M. E. Deconinck, B. M. Terhal
We show how one can solve the problem of discriminating between qubit states. We use the quantum state discrimination duality theorem and the Bloch sphere representation of qubits which allows for an easy geometric and analytical representation of the optimal guessing strategies.
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We propose a scheme by using the feed-forward control (FFC) to realize a better effect of discrimination of two nonorthogonal states after passing a noise channel based on the minimum- error (ME) discrimination. We show that the application of our scheme can highly improve the effect of discrimination compared with the ME discrimination without the FFC for any pair of nonorthogonal states and any degree of amplitude damping (AD). Especially, the effect of our optimal discrimination can reach that of the two initial nonorthogonal pure states in the presence of the noise channel in a determinist…
We derive a general lower bound on the minimum-error probability for {\it ambiguous discrimination} between arbitrary $m$ mixed quantum states with given prior probabilities. When $m=2$, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out a upper bound on th…
Unambiguous state discrimination of two mixed bipartite states via local operations and classical communications (LOCC) is studied and compared with the result of a scheme realized via global measurement. We show that the success probability of a global scheme for mixed-state discrimination can be achieved perfectly by the local scheme. In addition, we simulate this discrimination via a pair of pure entangled bipartite states. This simulation is perfect for local rather than global schemes due to the existence of entanglement and global coherence in the pure states. We also prove that LOCC pro…
Entangled states can help in quantum state discrimination by local operations and classical communication (LOCC). For example, a Bell state is necessary (and sufficient) to perfectly discriminate a set of either three or four Bell states by LOCC. In this paper, we consider the task of LOCC discrimination of the states of noisy Bell ensembles, where a given ensemble consists of the states obtained by mixing the Bell states with an arbitrary two-qubit state with nonzero probabilities. It is proved that a Bell state is required for optimal discrimination by LOCC, even though the ensembles do not …
In this paper, we address the problem of discriminating two given quantum operations. Firstly, based on the Bloch representation of single qubit systems, we give the exact minimum error probability of discriminating two single qubit quantum operations by unentangled input states. In particular, for the Pauli channels discussed in [Phys. Rev. A {\bf 71}, 062340 (2005)], we use a more intuitional and visual method to deal with their discrimination problem. Secondly, we consider the condition for perfect discrimination of two quantum operations. Specially, we get that two generalized Pauli channe…
We provide a simple proof for the necessity of conditions for discriminating with minimum error between a known set of quantum states.
Bayesian and Markovian classical feedforward for discriminating qubit channels
本文提出了一种基于贝叶斯和马尔可夫策略的量子比特通道区分方法，通过自适应协议和经典信息前馈实现多轮次区分，并展示了两种策略在有限参数区域内的性能差异。
The ability to uniquely identify a quantum state is integral to quantum science, but for non-orthogonal states, quantum mechanics precludes deterministic, error-free discrimination. However, using the non-deterministic protocol of unambiguous state discrimination (USD) enables error-free differentiation of states, at the cost of a lower frequency of success. We discriminate experimentally between non-orthogonal, high-dimensional states encoded in single photons; our results range from dimension $d=2$ to $d=14$. We quantify the performance of our method by comparing the total measured error rat…
We address the issue of the discrimination between two-qubit amplitude damping channels by exploring several strategies. For the single-shot, we show that the excited state does not always give the optimal input, and that side entanglement assistance has limited benefit. On the contrary, feedback assistance from the environment is more beneficial. For the two-shot, we prove the in-utility of entangled inputs. Then focusing on individual (local) measurements, we find the optimal adaptive strategy.
In this paper, we consider the problem of unambiguous discrimination between a set of mixed quantum states. We first divide the density matrix of each mixed state into two parts by the fact that it comes from ensemble of pure quantum states. The first part will not contribute anything to the discrimination, the second part has support space linearly independent to each other. Then the problem we consider can be reduced to a problem in which the strategy of set discrimination can be used in designing measurements to discriminate mixed states unambiguously. We find a necessary and sufficient con…
Laser light is widely used for communication and sensing applications, so the optimal discrimination of coherent states--the quantum states of light emitted by a laser--has immense practical importance. However, quantum mechanics imposes a fundamental limit on how well different coher- ent states can be distinguished, even with perfect detectors, and limits such discrimination to have a finite minimum probability of error. While conventional optical receivers lead to error rates well above this fundamental limit, Dolinar found an explicit receiver design involving optical feedback and photon c…
We address the problem of discriminating with minimal error probability two given quantum operations. We show that the use of entangled input states generally improves the discrimination. For Pauli channels we provide a complete comparison of the optimal strategies where either entangled or unentangled input states are used.
