{'fraction_non_alphanumeric': 0.06465402083889928, 'fraction_numerical': 0.008396626082973932, 'mean_word_length': 4.2257678500199445, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Incompatibility of local measurements provide advantage in local quantum state discrimination

A pack of quantum measurements that cannot be measured simultaneously is said to form a set of incompatible measurements. Every set of incompatible measurements have advantage over the compatible ones in a quantum state discrimination task where one prepares a state from an ensemble and sends it to another party, and the latter tries to detect the state using available measurements. We consider the local quantum state discrimination task where a sender prepares a bipartite state and sends the subsystems to two receivers. The receivers try to detect the sent state using locally incompatible measurements. We analyze the ratio of the probability of successfully guessing the state using incompatible measurements and the maximum probability of successfully guessing the state using compatible measurements. We find that this ratio is upper bounded by a simple function of robustnesses of incompatibilities of the local measurements. Interestingly, corresponding to every pair of sets of incompatible measurements, there exists at least one local state discrimination task where this bound can be achieved. We argue that the optimal local quantum state discrimination task does not present any"nonlocality", where the term is used in the sense of a difference between the ratios, of probabilities of successful detection via incompatible and compatible measurements, in global and local state discriminations. The results can be generalized to the regime of multipartite local quantum state distinguishing tasks.

In a few recent works, connections between minimum-error quantum state discrimination and incompatibility of measurements have been explored [1,5,[17][18][19][20]. The state discrimination task involves a sender, Alice, and a receiver, Bob. Alice prepares a quantum system in a particular state, taken from a particular ensemble and then sends the system to Bob, who is possibly at a distance. The ensemble appears at Alice with a certain given probability, and along with the quantum system, Alice may also send the information regarding the ensemble to Bob. The set of possible ensembles and their constituents are known to both the parties. After receiving the system, Bob tries to identify the state of the system through measurements, i.e., Bob tries to distinguish between the states of the ensemble. It is possible that Bob has access to a fixed set of measurements. Depending on the available set of measurements, Bob may not be able to identify the state of the system perfectly. In such a state discrimination problem, Bob can try to identify the state of the system through what is known as the minimum-error quantum state discrimination strategy [21][22][23][24], by minimizing the overall probability of error in guessing the state of the system.
In Refs. [17,18], it was shown that incompatible measurements provide an advantage over compatible ones in a state discrimination task. Thus, incompatible measurements can be regarded as resources in quantum state discrimination [19]. On the other hand, state discrimination tasks can be used to detect the incompatibility of measurements [5]. We mention here that a relation between quantum state discrimination and channel incompatibility has also been established [25,26].
In this work, we consider quantum state discrimination tasks where more than two parties are involved. Precisely, a sender, Alice, prepares a quantum system of more than one subsystem in a particular state and then sends the subsystems to two or more spatially separated parties. These parties try to identify the state of the system but they are not allowed to employ quantum communication between the spatially separated locations. In this situation, the allowed class of operations can be categorized into two groups, depending on the resources available: (i) local quantum operations (LO) [27][28][29] and (ii) local quantum operations and classical communication (LOCC) .
We establish connections for the two categories of local state discrimination tasks with the incompatibility of available local measurements. The spatially separated parties have access to sets of incompatible measurements, employing which they try to accomplish the given state discrimination task. We derive relations between the probability of successfully guessing (PSG) the state of the system using local incompatible measurements and incompatibility of the local measurements. We provide upper bounds, considering LO and LOCC separately, and analyzing a single round of measurements in the latter case, on the PSG and these upper bounds are the functions of incompatibility of local measurements. Interestingly, corresponding to every set of incompatible local measurements there exists at least one local state discrimination task in which this upper bound can be reached. The optimal state discrimination task which achieves this bound does not exhibit any "nonlocality".
The paper is organized as follows. In Sec. II, we present a formal definition of incompatible measurements and the robustness of incompatibility (ROI), a measure of incompatibility. Then, we consider different state discrimination tasks which we describe in detail in the beginning of Sec. III. In the same section, we establish the connection between ROI and PSG for LO. In Sec. IV, we find a similar connection between ROI and PSG for LOCC. We compare our bound with the bound, given in [18], in Sec. V. Finally, we conclude in Sec. VI.

