The question of the discrimination of the Bell states of two qudits (i.e., d-dimensional quantum systems) by means of passive linear optical elements and conditional measurements is discussed. A qudit is supposed to be represented by d optical modes containing exactly one photon altogether. From recent results of Calsamiglia it follows that there is no way how to distinguish the Bell states of two qudits for d>2 - not even with the probability of success lower than one - without any auxiliary photons in ancillary modes. Following the results of Carollo and Palma it is proved that it is impossible to distinguish even only one such a Bell state with certainty (i.e., with the probability of success equal to one), irrespective of how many auxiliary photons are involved. However, it is shown that auxiliary photons can help to discriminate the Bell states of qudits with the high probability of success: A Bell-state analyzer based on the idea of linear optics quantum computation that can achieve the probability of success arbitrarily close to one is described. It requires many auxiliary photons that must be first "combined" into entangled states.
We discuss a scheme in which sequential state-discrimination measurements are performed on qudits to determine the quantum state in which they were initially prepared. The qudits belong to a set of nonorthogonal quantum states and hence cannot be distinguished with certainty. Unambiguous state discrimination allows error-free measurements at the expense of occasionally failing to give a conclusive answer about the state of the qudit. Qudits have the potential to carry more information per transmission than qubits. We considered the situation in which Alice sends one of N qudits, where the dime…
We consider the unambiguous discrimination between two unknown qudit states in $n$-dimensional ($n\geqslant2$) Hilbert space. By equivalence of unknown pure states to known mixed states and with the Jordan-basis method, we demonstrate that the optimal success probability of the discrimination between two unknown states is independent of the dimension $n$. We also give a scheme for a physical implementation of the programmable state discriminator that unambiguously discriminate between two unknown states with optimal probability of success.
Multi-qudit systems are studied in tomographic probability representations of quantum qudit states. Results of calculations for Bell-type numbers within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states.
Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state holden by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this article, we show t…
Generalizations of the classic Bell inequality to higher dimensional quantum systems known as qudits are reputed to exhibit a higher degree of robustness to noise, but such claims are based on one particular noise model. We analyze the violation of the Collins-Gisin-Linden-Massar-Popescu inequality subject to more realistic noise sources and their scaling with dimension. This analysis is inspired by potential Bell inequality experiments with superconducting resonator-based qudits. We find that the robustness of the inequality to noise generally decreases with increasing qudit dimension.
A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. Using the qubit portrait method the Bell inequalities for two qubits and two qutrits are discussed in framework of probability representation of quantum mechanics. Semigroup of stochastic matrices is associated with tomographic probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as ansatz to provide a necessary condition f…
Bounds, expressed in terms of d and N, on full Bell locality of a quantum state for $N\geq 3$ nonlocally entangled qudits (of a dimension $d\geq 2$) mixed with white noise are known, to our knowledge, only within full separability of this noisy N-qudit state. For the maximal violation of general Bell inequalities by an N-partite quantum state, we specify the analytical upper bound expressed in terms of dilation characteristics of this state, and this allows us to find new general bounds in $d, N$, valid for all $d\geq 2$ and all $N\geq 3$, on full Bell locality under generalized quantum measur…
The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of U(n) group, we construct the Jordan basis to derive the analytical results for both the optimal unambi…
Recently, a kind of deterministic all-versus-nothing proof of Bell nonlocality induced from the qubit non-stabilizer state was proposed, breaking the tradition that deterministic all-versus-nothing proofs are always derived from stabilizer states. A trivial generalization to the qudit (d is even) version is by using a special basis map, but such a proof can still be reduced to the qubit version. So far, whether high dimensional non-stabilizer states can induce nontrivial deterministic all-versus-nothing proofs of Bell nonlocality remains unknown. Here we present an example induced from a speci…
We analyze the connection between Bell inequality violations and symmetric extendibility of quantum states. We prove that 2-qubit reduced states of multiqubit symmetric pure states do not violate the Bell Clauser-Horne-Shimony-Holt (CHSH) inequality. We then prove the more general converse that any 2-qubit state that violates the CHSH inequality cannot have a symmetric extension. We extend our analysis to qudits and provide a test for symmetric extendibility of 2-qudit states. We show that if a 2-qudit Bell inequality is monogamous, then any 2-qudit state that violates this inequality does not…
We show that the detection efficiencies required for closing the detection loophole in Bell tests can be significantly lowered using quantum systems of dimension larger than two. We introduce a series of asymmetric Bell tests for which an efficiency arbitrarily close to 1/N can be tolerated using N-dimensional systems, and a symmetric Bell test for which the efficiency can be lowered down to 61.8% using four-dimensional systems. Experimental perspectives for our schemes look promising considering recent progress in atom-photon entanglement and in photon hyperentanglement.
