv ⊗ T 1,A→B inv ⊗ T 1,A→B inv , similarly to (45). Moreover, we can derive the same result in the small deviation asymptotic setting with n samples.

Conclusion
In this paper, we have considered the hypothesis testing problem when the null hypothesis consists only of the required entangled state or is its neighbourhood. In order to treat the entanglement structure, we considered three settings concerning the range of accessible measurements as follows: M1: all measurements are allowed. M2: a measurement is forbidden if it requires quantum correlation between two distinct parties. M3: a measurement is forbidden if it requires quantum correlation between two distinct parties, or such among local samples. As a result, we found that there is no difference between the accuracies of M1 and M2 to first-order asymptotics. The protocol achieving the asymptotic bound has been proposed in the setting M2. In this setting, it is required to prepare two identical samples at the same time. However, it is difficult to prepare the two states from the same source. In order to avoid this difficulty, we proved that even if the two states are prepared from different sources, this proposed protocol works effectively. In particular, this protocol can be realized in the two-dimensional system if the four-valued Bell measurement can be realized. Moreover, concerning the finite samples case, we derived optimal testing for several examples. Thus, as has been demonstrated by Hayashi et al [24], it is a future target to demonstrate the proposed testing experimentally.
In this paper, the optimal test is constructed based on a continuous valued POVM. However, any realizable POVM is finite valued. Hence, it is desirable to construct the optimal test based on a finite-valued POVM. This problem has been partially discussed by Hayashi et al, and is more deeply discussed in another paper [30].
The obtained protocol is essentially equivalent to the following procedure based on quantum teleportation. First, we perform quantum teleportation from the system A to the system B, which succeeds when the true state is the required maximally entangled state. Next, we check whether the state of the system B is the initial state of the system A. Hence, an interesting relation between the obtained results and quantum teleportation is expected, Such a relation was partially treated by Virmani and Plenio [28], and will be treated in a forthcoming paper [31] more deeply.
As another problem, Acín et al [26] discussed the problem of testing whether a given n-i.i.d. state of an unknown pure state is the n-fold tensor product of a pure maximally entangled state (not the specific maximally entangled state) in the two-dimensional system. This problem is closely related to universal entanglement concentration [29]. Its d-dimensional case is a future problem.

Appendix A. Proof of lemmas 1 and 2
Assume that a set of tests satisfying the condition C is invariant for the action of G 2 . Let T ∈ T n α, be a test satisfying the condition C; then the test T def = ( f (g) † ) ⊗n T f (g) ⊗n also satisfies the condition C and belongs to the set T n α, . Since Hence, On the other hand, if the G 2 -invariant test T ∈ T n α, satisfies the condition C and Thus, we obtain the inequality opposite to (A.1). Therefore, the proof of lemma 1 is completed. Next, we proceed to prove lemma 2. Since the equation holds for ∀g ∈ G 2 , we obtain We choose a G 1 -invariant test T ∈ T n α, satisfying the condition C and β C α,n,G 1 ( σ ) = β(T, σ ). Then, Thus, we obtain the inequality opposite to (A.2), which yields lemma 2.

Appendix B. Basic properties of classical tests
In classical hypothesis testing, the Neyman-Pearson lemma plays a central role.

Lemma 6.
Assume that the null hypothesis involves one probability distribution P 0 and the alternative one involves another probability distribution P 1 . For any 1 > α > 0, we choose r and γ such that and define the testT P 0 ,P 1 ,α as Then, the test T P 0 ,P 1 ,α is the MP level-α test.
In classical statistics, the function P 0 (x)/P 1 (x) is called the likelihood ratio, which plays an important role.

Appendix C. Proof of proposition 1
Since the likelihood ratio P n (k)/P n q (k) is a monotonic decreasing function of k, the testT ,α equals the testT P n ,P n q ,α . Lemma 6 in appendix B guarantees that the testT ,α is the MP level-α test for the null hypothesis P n . Since a level-α test for the null hypothesis P n is a level-α test for the null hypothesis P n , Since the likelihood ratio P n p 0 (k)/P n p 1 (k) is a monotonic decreasing function of k for p 0 < p 1 , the testT ,α is a likelihood ratio test of P n p 0 and P n p 1 . Hence, corollary 2 guarantees that P n p 0 (T ,α ) P n p 1 (T ,α ). That is, the probability P n p (T ,α ) is a monotonic decreasing function of p. Since the definition of the testT ,α implies that P n (T ,α ) = 1 − α, P n p (T ,α ) 1 − α if p . In other words, the testT ,α is level-α for the null hypothesis P n . Hence, it follows from the inequality (C.1) that the testT ,α is level-α UMP test for the null hypothesis P n .

