From the perspective of quantum information theory, a system so simple as one restricted to just two nonorthogonal states can be surprisingly rich in physics. In this paper, we explore the extent of this statement through a review of three topics: (1) ``nonlocality without entanglement'' as exhibited in binary quantum communication channels, (2) the tradeoff between information gain and state disturbance for two prescribed states, and (3) the quantitative clonability of those states. Each topic in its own way quantifies the extent to which two states are ``quantum'' with respect to each other, i.e., the extent to which the two together violate some classical precept. It is suggested that even toy examples such as these hold some promise for shedding light on the foundations of quantum theory.
We discuss the unambiguous measurement of quantum nonorthogonal states in connection with the quantum cryptography. We show that checking a ratio of null one to signal is essential in detecting a certain kind of eavesdropping in the case of two nonorthogonal states quantum cryptography. We prove that it is not needed in the case of the four states quantum cryptography.
I demonstrate that, rather unexpectedly, there exist noisy quantum channels for which the optimal classical information transmission rate is achieved only by signaling alphabets consisting of nonorthogonal quantum states.
We provide a quantum gambling protocol using three (symmetric) nonorthogonal states. The bias of the proposed protocol is less than that of previous ones, making it more practical. We show that the proposed scheme is secure against non-entanglement attacks. The security of the proposed scheme against entanglement attacks is shown heuristically.
We give a (remote) quantum gambling scheme that makes use of the fact that quantum nonorthogonal states cannot be distinguished with certainty. In the proposed scheme, two participants Alice and Bob can be regarded as playing a game of making guesses on identities of quantum states that are in one of two given nonorthogonal states: if Bob makes a correct (an incorrect) guess on the identity of a quantum state that Alice has sent, he wins (loses). It is shown that the proposed scheme is secure against the nonentanglement attack. It can also be shown heuristically that the scheme is secure in th…
We present a simple proof of the minimum time for the quantum evolution between two arbitrary states. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the efficiency inequality even when the system pas…
A global measure of quantum correlations for tripartite nonorthogonal states is presented. It is introduced as the overall average of the pairwise correlations existing in all possible partitions. The explicit expressions for the global measure are derived for squared concurrence, entanglement of formation, quantum discord and its geometric variant. As illustration, we consider even and odd three-mode Schrödinger cat states based on Glauber coherent states. We also discuss limitations to sharing quantum correlations known as monogamy relations.
In practical quantum key distribution (QKD), weak coherent states as the photon sources have a limit in secure key rate and transmission distance because of the existence of multiphoton pulses and heavy loss in transmission line. Decoy states method and nonorthogonal encoding protocol are two important weapons to combat these effects. Here, we combine these two methods and propose a efficient method that can substantially improve the performance of QKD. We find a 79 km increase in transmission distance over the prior record using decoy states method.
We propose an oversimplified scheme to unambiguously discriminate nonorthogonal quantum field states inside high-Q cavities. Our scheme, which is based on positive operator-valued mea- sures (POVM) technique, uses a single three-level atom interacting resonantly with a single mode of a cavity-field and selective atomic state detectors. While the single three-level atom takes the role of the ancilla, the single cavity mode field represents the system we want to obtain information. The efficiency of our proposal is analyzed considering the nowadays achievements in the context of cavity QED.
We propose a scheme to enhance the fidelity of symmetric quantum cloning machine using a weak measurement. By adjusting the intensity of weak measurement parameter $p$, we obtain the copies with different optimal fidelity. Choosing proper value of $p$, we can obtain the perfect copies for initial qubits. In this paper, we focus on $1-2$ quantum cloning for two nonorthogonal states. Sets containing more than two linear independent states are also discussed briefly. Due to weak measurements being probabilistic, we obtain high fidelity at risk of probability. If the weak measurement successes, we…
We introduce a novel form of decoy-state technique to make the single-photon Bennett 1992 protocol robust against losses and noise of a communication channel. Two uninformative states are prepared by the transmitter in order to prevent the unambiguous state discrimination attack and improve the phase-error rate estimation. The presented method does not require strong reference pulses, additional electronics or extra detectors for its implementation.
