Optimal Success Probability Research Articles

Identification of malfunctioning quantum devices

We consider the problem of correctly identifying a malfunctioning quantum device that forms part of a network of N such devices, which can be considered as the quantum analog of classical anomaly detection. In the case where the devices in question are sources assumed to prepare identical quantum pure states, with the faulty source producing a different anomalous pure state, we show that the optimal probability of successful identification requires a global quantum measurement. We also put forth several local measurement strategies—both adaptive and nonadaptive—that achieve the same optimal probability of success in the limit where the number of devices to be checked is large. In the case where the faulty device performs a known unitary operation, we show that the use of entangled probes provides an improvement that even allows perfect identification for values of the unitary parameter that surpass a certain threshold. Finally, if the faulty device implements a known qubit channel, we find that the optimal probability for detecting the position of rank-one and rank-two Pauli channels can be achieved by product state inputs and separable measurements for any size of network, whereas for rank-three and general amplitude damping channels, optimal identification requires entanglement with N qubit ancillas. Published by the American Physical Society 2024

Conclusive Discrimination by $$N$$ Sequential Receivers between $$r\geq2$$ Arbitrary Quantum States

In the present paper, we develop a general mathematical framework for discrimination between $$r\geq2$$ quantum states by $$N\geq1$$ sequential receivers for the case in which every receiver obtains a conclusive result. This type of discrimination constitutes an $$N$$ -sequential extension of the minimum-error discrimination by one receiver. The developed general framework, which is valid for a conclusive discrimination between any number $$r\geq2$$ of quantum states, pure or mixed, of an arbitrary dimension and any number $$N\geq1$$ of sequential receivers, is based on the notion of a quantum state instrument, and this allows us to derive new important general results. In particular, we find a general condition on $$r\geq2$$ quantum states under which, within the strategy in which all types of receivers’ quantum measurements are allowed, the optimal success probability of the $$N$$ -sequential conclusive discrimination between these $$r\geq2$$ states is equal to that of the first receiver for any number $$N\geq2$$ of further sequential receivers and specify the corresponding optimal protocol. Furthermore, we extend our general framework to include an $$N$$ -sequential conclusive discrimination between $$r\geq2$$ arbitrary quantum states under a noisy communication. As an example, we analyze analytically and numerically a two-sequential conclusive discrimination between two qubit states via depolarizing quantum channels. The derived new general results are important both from the theoretical point of view and for the development of a successful multipartite quantum communication via noisy quantum channels.

Robust certification of unsharp instruments through sequential quantum advantages in a prepare-measure communication game

Communication games are one of the widely used tools that are designed to demonstrate quantum supremacy over classical resources in which two or more parties collaborate to perform an information processing task to achieve the highest success probability of winning the game. We propose a specific two-party communication game in the prepare-measure scenario that relies on an encoding-decoding task of specific information. We first demonstrate that quantum theory outperforms the classical preparation noncontextual theory and the optimal quantum success probability of such a communication game enables the semi-device-independent certification of qubit states and measurements. Further, we consider the sequential sharing of quantum preparation contextuality and show that, at most, two sequential observers can share the quantum advantage. The suboptimal quantum advantages for two sequential observers form an optimal pair which certifies a unique value of the unsharpness parameter of the first observer. Since the practical implementation inevitably introduces noise, we devised a scheme to demonstrate the robust certification of the states and unsharp measurement instruments of both the sequential observers.

Haystack hunting hints and locker room communication

AbstractWe want to efficiently find a specific object in a large unstructured set, which we model by a random ‐permutation, and we have to do it by revealing just a single element. Clearly, without any help this task is hopeless and the best one can do is to select the element at random, and achieve the success probability . Can we do better with some small amount of advice about the permutation, even without knowing the target object? We show that by providing advice of just one integer in , one can improve the success probability considerably, by a factor. We study this and related problems, and show asymptotically matching upper and lower bounds for their optimal probability of success. Our analysis relies on a close relationship of such problems to some intrinsic properties of random permutations related to the rencontres number.

Complete classification of steerability under local filters and its relation with measurement incompatibility

Quantum steering is a central resource for one-sided device-independent quantum information. It is manipulated via one-way local operations and classical communication, such as local filtering on the trusted party. Here, we provide a necessary and sufficient condition for a steering assemblage to be transformable into another via local filtering. We characterize the equivalence classes with respect to filters in terms of the steering equivalent observables (SEO), first proposed to connect the problem of steerability and measurement incompatibility. We provide an efficient method to compute the extractable steerability that is maximal via local filters and show that it coincides with the incompatibility of the SEO. Moreover, we show that there always exists a bipartite state that provides an assemblage with steerability equal to the incompatibility of the measurements on the untrusted party. Finally, we investigate the optimal success probability and rates for transformation protocols (distillation and dilution) in the single-shot scenario together with examples.

