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.
A class of self-similar sets of entangled quantum states is introduced, for which a recursive definition is provided. These sets, the "Bell gems," are defined by the subsystem exchange symmetry characteristic of the Bell states. Each Bell gem is shown to be an orthonormal basis of maximally entangled elements. A non-trivial example Bell gem is presented. Quantum circuits for producing the elements of this example from the computational basis states are provided.
In this paper, we present a generalized Bell inequality for mixed states. The distinct characteristic is that the inequality has variable bound depending on the decomposition of the density matrix. The inequality has been shown to be more refined than the previous Bell inequality. It is possible that a separable mixed state can violate the Bell inequality.
We derive Bell inequalities for graph states by generalizing the approach proposed by Ardehali [Phys. Rev. A 46, 5375 (1992)] for Greenberger-Horne-Zeilinger (GHZ) states. Using this method, we demonstrate that Bell inequalities with nonstabilizer observables are often superior to the optimal GHZ-Mermin-type (or stabilizer-type) Bell inequalities.
We employ a straightforward relation between mutually unbiased and Bell bases to extend the latter in terms of a direct construction for the former. We analyze in detail the properties of these new generalized Bell states, showing that they constitute an appropriate tool for testing entanglement in bipartite multiqudit systems.
We derive a multipartite generalized Bell inequality which involves the entire range of settings for each of the local observers. Especially, it is applied to show non-local behavior of a six-qubit mixture of Greenberger-Horne-Zeilinger correlations stronger than previous Bell inequalities. For certain noise admixture to the correlations an explicit local realistic model exists in the case of a standard Bell experiment. Bell experiments with many local settings reveal the non-locality of the state. It turns out that the new inequality is more stringent than many other Bell inequalities in the …
We show that the generalized Bell-type inequality, explicitly involving rotational symmetry of physical laws, is very efficient in distinguishing between true N-particle quantum correlations and correlations involving less particles. This applies to various types of generalized partial separabilities. We also give a rigorous proof that the new Bell inequalities are maximally violated by the GHZ states, and find a very handy description of the N-qubit correlation function.
The notion of equivalence of maximally entangled bases of bipartite d-dimensional Hilbert spaces is introduced. An explicit method of inequivalent bases construction is presented.
We propose a generalized structure of Bell inequalities for arbitrary d-dimensional bipartite systems, which includes the existing two types of Bell inequalities introduced by Collins-Gisin-Linden-Massar-Popescu [Phys. Rev. Lett. 88, 040404 (2002)] and Son-Lee-Kim [Phys. Rev. Lett. 96, 060406 (2006)]. We analyze Bell inequalities in terms of correlation functions and joint probabilities, and show that the coefficients of correlation functions and those of joint probabilities are in Fourier transform relations. We finally show that the coefficients in the generalized structure determine the cha…
We consider quantum systems, whose dynamical symmetry groups are semisimple Lie groups, which can be split or decay into two subsystems of the same symmetry. We prove that the only states of such a system that factorize upon splitting are the generalized coherent states. Since Bell's inequality is never violated by the direct product state, when the system prepared in the generalized coherent state is split, no quantum correlations are created. Therefore, the generalized coherent states are the unique Bell states, i.e., the pure quantum states preserving the fundamental classical property of s…
In this letter, we show that the measure of entanglement using the generalized Bell inequality and the distance between states coincide when we use the Hilbert-Schmidt norm. Our conclusions apply to the spin-1/2 Heisenberg chains with interaction between the first neighbors.
We study the problem of distinguishing maximally entangled quantum states by using local operations and classical communication (LOCC). A question of fundamental interest is whether any three maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are distinguishable by LOCC. In this paper, we restrict ourselves to consider the generalized Bell states. And we prove that any three generalized Bell states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are locally distinguishable.
We provide a method to describe quantum nonlocality for $n$-qubit systems. By treating the correlation function as an $n$-index tensor, we derive a generalized Bell inequality. Taking generalized Greenberger-Horne-Zeilinger (GHZ) state for example, we calculate quantum prediction under a series of measurement settings involving various angle parameters. We reveal the exact relationship between quantum prediction and the angle parameters. We show that there exists a set of optimal measurement settings and find the corresponding maximal quantum prediction for $n$-qubit generalized GHZ states. As…
An efficient and economical scheme is proposed for the perfect quantum teleportation of n-qubit non-maximally entangled state of generalized Bell-type. A Bell state is used as the quantum channel in the proposed scheme. It is also shown that the controlled teleportation of this n-qubit state can be achieved by using a GHZ state or a GHZ-like state as quantum channel. The proposed schemes are economical because for the perfect and controlled teleportation of n-qubit non-maximally entangled state of generalized Bell-type we only need a Bell state and a tripartite entangled state respectively. It…
A procedure is developed to automatically correct the bit flip and the arbitrary phase change errors in Generalized Bell states (GBS). The phase and parity characteristics of the GBS are encoded in ancilla bits without altering the state under consideration. The same information is then used to correct the state at a different point, through appropriate unitrary control operations. It is also shown that the distributed/indirect measurements on ancilla can yield the error syndrome. The restricted error correction circuit obtained for N-qubit entangled states of the Bell type, are generalized fo…
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…
Finite dimensional representations of extended Weyl-Heisenberg algebra are studied both from mathematical and applied viewpoints. They are used to define unitary phase operator and the corresponding eigenstates (phase states). It is also shown that the unitary depolarizers can be constructed in a general setting in terms of phase operators. Generation of generalized Bell states using the phase operator is presented and their expressions in terms of the elements of mutually unbiased bases are given.
