Quantum State Discrimination Problems Research Articles

Unextendible entangled bases and more nonlocality with less entanglement

We consider a general version of the phenomenon of more nonlocality with less entanglement, within the framework of the unambiguous (i.e., conclusive) quantum state discrimination problem under local quantum operations and classical communication. We show that although the phenomenon was obtained before for two qutrits, it can also be observed for two qubits, while still being at the single-copy level. We establish that the phenomenon is intrinsically connected to the concept of unextendible entangled bases, in the two-qubit case. In the process, we demonstrate a hierarchy of nonlocality among sets of two-qubit orthogonal pure states, where the "nonlocality" is in the sense of a difference between global and local abilities of quantum state discrimination. We present a complete characterization of two-qubit pure orthogonal state sets of cardinality three with respect to their nonlocality in terms of unambiguous local distinguishability, the status for other cardinalities being already known. The results are potentially useful for secure quantum communication technologies with an optimal amount of resources.

Asymptotic relative submajorization of multiple-state boxes

Pairs of states, or “boxes” are the basic objects in the resource theory of asymmetric distinguishability (Wang and Wilde in Phys Rev Res 1(3):033170, 2019. https://doi.org/10.1103/PhysRevResearch.1.033170), where free operations are arbitrary quantum channels that are applied to both states. From this point of view, hypothesis testing is seen as a process by which a standard form of distinguishability is distilled. Motivated by the more general problem of quantum state discrimination, we consider boxes of a fixed finite number of states and study an extension of the relative submajorization preorder to such objects. In this relation, a tuple of positive operators is greater than another if there is a completely positive trace nonincreasing map under which the image of the first tuple satisfies certain semidefinite constraints relative to the other one. This preorder characterizes error probabilities in the case of testing a composite null hypothesis against a simple alternative hypothesis, as well as certain error probabilities in state discrimination. We present a sufficient condition for the existence of catalytic transformations between boxes, and a characterization of an associated asymptotic preorder, both expressed in terms of sandwiched Rényi divergences. This characterization of the asymptotic preorder directly shows that the strong converse exponent for a composite null hypothesis is equal to the maximum of the corresponding exponents for the pairwise simple hypothesis testing tasks.

Erratum: Generalized quantum state discrimination problems [Phys. Rev. A 91 , 052304 (2015)

Erratum: Generalized quantum state discrimination problems [Phys. Rev. A 91 , 052304 (2015)

Quantum Stochastic Walk models for quantum state discrimination

Quantum Stochastic Walks (QSW) allow for a generalization of both quantum and classical random walks by describing the dynamic evolution of an open quantum system on a network, with nodes corresponding to quantum states of a fixed basis. We consider the problem of quantum state discrimination on such a system, and we solve it by optimizing the network topology weights. Finally, we test it on different quantum network topologies and compare it with optimal theoretical bounds.

Parametrization of quantum states based on the quantum state discrimination problem

A discrimination problem consists of N linearly independent pure quantum states $$\Phi =\{|\phi _i\rangle \}$$ and the corresponding occurrence probabilities $$\eta =\{\eta _i\}$$ . With any such problem, we associate, up to a permutation over the probabilities $$\{\eta _i\}$$ , a unique pair of density matrices $$\varvec{\rho _{_{T}}}$$ and $$\varvec{\eta _{{p}}}$$ defined on the N-dimensional Hilbert space $$\mathcal {H}_N$$ . The first one, $$\varvec{\rho _{_{T}}}$$ , provides a new parametrization of a generic full-rank density matrix in terms of the parameters of the discrimination problem, i.e., the mutual overlaps $$\gamma _{ij}=\langle \phi _i|\phi _j\rangle $$ and the occurrence probabilities $$\{\eta _i\}$$ . The second one, on the other hand, is defined as a diagonal density matrix $$\varvec{\eta _p}$$ with the diagonal entries given by the probabilities $$\{\eta _i\}$$ with the ordering induced by the permutation p of the probabilities. $$\varvec{\rho _{_{T}}}$$ and $$\varvec{\eta _{{p}}}$$ capture information about the quantum and classical versions of the discrimination problem, respectively. In this sense, when the set $$\Phi $$ can be discriminated unambiguously with probability one, i.e., when the states to be discriminated are mutually orthogonal and can be distinguished by a classical observer, then $$\varvec{\rho _{_{T}}} \rightarrow \varvec{\eta _{{p}}}$$ . Moreover, if the set lacks its independency and cannot be discriminated anymore, the distinguishability of the pair, measured by the fidelity $$F(\varvec{\rho _{_{T}}}, \varvec{\eta _{{p}}})$$ , becomes minimum. This enables one to associate with each discrimination problem a measure of discriminability defined by the fidelity $$F(\varvec{\rho _{_{T}}}, \varvec{\eta _{{p}}})$$ . This quantity, though distinct from the maximum probability of success, has the advantage of being easy to calculate, and in this respect, it can find useful applications in estimating the extent to which the set is discriminable.