Optimal unambiguous discrimination of Bell-like states with linear optics

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.

Enhancement of parameter-estimation precision by parity-time symmetric operation

We investigate precision of phase estimation without and with decoherence using a non-Hermitian operation, i.e. parity-time () symmetric operation. We derive the exact expressions of the optimal quantum Fisher information (QFI) and success probability of phase estimation for an exactly solving model consisting of a qubit interacting with a structured reservoir. Special attention is paid to the two schemes, i.e. pre-and post- symmetric operations, to show the performance of symmetric operation on QFI dynamics. By investigation, we find that pre- symmetric operator process could completely eliminate the effect of decoherence to improve the parameters estimation, while with the help of the post- symmetric operator process can potentially enhance and preserve the QFI by choosing appropriate non-Hermitian operator parameters.

General lower and upper bounds under minimum-error quantum state discrimination

For the optimal success probability under minimum-error discrimination between $r\geq2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the general bounds known in the literature. We also present the example where the new general analytical bounds, lower and upper, on the optimal success probability are tighter than most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to $r>2$ the form of the Helstrom bound. For $r=2$, each of our new bounds, lower and upper, reduces to the Helstrom bound.

Unambiguous State Discrimination with Intrinsic Coherence.

We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case with symmetrically distributed coherence. Making an extension to infinite-dimensional systems, we find that the coherence which is detrimental to state discrimination may become helpful and vice versa.

The postdoc variant of the secretary problem

The classical secretary problem involves sequentially interviewing a pool of n applicants with the aim of hiring exactly the best one in the pool—nothing less is good enough. The optimal decision strategy is easy to describe and the probability of success is 1/e. In this paper, we consider a minor variant of this classical problem. We wish to pick not the best but the second-best (the best is going to Harvard). In this case, an explicit solution can be given both for the optimal strategy and the associated optimal success probability. As n goes to infinity, the probability of success tends to 1/4. Apparently, it is easier to pick the best than the second best.

The universal unambiguous discriminator between two unknown qudit states

In this work, the unambiguous discrimination between two unknown states in d-dimensional Hilbert space is investigated. To achieve this goal, the Hilbert space of the states is divided into a series of subspaces, where the components of the two states projected in some of these subspaces can be unambiguously distinguished, and then the unambiguous discriminator between two unknown qudit states can be constructed. Moreover, the optimal success probability of the discrimination between two unknown states is obtained and the form of detection operators is given.

Role of fine-grained uncertainty in determining the limit of preparation contextuality

The optimal success probability of a communication game sets fundamental limitations on an operational theory. Quantum advantage of parity oblivious random access code (PORAC), a communication game, over classical resources reveals the preparation contextuality of quantum theory [Phys. Rev. Lett. 102, 010401 (2009)]. Optimal quantum advantage in the N-dit PORAC game for finite dimensions is an open problem. Here, we show that the degree of uncertainty allowed in an operational theory determines the amount of preparation contextuality. We connect the upper bound of fine-grained uncertainty relation to the success probability of PORAC game played with the quantum resource. Subsequently, we find the optimal success probability for the 2-dit PORAC game using MUBs for the decoding strategy. Finally, we also derive an upper bound on quantum advantage for the N-dit PORAC game.

Two-sequential conclusive discrimination between binary coherent states via indirect measurements

A general scenario for an N-sequential conclusive state discrimination introduced recently in Loubenets and Namkung [arXiv:2102.04747] can provide a multipartite quantum communication realizable in the presence of a noise. In the present article, we propose a new experimental scheme for the implementation of a sequential conclusive discrimination between binary coherent states via indirect measurements within the Jaynes–Cummings interaction model. We find that if the mean photon number is less than 1.6, then, for our two-sequential state discrimination scheme, the optimal success probability is larger than the one presented in Fields, Bergou, and Varga (2020, IEEE Int. Conf. Quant. Comp. Eng.). We also show that, if the mean photon number is almost equal to 1.2, then the optimal success probability nearly approaches the Helstrom bound.

Efficient entanglement distillation for quantum channels with polarization mode dispersion

Quantum entanglement shared by remote network nodes serves as a valuable resource for promising applications in distributed computing, cryptography, and sensing. However, distributing high-quality entanglement via fiber-optic routes could be challenging due to the various decoherence mechanisms in fibers. In particular, one of the primary polarization decoherence mechanisms in optical fibers is polarization mode dispersion (PMD), which is the distortion of optical pulses by randomly varying birefringences. To mitigate the effect of decoherence in entangled particles, quantum entanglement distillation (QED) algorithms have been proposed. One particular class, the recurrence QED algorithms, stands out because it has relatively relaxed requirements both on the size of the quantum circuits involved and on the initial quality of entanglement between particles. However, because the number of required particles grows exponentially with the number of distillation rounds, an efficient recurrence algorithm needs to converge quickly. We present a recurrence QED algorithm designed for photonic qubit pairs affected by PMD-degraded channels. Our proposed algorithm achieves the optimal fidelity as well as the optimal success probability (conditioned on the fact that optimal fidelity is achieved) in every round of distillation. The attainment of the maximal fidelity improves the convergence speed of fidelity with respect to the number of distillation rounds from linear to quadratic and, hence, significantly reduces the number of rounds. Combined with the fact that the optimal success probability is achieved, the proposed algorithm provides an efficient method to distribute entangled states with a high fidelity via optical fibers.

