Quantum decision theory-driven neural networks for non-orthogonal quantum state discrimination

Nonorthogonal quantum state discrimination (QSD) plays an important role in quantum information and quantum communication. In addition, compared to Hermitian quantum systems, parity-time-($\mathcal{PT}$-)symmetric non-Hermitian quantum systems exhibit novel phenomena and have attracted considerable attention. Here, we experimentally demonstrate QSD in a $\mathcal{PT}$-symmetric system (i.e., $\mathcal{PT}$-symmetric QSD), by having quantum states evolve under a $\mathcal{PT}$-symmetric Hamiltonian in a lossy linear optical setup. We observe that two initially nonorthogonal states can rapidly evolve into orthogonal states, and the required evolution time can even be vanishing provided the matrix elements of the Hamiltonian become sufficiently large. We also observe that the cost of such a discrimination is a dissipation of quantum states into the environment. Furthermore, by comparing $\mathcal{PT}$-symmetric QSD with optimal strategies in Hermitian systems, we find that at the critical value, $\mathcal{PT}$-symmetric QSD is equivalent to the optimal unambiguous state discrimination in Hermitian systems. We also extend the $\mathcal{PT}$-symmetric QSD to the case of discriminating three nonorthogonal states. The QSD in a $\mathcal{PT}$-symmetric system opens a new door for quantum state discrimination, which has important applications in quantum computing, quantum cryptography, and quantum communication.

The problem of non-orthogonal state discrimination underlies crucial quantum information tasks, such as cryptography and computing protocols. Therefore, it is decisive to find optimal scenarios for discrimination among quantum states. We experimentally investigate the strategy for the optimal discrimination of two non-orthogonal states considering a fixed rate of inconclusive outcomes (FRIO). The main advantage of the FRIO strategy is to interpolate between unambiguous and minimum error discrimination by solely adjusting the rate of inconclusive outcomes. We present a versatile experimental scheme that performs the optimal FRIO measurement for any pair of generated non-orthogonal states with arbitrary a priori probabilities and any fixed rate of inconclusive outcomes. Considering different values of the free parameters in the FRIO protocol, we implement it upon qubit states encoded in the polarization mode of single photons generated in the spontaneous parametric down-conversion process. Moreover, we resort to a newfangled double-path Sagnac interferometer to perform a three-outcome non-projective measurement required for the discrimination task, showing excellent agreement with the theoretical prediction. This experiment provides a practical toolbox for a wide range of quantum state discrimination strategies using the FRIO scheme, which can significantly benefit quantum information applications and fundamental studies in quantum theory.

The problem of non-orthogonal state discrimination underlies crucial quantum information tasks, such as cryptography and computing protocols. Therefore, it is decisive to find optimal scenarios for discrimination among quantum states. We experimentally investigate the strategy for the optimal discrimination of two non-orthogonal states considering a fixed rate of inconclusive outcomes (FRIO). The main advantage of the FRIO strategy is to interpolate between unambiguous and minimum error discrimination by solely adjusting the rate of inconclusive outcomes. We present a versatile experimental scheme that performs the optimal FRIO measurement for any pair of generated non-orthogonal states with arbitrary a priori probabilities and for any fixed rate of inconclusive outcomes. Considering different values of the free parameters in the FRIO protocol, we implement it upon qubit states encoded in the polarization mode of single photons generated in the spontaneous parametric down-conversion process. Moreover, we resort to a newfangled double-path Sagnac interferometer to perform a three-outcome non-projective measurement required for the discrimination task, showing excellent agreement with the theoretical prediction. This experiment provides a practical toolbox for a wide range of quantum state discrimination strategies using the FRIO scheme, which can greatly benefit quantum information applications and fundamental studies in quantum theory.

This paper introduces remarkable achievement in theory on non-orthogonal state in quantum optics that can describe macroscopic quantum effect, and gives a survey of theorems in quantum information science based on non-orthogonal state. Then it is shown that these provide potential applications to Quantum Methodology such as quantum reading, quantum imaging and to Quantum Enigma Cipher which is a general model of physical cipher

Quantum mechanics forbids deterministic discrimination among nonorthogonal states. Nonetheless, the capability to distinguish nonorthogonal states unambiguously is an important primitive in quantum information processing. In this work, we experimentally implement generalized measurements in an optical system and demonstrate the first optimal unambiguous discrimination between three nonorthogonal states, with a success rate of 55%, to be compared with the 25% maximum achievable using projective measurements. Furthermore, we present the first realization of unambiguous discrimination between a pure state and a nonorthogonal mixed state.