We review classical properties of harmonic-oscillator coherent states. Then we discuss which of these classical properties are preserved under the group-theoretic generalization of coherent states. We prove that the generalized coherent states of quantum systems with Lie-group symmetries are the unique Bell states, i.e., the pure quantum states preserving the fundamental classical property of satisfying Bell's inequality upon splitting.
In the paper [Phys. Rev. A 65, 052331(2002)], an entanglement-based quantum key distribution protocol for d-level systems was proposed. However, in this Comment, it is shown that this protocol is insecure for a special attack strategy.
Several authors have recently claimed that Bell's inequalities (BI) do not apply to certain types of generalized local hidden variables (HV) models. These claims are rejected, by means of a proof of BI valid for a very broad class of local HV models (deterministic or stochastic, with or without memory, time dependent or time independent, setting dependent or setting independent, and even allowing classical communication between the wings of the experiment). The precise meaning of the locality requirement is clarified during the proof. The exact roles of the two assumptions that amount to Bell'…
Local distinguishability of sets of generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set of GBSs can be locally transformed to a certain type of GBS set that is easily distinguishable within local operations and one-way classical communication. We then show that, if the space dimension $d$ is a prime, these conditions are necessary and sufficient for sets of $d$ GBSs in $\mathbb{C}^d \otimes \mathbb{C}^d$ to be locally distinguishable. Thus we obtain a simple computable criterion for local distinguishability of sets of $d$ GBSs in prime dimension $d$.
We derive a Bell inequality based on a generalized quasiprobability function which is parameterized by one non-positive real value. Two types of known Bell inequalities formulated in terms of the Wigner and Q functions are included as limiting cases. We investigate violations of our Bell inequalities for single photon entangled states and two-mode squeezed vacuum states when varying the detector efficiency. We show that the Bell inequality for the Q function allows the lowest detection efficiency for violations of local realism.
In a recent comment \cite{ch1} it has been claimed that an entangled-based quantum key distribution protocol proposed in \cite{zhang} and its generalization to d-level systems in \cite{v1} are insecure against an attack devised by the authors of the comment. We invalidate the arguments of the comment and show that the protocols are still secure.
The phenomenon of quantum steering in bipartite quantum systems can be reduced to the question whether or not the first party can perform measurements such that the conditional states on the second party can be explained by a local hidden state model. Clearly, the answer to this depends on the measurements which the first party is able to perform. We introduce a local hidden state model explaining the conditional states for all generalized measurements on Bell-diagonal states of two qubits. More precisely, it is known for the restricted case of projective measurements and Bell-diagonal states …
In this article, we analyse the relationship between the Bell violation and the secure key rate of entanglement assisted quantum key distribution (QKD) protocols. Specifically, we address the question whether Bell violation is necessary or sufficient for secure communication. We construct a class of states which do not show Bell violation, however, which can be used for secure communication after local filtering. Similarly, we identify another class of states which show Bell violation but can not be used for generating secure key even after local filtering. The existence of these two classes o…
We consider the entanglement of orthogonal generalized Bernoulli states in two separate single-mode high-$Q$ cavities. The expectation values and the correlations of the electric field in the cavities are obtained. We then define, in each cavity, a dichotomic operator expressible in terms of the field states which can be, in principle, experimentally measured by a probe atom that ``reads'' the field. Using the quantum correlations of couples of these operators, we construct a Bell's inequality which is shown to be violated for a wide range of the degree of entanglement and which can be tested …
Designing a quantum key agreement (QKA) protocol is always a challenging task, because both the security and the fairness properties have to be considered simultaneously. Recently, Zhu et al. (Quantum Inf Process 14(11): 4245-4254) pointed out that Shukla et al.'s QKA protocol (Quantum Inf Process 13(11): 2391-2405) has some security flaws (which lead to the Participant Attack). Moreover, they proposed an improvement to avoid these weaknesses. However this study points out that the improved protocol also suffers from a colluding attack, i.e., two dishonest participants in the protocol can coll…
We study the mathematical structures and relations among some quantities in the theory of quantum entanglement, such as separability, weak Schmidt decompositions, Hadamard matrices etc.. We provide an operational method to identify the Schmidt-correlated states by using weak Schmidt decomposition. We show that a mixed state is Schmidt-correlated if and only if its spectral decomposition consists of a set of pure eigenstates which can be simultaneously diagonalized in weak Schmidt decomposition, i.e. allowing for complex-valued diagonal entries. For such states, the separability is reduced to t…
A quantum key distribution protocol with classical Bob based on polarization entangled photon pairs is presented. It approximates a single photon and exploited the inherent randomness of quantum measurements to attain highly secure keys and high efficiency of the transmission.
