Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d times d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations. This is a shortened version of a paper [1] recently submitted to Communications in Mathematical Physics; here the proofs have been omitted.
Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d times d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations. This is a shortened version of a paper [1] recently submitted to Communications in Mathematical Physics; here the proofs have been omitted.
We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability, thus establishing the associated Chernoff exponent. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite asymptotic separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB.
Quantum data hiding asks how much distinguishing power can be lost when global measurements are restricted to local measurements and classical communication. In this work, we establish sharp results and improved bounds for several natural classes of restricted measurements. For bipartite systems on Cn⊗Cm, we prove that the optimal data-hiding ratios against separable and LOCC measurements are both min{n,m}. This result follows from a stronger result that, for every 2≤p≤∞, the largest centered Schatten p-ball whose associated binary measurements are implementable by finite-round LOCC has radius min{n,m}2/p−1. This strengthens the classic separable-ball theorems, while also providing an explicit finite-round LOCC implementation. For Alice-first one-way LOCC with Alice's local dimension equal to n, we prove that the optimal ratio is (1+o(1))n, with the upper bound obtained from a Gaussian rank-one POVM. For local operations without communication, we improve the universal upper bound to (π3/4+o(1))min{n,m}.
In this paper, we consider the conditions under which a given ensemble of two-qubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the perfect distinguishability problem of two-qubit ensembles-both for separable operations and LOCC-by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state. Then, for the well-known task of minimum error discrimination, it is shown that almost all two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that any two pure states can be distinguished optimally by LOCC. Special attention is given to ensembles that lack entanglement, and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by LOCC, thus providing new examples of the phenomenon known as non-locality without entanglement. We next consider an example of N parties who each share the same state but who are ignorant of its identity. The state is drawn from the rotationally invariant trine ensemble, and we establish a tight connection between the N-copy ensemble and Shor's lifted single-copy ensemble. For any finite N, we prove that optimal identification of the states cannot be achieved by LOCC; however, as N→∞, LOCC can indeed discriminate the states optimally. This is the first result of its kind. Finally, we turn to the task of unambiguous discrimination and derive new lower bounds on the LOCC inconclusive probability for symmetric states. When applied to the double trine ensemble, this leads to a rather different distinguishability character than when the minimum error probability is considered.
Quantum state discrimination involves identifying a given state out of a set of possible states. When the states are mutually orthogonal, perfect state discrimination is always possible using a global measurement. In the case of multipartite systems when the parties are constrained to use multiple rounds of local operations and classical communication (LOCC), perfect state discrimination is often impossible even with the use of asymptotic LOCC, wherein an error is allowed but must vanish in the limit of an infinite number of rounds. Utilizing our recent results on asymptotic LOCC, we derive a lower bound on the error probability for LOCC discrimination of any given set of mutually orthogonal pure states. Informed by the insights gained from this lower bound, we are able to prove necessary conditions for perfect state discrimination by asymptotic LOCC. We then illustrate by example the power of these necessary conditions in significantly simplifying the determination of whether perfect discrimination of a given set of states can be accomplished arbitrarily closely using LOCC. The latter examples include a proof that perfect discrimination by asymptotic LOCC is impossible for any minimal unextendible product basis (UPB), where minimal means that for the given multipartite system no UPB with a smaller number of states can exist. We also give a simple proof that what has been called strong nonlocality without entanglement is considerably stronger than had previously been demonstrated.
Quantum data hiding concerns pairs of states which are highly distinguishable with global measurements, yet nearly indistinguishable when restricted to local operations and classical communication. More than two decades after Eggeling and Werner introduced a multipartite hiding scheme based on Werner states, the optimal dependence of its uniform security guarantee on the number of parties and local dimension remained unknown. We resolve this problem by showing that the distinguishing bias of any pair of n-qudit Werner states is O(n2/d) under measurements whose effects remain positive under partial transposition (PPT) across every bipartition. An explicit pair attains this scaling using only nonadaptive local measurements, establishing worst-case optimality and showing that the PPT relaxation preserves the optimal dependence on both the number of parties and the local dimension. At fixed security, this extends the certified hiding regime from n=O(d1/4) to n=O(d). Beyond data hiding, the same bound implies that testing any nontrivial unitarily invariant property with adaptive single-copy measurements requires Ω(d) copies, yielding separations for any property with dimension-independent sample complexity under collective measurements. Our proof reduces the distinguishing bias to trace norms of partially transposed operators and analyzes them using mixed Schur-Weyl duality, demonstrating the utility of representation theoretic methods developed for port-based teleportation to data hiding and quantum property testing.
Entanglement and nonlocality play a fundamental role in quantum computing. To understand the interplay between these phenomena, researchers have considered the model of local operations and classical communication, or LOCC for short, which is a restricted subset of all possible operations that can be performed on a multipartite quantum system. The task of distinguishing states from a set that is known a priori to all parties is one of the most basic problems among those used to test the power of LOCC protocols, and it has direct applications to quantum data hiding, secret sharing and quantum channel capacity. The focus of this thesis is on state distinguishability problems for classes of quantum operations that are more powerful than LOCC, yet more restricted than global operations, namely the classes of separable and positive-partial-transpose (PPT) measurements. We \nbuild a framework based on convex optimization (on cone programming, in particular) to study such problems. Compared to previous approaches to the problem, the method described in this thesis provides precise numerical bounds and quantitative analytic results. By combining the duality theory of cone programming with the channel-state duality, we also establish a novel connection between the state distinguishability problem and the study of positive linear maps, which is a topic of independent interest in quantum information theory. \n \nWe apply our framework to several questions that were left open in previous works regarding the distinguishability of maximally entangled states and unextendable product sets. First, we exhibit small sets of orthogonal maximally entangled states in that are not perfectly distinguishable by LOCC. As a consequence of this, we show a gap between the power of PPT and separable measurements for the task of distinguishing sets consisting only of maximally entangled states. Furthermore, we prove tight bounds on the entanglement cost that is necessary to distinguish any sets of Bell states, thus showing that quantum teleportation is optimal for this task. Finally, we provide an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements, along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements.