Quantum state discrimination, which seeks to correctly determine the state of a quantum system among a set of possible states through a finite number of measurements, has become fundamental for the increasing development of quantum technologies due to its underlying relation to various quantum information processing and communication tasks. In this work, we thoroughly explore the problem of distinguishing between two and three finite photon ensembles with identical compressed density operators, with a main focus on local solutions, i.e., with no entangling operations among ensemble photons. Using semidefinite programming, we provide optimal analytical and numerical discrimination strategies in the very low photon-number regime. Then, multiple approaches to an interferometry-based solution for the three-ensemble problem are analyzed based on existing experimental literature in single-photon polarization state discrimination. However, none of these approaches outperform a trivial extension to a well-known local solution of the two-ensemble problem given by d'Espagnat. An in-depth discussion on the local distinguishability of the three ensembles is given throughout the text. The results from these analyses suggest that more successful discrimination protocols require performing entangling operations among ensemble photons in order to exploit the inner correlations within each ensemble.
Quantum state discrimination, which seeks to correctly determine the state of a quantum system among a set of possible states through a finite number of measurements, has become fundamental for the increasing development of quantum technologies due to its underlying relation to various quantum information processing and communication tasks. In this work, we thoroughly explore the problem of distinguishing between two and three finite photon ensembles with identical compressed density operators, with a main focus on local solutions, i.e., with no entangling operations among ensemble photons. Using semidefinite programming, we provide optimal analytical and numerical discrimination strategies in the very low photon-number regime. Then, multiple approaches to an interferometry-based solution for the three-ensemble problem are analyzed based on existing experimental literature in single-photon polarization state discrimination. However, none of these approaches outperform a trivial extension to a well-known local solution of the two-ensemble problem given by d'Espagnat. An in-depth discussion on the local distinguishability of the three ensembles is given throughout the text. The results from these analyses suggest that more successful discrimination protocols require performing entangling operations among ensemble photons in order to exploit the inner correlations within each ensemble.
A key concept of quantum information theory is that accessing information encoded in a quantum system requires us to discriminate between several possible states the system could be in. A natural generalization of this problem, namely, quantum sequence discrimination, appears in various quantum information processing tasks, the objective being to determine the state of a finite sequence of quantum states. Since such a sequence is a composite quantum system, the fundamental question is whether an optimal measurement is local, i.e., comprising measurements on the individual members, or collective, i.e., requiring joint measurement(s). In some known instances of this problem, the optimal measurement is local, whereas in others, it is collective. But, so far, a definite prescription based solely on the problem description has been lacking. In this paper, we prove that if the members of a given sequence are secretly and independently drawn from an ensemble or even from different ensembles, the optimum success probability is achievable by fixed local measurements on the individual members of the sequence, and no collective measurement is necessary. This holds for both minimum-error and unambiguous state discrimination paradigms.
Two pure orthogonal quantum states can be perfectly distinguished by sequential local action of multiple pairs of parties. However, this process typically leads to the complete dissolution of entanglement in the states being discriminated. We propose a protocol that allows an arbitrary number of pairs of parties to distinguish between any two orthogonal, entangled, two-qubit pure states using local quantum operations and classical communication, with a success probability greater than that of random guessing, while ensuring that at each step, the individual ensemble states retain a finite amount of entanglement. Our protocol employs the minimum-error state discrimination approach. For demonstrating the retention of entanglement in the ensemble states at each step, we use logarithmic negativity as well as the concept of entanglement witnessing. For a large family of sets of the two states, the success probability of discrimination can be as close as required to unity, while sustaining a finite amount of entanglement in each step.
Quantum state discrimination is one of the most fundamental tasks in quantum information since it describes the read-out process of classical information from quantum states in quantum communications.In quantum state discrimination, a given test state is guaranteed to be identical to one of a set of candidate quantum states.The task is to determine the identity of the test state by performing an appropriate quantum measurement based on the classical descriptions of the candidate states.Quantum state discrimination also has several connections to the principles of quantum mechanics such as the no-cloning theorem and the no-signaling principle.One of fundamental properties of quantum mechanics is that the exact classical description of unknown quantum states cannot be retrieved from a single copy of the physical state.Quantum state discrimination in the case that the candidate states are given as two unknown physical states (reference states) but their classical descriptions are not given, has been investigated.The two reference states are labeled one and two, respectively.The task is to determine which of the two reference states is equivalent to the test state, whereas the descriptions of the states are not of interest.We call such a task as equivalence determination of quantum states.Discrimination of quantum operations has been also investigated and it has been shown that there is an intrinsic difference between discrimination of quantum state and discrimination of quantum operations.We analyze a unitary operation version of the equivalence determination task by using the concept of higher-order quantum computation to investigate properties of discrimination of other quantum objects.We consider three black-boxes (test box and reference boxes) implementing unknown unitary operations given as physical systems.The test box is guaranteed to implement one of the two unitary operations given by the reference boxes.Equivalence determination of unitary operations is a task to determine the reference box implementing the same unitary operation as the one by the test box.We assume that the unitary operations of the reference boxes are randomly sampled from the Haar measure of SU(2).We first consider the case that each of three black-boxes can be used only once both in parallel and ordered uses of the black-boxes.We show that the optimal success probability for equivalence determination is 7/8 by obtaining analytical solutions of the corresponding semidefinite programmings (SDPs) both in the parallel and ordered strategies.The optimal success probability can be achieved even without using one of the reference boxes.We also showed that entanglement of an initial state across the systems on which the reference boxes act and the test box acts is necessary to achieve the optimal success probability.We then consider the case that the multiple uses of the reference boxes are allowed.We consider the parallel strategies up to four uses of each of the reference boxes and obtain the optimal success probabilities by numerically solving the semidefinite programmings.We numerically show that the ordered strategies give improvement over the parallel strategies and the optimal success probability varies depending on the order of the black-boxes when one reference box is used once and the other reference box is used twice.This result indicates that an appropriate order of the black-boxes under the ordered strategies is necessary to obtain improvement over the parallel strategies. Chapter 1 Introduction Notations• H i , K j : Hilbert spaces labeled with i and j.• L(H): The set of linear operators on H.• L(H, K): The set of linear operators from H to K.• S(H) : The set of positive semidefinite linear operation of unit trace on H • Tr H : The partial trace over H • U † : The conjugate transpose of U • {Π i }: The positive operator-valued measure • { Π i }: The quantum tester • δ ij : The Kronecker delta, equal to 1 if i = j and 0 otherwise • dµ(•) : The Haar measure, the uniform distribution of the space of unitary operations • |I⟩⟩ : Unnormalized maximally entangled vector defined as |I⟩⟩ = i |i⟩|i⟩ • M : A quantum channel • M : Choi operator of a quantum channel M• U j : The irreducible subspace of the total angular momentum j.