Generalized bipartite quantum state discrimination problems with sequential measurements

We investigate an optimization problem of finding quantum sequential measurements, which forms a wide class of state discrimination problems with the restriction that only sequential measurements are allowed. Sequential measurements from Alice to Bob on a bipartite system are considered. Using the fact that the optimization problem can be formulated as a problem with only Alice's measurement and is convex programming, we derive its dual problem and necessary and sufficient conditions for an optimal solution. In the problem we address, the output of Alice's measurement can be infinite or continuous, while sequential measurements with a finite number of outcomes are considered. It is shown that there exists an optimal sequential measurement in which Alice's measurement with a finite number of outcomes as long as a solution exists. We also show that if the problem has a certain symmetry, then there exists an optimal solution with the same type of symmetry. A minimax version of the problem is considered, and necessary and sufficient conditions for a minimax solution are derived. An example in which our results can be used to obtain an analytical expression for an optimal sequential measurement is finally provided.

Finding Optimal Solutions for Generalized Quantum State Discrimination Problems

We try to find an optimal quantum measurement for generalized quantum state discrimination problems, which include the problem of finding an optimal measurement maximizing the average correct probability with and without a fixed rate of inconclusive results and the problem of finding an optimal measurement in the Neyman-Pearson strategy. We propose an approach in which the optimal measurement is obtained by solving a modified version of the original problem. In particular, the modified problem can be reduced to one of finding a minimum error measurement for a certain state set, which is relatively easy to solve. We clarify the relationship between optimal solutions to the original and modified problems, with which one can obtain an optimal solution to the original problem in some cases. Moreover, as an example of application of our approach, we present an algorithm for numerically obtaining optimal solutions to generalized quantum state discrimination problems.

Simple, Near-Optimal Quantum Protocols for Die-Rolling

Die-rolling is the cryptographic task where two mistrustful, remote parties wish to generate a random D-sided die-roll over a communication channel. Optimal quantum protocols for this task have been given by Aharon and Silman (New Journal of Physics, 2010) but are based on optimal weak coin-flipping protocols that are currently very complicated and not very well understood. In this paper, we first present very simple classical protocols for die-rolling that have decent (and sometimes optimal) security, which is in stark contrast to coin-flipping, bit-commitment, oblivious transfer, and many other two-party cryptographic primitives. We also present quantum protocols based on the idea of integer-commitment, a generalization of bit-commitment, where one wishes to commit to an integer. We analyze these protocols using semidefinite programming and finally give protocols that are very close to Kitaev’s lower bound for any D ≥ 3 . Lastly, we briefly discuss an application of this work to the quantum state discrimination problem.

Solutions to the mean king’s problem: Higher-dimensional quantum error-correcting codes

Mean king’s problem is a kind of quantum state discrimination problems. In the problem, we try to discriminate eigenstates of noncommutative observables with the help of classical delayed information. The problem has been investigated from the viewpoint of error detection and correction. We construct higher-dimensional quantum error-correcting codes against error corresponding to the noncommutative observables. Any code state of the codes provides a way to discriminate the eigenstates correctly with the classical delayed information.

Erratum: Generalized quantum state discrimination problems [Phys. Rev. A91, 052304 (2015)

Erratum: Generalized quantum state discrimination problems [Phys. Rev. A<b>91</b>, 052304 (2015)

Generalized quantum state discrimination problems

We address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average success probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used.

Class of unambiguous state discrimination problems achievable by separable measurements but impossible by local operations and classical communication

We consider an infinite class of unambiguous quantum state discrimination problems on multipartite systems, described by Hilbert space $\cal{H}$, of any number of parties. Restricting consideration to measurements that act only on $\cal{H}$, we find the optimal global measurement for each element of this class, achieving the maximum possible success probability of $1/2$ in all cases. This measurement turns out to be both separable and unique, and by our recently discovered necessary condition for local quantum operations and classical communication (LOCC), it is easily shown to be impossible by any finite-round LOCC protocol. We also show that, quite generally, if the input state is restricted to lie in $\cal{H}$, then any LOCC measurement on an enlarged Hilbert space is effectively identical to an LOCC measurement on $\cal{H}$. Therefore, our necessary condition for LOCC demonstrates directly that a higher success probability is attainable for each of these problems using general separable measurements as compared to that which is possible with any finite-round LOCC protocol.

Dimension witnesses and quantum state discrimination.

Dimension witnesses allow one to test the dimension of an unknown physical system in a device-independent manner, that is, without placing assumptions about the functioning of the devices used in the experiment. Here we present simple and general dimension witnesses for quantum systems of arbitrary Hilbert space dimension. Our approach is deeply connected to the problem of quantum state discrimination, hence establishing a strong link between these two research topics. Finally, our dimension witnesses can distinguish between classical and quantum systems of the same dimension, making them potentially useful for quantum information processing.