In this work, we present a new class of genuine multipartite Bell inequalities, that is particularly designed for multipartite device-independent (DI) quantum key distribution (QKD), also called DI conference key agreement. We prove the classical bounds of this inequality, discuss how to maximally violate it and show its usefulness by calculating achievable conference key rates via the violation of this Bell inequality. To this end, semidefinite programming techniques based on [Nat. Commun. 2, 238 (2011)] are employed and extended to the multipartite scenario. Our Bell inequality represents a …
We propose a new scheme in which perfect transmission of 1-qubit information is achieved with high success using purposefully delayed sharing of non-maximally entangled 2-qubit resource and repeated generalized Bell-state measurements (GBSM). Alice possesses initially all qubits and she makes repeated GBSM on the pair of qubits, consisting of (1) the qubit of information state and (2) one of the two entangled resource qubits (taken alternately) until transmission with perfect fidelity is indicated. Alice then sends to Bob, the qubit not used in the last GBSM and also the result of this GBSM an…
We consider general settings of Bell inequality experiments with many parties, where each party chooses from a finite number of measurement settings each with a finite number of outcomes. We investigate the constraints that Bell inequalities place upon the correlations possible in a local hidden variable theories using a geometrical picture of correlations. We show that local hidden variable theories can be characterized in terms of limited computational expressiveness, which allows us to characterize families of Bell inequalities. The limited computational expressiveness for many settings (ea…
We proposed a new scheme for quantum key distribution based on entanglement swapping. By this protocol \QTR{em}{Alice} can securely share a random quantum key with \QTR{em}{Bob}, without transporting any particle.
In device-independent quantum key distribution (DIQKD) the security is not based on any assumptions about the intrinsic properties of the devices and the quantum signals, but on the violation of a Bell inequality. We introduce a DIQKD scenario in which an optimal Bell inequality is constructed from the performed measurement data, rather than fixing beforehand a specific Bell inequality. Our method can be employed in a general way, for any number of measurement settings and any number of outcomes. We provide an implementable DIQKD protocol and perform finite-size security key analysis for colle…
We report on a complete free-space field implementation of a modified Ekert91 protocol for quantum key distribution using entangled photon pairs. For each photon pair we perform a random choice between key generation and a Bell inequality. The amount of violation is used to determine the possible knowledge of an eavesdropper to ensure security of the distributed final key.
A proof of Bell's theorem without inequalities and involving only two observers is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin. This proof is generalized to obtain an inequality-free proof of Bell's theorem for a set of n Bell states (with n odd) shared between two distant observers. A generalized CHSH inequality is formulated for n Bell states shared symmetrically between two observers and it is shown that quantum mechanics violates this inequality by an amount that grows exponentially with increasing n.
The time-reversed version of entanglement-based quantum key distribution (QKD), called measurement-device-independent QKD (mdiQKD), was originally introduced to close arbitrary security loopholes of measurement devices. Here we show that the mdiQKD has another advantage which should be distinguished from the entanglement-based QKD. In particular, an all-photonic adaptive Bell measurement, based on the concept of quantum repeaters, can be installed solely in the mdiQKD, which leads to a square root improvement in the key rate. This Bell measurement also provides a similar improvement in the sin…
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the tool of Bayesian networks, a graphical model used to reason about probabilistic causation. Indeed, Bell's theorem concerns a particular kind of a Bayesian network and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalized Bell scenarios involving more complex Bay…
In a recent paper [J. Opt. B: Quantum Semiclass. Opt. 5 (2003) 155-157], a quantum key distribution scheme based on entanglement swapping was proposed, which exhibited two improvements over the previous protocols. In this Comment, it is shown that the scheme has no properties as been discussed.
We clarify the argument on the how (nonlocal) degenerate Bell measurement can be replaced by local measurements in the modified Lo-Chau quantum key distribution protocol. Discussing security criterion for users, we describe how eavesdropper's refined information on the final state is not helpful. We argue that current discussions on the equivalence of the Bell and the local measurements are not clear. We show how the problem of equivalence can be resolved using the fact that eavesdropper's refined information is not helpful for her.