A linear optics-based scheme to implement various quantum information processing tasks is of paramount importance due to ease of implementation and low noise. Many information-theoretic tasks depend on the successful discrimination of Bell states. A no-go theorem has been proved in literature which tells that it is not possible to perfectly discriminate among the four Bell states by restricting measurement apparatus to linear optical elements. The success probability is only 50%. Through using extra resources such as hyper entanglement, ancillary entanglement, and even a minimum amount of non-linearity complete Bell-state discrimination can be achieved. The success probability for Bell-like state discrimination is only 25%. We find that this can be boosted up to 50% using hyperentanglement in polarization, momentum, or OAM degrees of freedom of the photons which is in contrast to the Bell-state discrimination scenario where 100% can be achieved. Furthermore, we find that by using correlation in time of the photons all four Bell states can be distinguished with 100% success probability while for the Bell-like state discrimination, it strictly lies between 25% and 50% depending on the state parameter with only three Bell-like states being distinguishable. We also observe a similar contrast when we use ancillary entangled photons. While the success probability for all four Bell-state discrimination increases as 1−2N1 where N is the number of ancillary photons for Bell-like states it depends again on the state parameters and can be less than 25% in some cases. Also adding further ancillary photons decreases the success probability. We then show that using non-linear gadgets namely SFG 100% success probability can be achieved even for Bell-like state discrimination.
Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems, they are technologically demanding. Therefore, determining the constraints posed by linear optics and optimizing linear optical operations for specific tasks under those constraints, without the use of ancillae, can facilitate their potential implementation. Here, we consider the task of unambiguously discriminating between Bell-like states using linear optics and without the use of ancillary photons. This is a basic problem relevant in diverse settings, for example, in the measurement of the output of an entangling quantum circuit or for entanglement swapping at a quantum repeater station. While it is known that exact Bell states of two qubits can be discriminated with an optimal success probability of 50%, we find, surprisingly, that for Bell-like states the optimal probability can be only 25%. We analyze a set of Bell-like states in terms of their distinguishability, entanglement as measured by concurrence, and parameters of the beam-splitter network used for unambiguous discrimination. Further, we provide the linear optical configuration comprising single-photon detectors and beam splitters with input-state-dependent parameters that achieves optimal discrimination in the Bell-like case.
Discrimination between quantum states is a fundamental task in quantum information theory. Given two arbitrary tensor-product quantum states (TPQS) ρ±=ρ±(1)⊗⋯⊗ρ±(N), determining the joint N-system measurement to optimally distinguish between the two states is a hard problem. Thus, there is great interest in identifying local measurement schemes that are optimal or close-to-optimal. In this work, we focus on distinguishing between two general TPQS. We begin by generalizing previous work by Acin et al. (Phys. Rev. A 71, 032338) to show that a locally greedy (LG) scheme using Bayesian updating can optimally distinguish between two states that can be written as tensor products of arbitrary pure states. Then, we show that even in the limit of large N the same algorithm cannot distinguish tensor products of mixed states with vanishing error probability. This poor asymptotic behavior occurs because the Helstrom measurement becomes trivial for sufficiently biased priors. Based on this, we introduce a modified locally greedy (MLG) scheme with strictly better performance. In the second part of this work, we compare these simple local schemes with a general dynamic programming (DP) approach that finds the optimal series of local measurements to distinguish the two states. When the subsystems are non-identical, we demonstrate that the ordering of the systems affects performance and we extend the DP technique to determine the optimal ordering adaptively. Finally, in contrast to the binary optimal collective measurement, we show that adaptive protocols on sufficiently large (e.g., qutrit) subsystems must contain non-binary measurements to be optimal. (The code that produced the simulation results in this paper can be found at: https://github.com/SarahBrandsen/AdaptiveStateDiscrimination)