Quantum query complexity of symmetric oracle problems

We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t-query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn. More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G, we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.

Unambiguous discrimination of fermionic states through local operations and classical communication

The paper studies unambiguous discrimination of fermionic states through local operations and classical communication (LOCC). In the task of unambiguous discrimination, no error is tolerated but an inconclusive result is allowed. We show that contrary to the quantum case, it is not always possible to distinguish two fermionic states through LOCC unambiguously with the same success probability as if global measurements were allowed. Furthermore, we prove that we can overcome such a limit through an ancillary system made of two fermionic modes, independently of the dimension of the system, prepared in a maximally entangled state: in this case, LOCC protocols achieve the optimal success probability.

Unambiguous State Discrimination with Intrinsic Coherence

We investigate the simulation of pure-mixed (quantum filtering) and mixed-mixed state discriminations, by discriminating a pair of pure states and comparing the optimal success probability with the ones for pure-mixed and mixed-mixed discriminations. It is shown thatunder the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to the simulation of filtering, the coherence is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case with symmetrically distributed coherence. This result is due to the different distribution of the intrinsic coherence in the two schemes. Extending this problem to infinite-dimensional systems, we find that the coherence which is detrimental to state discrimination may become helpful and vice versa.

A three-player coherent state embezzlement game

We introduce a three-player nonlocal game, with a finite number of classical questions and answers, such that the optimal success probability of1in the game can only be achieved in the limit of strategies using arbitrarily high-dimensional entangled states. Precisely, there exists a constant0<c≤1such that to succeed with probability1−εin the game it is necessary to use an entangled state of at leastΩ(ε−c)qubits, and it is sufficient to use a state of at mostO(ε−1)qubits. The game is based on the coherent state exchange game of Leung et al.\ (CJTCS 2013). In our game, the task of the quantum verifier is delegated to a third player by a classical referee. Our results complement those of Slofstra (arXiv:1703.08618) and Dykema et al.\ (arXiv:1709.05032), who obtained two-player games with similar (though quantitatively weaker) properties based on the representation theory of finitely presented groups andC∗-algebras respectively.

Generalized sequential state discrimination for multiparty QKD and its optical implementation

Sequential state discrimination is a strategy for N separated receivers. As sequential state discrimination can be applied to multiparty quantum key distribution (QKD), it has become one of the relevant research fields in quantum information theory. Up to now, the analysis of sequential state discrimination has been confined to special cases. In this report, we consider a generalization of sequential state discrimination. Here, we do not limit the prior probabilities and the number of quantum states and receivers. We show that the generalized sequential state discrimination can be expressed as an optimization problem. Moreover, we investigate a structure of generalized sequential state discrimination for two quantum states and apply it to multiparty QKD. We demonstrate that when the number of receivers is not too many, generalized sequential state discrimination for two pure states can be suitable for multiparty QKD. In addition, we show that generalized sequential state discrimination for two mixed states can be performed with high optimal success probability. This optimal success probability is even higher than those of quantum reproducing and quantum broadcasting strategy. Thus, generalized sequential state discrimination of mixed states is adequate for performing multiparty QKD. Furthermore, we prove that generalized sequential state discrimination can be implemented experimentally by using linear optics. Finally, we analyze the security of multiparty QKD provided by optimal sequential state discrimination. Our analysis shows that the multiparty QKD guarantees nonzero secret key rate even in low channel efficiency.

Optimal entanglement-assisted almost-random access codes

Entanglement-assisted random access codes (EARACs) compress a large amount of bits, such that any chosen one of them can be recovered with a large probability of success. They have been used for information theoretic treatments of semi-device-independent randomness certification and other communication complexity problems. For high-dimensional problems, the access request distribution can be used to obtain better performance, which until now could only be achieved through numerical methods. This paper makes an analytical account of access request distribution for arbitrary input sizes. We introduce entanglement-assisted almost-random access codes (EAARACs), which maximize the average probability of access under an arbitrary distribution of requests, thus refining EARACs and generalizing the methods used in the study of low-dimensional communication complexity problems. We analytically find the optimal probability of success for EAARACs under arbitrary request distributions and provide an illustrative numerical example.