We study the discrimination of N mixed quantum states in an optimal measurement that maximizes the probability of correct results while the probability of inconclusive results is fixed at a given value. After considering the discrimination of N states in a d-dimensional Hilbert space, we focus on the discrimination of qubit states. We develop a method to determine an optimal measurement for discriminating arbitrary qubit states, taking into account that often the optimal measurement is not unique and the maximum probability of correct results can be achieved by several different measurements. …
We consider the problem of discriminating finite-dimensional quantum processes, also called quantum supermaps, that can consist of multiple time steps. Obtaining the ultimate performance for discriminating quantum processes is of fundamental importance, but is challenging mainly due to the necessity of considering all discrimination strategies allowed by quantum mechanics, including entanglement-assisted strategies and adaptive strategies. In the case in which the processes to be discriminated have internal memories, the ultimate performance would generally be more difficult to analyze. In thi…
Based on our previous publication [U. Herzog and J. A. Bergou, Phys.Rev. A 71, 050301(R) (2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure p…
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 discrimination of weak coherent states of light with significant overlaps by nondestructive measurements on the light states through measuring atomic states that are entangled to the coherent states via dipole coupling. In this way, the problem of measuring and discriminating coherent light states is shifted to finding the appropriate atom-light interaction and atomic measurements. We show that this scheme allows us to attain a probability of error extremely close to the Helstrom bound, the ultimate quantum limit for discriminating binary quantum states, through the simple Jaynes-…
It is known that a party with access to a Deutschian closed timelike curve (D-CTC) can perfectly distinguish multiple non-orthogonal quantum states. In this paper, we propose a practical method for discriminating multiple non-orthogonal states, by using a previously known quantum circuit designed to simulate D-CTCs. This method relies on multiple copies of an input state, multiple iterations of the circuit, and a fixed set of unitary operations. We first characterize the performance of this circuit and study its asymptotic behavior. We also show how it can be equivalently recast as a local, ad…
Discriminating between orthogonal quantum systems without destroying their entanglement is of interest to quantum computation and communication. In this paper, we explicate the schemes for the non-destructive discrimination (NDD) of 16 orthogonal four qubit cluster states and 32 orthogonal five qubit cluster states respectively. This technique will find its applications in various branches of quantum information theory.
Entanglement cost of discriminating quantum states under locality constraints
本文研究局部约束下区分量子态的纠缠代价，引入谱PPT距离量化测量局域性并给出下界，同时建立上界，证明单个Bell态可辅助纯态最优区分任意其他态。
In this thesis we study the problem of unambiguously discriminating two mixed quantum states. We first present reduction theorems for optimal unambiguous discrimination of two generic density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank $r$ in a 2$r$-dimensional Hilbert space. These reduction theorems also allow us to reduce USD problems to simpler ones for which the solution might be known. As an application, we consider the unambiguous comparison of $n$ linearly independent pure states with a simple symmetry. Moreover, lower bound…
Enhanced quantum state discrimination under general measurements with entanglement and nonorthogonality restrictions
本文研究了在资源受限条件下，通过非正测量实现量子态鉴别误差低于Helstrom界限的可能性，展示了无需初始纠缠即可实现有效非正测量的方法。
We consider the problem of optimally discriminating two Pauli channels in the minimax strategy, maximizing the smallest of the probabilities of correct identification of the channel. We find the optimal input state at the channel and show the conditions under which using entanglement strictly enhances distinguishability. We finally compare the minimax strategy with the Bayesian one.
We consider the problem of discriminating among a set of unitaries by means of measurements performed on the state undergoing the transformation. We show that use of entangled probes improves the discrimination in the two following cases: i) for a set of unitaries that are the UIR of a group and, ii) for any pair of transformations provided that multiple uses of the channel are allowed.