This study proposes a simple and efficient one-out-of-two quantum oblivious transfer (QOT) protocol based on nonorthogonal states. The nonorthogonal property grants quantum bit immunity to some operations in order to achieve the irreversible goal of discarding a message, resulting in a one-out-of-two selection effect. In addition, it can also prevent entangled cheating from an illegal agent. The resulting QOT protocol is therefore built directly on quantum resources, rather than on a two-level structure which must first create two classical keys using the quantum resources (all-or-nothing QOT)…
Quantum mechanics forbids deterministic discrimination among non-orthogonal 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 p…
We suggest here a two-point eavesdropping strategy to two nonorthogonal states protocol of quantum key distribution over a fiber-optic channel. Suppose that the single-photon sources and detectors of Alice, Bob and Eves are ideal ones, the two-point attack can break the two nonorthogonal states protocol if the distance between Alice and Bob is sufficiently long. For single-photon type B92 protocol, the upper bound of distance is about 30 kilometers. It is interesting that Bennett's original multi-photon protocol is secure against both two-point attack and beam splitting attack, even if the pro…
In this work we propose a probabilistic method which allows an unambiguous modification of two non-orthogonal quantum states. We experimentally implement this protocol by using two-photon polarization states generated in the process of spontaneous parametric down conversion. In the experiment, for codifying initial quantum states, we consider single photon states and heralded detection. We show that the application of this protocol to entangled states, it allows a fine control of the amount of entanglement of the initial state.
A class of quantum protocols of bit commitment is constructed based on the nonorthogonal states coding and the correlation immunity of some Boolean functions. The binding condition of these protocols is guaranteed mainly by the law of causality and the concealing condition is guaranteed by the indistinguishability between nonorthogonal quantum states and the correlation immunity of Boolean functions. We also give out an oblivious transfer protocol based on two-nonorthogonal states coding and build a bit commitment protocol on top of it. The relationship between these protocols and the well kno…
Quantum information refers to the distinctive information-processing properties of quantum systems, which arise when information is stored in or retrieved from nonorthogonal quantum states. More information is required to prepare an ensemble of nonorthogonal quantum states than can be recovered from the ensemble by measurements. Nonorthogonal quantum states cannot be distinguished reliably, cannot be copied or cloned, and do not lead to exact predictions for the results of measurements. These properties contrast sharply with those of information stored in the microstates of a classical system.
Improved Measurement Cost Scaling in the Nonorthogonal Quantum Eigensolver
Berkeley Center for Quantum Information and Computation（伯克利量子信息与计算中心）；Department of Chemistry, University of California, Berkeley（加州大学伯克利分校化学系）；Challenge Institute for Quantum Computation（挑战量子计算研究所）
Nonlocality without entanglement (NLWE) is a nonlocal quantum phenomenon that arises in separable state discrimination. We show that the availability of the postmeasurement information about the prepared subensemble can affect the occurrence of NLWE in discriminating nonorthogonal nonentangled states. We provide a two-qubit state ensemble consisting of four nonorthogonal separable pure states and show that the postmeasurement information about the prepared subensemble can annihilate NLWE. We also provide another two-qubit state ensemble consisting of four nonorthogonal separable states and sho…
The problem of unambiguously distinguishing among nonorthogonal but linearly independent quantum states can be solved by mapping the set of nonorthogonal quantum states onto a set of orthogonal ones, which can then be distinguished without error. Such nonunitary transformations can be performed conditionally on quantum systems; a unitary transformation is carried out on a larger system of which the system of interest is a subsytem, a measurement is performed, and if the proper result is obtained, the desired nonunitary transformation will have been performed on the subsystem. We show how to co…
Nonorthogonal variational quantum simulation for quantum chemistry
Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（合肥微尺度物质科学国家研究中心和中国科学技术大学物理学院）；Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China（上海量子科学研究中心和中国科学院量子信息重点实验室（中国科学技术大学））；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室（中国科学技术大学））；Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University（教育部理论化学与计算化学重点实验室和北京师范大学化学学院）；Center on Frontiers of Computing Studies, School of Computer Science, Peking University（前沿计算研究中心和北京大学生命科学学院）
Roa et al. showed that quantum state discrimination between two nonorthogonal quantum states does not require quantum entanglement but quantum dissonance only. We find that quantum coherence can also be utilized for unambiguous quantum state discrimination. We present a protocol and quantify the required coherence for this task. We discuss the optimal unambiguous quantum state discrimination strategy in some cases. In particular, our work illustrates an avenue to find the optimal strategy for discriminating two nonorthogonal quantum states by measuring quantum coherence.