Greenberger-Horne-Zeilinger states are intuitively known to be the most non-classical ones. They lead to the most radically nonclassical behavior of three or more entangled quantum subsystems. However, in case of two-dimensional systems, it has been shown that GHZ states lead to more robustness of Bell nonclassicality in case of geometrical inequalities than in case of Mermin inequalities. We investigate various strategies of constructing geometrical Bell inequalities (BIs) for GHZ states for any dimensionality of subsystems.
The concept of asymptotic correctability of Bell-diagonal quantum states is generalised to elementary quantum systems of higher dimensions. Based on these results basic properties of quantum state purification protocols are investigated which are capable of purifying tensor products of Bell-diagonal states and which are based on $B$-steps of the Gottesman-Lo-type with the subsequent application of a Calderbank-Shor-Steane quantum code. Consequences for maximum tolerable error rates of quantum cryptographic protocols are discussed.
We introduce two types of statistical quasi-separation between local observables to construct two-party Bell-type inequalities for an arbitrary dimensional systems and arbitrary number of measurement settings per site. Note that, the main difference between statistical quasi-separations and the usual statistical separations is that the former are not symmetric under exchange of the two local observables, whereas latter preserve the symmetry. We show that a variety of Bell inequalities can be derived by sequentially applying triangle inequalities which statistical quasi-separations satisfy. A s…
We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased and a composite state is maximally entangled. This feature is similar to Clauser-Horne-Shimony-Holt inequality for two qubits but is in contrast with the two types of inequalities, Collins-Gisin-Linden-Massar-Popescu and Son-Lee-Kim, for high-dimensional systems. The generalization to aribitrary prime-dimensional systems is discussed.
Following the linear programming prescription of Ref. \cite{PRA72}, the $d\otimes d$ Bell diagonal entanglement witnesses are provided. By using Jamiolkowski isomorphism, it is shown that the corresponding positive maps are the generalized qudit Choi maps. Also by manipulating particular $d\otimes d$ Bell diagonal separable states and constructing corresponding bound entangled states, it is shown that thus obtained $d\otimes d$ BDEW's (consequently qudit Choi maps) are non-decomposable in certain range of their parameters.
We study the maximum-confidence (MC) measurement strategy for discriminating among nonorthogonal symmetric qudit states. Restricting to linearly dependent and equally likely pure states, we find the optimal positive operator valued measure (POVM) that maximizes our confidence in identifying each state in the set and minimizes the probability of obtaining inconclusive results. The physical realization of this POVM is completely determined and it is shown that after an inconclusive outcome, the input states may be mapped into a new set of equiprobable symmetric states, restricted, however, to a …
Geometrical Bell Inequalities (GBIs) are the strongest known Bell inequalities for collections of qubits. However, their generalizations to other systems is not yet fully understood. We formulate GBIs for an arbitrary number $N$ of observers, each of which possesses a particle of an arbitrary dimension $d$. The whole $(d-1)$-parameter family of local observables with eigenbases unbiased to the computational basis is used, but it is immediate to use a discrete subset of them. We argue analytically for qutrits and numerically for other systems that the violations grows exponenetially with $N$. W…
We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. The tight structure of Bell inequalities for three $d$-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.
Using the generalized Bell states and controlled not gates, we introduce an enatanglement-based quantum key distribution (QKD) of d-level states (qudits). In case of eavesdropping, Eve's information gain is zero and a quantum error rate of (d-1)/d is introduced in Bob's received qudits, so that for large d, comparison of only a tiny fraction of received qudits with the sent ones can detect the presence of Eve.
I derive separability inequalities for Bell correlations of observables in arbitrary pure or mixed $N$ Qudit states in $D^N$-dimensional state space. I find states (a continuum of states if $D>3$) including maximally entangled states which violate these inequalities by a factor $2^{N-1}$ ; local reality Bell inequalities are much weaker, their maximum violation being by a factor $2^{(N-1)/2}$. The separability inequalities allow tests of entanglement of unknown states using only the measured correlations .
The aim of this thesis is to investigate quantum entanglement and quantum nonlocality of bipartite finite-dimensional systems (bipartite qudits). Entanglement is one of the most fascinating non-classical features of quantum theory, and besides its impact on our view of the world, it can be exploited for applications such as quantum cryptography and quantum computing. (...) Although entanglement and nonlocality are ordinarily regarded as one and the same, under close consideration this cannot be taken for granted. The reason for this is that entanglement is defined by the mathematical structure…
The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from $d=2$ to $d=7$. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective, by directly addressing the space of joint proba…
We make use of local operations with two ancilla bits to deterministically distinguish all the four Bell states, without affecting the quantum channel containing these Bell states.