II. INCOMPATIBLE MEASUREMENTS AND ROBUSTNESS OF INCOMPATIBILITY
A set of measurements {M x } x is called compatible or incompatible depending on their joint measurability. If {M x } x can be measured simultaneously using a parent measurement, G, we say that it is compatible. We denote the measurement operators, associated with different outcomes of a measurement M x and G, by {M a|x } a and {G λ } λ respectively. The measurement operators corresponding to the measurements {M x } x can be expressed in terms of G as where p(a|x, λ) is a conditional probability distribution, being the identity operator. Measurements which are not compatible (i.e., not jointly measurable) are called incompatible measurements. To quantify incompatibility of a set of measurements, {M k } k , the robustness of incompatibility (ROI), denoted I M , was introduced in the literature (for example, see Ref. [18]). ROI can be defined by the minimal amount of noise that is required to be mixed with a set of incompatible measurement to make it compatible, i.e., where M k = {M c|k } c . Therefore, by noise, we mean an arbitrary set of measurements {Λ k } k , Λ k = {Λ c|k } c , which is mixed with the set of measurements {M k } k .

III. CONNECTION OF INCOMPATIBILITY WITH STATE DISCRIMINATION PROBLEM
In several articles [5,[17][18][19], it has been shown that incompatible measurement can provide advantage over the compatible ones in certain state discrimination tasks. In this work, we consider the following state discrimination tasks.
State discrimination using LO.-If the spatially separated parties are restricted to perform local operations only, on their subsystems, in order to accomplish the given state discrimination task, then it is known as state discrimination by LO. We note that in this scenario, classical communication among the parties during the local operations within a round or between the rounds, is not allowed, however, after all the local operations are accomplished, the parties can discuss the measurement outcomes to identify the state of the system [27][28][29].
State discrimination using LOCC.-In the above setting, if the parties are allowed to avail any sequence of classical communication between the local measurements in order to accomplish the given state discrimination task, then it is called state discrimination by LOCC . Thus, in this case, one party can choose her/his measurement, based on the measurement outcomes of the other parties.
State discrimination task with pre-measurement information.-In this task, Alice chooses an ensemble, E y , of bipartite states with probability q(y). Then she prepares a quantum system in a bipartite state, ρ b|y , taken from E y , with probability q(b|y) and sends the subsystems to Bob1 and Bob2 along with the information of y. Bob1 and Bob2 have sets of measurements {M k } k and {N l } l respectively, using which Bob1 and Bob2 want to identify the state of the system. We refer to this as SD1.
State discrimination task with post-measurement information.-This task is the same as the preceding one except that in this case, Bob1 and Bob2 do not have any information about y, prior to the measurement. After the performance of measurements, Alice informs them about the particular ensemble, {E y }, and then depending on y and the measurement outcomes, Bob1 and Bob2 make a guess about ρ b|y . This state discrimination task will be referred to as SD2.
In this section, we consider SD1 and provide a relation between the probability of successfully guessing (PSG) the state of the system, using the local measurements {M k } k and {N l } l , and the incompatibility of these measurements. We first consider the case where only local operations (LO) are allowed. In this case Bob1 and Bob2 apply disjoint measurements, i.e., they do not communicate in between their measurements, but after getting all the measurements done, they are allowed to discuss the measurement outcomes with each other, to together make the final conclusion regarding the state to be identified.