In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to a…
We propose an optimal discrimination scheme for a case of four linearly independent nonorthogonal symmetric quantum states, based on linear optics only. The probability of discrimination is in agreement with the optimal probability for unambiguous discrimination among N symmetric states [Phys. Lett. A \textbf{250}, 223 (1998)]. The experimental setup can be extended for the case of discrimination among $2^M$ nonorthogonal symmetric quantum states.
We consider quantum circuits with time travel designed for distinguishing specific nonorthogonal quantum states in two most popular models: Deutschs and postselected. We modify them by a presence of weakly coupled thermal environment. Using the Davies approximation we study how the thermal noise affects an ability of the circuits to distinguish nonorthogonal quantum states. We show that for purely dephasing environment a paradoxial power of such circuits remains preserved. We also present a physics based argument for conditions of validity of the maximum entropy rule introduced by David Deutsc…
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 e…
The impossibility of deterministic and error-free discrimination among nonorthogonal quantum states lies at the core of quantum theory and constitutes a primitive for secure quantum communication. Demanding determinism leads to errors, while demanding certainty leads to some inconclusiveness. One of the most fundamental strategies developed for this task is the optimal unambiguous measurement. It encompasses conclusive results, which allow for error-free state retrodictions with the maximum success probability, and inconclusive results, which are discarded for not allowing perfect identificati…
A Markovian open quantum system which relaxes to a unique steady state $ρ_{ss}$ of finite rank can be decomposed into a finite physically realizable ensemble (PRE) of pure states. That is, as shown by Karasik and Wiseman [Phys. Rev. Lett. 106, 020406 (2011)], in principle there is a way to monitor the environment so that in the long time limit the conditional state jumps between a finite number of possible pure states. In this paper we show how to apply this idea to the dynamics of a double quantum dot arising from the feedback control of quantum transport, as previously considered by one of u…
Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation…
We study the extent to which ψ-epistemic models for quantum measurement statistics---models where the quantum state does not have a real, ontic status---can explain the indistinguishability of nonorthogonal quantum states. This is done by comparing the overlap of any two quantum states with the overlap of the corresponding classical probability distributions over ontic states in a ψ-epistemic model. It is shown that in Hilbert spaces of dimension $d \geq 4$, the ratio between the classical and quantum overlaps in any ψ-epistemic model must be arbitrarily small for certain nonorthogonal states,…
The Extended Wigner's Friend thought experiment, which involves a quantum system with an agent who draws conclusions based on the results of a measurement of a quantum state provided in two nonorthogonal versions by another agent, led its designers to the conclusion that quantum theory cannot consistently explain the use of itself. It has also been suggested that this thought experiment is equivalent to entangled state (Bell-type) experiments. This study indicates that the assumption of the first Wigner's friend's freedom of choice, regarding how to prepare a quantum state in one of the two av…
Quantum information is a rapidly advancing area of interdisciplinary research. It may lead to real-world applications for communication and computation unavailable without the exploitation of quantum properties such as nonorthogonality or entanglement. We review the progress in quantum information based on continuous quantum variables, with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagnetic field.
This paper addresses the question whether a variant of a modal interpretation is conceivable that could accommodate property ascriptions associated with nonorthogonal resolutions of the unity and nonorthogonal families of relative states as they occur in imperfect or genuinely unsharp measurements. I will review a recent formulation of the quantum measurement problem in the form of an insolubility theorem that incorporates the case of unsharp object observables as well as certain types of unsharp pointers. In addition to demonstrating the necessity for some modification of quantum mechanics, t…
A scheme for teleporting an unknown two-particle entangled state via W class states is proposed. In this scheme, the W class entangled states are considered as quantum channels. It is shown that by means of optimal discrimination between two nonorthogonal quantum states, probabilistic teleportation of the two-particle entangled state can be achieved.