Quantum channel discrimination is used to test quantum field theory in non-inertial frames. We search for optimal strategies which can best see the thermality of the Unruh effect. We find that the usual strategy of counting particles in the vacuum can be improved, thereby enhancing the discrimination. Coherent state probes, which are practical and feasible, give exponential improvement in the discrimination of the Unruh channel and come very close to optimal. In particular, we show that using a short pulse laser, the accelerations required to test the Unruh effect can be reduced by at least th…
Discrimination of bosonic dephasing quantum channels
本研究探讨了区分两个玻色子退相量子信道的问题，证明了无歧义区分不可行，并在有或无输入能量约束下优化了错误概率，且将结果推广至多次区分。
The theory of generalised measurements is used to examine the problem of discriminating unambiguously between non-orthogonal pure quantum states. Measurements of this type never give erroneous results, although, in general, there will be a non-zero probability of a result being inconclusive. It is shown that only linearly-independent states can be unambiguously discriminated. In addition to examining the general properties of such measurements, we discuss their application to entanglement concentration.
When discriminating between two pure quantum states, there exists a quantitative tradeoff between the information retrieved by the measurement and the disturbance caused on the unknown state. We derive the optimal tradeoff and provide the corresponding quantum measurement. Such an optimal measurement smoothly interpolates between the two limiting cases of maximal information extraction and no measurement at all.
We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities.
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condi…
We consider the optimal discrimination of nonorthogonal qubit states with post-measurement information and provide an analytic structure of the optimal measurements. We also show that there is always a null optimal measurement when post-measurement information is given. Further, in discriminating four states using post-measurement information, we analytically provide the optimal probability of correct guessing and show that the uniqueness of optimal measurement is equivalent to the non-existence of non-null optimal measurement with post-measurement information.
We study the problem of discriminating between non-orthogonal quantum states with least probability of error. We demonstrate that this problem can be simplified if we solve for the error itself rather than solving directly for the optimal measurement. This method enables us to derive solutions directly and thus make definite statements about the uniqueness of an optimal strategy. This approach immediately leads us to a state-discrimination analogue of Davies Theorem. In the course of this, a complete solution for distinguishing equally likely pure qubit states is presented.
Geometric Bloch Vector Solution to Minimum Error Discriminations of Mixed Qubit States
本研究针对混合量子比特态的最小误差辨识问题，提出基于几何布洛赫矢量的四步结构化方法，引入拉格朗日算子与几何POVM类别，可求解任意先验概率下N个混合量子比特态的所有最优辨识解，含4量子比特态的新结果。
We consider the problem of discriminating between two quantum coherent states by interpreting a single state like being a collection of several successive copies of weaker coherent states. By means of recent results on multiple-copy state discrimination, it is possible to give a reinterpretation of the Dolinar receiver, and carry out a quite straightforward analysis of its behavior. We also propose and investigate a suboptimal detection scheme derived from the Dolinar's architecture, which is shown to slightly outperform some other near-optimal schemes available in literature.
Nondestructive discrimination of Bell states between distant parties
本文提出利用预共享纠缠资源对远距离各方持有的贝尔态进行无损区分的方案，证明无辅助资源时成功概率上限为1/4，并在IonQ量子计算机上验证其可超越经典界限。
The necessary and sufficient conditions for minimization of the generalized rate error for discriminating among $N$ pure qubit states are reformulated in terms of Bloch vectors representing the states. For the direct optimization problem an algorithmic solution to these conditions is indicated. A solution to the inverse optimization problem is given. General results are widely illustrated by particular cases of equiprobable states and $N=2,3,4$ pure qubit states given with different prior probabilities.
In this article, by treating minimum error state discrimination as a complementarity problem, we obtain the geometric optimality conditions. These can be used as the necessary and sufficient conditions to determine whether every optimal measurement operator can be nonzero. Using these conditions and an inductive approach, we demonstrate a geometric method and the intrinsic polytope for N-qubit mixed-state discrimination. When the intrinsic polytope becomes a point, a line segment, or a triangle, the guessing probability, the necessary and sufficient condition for the exact solution, and the op…
We consider a state discrimination problem which deals with settings of minimum-error and unambiguous discrimination systematically by introducing a margin for the probability of an incorrect guess. We analyze discrimination of three symmetric pure states of a qubit. The measurements are classified into three types, and one of the three types is optimal depending on the value of the error margin. The problem is formulated as one of semidefinite programming. Starting with the dual problem derived from the primal one, we analytically obtain the optimal success probability and the optimal measure…
We revisit the problem of discriminating orthogonal quantum states within the local quantum operation and classical communication (LOCC) paradigm. Our particular focus is on the asymptotic situation where the parties have infinite resources and the protocol may become arbitrarily long. Our main result is a necessary condition for perfect asymptotic LOCC discrimination. As an application, we prove that for complete product bases, unlimited resources are of no advantage. On the other hand, we identify an example, for which it still remains undecided whether unlimited resources are superior.