Quantum state discrimination bounds for finite sample size

In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of the two given and completely known states, ρ or σ. In general, it is not possible to decide the identity of the true state with certainty, and the optimal measurement strategy depends on whether the two possible errors (mistaking ρ for σ, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies, and the decay rate is given by a certain statistical distance between ρ and σ (the Chernoff distance, the Hoeffding distances, and the relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a system, and therefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errors, and the mixed error probabilities related to the Chernoff and the Hoeffding errors.

Quantum state discrimination with general figures of merit

We solve the problem of quantum state discrimination with ``general (symmetric) figures of merit'' for an even number of symmetric quantum bits with use of the no-signaling principle. It turns out that conditional probability has the same form for any figure of merit. Optimal measurement and corresponding conditional probability are the same for any monotonous figure of merit.

SHARING OF D-DIMENSIONAL QUANTUM STATES

We propose a scheme for the deterministic sharing arbitrary qudit states among three distant parties and characterize the set of ideal quantum channels. We also show that the use of non-ideal quantum channels for quantum state sharing can be related to the problem of quantum state discrimination. This allows us to formulate a protocol which leads to perfect quantum state sharing with a finite success probability.

Quantum algorithm for the asymmetric weight decision problem and its generalization to multiple weights

As one of the applications of Grover search, an exact quantum algorithm for the symmetric weight decision problem of a Boolean function has been proposed recently. Although the proposed method shows a quadratic speedup over the classical approach, it only applies to the symmetric case of a Boolean function whose weight is one of the pair {0 < w 1 < w 2 < 1, w 1 + w 2 = 1}. In this article, we generalize this algorithm in two ways. Firstly, we propose a quantum algorithm for the more general asymmetric case where {0 < w 1 < w 2 < 1}. This algorithm is exact and computationally optimal. Secondly, we build on this to exactly solve the multiple weight decision problem for a Boolean function whose weight as one of {0 < w 1 < w 2 < · · · < w m < 1}. This extended algorithm continues to show a quantum advantage over classical methods. Thirdly, we compare the proposed algorithm with the quantum counting method. For the case with two weights, the proposed algorithm shows slightly lower complexity. For the multiple weight case, the two approaches show different performance depending on the number of weights and the number of solutions. For smaller number of weights and larger number of solutions, the weight decision algorithm can show better performance than the quantum counting method. Finally, we discuss the relationship between the weight decision problem and the quantum state discrimination problem.

Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers

We address the limit of the Gaussian operations and classical communication in the problem of quantum state discrimination. We show that the optimal Gaussian strategy for the discrimination of the binary phase shift keyed (BPSK) coherent signal is a simple homodyne detection. We also propose practical near-optimal quantum receivers that beat the BPSK homodyne limit in all areas of the signal power. Our scheme is simple and does not require realtime electrical feedback.

State discrimination with error margin and its locality

There are two common settings in a quantum-state discrimination problem. One is minimum-error discrimination where a wrong guess (error) is allowed and the discrimination success probability is maximized. The other is unambiguous discrimination where errors are not allowed but the inconclusive result ``I do not know'' is possible. We investigate a discrimination problem with a finite margin imposed on the error probability. The two common settings correspond to the error margins 1 and 0. For arbitrary error margin, we determine the optimal discrimination probability for two pure states with equal occurrence probabilities. We also consider the case where the states to be discriminated are multipartite and show that the optimal discrimination probability can be achieved by local operations and classical communication.

Phase-remapping attack in practical quantum-key-distribution systems

Quantum key distribution (QKD) can be used to generate secret keys between two distant parties. Even though QKD has been proven unconditionally secure against eavesdroppers with unlimited computation power, practical implementations of QKD may contain loopholes that may lead to the generated secret keys being compromised. In this paper, we propose a phase-remapping attack targeting two practical bidirectional QKD systems (the ``plug-and-play'' system and the Sagnac system). We showed that if the users of the systems are unaware of our attack, the final key shared between them can be compromised in some situations. Specifically, we showed that, in the case of the Bennett-Brassard 1984 (BB84) protocol with ideal single-photon sources, when the quantum bit error rate (QBER) is between 14.6% and 20%, our attack renders the final key insecure, whereas the same range of QBER values has been proved secure if the two users are unaware of our attack; also, we demonstrated three situations with realistic devices where positive key rates are obtained without the consideration of Trojan horse attacks but in fact no key can be distilled. We remark that our attack is feasible with only current technology. Therefore, it is very important to be aware of our attack in order to ensure absolute security. In finding our attack, we minimize the QBER over individual measurements described by a general POVM, which has some similarity with the standard quantum state discrimination problem.