The discrimination of two nonorthogonal states is a fundamental element for secure and efficient communication. Quantum measurements of nonorthogonal coherent states can enhance information transfer beyond the limits of conventional technologies. We demonstrate a strategy for binary state discrimination based on optimized single-shot measurements with photon number resolving (PNR) detection with finite number resolution. This strategy enables a high degree of robustness to noise and imperfections while being scalable to high rates and in principle allows for surpassing the quantum noise limit …
In contradistinction with some plausible statements of the information theory, we point out the possibility of the zero energy quantum information processing. Particularly, we investigate the rate of the entanglement formation in the operation of the quantum "oracles" employing the "quantum parallelism", and we obtain that the relative maximum of the rate of the operation distinguishes the zero average energy of interaction in the composite system "input register + output register". This result is reducible to neither of the previously obtained bounds, and therefore represents a new bound for …
We have investigated the problem of discriminating between nonorthogonal quantum states with least probability of error. We have determined that the best strategy for some sets of states is to make no measurement at all, and simply to always assign the most commonly occurring state. Conditions which describe such sets of states have been derived.
We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mechanics for distinguishing nonorthogonal states and speeding up hard computations. If a CTC-assisted computer is presented with a labeled mixture of states to be distinguished--the most natural formulation--we show that the CTC is of no use. The apparent contradiction with recent claims that CTC-assisted computers can perfectly distinguish nonorthogonal states is resolved by noting that CTC-assisted evolution is nonlinear, so the output of such a computer on a mixture of inputs is not a convex combi…
When classical information is sent through a quantum channel of nonorthogonal states, there is a possibility that transmittable classical information exceeds a channel capacity in a single use of the initial channel by extending it into multi-product channel. In this paper, it is shown that this remarkable feature of a quantum channel, so-called superadditivity, appears even in as low as the third extended coding of the simplest binary input channel. A physical implementation of this channel is indicated based on cavity QED techniques.
We introduce the quantum quincunx, which physically demonstrates the quantum walk and is analogous to Galton's quincunx for demonstrating the random walk. In contradistinction to the theoretical studies of quantum walks over orthogonal lattice states, we introduce quantum walks over nonorthogonal lattice states (specifically, coherent states on a circle) to demonstrate that the key features of a quantum walk are observable albeit for strict parameter ranges. A quantum quincunx may be realized with current cavity quantum electrodynamics capabilities, and precise control over decoherence in such…
In quantum information processing, {using a receiver device to differentiate between two nonorthogonal states leads to a quantum error probability. The minimum possible error is} known as the Helstrom bound. In this work we study and compare quantum limits for states which generalize the Glauber-Sudarshan coherent states, like non-linear, Perelomov, Barut-Girardello, and (modified) Susskind-Glogower coherent states. For some of these, we show that the Helstrom bound can be significantly lowered and even vanish in specific regimes.
Quantum mechanics forbids perfect discrimination among nonorthogonal states through a single shot measurement. To optimize this task, many strategies were devised that later became fundamental tools for quantum information processing. Here, we address the pioneering minimum-error (ME) measurement and give the first experimental demonstration of its application for discriminating nonorthogonal states in high dimensions. Our scheme is designed to distinguish symmetric pure states encoded in the transverse spatial modes of an optical field; the optimal measurement is performed by a projection ont…
We present a scheme for multiparty quantum secret sharing of a private key with pure entangled states and decoy photons. The boss, say Alice uses the decoy photons, which are randomly in one of the four nonorthogonal single-photon states, to prevent a potentially dishonest agent from eavesdropping freely. This scheme requires the parties of communication to have neither an ideal single-photon quantum source nor a maximally entangled one, which makes this scheme more convenient than others in a practical application. Moreover, it has the advantage of having high intrinsic efficiency for qubits …
We demonstrate superadditivity in the communication capacity of a binary alphabet consisting of two nonorthogonal quantum states. For this scheme, collective decoding is performed two transmissions at a time. This improves upon the previous schemes of Sasaki et al. [Phys. Rev. A 58, 146 (1998)] where superadditivity was not achieved until a decoding of three or more transmissions at a time. This places superadditivity within the regime of a near-term laboratory demonstration. We propose an experimental test based upon an alphabet of low photon-number coherent states where the signal decoding i…