Often it is important to consider the expansion of a quantum state ⋎ ψ) in terms of physically meaningful basis states. For example, molecular orbitals can be expressed as linear combinations of atomic orbitals, or vibrational states can be expressed as super positions of local or normal mode eigenstates. In such expansions, it then becomes desirable to determine how much “character” a quantum state has in one of these basis states. One way of accimplishing this task is to calculate the projected probability of |ψ) on basis state |j). In this paper, we consider this general quantum mechanical problem. If the basis states are orthonormal, then the projected probability of|ψ) on |j) is of course | |2. However, if the basis states are not orthogonal, then this result is no longer valid and one must develop a more general theory to calculate these projected probabilities. An earlier paper used one-dimensional projection operators to initiate this theory and gave closed form results for the case of two non-orthogonal basis states [1]. One- and many-dimensional projection operators, together with linear algebraic techniques, are used to extend this theory to the n non-orthogonal basis state case. Explicit closed form results are given for the two- and three-state cases, and a general algorithm is developed for the case of four or more basis states. Application of the theory is made to atomic populations in three- to six-atom molecules, and comparisons are made to the related work of Mulliken.

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).

Quantum cryptography is a paradigm for the establishment of secret keys and data confidentiality, which represents an alternative in the quantum era because its security properties are based on the principles of quantum physics. Unfortunately, errors that occur during transmission and detection of quantum states have made it difficult to implement this technology globally. However, a new cryptographic key quantum distribution scheme based on non-orthogonal state pairs has recently been published which considerably outperforms known schemes. This article describes the fundamentals of this protocol which are represented as an algorithm and the pseudo-code of the most relevant functions of the system is shown; The current development of the software for the distillation of non-orthogonal quantum states by means of binary frames is presented, which demonstrates the transmission control, reconciliation and privacy amplification of the shared secret bits. Likewise, we present the results obtained from the computer system and its interpretation in relation to the efficiency of the protocol, which exceeds 50% channel error rates and a quadratic growth of the length of the secret key as a function of the number of double detection events. Objectives: Demonstrate the effectiveness of the non-orthogonal state distillation protocol through binary frames using the software developed. Methodology: For the development of this project, the following methodology has been carried out (see Figure 1). Contribution: The results of this software guide tests for quantum distillation in an experimental communications environment in order to provide a useful solution in the era of quantum information transmission and communication technologies.

Quantum computing can solve problems that are difficult to solve in classical computing, expanding the range of problems that can be effectively computed within the allowable range of classical physical principles, and posing a challenge to the extended Church-Turing thesis in classical computing. Here, we discuss an interesting question: how to achieve more powerful computers by breaking through the limitations of physical principles, further enhancing the capabilities of quantum computers. To extend quantum computing, novel operations related to relativistic physics are a crucial candidate. Among them, the concept of closed time-like curve has aroused widespread interest, and it introduces the ability for time travel. Mathematically, quantum state along the closed time-like curve is determined through self-consistent equations, which has been demonstrated in simulations. Here, we consider a novel manipulation capability that allows quantum computing to achieve time-travelling quantum control gate. This is an intuitive extension of the graphical language of quantum circuits. Explaining quantum circuits as tensor networks, we first explain how to experimentally simulate the output of such a circuit in a system without time-travel capability. Then, we take an example to demonstrate an extended quantum algorithm that can efficiently solve SAT problems, indicating that with the involvement of time-travelling quantum gates, the computational complexity class P = NP. We also anticipate that the time-travelling quantum gates will play a facilitating role in accomplishing other quantum tasks, including achieving deterministic non-orthogonal quantum state discrimination, and quantum state cloning. Our results contribute to a more in-depth understanding of the relationship between computation and physical principles.

As a fundamental and challenging problem in quantum information processing, quantum state discrimination (QSD) has recently been revisited from the perspective of non-Hermitian (NH) physics. The existing work about QSD in NH systems is mostly limited to PT-symmetric or pseudo-Hermitian systems with real spectra, while generic NH Hamiltonians possess complex spectra. To explore the underlying physics of quantum state discrimination in NH systems, we first demonstrate that the unambiguous discrimination of non-orthogonal quantum states can be realized with P-pseudo-Hermitian and PT-symmetric Hamiltonians in the broken phase and the required evolution time can be arbitrarily small. We further extend the regime to generic non-Hermitian Hamiltonians and demonstrate the feasibility of unambiguous discrimination. Under the same energy constraint, we establish the criteria for constructing P-pseudo-Hermitian Hamiltonians that enable unambiguous discrimination of quantum states with smaller angular separation or within shorter evolution time than any fixed PT-symmetric system. Furthermore, we also discuss the potential impact of exceptional points (EPs) on QSD. Our results suggest that the non-orthogonal eigenstates of NH Hamiltonians, rather than PT symmetry or pseudo-Hermitian symmetry, are fundamental to unambiguous discrimination and information flow in NH systems.