A. Upper bound on the guessing probability using local operations
We consider here the case where Bob1 and Bob2 tries to discriminate the state using LO, and first the case where they have the knowledge of y prior to the measurement. The set of measurements available to Bob1 and Bob2 are locally incompatible. After receiving the state ρ b|y , Bob1 (Bob2) chooses a measurement, M k (N l ), with probability p(k|y) (p(l|y)). The probability of guessing the state correctly using these measurements is given by where M c|k and N d|l are the measurement operators, corresponding to the outcomes c, d, associated with the measurements M k and N l respectively. After getting the outcomes c and d, Bob1 and Bob2 call for a guess regarding the value of b, and this is according to the probability, p(b|c, d, y). Then, the optimal PSG using measurement M k and N l is On the other hand, if Bob1 and Bob2 had locally compatible measurements, and had got the information of y only after performing the measurement, then the PSG would have been Then the maximum PSG using locally compatible measurements in SD2 can be written as where the maximization is taken over the set of locally compatible measurements, CM, and the probabilities, p(k), p(l), p(c, d, y).
From now on, whenever a set of locally incompatible measurements will be used for state discrimination, we will consider that y is known to Bob1 and Bob2 prior to the measurement, and in state discrimination tasks using compatible measurements, we will assume pre-measurement information about y is not available [5,[17][18][19].
Let ROIs of the local measurements {M k } k and {N l } l be I M and I N respectively. Moreover, let the optimization, shown in Eq. (2), be attained by Then, using Eq. (2), one can find: We take the tensor product of these two inequations, to find: We now multiply both sides of the above inequality by q(y)q(b|y)p(k|y)p(l|y)p(b|c, d, y)ρ b|y , and then sum over the parameters c, d, k, l, b, y. Thereafter, taking trace, it becomes: We substitute k,l,c,d p(k|y)p(l|y)p(b|c, d, y)p * (c|k, λ)p * (d|l, ν) by a new conditional probability distribution, p(b|λ, ν, y). Thus we have The expression y,b,λ,ν q(y)q(b|y)p(b|λ, ν, y)tr[ρ b|y G * λ ⊗H * ν ] represents the PSG using a single pair of local measurements, viz., {G λ } λ and {H ν } ν . Hence, it would be less than P C g,LO ({E y }) which is the PSG, optimized over all compatible measurements. So we can write The above relation holds for all probability distributions p(k|y), p(l|y), and p(b|c, d, y). Hence it also holds if we maximize the left hand side of the above inequality with respect to these probabilities. Therefore we get Instead of this bipartite state discrimination task, we can also consider an n-partite state discrimination task, where Alice prepares an n-partite system and then sends the subsystems to Bob1, Bob2, etc. After receiving the subsystems, Bob1, Bob2, . . ., Bobn tries to identify the state using a set of local measurements, say {O 1 Using the same technique as described in the above, it is possible to show the following: B. Bound can be achieved Here we show that corresponding to every pair of incompatible measurements {M k } k and {N l } l , there exists at least one state discrimination task for which this upper bound can be achieved. Before going into the actual proof, let us first state the semi-definite program (SDP), through which ROI of a set of measurements, can be expressed. The forms of the primal SDPs to determine the ROIs of the measurements {M k } k , and {N l } l are given by Here, s = 1 + r, where r is defined in (2).G c = sG c and the positivity of Λ c|k in (2), leads to the inequality c D c (c|k)G c ≥ M c|k , where p(c|k, λ) = c D c (c|k)p(c|λ), c = c 1 c 2 . . . c n , a string of outcomes and D c (c|k) = δ c,c k . A similar situation is true for Eq. (15). The corresponding dual SDPs can be expressed as w ck ≥ 0, tr[X] = 1, where w ck , X, z dl , and Y are the dual variables. See Ref. [18] for a more detailed treatment of these primal and dual problems. We consider the dual variables w * ck , X * , z * dl and Y * for which the optimizations in Eqs. (16) and (17) are achieved and write Let us now introduce some new variables, given by The dual variables w * ck and z * dl are positive, hermitian operators. So, ρ * cd|kl is a quantum state. We now state the corresponding state discrimination task: Alice can choose an ensemble E * kl with probability q * (kl) which consists of bipartite states ρ * cd|kl . The probability of choosing a state ρ * cd|kl from E kl is q * (cd|kl) = q * (cd,kl) q * (kl) , q * (kl) = cd q * (cd, kl). Then, Alice prepares a quantum system