Quantum superpositions of coherent states are produced both in microwave and optical domains, and are considered realizations of the famous "Schrödinger cat" state. The recent progress shows an increase in the number of components and the number of modes involved. Our work creates a theoretical framework for treatment of multicomponent two-mode Schrödinger cat states. We consider a class of single-mode states, which are superpositions of $N$ coherent states lying on a circle in the phase space. In this class we consider an orthonormal basis created by rotationally-invariant circular states (RI…
We propose a framework to analyze linear optical circuits based on an analogy with stabilizer formalism in quantum circuits, which provides efficiently computable formulas related to state discriminations. Hence, we analyze a Bell-state discrimination scheme with linear optics and ancillary single photons. With an increasing number of ancilla photons, the success probability of Bell-state discrimination has a maximum of $\frac{403}{512} \simeq 0.787$ at $28$ ancilla photons. By contrast, the corresponding two-qubit measurement asymptotically approaches a maximally entangling measurement.
For an even qudit dimension $d\geq 2,$ we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell-Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value $3/2$ - the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an arbitrary even $d\geq 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results ar…
A complete and deterministic Bell state measurement was realized by a simple linear optics experimental scheme which adopts 2-photon polarization-momentum hyperentanglement. The scheme, which is based on the discrimination among the single photon Bell states of the hyperentangled state, requires the adoption of standard single photon detectors. The four polarization Bell states have been measured with average fidelity $F=0.889\pm0.010$ by using the linear momentum degree of freedom as the ancilla. The feasibility of the scheme has been characterized as a function of the purity of momentum enta…
The class of d-setting, d-outcome Bell inequalities proposed by Ji and collaborators [Phys. Rev. A 78, 052103] are reexamined. For every positive integer d > 2, we show that the corresponding non-trivial Bell inequality for probabilities provides the maximum classical winning probability of the Clauser-Horne-Shimony-Holt-like game with d inputs and d outputs. We also demonstrate that the general classical upper bounds given by Ji et al. are underestimated, which invalidates many of the corresponding correlation inequalities presented thereof. We remedy this problem, partially, by providing the…
The so-called quasi-Bell entangled coherent states in a thermal environment are studied. In the analysis, we assume thermal noise affects only one of the two modes of each state. First the matrix representation of the density operators of the quasi-Bell entangled coherent states in a thermal environment is derived. Secondly we investigate the entanglement property of one of the quasi-Bell entangled coherent states with thermal noise. At that time a lower bound of the entanglement of formation for the state is computed. Thirdly the minimax discrimination problem for two cases of the binary set …
We specify the local quasi hidden variable (LqHV) model reproducing the probabilistic description of all N-partite joint von Neumann measurements on an N-qudit state. Via this local probability model, we derive a new upper bound on the maximal violation by an N-qudit state of N-partite Bell inequalities of any type (either on correlation functions or on joint probabilities) for S observables per site. This new upper bound not only improves for all N, S and d the corresponding results available for general Bell inequalities in the literature but also, for the N-qubit case with two observables p…
The original versions (1 and 2) of this paper paper contain a fatal error. All my attempts to patch the error have failed. As a service to the community I explain the error in some detail.