Intermediate states are known from intercept/resend eavesdropping in the BB84 quantum cryptographic protocol. But they also play fundamental roles in the optimal eavesdropping strategy on BB84 and in the CHSH inequality. We generalize the intermediate states to arbitrary dimension and consider intercept/resend eavesdropping, optimal eavesdropping on the generalized BB84 protocol and present a generalized CHSH inequality for two entangled quNits based on these states.
A new protocol for quantum key distribution based on entanglement swapping is presented. In this protocol, both certain key and random key can be generated without any loss of security. It is this property differs our protocol from the previous ones. The rate of generated key bits per particle is improved which can approach six bits (4 random bits and 2 certain bits) per four particles.
Any Quantum Key Distribution (QKD) protocol consists first of sequences of measurements that produce some correlation between classical data. We show that these correlation data must violate some Bell inequality in order to contain distillable secrecy, if not they could be produced by quantum measurements performed on a separable state of larger dimension. We introduce a new QKD protocol and prove its security against any individual attack by an adversary only limited by the no-signaling condition.
We present Bell-type tests of nonclassicality and non-Gaussianity for single-mode fields employing a generalized quasiprobability function. Our nonclassicality tests are based on the observation that two orthogonal quadratures in phase space (position and momentum) behave as independent realistic variables for a coherent state. Taking four (three) points at the vertices of a rectangle (right triangle) in phase space, our tests detect every pure nonclassical Gaussian state and a range of mixed Gaussian states. These tests also set an upper bound for all Gaussian states and their mixtures, which…
We consider information-theoretic inequalities of the Bell type in the presence of decoherence. It is natural that too strong coupling with the environment can prevent an observation of quantum correlations. In this regard, the use of various entropic functions may give additional capabilities to reveal desired correlations. It was already shown that the Bell and Leggett-Garg inequalities in terms conditional Tsallis entropies are more sensitive in the cases of detection inefficiencies. In this paper, we study capabilities of generalized conditional entropies of the Tsallis type in analyzing t…
In this paper, we study the local unitary classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we firstly introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all of pairs in $d\otimes d$ quantum system into $\prod_{j=1}^{n} (k_{j} + 1) $ LU-inequivalent pairs when the prime facto…
This paper proposes a generalized Bell-like inequality (GBI) for multiparticle entangled Schrödinger-cat--states of arbitrary spin-$s$. Based on quantum probability statistics the GBI and violation are formulated in an unified manner with the help of state density operator, which can be separated to local and non-local parts. The local part gives rise to the inequality, while the non-local part is responsible for the violation. The GBI is not violated at all by quantum average except the spin-$1/2$ entangled states. If the measuring outcomes are restricted in the subspace of spin coherent stat…
Quantum secret-sharing protocols involving N partners (NQSS) are key distribution protocols in which Alice encodes her key into $N-1$ qubits, in such a way that all the other partners must cooperate in order to retrieve the key. On these protocols, several eavesdropping scenarios are possible: some partners may want to reconstruct the key without the help of the other ones, and consequently collaborate with an Eve that eavesdrops on the other partners' channels. For each of these scenarios, we give the optimal individual attack that the Eve can perform. In case of such an optimal attack, the a…
The Ekert 91 quantum key distribution (QKD) protocol appears to be secure whatever devices legitimate users adopt for the protocol, as long as the devices give a result that violates Bell's inequality. However, this is not the case if they ignore non-detection events because Eve can make use of the detection-loophole, as Larrson showed. We show that even when legitimate users take into account non-detection events Eve can successfully eavesdrop if the QKD system has been appropriately designed by the manufacturer. A loophole utilized here is that of `free-choice' (or `real randomness'). Local …
Nonlocality, one of the most remarkable aspects of quantum mechanics, is closely related to Bayesian game theory. Quantum mechanics can offer advantages to some Bayesian games, if the payoff functions are related to Bell inequalities in some way. Most of these Bayesian games that have been discussed are common interest games. Recently the first conflicting interest Bayesian game is proposed in Phys. Rev. Lett. 114, 020401 (2015). In the present paper we present three new conflicting interest Bayesian games where quantum mechanics offers advantages. The first game is linked with Cereceda inequa…
Generalized quantum discord $(D_q)$, Einstein-Podolsky-Rosen steering $(S)$, entanglement $(E)$, and Bell nonlocality ($N$), are logically distinct quantifiers of quantum correlations. All these measures capture nonclassical aspects of quantum states and play some role as resources in quantum information processing. In this work, we look for the hierarchy satisfied by these quantum correlation witnesses for a class of two-qubit states. We show that $N \triangleright S\triangleright E\triangleright D_q$, meaning that nonlocality implies steering, which in turn implies entanglement, which then i…
We make a brief comment on measurement of quantum operators with degenerate eigenstates and apply to quantum teleportation. We also try extending the quantum teleportation by Bennett et al [5] to more general situation by making use of generalized Bell states.