It is shown that generalized measurements, required for optimally discriminating between nonorthogonal joint polarization states of two indistinguishable photons, can be realized with the help of polarization-dependent two-photon absorption and by means of sum-frequency generation. Optimization schemes are investigated with respect to minimizing the error probability in inferring the states, as well as with respect to maximizing the probability of success for unambiguous discrimination. Moreover, an implementation of error-minimizing discrimination between N symmetric single-photon states is s…
We investigate the performance of a Kennedy receiver, which is known as a beneficial tool in optical coherent communications, to the quantum state discrimination of the two superpositions of vacuum and single photon states corresponding to the $\hatσ_x$ eigenstates in the single-rail encoding of photonic qubits. We experimentally characterize the Kennedy receiver in vacuum-single photon two-dimensional space using quantum detector tomography and evaluate the achievable discrimination error probability from the reconstructed measurement operators. We furthermore derive the minimum error rate ob…
We show how to optimally discriminate between K distinct quantum states, of which N copies are available, using one-at-a-time interactions with each of the N copies. While this task (famously) requires joint measurements on all N copies, we show that it can be solved with one-at-a-time "coherent measurements" performed by an apparatus with log(K) qubits of quantum memory. We apply the same technique to optimal discrimination between K distinct N-particle matrix product states of bond dimension D, using a coherent measurement apparatus with log(K) + log(D) qubits of memory.
We address the problem of unambiguous discrimination and identification among quantum observables. We set a general framework and investigate in details the case of qubit observables. In particular, we show that perfect discrimination with two shots is possible only for sharp qubit observables (e.g. Stern-Gerlach apparatuses) associated with mutually orthogonal directions. We also show that for sharp qubit observables associated to nonorthogonal directions unambiguous discrimination with an inconclusive result is always possible.
Fast and accurate measurement is a highly desirable, if not vital, feature of quantum computing architectures. In this work we investigate the usefulness of adaptive measurements in improving the speed and accuracy of qubit measurement. We examine a particular class of quantum computing architectures, ones based on qubits coupled to well controlled harmonic oscillator modes (reminiscent of cavity-QED), where adaptive schemes for measurement are particularly appropriate. In such architectures, qubit measurement is equivalent to phase discrimination for a mode of the electromagnetic field, and w…
A photon detector combining the two avalanche photon diodes (APD) has been demonstrated for qubit discrimination in 1550 nm. Spikes accompanied with the signals in gated-mode were canceled by balanced output from the two APDs. The spike cancellation enabled one to reduce the threshold in the discriminators, and thus the gate pulse voltage. The dark count probability and afterpulse probability were reduced to 7x10^-7 and 10^-4, respectively, without affecting the detection efficiency (11 %) at 178 K.
We show a geometric formulation for minimum-error discrimination of qubit states, that can be applied to arbitrary sets of qubit states given with arbitrary a priori probabilities. In particular, when qubit states are given with equal \emph{a priori} probabilities, we provide a systematic way of finding optimal discrimination and the complete solution in a closed form. This generally gives a bound to cases when prior probabilities are unequal. Then, it is shown that the guessing probability does not depend on detailed relations among given states, such as angles between them, but on a property…
It is well known that the number of entanglement classes in SLOCC (stochastic local operations and classical communication) classifications increases with the number of qubits and is already infinite for four qubits. Bearing in mind the rapid evolution of experimental technology, criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore in the focus of intense theoretical research. In this article we develop a general criterion for the discrimination of pure N-partite entangled states in terms of polynomial SL(d,C) invariants.…
Discrimination of unitary operations is fundamental in quantum computation and information. A lot of quantum algorithms including the well-known Deutsch-Jozsa algorithm, Simon's algorithm, and Grover's algorithm can essentially be regarded as discriminating among individual, or sets of unitary operations (oracle operators). The problem of discriminating between two unitary operations $U$ and $V$ can be described as: Given $X\in\{U, V\}$, determine which one $X$ is. If $X$ is given with multiple copies, then one can design an adaptive procedure that takes multiple queries to $X$ to output the i…
The discrimination of quantum measurements is an important subject of quantum information processes. In this paper we present a novel protocol for local quantum measurement discrimination with multi-qubit entanglement systems. It is shown that, in the case of Bob did not know Alice's result of measurement, if both two observers (Alice and Bob) agreed in advance that Alice should measure her qubits before an appointed time t, the local discrimination of two different kinds of measurement can be completed via numerous eight-qubit GHZ entangled states and selective projective measurement.