in the state ρ * cd|kl and the subsystems are sent to Bob1 and Bob2. The task of Bob1 and Bob2 is to identify cd. To complete the task successfully, Bob1 and Bob2 choose the measurements M k ′ and N l ′ from the sets of measurements {M k } k and {N l } l with probability p(k ′ |kl) and p(l ′ |kl) (in case of SD1) or with probability p(k ′ ) and p(l ′ ) (in case of SD2) respectively. The operators associated with the measurements M k ′ and N l ′ are given by {M c ′ |k ′ } c ′ and {N d ′ |l ′ } d ′ . We first consider SD1 with a specific strategy, i.e., p(k ′ |kl) = δ kk ′ , p(l ′ |kl) = δ ll ′ and p(cd|c ′ , d ′ , kl) = δ cc ′ δ dd ′ . The probability of guessing the state of the system correctly using this particular strategy, can be denoted byP SD1 The last line in the above equation is written using Eq. (18). However, the maximum PSG using compatible measurements without having pre-measurement information is given by Here the maximization is taken over the compatible measurements {G k ′ } and {H l ′ }, and the probability distributions The summation is taken over the set of variables τ = {c, c ′ , d, d ′ , k, k ′ , l, l ′ }. Let the above maximization be achieved for the set of measurements {G * k ′ } and {H * l ′ }, and probabilities We can substitute p * (cd|c ′ , d ′ , kl) by p * (c|c ′ , k)p * (d|d ′ , l) in the above equation. [See Appendix A for a proof.] Let us now decompose p * (c|c ′ , k) and p * (d|d ′ , l) in terms of deterministic probabilities, i.e., p * (c|c ′ , k) = c p * (c|c ′ )D c (c|k) and p * (d|d ′ , l) = d p * (d|d ′ )E d (d|l). c is defined after SDP (15). d can also be defined in a similar way. Using these decompo-sitions, we can write From the dual SDP formulations, i.e., from (16) and (17), we have Using these we can write If we take the sum over c ′ , d ′ , k ′ , l ′ , c, and d on both sides of the above inequality then, since tr[X] = tr[Y] = 1, the left hand side will become equal to unity, and the right hand side will become equal to M * N * P C LO ({E kl })) (see Eq. (23)). Thus we get Using inequality (20), along with the above inequality, we get But the LHS of the above expression is upper bounded by (1 + I M )(1 + I N ), see (12). Therefore, we conclude We want to emphasize the fact that for a general set of ensembles, {E y } y , along with a pair of sets of local measurements, say  (28) is valid for at least one state discrimination task, it may not be true for an arbitrary state discrimination task. But what we have proved is that corresponding to every {M k } k and {N l } l , there exists at least one state discrimination task, it is possible to get the maximum advantage of using locally incompatible measurements. We can generalize Eq. (28) for n-partite state discrimination tasks, and it is possible to write In the succeeding section, we discuss about the case where both local quantum operations and classical communication are allowed to use for accomplishing the task of state discrimination.

IV. STATE DISCRIMINATION USING LOCAL OPERATIONS AND CLASSICAL COMMUNICATION
Let us now consider the same couple of state discrimination tasks (SD1 and SD2) as described in the preceding section, the only difference being that Bob1 and Bob2 now have the facility of using classical communication between the local measurements. Nevertheless, only one round of measurements is considered here, i.e., Bob1 first chooses his measurement, M k , with probability p(k|y) (for SD1) or p(k) (for SD2) and Bob2 chooses his measurement, N l , after getting the knowledge of Bob1's measurement outcome (say, M c|k clicked), with probability p(l|c, y) (for SD1) or p(l|c) (for SD2). In this scenario, the the maximum PSG in case of SD1, using a particular set of measurements, {M k } and {N l }, is given by In case of SD2, the maximum PSG using compatible measurements is where the maximization has to be taken over the set of parameters {{G k }, {H l }, p(k), p(l|c), p(b|c, d, y)}. Suppose again that ROIs of the measurements {M k } k and {N l } l are I M and I N . Now multiplying both side of the inequality of (8) by q(y)q(b|y)p(k|y)p(l|c, y)p(b|c, d, y)ρ b|y , taking trace, and summing over c, d, k, l, b, y, we get c,d,k,l,b,y q(y)q(b|y)p(k|y)p(l|c, y)p(b|c, d, y)tr[ρ b|y M c|k ⊗ N d|l ] ≤ (1 + I M )(1 + I N ) c,d,k,l,b,y,λ q(y)q(b|y)p(k|y)p(l|c, y)p(b|c, d, y) p * (c|k, λ)p * (d|l, ν)tr[ρ b|y G * λ ⊗ H * ν ]. (30) By defining a new probability distribution p(b|λ, ν, y) = c,d,k,l p(b|c, d, y)p(k|y)p(l|c, y)p * (c|k, λ)p * (d|l, ν), we have c,d,k,l,b,y q(y)q(b|y)tr[ρ b|y M c|k ⊗ N d|l ]p(k|y)p(l|c, y)p(b|c, d, y) ≤ (1 + I M ) (1 + I N ) b,y,λ,ν q(y)q(b|y)tr[ρ b|y G * λ ⊗ H * ν ]p(b|λ, ν, y).
