More than two decades ago, Bennett \emph{et al.} [Phys. Rev. A \textbf{59}, 1070 (1999)] asked whether perfect local discrimination of orthogonal quantum states can require more than two copies. This question was subsequently answered for adaptive protocols that process the copies separately [Phys. Rev. Lett. \textbf{126}, 210505 (2021)], but remained open when each laboratory is allowed to process its local copies collectively. Here we resolve this stronger setting and identify collective access across repeated local inputs as a distinct resource. For every fixed odd-prime local dimension, there exist complete maximal-stabilizer eigenbases whose copy complexity under individual-copy separable measurements diverges with system size. Under collective processing this behavior changes sharply: every maximal-stabilizer eigenbasis in odd-prime local dimension is perfectly decoded by one-round collective LOCC using at most three copies. Collective processing, however, does not remove multicopy hardness in general. For every fixed local dimension $d\ge2$, we prove the existence, within an explicit phase family, of complete bases whose copy complexity remains unbounded even under collective separable measurements, with a square root of the number of subsystems as lower-bound scale. Thus sample number, spatial measurement power, and coherent access across repeated local inputs are distinct resources in distributed quantum measurement.
More than two decades ago, Bennett \emph{et al.} [Phys. Rev. A \textbf{59}, 1070 (1999)] asked whether perfect local discrimination of orthogonal quantum states can require more than two copies. This question was subsequently answered for adaptive protocols that process the copies separately [Phys. Rev. Lett. \textbf{126}, 210505 (2021)], but remained open when each laboratory is allowed to process its local copies collectively. Here we resolve this stronger setting and identify collective access across repeated local inputs as a distinct resource. For every fixed odd-prime local dimension, there exist complete maximal-stabilizer eigenbases whose copy complexity under individual-copy separable measurements diverges with system size. Under collective processing this behavior changes sharply: every maximal-stabilizer eigenbasis in odd-prime local dimension is perfectly decoded by one-round collective LOCC using at most three copies. Collective processing, however, does not remove multicopy hardness in general. For every fixed local dimension d≥2, we prove the existence, within an explicit phase family, of complete bases whose copy complexity remains unbounded even under collective separable measurements, with a square root of the number of subsystems as lower-bound scale. Thus sample number, spatial measurement power, and coherent access across repeated local inputs are distinct resources in distributed quantum measurement.
Ensembles of composite quantum states can exhibit nonlocal behavior in the sense that their optimal discrimination may require global operations. Such an ensemble containing N pairwise orthogonal pure states, however, can always be perfectly distinguished under an adaptive local scheme if (N-1) copies of the state are available. In this Letter, we provide examples of orthonormal bases in two-qubit Hilbert space whose adaptive discrimination require three copies of the state. For this composite system, we analyze multicopy adaptive local distinguishability of orthogonal ensembles in full generality which, in turn, assigns varying nonlocal strength to different such ensembles. We also come up with ensembles whose discrimination under an adaptive separable scheme require less numbers of copies than adaptive local schemes. Our construction finds important application in multipartite secret sharing tasks and indicates toward an intriguing superadditivity phenomenon for locally accessible information.
Perfect discrimination of an orthogonal 2⊗2 basis may require collective separable measurements on two copies
We consider an orthogonal basis of C2⊗C2 with three entangled states and one product state. This basis is not perfectly distinguishable by LOCC or even by separable measurements when a single copy of the unknown state is available. We show that with two copies no adaptive separable protocol, which measures the copies one at a time, distinguishes the basis perfectly. We then show that a collective separable measurement on the two copies does. Whether two copies suffice for LOCC remains open.