Profitable entanglement for channel discrimination
本文研究了辅助纠缠在区分两个一般量子比特信道中的作用，通过构造性方法分别分析了极值和非极值信道，确定了纠缠提升成功概率的精确条件。
Measurement for qubits plays a key role in quantum computation. Current methods for classifying states of single qubit in a superconducting multi-qubit system produce fidelities lower than expected due to the existence of crosstalk, especially in case of frequency crowding. Here, We make the digital signal processing (DSP) system used in measurement into a shallow neural network and train it to be an optimal classifier to reduce the impact of crosstalk. The experiment result shows the crosstalk-induced readout error deceased by 100% after a 3-second optimization applied on the 6-qubit supercon…
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 …
The problem of discriminating with minimum error between two mixed quantum states is reviewed, with emphasize on the detection operators necessary for performing the measurement. An analytical result is derived for the minimum probability of errors in deciding whether the state of a quantum system is either a given pure state or a uniform statistical mixture of any number of mutually orthogonal states. The result is applied to two-qubit states, and the minimum error probabilities achievable by collective and local measurements on the qubits are compared.
Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. A programmable discrimination machine performs this task when the pair of possible states is not a priori known, but instead the two possible states are provided through two respective program ports. We study optimal programmable discrimination machines for general qubit states when several copies of states are available in the data or program ports. Two scenarios are considered: one in which the purity of the possi…
We study discrimination of m quantum measurements in the scenario when the unknown measurement with n outcomes can be used only once. We show that ancilla-assisted discrimination procedures provide a nontrivial advantage over simple (ancilla-free) schemes for perfect distinguishability and we prove that inevitably m <= n. We derive necessary and sufficient conditions of perfect distinguishability of general binary measurements. We show that the optimization of the discrimination of projective qubit measurements and their mixtures with white noise is equivalent to the discrimination of specific…
We discuss sequential unambiguous state-discrimination measurements performed on the same qubit. Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, who also measures it. The object in both cases is to determine which state Alice sent. In an unambiguous state discrimination measurement, we never make a mistake, i.e. misidentify the state, but the measurement may fail, in which case we gain no information about which state was sent. We find that there is a nonzero probability for both Bob and Charlie to identify the stat…
We show that a unitary operation (quantum circuit) secretely chosen from a finite set of unitary operations can be determined with certainty by sequentially applying only a finite amount of runs of the unknown circuit. No entanglement or joint quantum operations is required in our scheme. We further show that our scheme is optimal in the sense that the number of the runs is minimal when discriminating only two unitary operations.
Ensembles of composite quantum states can exhibit nonlocal behaviour 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 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 3 copies of the state. For this composite system we analyze multi-copy adaptive local distinguishability of orthogonal ensembles in full generality w…
Distinguishability is a fundamental and operational task generally connected to information applications. In quantum information theory, from the postulates of quantum mechanics it often has an intrinsic limitation, which then dictates and also characterises capabilities of related information tasks. In this work, we consider discrimination between bipartite two-qubit unitary transformations by local operations and classical communication (LOCC) and its relations to entangling capabilities of given unitaries. We show that a pair of entangling unitaries which do not contain local parts, if they…
We consider the problem of discriminating macromolecular structures in an electron microscope, through a specific beam shaping technique. Our approach is based on maximizing the which-molecule information extracted from the state of each electron. To this aim, the optimal observables are derived within the framework of quantum state discrimination, which allows one to fully account from the quantum character of the probe. We simulate the implementation of such optimal observable on a generalized orbital angular momentum (OAM) sorter and benchmark its performance against the best known real spa…
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