Evaluating the amount of information obtained from non-orthogonal quantum states is an important topic in the field of quantum information. The commonly used evaluation method is Holevo bound, which only provides a loose upper bound for quantum measurement. In this paper, we provide a theoretical study of the positive operator-valued measure (POVM) for discriminating nonorthogonal states. We construct a generalized POVM measurement operation, and derive the optimal one for state discrimination by Lagrange multiplier method. With simulation, we find that the optimal POVM measurement provides a …
We consider the problem of a state determination for a two-level quantum system which can be in one of two nonorthogonal mixed states. It is proved that for the two independent identical systems the optimal combined measurement (which considers the pair as one system) cannot be less optimal than the corresponding sequential one (local measurements, accompanying by transfer of classical information). The case of equality is achieved only when the mixed states have the same eigenvalues or the same eigenvectors. Further, we consider a case then the two systems are entangled: measurement of one sy…
We present an optical implementation of two programmable quantum measurement devices. The first one serves for unambiguous discrimination of two nonorthogonal states of a qubit. The particular pair of states to be discriminated is specified by the quantum state of a program qubit. The second device can perform von Neumann measurements on a single qubit in any basis located on the equator of the Bloch sphere. Again, the basis is selected by the state of a program qubit. In both cases the data and program qubits are represented by polarization states of photons. The experimental apparatus exploi…
Here we show that, in principle it is possible to clone (measure) a single arbitrary unknown quantum state of a spin-$\frac{1}{2}$ particle (an electron) with arbitrary precision and with success probability tending to one, using protective measurement. We first transfer the information from spin to spatial degree of freedom (d.o.f) of system electron, then trap it in a double well potential, and finally measure it protectively using a probe electron (which donot get entangled with system electron, but still extracts expectation value of an observable from a single quantum system (system elect…
The possibility to save and process information in fundamentally indistinguishable states is the quantum mechanical resource that is not encountered in classical computing. I demonstrate that, if energy constraints are imposed, this resource can be used to accelerate information-processing without relying on entanglement or any other type of quantum correlations. In fact, there are computational problems that can be solved much faster, in comparison to currently used classical schemes, by saving intermediate information in nonorthogonal states of just a single qubit. There are also error corre…
We present for the first time, a bidirectional Quantum Key Distribution protocol with minimal encoding operations derived from the use of two `nonorthogonal' unitary transformations selected from two mutually unbiased unitary bases; which are indistinguishable in principle for a single use. Along with its decoding procedure, it is a stark contrast to its `orthogonal encoding' predecessors. Defining a more relevant notion of security threshold for such protocols, the current protocol outperforms its predecessor in terms of security as the maximal amount of information an eavesdropper can glean …
The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the ``Heisenberg uncertainty principle.'' A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for…
A scheme for implementing the discrete-time quantum walk on the Bloch sphere is proposed, which is closely related to the SU(2) group. A spin cluster serves as the walker, whereas its location on the Bloch sphere is described by the spin coherent state. An additional spin that interacts with the spin cluster plays the role of a coin, whose state determines the rotation of the spin cluster. The Wigner function is calculated to visualize the movement of the walker on the Bloch sphere, with which the probability distribution and the standard deviation are also achieved. The quadratic enhancement …
Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmet…
Experimental demonstration of optimal measurement for unambiguously discriminating asymmetric qudit states
Center for Quantum Technology, Korea Institute of Science and Technology (KIST)（韩国科学技术院量子技术中心）；Division of Quantum Information, KIST School, University of Science and Technology（韩中科技大学KIST学院量子信息系）；Department of Applied Physics, Kyung Hee University（庆熙大学应用物理系）
We present two efficient quantum key distribution schemes over two different collective-noise channels. The accepted hypothesis of collective noise is that photons travel inside a time window small compared to the variation of noise. Noiseless subspaces are made up of two Bell states and the spatial degree of freedom is introduced to form two nonorthogonal bases. Although these protocols resort to entangled states for encoding the key bit, the receiver is only required to perform single-particle product measurements and there is no basis mismatch. Moreover, the detection is passive as the rece…
A recent paper from Brun et al. has argued that access to a closed timelike curve (CTC) would allow for the possibility of perfectly distinguishing nonorthogonal quantum states. This result can be used to develop a protocol for instantaneous nonlocal signaling. Several commenters have argued that nonlocal signaling must fail in this and in similar cases, often citing consistency with relativity as the justification. I argue that this objection fails to rule out nonlocal signaling in the presence of a CTC. I argue that the reason these authors are motivated to exclude the prediction of nonlocal…
The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the ``Heisenberg uncertainty principle.'' A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for…