Quantum measurement and control science provides a toolkit for implementing quantum information protocols, overcoming noisy operation and minimizing the use of costly quantum resources. The importance of finding optimal measurement and control strategies is revealed in the task of quantum state discrimination, a characteristic feature of quantum mechanics and a primitive for quantum information science and technology. When a quantum system is prepared in one of two known nonorthogonal quantum states, no measurement can deterministically tell which state the system is in. Two strategies can be applied to discriminate non-orthogonal states efficiently. Unambiguous state discrimination (USD) is strategy that provides a guess which is either correct, or inconclusive [1]. Alternatively, minimum error discrimination (MED) strategy will always provide a conclusive answer, which can sometimes be incorrect [2]. Perfect nonorthogonal state discrimination is still impossible even when multiple copies of the system are available, although different strategies increase the chance of having conclusive (for USD), or correct (for MED) result. Multiple-copy measurement strategies can be either collective, when a single measurement involves all the copies of the system, or local, when each copy of the system is measured separately. The latter can be further divided into fixed, where the measurement applied to each copy is the same, or adaptive, where the measurement on each subsequent copy depends on the outcomes of the previous measurements [3,4]. A multiple-copy MED strategy is defined by its goal of minimizing the average error for fixed resources, i.e. the number of copies of the system. An alternative approach we consider, is to minimize the average resources required, while keeping errors below a given bound [5]. This approach is central for fault-tolerant quantum computing and has been applied to a number of quantum control strategies [6,7]. We call the corresponding state discrimination task the guaranteed bounded error discrimination (GBED) task. Intuitively, one may assume that the multiple-copy strategies optimal for MED would be also optimal for GBED. We experimentally apply two known local non-adaptive strategies, previously considered for the MED task, to the two-state GBED problem [5]. We then derive and experimentally demonstrate a new local strategy, designed specifically for the GBED task that outperforms other strategies. Moreover, it performs usually better than, and in the regime of small error, scales better than, the theoretical performance of the optimal adaptive strategy for the MED task. The discovered reversal in the performance of schemes when swapping the task definition from performance-maximization to resource-minimization, is similar to that previously observed in state purification [6], suggesting that this phenomenon is a generic one. [1] I. D. Ivanovic, Phys. Lett. A 123, 257 (1987). [2] C. W. Helstrom, Quantum detection and estimation theory (1976). [3] A. Acín, et al., Phys. Rev. A 71, 032338 (2005). [4] B. L. Higgins, et al., Phys. Rev. Lett. 103, 220503 (2009). [5] S. Slussarenko, et al., PRL 118, 030502 (2017). [6] H. M. Wiseman and J. F. Ralph. New J. Phys. 8, 90 (2006). [7] J. Combes, et al., Phys. Rev. Lett. 100, 160503 (2008).
Can measurements chosen adaptively, in response to earlier outcomes, be exponentially more informative than measurements fixed in advance? In many quantum settings, adaptivity helps at most polynomially; known exponential advantages require coherent measurements across copies or restricted measurements. We show an exponential advantage measuring one copy at a time, with no quantum memory kept between state copies, even when the nonadaptive learner may perform arbitrary global measurements on each copy. We construct explicit families of multi-qubit states in which a scalar parameter is encoded in one of exponentially many candidate observables, with an auxiliary marker indicating which one. An adaptive protocol identifies the relevant observable and then measures it using only single-qubit measurements and one round of adaptation. At fixed accuracy, it needs logarithmically many copies as the qubit count grows, whereas every nonadaptive protocol needs exponentially many. The same mechanism yields an exponential gap in local asymptotic minimax mean squared error constants, tight up to the exponential base. The states are separable and classical in label-dependent product bases; no entanglement or other nonclassical correlations are required. The gap comes from measurement incompatibility: the candidate observables cannot be measured jointly, so a fixed measurement must divide its information budget among them, while adaptivity selects the informative one. Accordingly, the separation disappears when the candidate observables commute, and cannot occur with fully observed classical samples. Measurement incompatibility thus emerges as a resource for quantum learning, distinct from entanglement and quantum memory, that classical feedback alone can exploit.
Quantum nonlocality, a form of nonclassicality, serves as a crucial resource in quantum information processing. Nonlocality in quantum systems is exemplified by the perfect discrimination of orthogonal quantum states through single joint measurements, which is unattainable using local operations and classical communication (LOCC). Bandyopadhyay [S. Bandyopadhyay, Phys. Rev. Lett. 106, 210402 (2011)] established that, with LOCC alone, at most $N\ensuremath{-}1$ copies of N orthogonal quantum states are required for perfect discrimination. Here, we present the experimental verification of orthogonal quantum state discrimination employing LOCC. Our results demonstrate that up to three copies suffice for the perfect distinction of four orthogonal two-qubit states via LOCC. The verified method of perfectly distinguishing orthogonal quantum state sets offers potential applications in quantum information processes, including quantum secret sharing and quantum teleportation.
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.