Appendix E. Proof of (9)
For a fixed density matrix σ on H A,B , we define a density matrix σ q by Then, the state σ belongs to the null hypothesis S . Let T be a U (1)-invariant test at level-α.

A,B
. Since is commutative with the projection P n k,|φ 0 A,B , we obtain In the following, we focus on hypothesis testing of the null hypothesis σ n and the alternative hypothesis σ n p because the test T is a level-α test with the null hypothesis σ n at least. Since P n k,|φ 0 A,B σ ⊗n p P n k,|φ 0 A,B does not depend on q, two states σ n and σ n p commute. So, there exists a common basis {e k,l } diagonalizing them. As they are written as σ = k l P k,l 0 |e k,l e k,l | and σ p = k l P k,l 1 |e k,l e k,l |, our problem is essentially equivalent to classical hypothesis testing of the null hypothesis P 0 def = (P k,l 0 ) and the alternative hypothesis P 1 def = (P k,l 1 ). Since the likelihood ratio is given by the ratio ( k (1 − ) n−k / p k (1 − p) n−k ), we have T n ,α = k lT n ,α (k)|e k,l e k,l |, whereT n ,α (k) is given in (6). Hence, lemma 6 in appendix B guarantees that Tr T σ n p Tr T n ,α σ n p because the test T is a level-α test of the null hypothesis σ n . Since T n ,α is U (1)-invariant, equation (E.1) guarantees that Tr T σ ⊗n Tr T n ,α σ ⊗n .

Appendix F. Proof of lemma 3
Let T be a one-way LOCC ( When Alice observes the data i, Bob's state is (1/Tr M i )M i . Since this test is at level-0, where P i is the projection to the range of the matrix M i .

Appendix G. Proof of lemma 4
It follows from condition (19) that i p i = 1. We choose the vector ϕ def = √ d i p i u i ⊗ u i . Since the function x → |x| 2 is convex, we obtain Hence, On the other hand, Appendix H. Proof of (15) The representation space H A ⊗ H B of an SU (d)-action can be irreducibly decomposed into two subspaces: one is the one-dimensional space φ 0 A,B spanned by φ 0 A,B . The other is its orthogonal The equation φ 0 A,B |T 1,A→B inv |φ 0 A,B = 1 implies that t 0 = 1. Its trace can be calculated as

Lemma 7. If the test T is separable on the space H A ⊗ H B , then
where d is the dimension of H A .
Proof. Since T is separable, T has the form T = l |u i ⊗ u i u i ⊗ u i |. For any two vectors u, u , the Schwarz inequality yields Hence, we have Taking the sum, we obtain (I.1).

Lemma 8. A state u ∈ H A 1 ⊗ H A 2 is maximally entangled if and only if
Proof. The condition (J.1) is equivalent to the condition that φ 0 A 1 ,B 1 |u ⊗ u equals constant times |φ 0 A 2 ,B 2 . When we choose the matrix U as u = i, j A 2 , this condition equals the condition that U I U † is a constant matrix. Thus, if and only if u is maximally entangled, U is unitary, which is equivalent to the condition (J.1). Similarly, we can show that if and only if u is maximal entangled, the condition (J.2) holds. Hence, the desired argument is proved.
The relations (14) and (16) is an eigenvector of T (M) for the eigenvalue 1. Hence, lemma 8 implies (26). On the other hand, Since 0 P T (M)P I , we obtain (28). Next, we consider the case of M = M 2 cov . The test T (M 2 cov ) is invariant with respect to the following action, i.e.
Since the subspace φ 0 ⊥ is an irreducible subspace, the equation (26) implies that where t is a constant. Since the equation (17) implies that Tr T (M 2 cov ) = d 2 , we obtain Indeed, from the SU (d)-invariance, this value depends only on the inner product r def = | u 1 x , u 2 x | 2 . Hence, we can denote it by f (r ). Without loss of generality, we can assume that u 1 x = |0 and The group SU (d) has the subgroup: Hence, We put Hence, putting p(x) Denoting the projection to the symmetric subspace of H ⊗2 A by P S , we obtain The equality holds if p(x) = 1/d for all x. That is, if T = T 1→2 inv , the equality holds. Therefore, we obtain (35). Letting Hence, (M.2) By using the notations can be calculated as Similarly to (M.2), by focusing on the elements g 1,2 , g 2,3 , g 3,1 of SU (d) such that we can prove