We study local-realistic inequalities, Bell-type inequalities, for bipartite pure states of finite dimensional quantum systems -- qudits. There are a number of proposed Bell-type inequalities for such systems. Our interest is in relating the value of Bell-type inequality function with a measure of entanglement. Interestingly, we find that one of these inequalities, the Son-Lee-Kim inequality, can be used to measure entanglement of a pure bipartite qudit state and a class of mixed two-qudit states. Unlike the majority of earlier schemes in this direction, where number of observables needed to c…
Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems they are technologically demanding. Therefore, determining the constraints posed by linear optics and optimizing linear optical operations for specific tasks under those constraints, without the use of ancillas, can facilitate their potential implementation. Here, we consider the task …
In this work, we study the local distinguishability of maximally entangled states (MESs). In particular, we are concerned with whether any fixed number of MESs can be locally distinguishable for sufficiently large dimensions. Fan and Tian \emph{et al.} have already obtained two satisfactory results for the generalized Bell states (GBSs) and the qudit lattice states when applied to prime or prime power dimensions. We construct a general twist-teleportation scheme for any orthonormal basis with MESs that is inspired by the method used in [Phys. Rev. A \textbf{70}, 022304 (2004)]. Using this tele…
We address the problem of unambiguously identifying the state of a probe qudit with the state of one of d reference qudits. The reference states are assumed pure and linearly independent but we have no knowledge of them. The state of the probe qudit is assumed to coincide equally likely with either one of the d unknown reference states. We derive the optimum measurement strategy that maximizes the success probability of unambiguous identification and find that the optimum strategy is a generalized measurement. We give both the measurement operators and the optimum success probability explicitl…
We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by 3/2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3/2 for violation of the original Bell inequality under Alice and Bob spin measurem…
We studied the distinguishability of generalized Bell states under local operations and classical communication. We introduced the concept of maximally commutative set (MCS), subset of generalized Pauli matrices whose elements are mutually commutative and there is no other generalized Pauli matrix that is commute with all the elements of this set. We found that MCS can be considered as a detector for local distinguishability of set $\mathcal{S}$ of generalized Bell states. In fact, we got an efficient criterion. That is, if the difference set of $\mathcal{S}$ is disjoint with or completely con…
I give a first characterization of the class of generalized measurements that can be exactly realized on a pair of qudits encoded in indistinguishable particles, by using only linear elements and particle detectors. Two immediate results follow from this characterization. (i) The Schmidt number of each POVM element cannot exceed the number of initial particles. This rules out any possibility of performing perfect Bell-measurements for qudits. (ii) The maximum probability of performing a generalized incomplete Bell-measurement is 1/2.
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 …
Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups $IGL(n, \mathbb{R})$ and $GL(n, \mathbb{R})$ respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities ar…
Using the necessary and sufficient conditions, minimum error discrimination among two sets of similarity transformed equiprobable quantum qudit states is investigated. In the case that the unitary operators are generating sets of two irreducible representations, the optimal set of measurements and the corresponding maximum success probability of discrimination are determined in closed form. In the case of reducible representations, there exists no closed-form formula in general, but the procedure can be applied in each case accordingly. Finally, we give the maximum success probability of optim…
We consider the possibility of performing linear optical quantum computation making use of extra photonic degrees of freedom. In particular we focus on the case where we use photons as quadbits. The basic 2-quadbit cluster state is a hyper-entangled state across polarization and two spatial mode degrees of freedom. We examine the non-deterministic methods whereby such states can be created from single photons and/or Bell pairs, and then give some mechanisms for performing higher-dimensional fusion gates.
A scheme for distributed quantum measurement that allows nondestructive or indirect Bell measurement was proposed by Gupta et al., (Int. J. Quant. Infor. \textbf{5} (2007) 627) and subsequently realized experimentally using an NMR-based three-qubit quantum computer by Samal et al., (J. Phys. B, \textbf{43} (2010) 095508). In the present work, a similar experiment is performed using the five-qubit super-conductivity-based quantum computer, which has been recently placed in cloud by IBM Corporation. The experiment confirmed that the Bell state can be constructed and measured in a nondestructive …
We provide a simple class of 2-qudit states for which one is able to formulate necessary and sufficient conditions for separability. As a byproduct we generalize well known construction provided by Horodecki et al. for d=3. It is hoped that these states with known separability/entanglement properties may be used to test various notions in entanglement theory.
The propagation of macroscopic entangled states over large distances in the presence of loss is of fundamental interest and may have practical applications as well. Here we describe two different techniques in which state discrimination can be used to violate Bell's inequality with macroscopic phase-entangled coherent states. We find that Bell's inequality can be violated by these macroscopic states over a distance of approximately 400 km in commercially-available optical fibers.
Too much noise kills entanglement. This is the main problem in its production and transmission. We use a handy approach to indicate noise resistance of entanglement of a bi-partite system described by $d\times d$ Hilbert space. Our analysis uses a geometric approach based on the fact that if a scalar product of a vector $\vec{s}$ with a vector $\vec {e}$ is less than the square of the norm of $\vec{e}$, then $\vec{s}\neq\vec{e}$. We use such concepts for correlation tensors of separable and entangled states. As a general form correlation tensors for pairs of qudits, for $d>2$, is very difficul…
It is known that it is impossible to unambiguously distinguish the four Bell states encoded in pairs of photon polarizations using only linear optics. However, hyperentanglement, the simultaneous entanglement in more than one degree of freedom, has been shown to assist the complete Bell analysis of the four Bell states (given a fixed state of the other degrees of freedom). Yet introducing other degrees of freedom also enlarges the total number of Bell-like states. We investigate the limits for unambiguously distinguishing these Bell-like states. In particular, when the additional degree of fre…
