We study the distinguishability associated to either a single POVM or a family of POVMs. These norms quantify how quantum measurements perform in the task of discriminating two quantum states. We address the following questions regarding their behaviour in the high-dimensional regime:
(i) How many POVMs are needed to approximate the family of all POVMs on C^d? We show that the answer is exp(O(d^2)).
(ii) How many outcomes are required for a POVM to emulate the uniform POVM on C^d? Improving on previous results, we show that O(d^2) outcomes are necessary and sufficient. Moreover, using the concept of zonoid and approximation results from Banach space theory, we show that this property extends (in a weaker form) to an arbitrary POVM.
(iii) On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to separable measurements implies a substantial loss.
We study the distinguishability associated to either a single POVM or a family of POVMs. These norms quantify how quantum measurements perform in the task of discriminating two quantum states. We address the following questions regarding their behaviour in the high-dimensional regime:
(i) How many POVMs are needed to approximate the family of all POVMs on C^d? We show that the answer is exp(O(d^2)).
(ii) How many outcomes are required for a POVM to emulate the uniform POVM on C^d? Improving on previous results, we show that O(d^2) outcomes are necessary and sufficient. Moreover, using the concept of zonoid and approximation results from Banach space theory, we show that this property extends (in a weaker form) to an arbitrary POVM.
(iii) On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to separable measurements implies a substantial loss.
We study the distinguishability norms associated to families of locally restricted POVMs on multipartite systems. These norms (introduced by Matthews, Wehner and Winter) quantify how quantum measurements, subject to locality constraints, perform in the task of discriminating two multipartite quantum states. We mainly address the following question regarding the behaviour of these distinguishability norms in the highdimensional regime: On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to local measurements and classical communication, or even just to separable measurements, implies a substantial loss. We also provide examples of state pairs which can be perfectly distinguished by local measurements if (one-way) classical communication is allowed between the parties, but very poorly without it. Finally, we study how many POVMs are needed to distinguish almost perfectly any pair of states on C^d, showing that the answer is exp(Θ(d^2 )).
In this paper, we establish a connection between zonoids (a concept from classical convex geometry) and the distinguishability norms associated to quantum measurements or POVMs (Positive Operator-Valued Measures), recently introduced in quantum information theory. This correspondence allows us to state and prove the POVM version of classical results from the local theory of Banach spaces about the approximation of zonoids by zonotopes. We show that on Cd , the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM having only O(d2) outcomes. We also show that similar (but weaker) approximation results actually hold for any POVM on Cd . By considering an appropriate notion of tensor product for zonoids, we extend our results to the multipartite setting: we show, roughly speaking, that local POVMs may be sparsified locally. In particular, the local uniform POVM on Cd1⊗⋯⊗Cdk can be approximated by a discrete POVM which is local and has O(d12×⋯×dk2) outcomes.
We study the distinguishability of bipartite quantum states by positive operator-valued measures with positive partial transpose (PPT POVMs). The contributions of this paper include: 1) we give a negative answer to an open problem of showing a limitation of a previous known method for detecting nondistinguishability; 2) we show that a maximally entangled state and its orthogonal complement, no matter how many copies are supplied, cannot be distinguished by the PPT POVMs, even unambiguously. This result is much stronger than the previous known ones; and 3) we study the entanglement cost of distinguishing quantum states. It is proved that √{2/3}|00〉+√{1/3}|11〉 is sufficient and necessary for distinguishing three Bell states by the PPT POVMs. An upper bound of entanglement cost of distinguishing a d ⊗ d pure state and its orthogonal complement is obtained for separable operations. Based on this bound, we are able to construct two orthogonal quantum states, which cannot be distinguished unambiguously by separable POVMs, but finite copies would make them perfectly distinguishable by local operations and classical communication. We further observe that a two-qubit maximally entangled state is always enough for distinguishing a d ⊗ d pure state and its orthogonal complement by the PPT POVMs, no matter the value of d. In sharp contrast, an entangled state with Schmidt number at least d is always needed for distinguishing such two states by separable POVMs. As an application, we show that the entanglement cost of distinguishing a d ⊗ d maximally entangled state and its orthogonal complement must be a maximally entangled state for d=2, which implies that teleportation is optimal, and in general, it could be chosen as O(logd/d).
The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measurements yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two bipartite quantum states, employing positive operator-valued measures (POVMs) with positive partial transpose (PPT) to achieve optimal success probability through general entangled measurements. We first introduce two quantities called the spectral PPT-distance and relative spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show these quantities are related to the entanglement cost of optimal discrimination by PPT POVMs. Following this, we establish bounds and develop SDP hierarchies to estimate the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging these results, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measurements yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two bipartite quantum states, employing positive operator-valued measures (POVMs) with positive partial transpose (PPT) to achieve optimal success probability through general entangled measurements. First, we introduce an efficiently computable quantity called the spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show that it can be a lower bound for the entanglement cost of optimal discrimination by PPT POVMs. Second, we establish an upper bound on the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging this result, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
We study the distinguishability of quantum states under local operations with classical communication (LOCC), separable, and positive-partial-transpose (PPT) measurements, focusing on locally diagonal orthogonally invariant (LDOI) states -- those invariant under local diagonal orthogonal twirling. This class includes many important families such as Werner states, isotropic states, X-states, and Dicke states. We show that optimal PPT and separable measurements for distinguishing LDOI states can always be taken to be LDOI, and the LOCC supremum can be approached by LDOI LOCC POVMs, enabling a dimensional reduction from n4 to O(n2) in the associated optimization problems. We establish efficiently computable bounds on the distinguishability of orthonormal LDOI bases and prove that for a broad class of such bases -- including all two-qubit cases -- the LOCC supremum equals the PPT and separable optima. More generally, we show the gap between PPT and LOCC distinguishability is at most (n−2)/(2n2) for local dimension n.