The expression, b,y,λ,ν q(y)q(b|y)tr[ρ b|y G * λ ⊗ H * ν ]p(b|λ, ν, y), in the RHS of the above inequality represents the success probability using a particular setting of local measurements. Thus, it is less than or equal to the maximum PSG using compatible local measurements, i.e., c,d,k,l,b,y q(y)q(b|y)tr[ρ b|y M c|k ⊗ N d|l ]p(k|y)p(l|c, y)p(b|c, d, y) Maximizing the LHS of the above equation over the probability distributions p(k|y), p(l|c, y), and p(b|c, d, y), we have i.e., The RHS in the above bound is the same as in the bound given in (12). Thus, in the present setup, we do not have an improved bound by using classical communication.
Achievability and more parties. Considering the same set of ensembles, {E kl }, with states ρ cd|kl , and same probability distributions q(cd, kl) as in Eq. (19) it can be easily shown that in this situation also, it is possible to reach the bound (33). The case of more parties here is again similar to that in the preceding section.

V. THE LOCAL BOUNDS VS THE GLOBAL ONE
Let us now consider a state discrimination task that is different from the ones considered until now, where only two parties are involved, say Alice and Bob. Here, Alice chooses an ensemble E y with probability q(y). She then prepares a quantum system in a state ρ b|y , taken from E y with probability q(b|y) and sends the system to Bob along with the value of y. Bob's task is to identify b. Bob has a set of measurements {Q x } x . Depending on y, Bob chooses a particular measurement, Q x from a set of measurements, {Q x } x with probability p(x|y). The maximum PSG using the set of measurements {Q x } x is denoted by P I ({E y }, {Q x }). The maximum PSG optimized over the set of compatible measurements, when no information is available about y until the measurement is performed, is P C ({E y }). It is known that [18]. In this section, we want to compare this "global" bound with the "local" ones obtained in (12) and in (33).
Let incompatibility of the set of global measurements {M k ⊗ N l } be I M⊗N .
It can be shown that 1 + I M⊗N = (1 + I M )(1 + I N ).
[See Appendix B for a proof.] Since LO is a subset of LOCC and LOCC is a subset of separable operations [31,75] For instance, let us consider that Alice has only one ensemble E 0 , i.e q(y) = δ y,0 . The ensemble consists of equally probable two-qubit maximally entangled states. Since the states are orthogonal, they can be globally distinguished (by measuring onto the basis of the states). Thus we have P C (E 0 ) = 1. But even if classical communication is allowed, they can never be deterministically distinguished using local operations [35]. Thus P C (E 0 ) < P C (E 0 ). Hence, we can say the bounds given in (12) and (33) restrict the PSG using incompatible measurements more than what would in general be possible via the previously known global bound.
Absence of nonlocality in optimal local state discrimination. Since 1 + I M⊗N = (1 + I M )(1 + I N ), we see the the bounds on the ratios of the probabilities, i.e. on P I LO/LOCC /P C LO/LOCC , given in (12) and (33) are equal with the global bound on P I LO/LOCC /P C LO/LOCC presented in [18]. For each set of local measurements, there exists a corresponding state discrimination task where the bounds are achievable. This indicates that there is no "nonlocality" present in the ratio of the success probabilities using incompatibility measures, in the optimal state discrimination process. Here, "nonlocality" is being used in the sense of a difference between the ratios of the probabilities, in global and local state distinguishability. Note however that nonlocality in the individual probabilities might still be present which might have canceled out at the time of taking the ratios.

VI. CONCLUSION
Incompatibility of observables is a signature quantum mechanical property, which is active in almost all quantum tasks. It was known that incompatibility can be used as a resource in quantum state discrimination protocols. In this paper, we have considered local quantum state discrimination tasks, where in one case, only local quantum operations were allowed, and in the other, unidirectional classical communication was allowed along with the local operations. We have presented an upper bound on the ratio of the probability of successfully guessing the sent quantum state using incompatible measurements and the maximum probability of the same using any set of compatible ones. This upper bound is same for both local operations and local operations assisted by unidirectional classical communication, and is an achievable bound in at least one local quantum state discrimination exercise. We have compared the local bound with the existing global bound. We have shown that the optimal local quantum state discriminations do not reveal any nonlocality in the ratios of the probabilities between incompatible and compatible measurements.

ACKNOWLEDGMENTS
We acknowledge partial support from the Department of Science and Technology, Government of India, through QuEST Grant No. DST/ICPS/QUST/Theme-3/2019/120. In Sec. III B, we have considered a particular set of ensembles, {E kl } kl , having particular set of states, {ρ cd|kl } cd . Alice chooses an ensemble E kl with probability q * (kl). She then chooses a state, ρ cd|kl with probability q * (cd|kl) from E kl . The mathematical forms of the variables are given below (readers can also take a look at Eqs. (18) and (19)): We can define two other sets of ensembles, E 1 = {E k } k and E 2 = {E l } l , where each of the ensembles have states, {ρ c|k } c and {ρ d|l } d . Now Alice independently chooses the ensembles E k and E l with probabilities q(k) and q(l) from the sets E 1 and E 2 respectively. And then from these ensembles, she chooses the states ρ c|k and ρ d|l with probabilities q(c|k) and q(d|l) respectively. We define these states and the probabilities in the following way: It is clearly visible that we have defined these new set of states in such a way that ρ cd|kl = ρc|k ⊗ ρd|l. The probabilities are also related: q * (cd, kl) = q * (c, k)q * (d, l), therefore q * (kl) = q * (k)q * (l) and q * (cd|kl) = q * (c|k)q * (d|l). Thus choosing ρ cd|kl from a chosen ensemble E kl is equivalent of independently choosing ρ c|k and ρ d|l from ensembles E k and E l . Let us consider a particular ensemble E kl and particular measurements M k ′ and N l ′ . If we measure M k ′ and N l ′ on a state ρ cd|kl then the probability of getting outcome c ′ and d ′ is denoted by p(c ′ , d ′ |cd, kl). Then from Bayes' theorem, we know that if the measurement outcomes are c ′ and d ′ , then the probability that the state on which the measurement have been done is ρ cd|kl is given by p(cd|c ′ , d ′ , kl) = p(c ′ , d ′ |cd, kl)q * (cd, kl) cd p(c ′ , d ′ |cd, kl)q * (cd, kl) = tr ρ cd|kl M c ′ |k ′ ⊗ N d ′ |l ′ q * (c, k)q * (d, l) cd tr ρ cd|kl M c ′ |k ′ ⊗ N d ′ |l ′ q * (c, k)q * (d, l) = p(c|c ′ , k)p(d|d ′ , l).

B. Relation between global and local incompatibility
Let us suppose that for two given sets of measurements {M k } and {N l }, the primal SDPs, defined in Eqs. (14) and (15) c,dG Incompatibility of the global measurement {M k ⊗ N l } can be derived using the following primal SDP: s.t.