Unambiguous (non-orthogonal) state discrimination (USD) has a fundamental importance in quantum information and quantum cryptography. Various aspects of two-state and multiple-state USD are studied here using singular value decomposition of the evolution operator that describes a given state discriminating system. In particular, we relate the minimal angle between states to the ratio of the minimal and maximal singular values. This is supported by a simple geometrical picture in two-state USD. Furthermore, by studying the singular vectors population we find that the minimal angle between input vectors in multiple-state USD is always larger than the minimal angle in two-state USD in the same system. As an example we study what pure states can be probabilistically transformed into maximally entangled pure states in a given system .

In this paper, the finite square well and its application are investigated, namely the quantum state discrimination. The finite square well is treated in all standard textbooks on introductory quantum mechanics. It is used as a simple ‘model of departure’ in many areas of physics. In atomic and molecular physics, it may be used as a model of an electron moving in the mean field of a linear molecule It also arises as the partial wave radial equation for a spherically symmetric, finite square-well potential. The Schrodinger equation for finite square well is solved. The domain is divided into three regions by the existing potential V 0, so for convenience, those three regions are named Region I, Region II, and Region III, respectively. Specifically, the potential of Region I and III are V 0, and that of Region II is 0. Also, a constant is needed to make this wave function normalized. Then since it is a wave function, it is necessary to make sure the function is continuous and differentiable everywhere within the domain. Besides, the wave function needs to be either odd or even just like infinite square well. After that, the wave functions for the three intervals can be obtained, and the exact quantum state is distinguished from a group of different quantum states. In the end, two wave functions are obtained; one for even form and one for odd form. Then all the energy level of the corresponding function with different wavelength needs to be found and listed. Local discrimination of orthogonal quantum states has attracted much attention during the last twenty years. The results are applied to the quantum-information task of state discrimination, by using the obtained six states in finite square well. It is assumed that all the quantum states are locally distinguishable, and the six states are distinguished using the hypothesis of quantum measurements.

Recently the performance of the quantum key distribution (QKD) is substantially improved by the decoy state method and the non-orthogonal encoding protocol, separately. In this paper, a practical non-orthogonal decoy state protocol with a heralded single photon source (HSPS) for QKD is presented. The protocol is based on 4 states with different intensities, i.e. one signal state and three decoy states. The signal state is for generating keys; the decoy states are for detecting the eavesdropping and estimating the fraction of single-photon and two-photon pulses. We have discussed three cases of this protocol, i.e. the general case, the optimal case and the special case. Moreover, the final key rate over transmission distance is simulated. For the low dark count of the HSPS and the utilization of the two-photon pulses, our protocol has a higher key rate and a longer transmission distance than any other decoy state protocol.

We analyze possible connections between quantum-inspired classifications and support vector machines. Quantum state discrimination and optimal quantum measurement are useful tools for classification problems. In order to use these tools, feature vectors have to be encoded in quantum states represented by density operators. Classification algorithms inspired by quantum state discrimination and implemented on classic computers have been recently proposed. We focus on the implementation of a known quantum-inspired classifier based on Helstrom state discrimination showing its connection with support vector machines and how to make the classification more efficient in terms of space and time acting on quantum encoding. In some cases, traditional methods provide better results. Moreover, we discuss the quantum-inspired nearest mean classification.

In quantum information and quantum computing, the carrier of information is some quantum system and information is encoded in its state. After processing the state in the quantum processor, the information has to be read out. Clearly, this task is equivalent to determining the final state of the system. We begin by briefly reviewing various possible state discrimination strategies that are optimal with respect to some reasonable criteria and report on recent advances in the unambiguous discrimination of mixed quantum states. This strategy has been successfully applied to devise a class of novel probabilistic quantum algorithms and has been demonstrated experimentally, using a linear optical implementation via generalized interferometers. In the second part we present a scheme for communication via completely unknown quantum states. In this context we discuss programmable quantum state discriminators that are universal, i.e. perform optimally on average, independently of the actual states used for the communication scheme. We conclude with a discussion of possible experimental implementations of the proposed device.