Appendix N. Proof of (37)
Let T be an SU (d)-invariant separable level-α test among A 1 , . . . , A n and B 1 , . . . , B n for the null hypothesis S 0 . Then, T has the following form: It follows from the SU (d)-invariance of T that The concavity of the function x → log x implies that log Tr log Tr gu ⊗ gu |σ |gu ⊗ gu ν(dg).
Denoting the RHS by C, we obtain Tr T σ ⊗n k a k e C = e C k a k = e C d n (1 − α).

Appendix O. Proof of (41) and (40)
Thus, the test T has the form where and µ is an arbitrary probability measure. Since our purpose is calculating the minimum value of the second error probability Tr T σ ⊗2 , we can assume that the second term of (20) is 0 without loss of generality. Therefore, lemma 4 implies that Moreover, SU (2) × U (1)-invariance guarantees that T = U (g, θ) ⊗2 T (U (g, θ) ⊗2 ) † for ∀g ∈ SU (2) and ∀θ ∈ R. Hence, Taking the integral, we obtain Therefore, the RHS can be written using projections of irreducible spaces with respect to the action of the group SU (2) × U (1). Indeed, the tensor product space can be decomposed to the direct sum product of the following irreducible spaces with respect to the action of the group SU (2) × U (1): where |i, j 1,2 denotes the vector φ i A 1 ,B 1 ⊗ φ j A 2 ,B 2 and ω = (−1 + √ 3i)/2. The meaning of this notation is as follows. The superscript k = 0, 1, 2 denotes the U (1)-action, i.e. the element e iθ acts on this space as e kθ i . The subscript l = 1, 3, 5 denotes the dimension of the space. In the spaces labelled as , the action |i, j 1,2 → | j, i 1,2 is described as the action of the constant 1. But, in the spaces labelled as , it is described as the action of the constant −1. In the following, for simplicity, we abbreviate the projections to the subspaces k l and k l as k l and k l , respectively. Hence, we obtain 1 2π 2π 0 SU (2) (U (g, θ ) ⊗2 ) † σ ⊗2 U (g, θ) ⊗2 ν(dg)dθ In order to calculate the quantities Tr σ ⊗2 k l and Tr σ ⊗2 k l , we describe the matrix elements of σ on the basis of φ 0 A,B , . . . , φ 3 A,B by x i, j def = φ i A,B |σ |+ i A,B . For convenience, we treat this matrix by use of the notation where a is a real number, b is a three-dimensional complex-valued vector, C is a 3 × 3 Hermitian matrix. Thus, the quantities Tr σ ⊗2 k l and Tr σ ⊗2 k l are evaluated as x i,i x j, j − |x i, j | 2 = 1 2 (Tr C) 2 − (Tr C 2 ) , where C is the complex conjugate of C. As is proved later, the inequalities

47
On the other hand, we focus on the following basis of the space H A 1 ⊗ H A 2 : The other space H B 1 ⊗ H B 2 is spanned by the complex conjugate basis: Using this basis, the irreducible subspaces of H where |i, j A,B denotes the vector ϕ i A 1 ,A 2 ⊗ ϕ j B 1 ,B 2 . In the following, we denote the vectors u x ∈ H A 1 ,A 2 and u x ∈ H B 1 ,B 2 using scalars a x ,a x and three-dimensional vectors w x , w x as Therefore, we obtain (40), which implies the part of (41). In order to prove (O.3), we denote the eigenvalues of C by λ 1 , λ 2 , λ 3 in decreasing order, i.e. λ 1 > λ 2 > λ 3 . First, we prove that aλ 1 |b| 2 as follows. Let s be an arbitrary real number. Then, 0 (s, b)|σ |(s, b) = as 2 + 2 s b 2 + b|C|b .