A. Wojcik, A. Grudka, R. W. Chhajlany
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.
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We computationally investigate the complete polytope of Bell inequalities for 2 particles with small numbers of possible measurements and outcomes. Our approach is limited by Pitowsky's connection of this problem to the computationally hard NP problem. Despite this, we find that there are very few relevant inequivalent inequalities for small numbers. For example, in the case with 3 possible 2-outcome measurements on each particle, there is just one new inequality. We describe mixed 2-qubit states which violate this inequality but not the CHSH. The new inequality also illustrates a sharing of b…
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…
On the representations of Bell's operators in Quantum Mechanics
该研究发现维数大于2的希尔伯特空间中Bell-CHSH不等式的Bell算符存在幺正不等价表示，不同表示违反程度不同，最大违反由Tsirelson界给出，源于空间模式配对机制。
Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are crucial to many conceptual and practical aspects of quantum theory. Here, we develop their role in quantum nonlocality by: i) introducing families of Bell inequalities that are maximally violated by $d$-dimensional MUBs and SICs respectively, ii) proving device-independent certification of natural operational notions of MUBs and SICs, and iii) using MUBs and SICs to develop optimal-rate and nearly optimal-rate protocols for device independent quantum key distribution and device-independent quantum rando…
The only information available about an alleged source of entangled quantum states is the amount $S$ by which the Clauser-Horne-Shimony-Holt (CHSH) inequality is violated: nothing is known about the nature of the system or the measurements that are performed. We discuss how the quality of the source can be assessed in this black-box scenario, as compared to an ideal source that would produce maximally entangled states (more precisely, any state for which $S=2\sqrt{2}$). To this end, we introduce several inequivalent notions of fidelity, each one related to the use one can make of the source af…
Uncountably many inequivalent maximally entangled measurements for two qutrits
研究两个三量子比特最大纠缠测量，给出单参数族测量基，证明其在局域酉操作下不等价，由三维对称信息完备态集构建，还构造了野生误差基首个例子，并讨论了不同测量在量子信息场景下的性能差异。
Experimental Demonstration of Inequivalent Mutually Unbiased Bases
通过高精度光子学系统上的估计问题，实验演示了维度4中不等价互不偏基的操作区别，表明其信息提取能力不同。
All Entangled States are Nonlocal and Self-Testable in the Broadcast Scenario
该研究证明在广播场景中，所有纠缠态均为非本地且可自测，揭示了量子理论中纠缠与非本地性的关系。
Mutually unbiased bases (MUBs) provide a standard tool in the verification of quantum states, especially when harnessing a complete set for optimal quantum state tomography. In this work, we investigate the detection of entanglement via inequivalent sets of MUBs, with a particular focus on unextendible MUBs. These are bases for which an additional unbiased basis cannot be constructed and, consequently, are unsuitable for quantum state verification. Here, we show that unextendible MUBs, as well as other inequivalent sets in higher dimensions, can be more effective in the verification of entangl…
How can one discriminate different inequivalent classes of multiparticle entanglement experimentally? We present an approach for the discrimination of an experimentally prepared state from the equivalence class of another state. We consider two possible measures for the discrimination strength of an observable. The first measure is based on the difference of expectation values, the second on the relative entropy of the probability distributions of the measurement outcomes. The interpretation of these measures and their usefulness for experiments with limited resources are discussed. In the cas…
We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the n…
We established a physically utilizable Bell inequality based on the Peres-Horodecki criterion. The new quadratic probabilistic Bell inequality naturally provides us a necessary and sufficient way to test all entangled two-qubit or qubit-qutrit states including the Werner states and the maximally entangled mixed states.
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.
Special features of the Weyl-Heisenberg Bell basis imply unusual entanglement structure of Bell-diagonal states
本文研究Weyl–Heisenberg标准Bell基的群结构，揭示其导致Pauli信道与twirl信道等价，并使Bell对角态具有最高的PPT及PPT纠缠态占比。
We present a crytographic protocol based upon entangled qutrit pairs. We analyse the scheme under a symmetric incoherent attack and plot the region for which the protocol is secure and compare this with the region of violations of certain Bell inequalities.
Using a generalization of Cayley's hyperdeterminant as a new measure of tripartite fermionic entanglement we obtain the SLOCC classification of three-fermion systems with six single particle states. A special subclass of such three-fermion systems is shown to have the same properties as the well-known three-qubit ones. Our results can be presented in a unified way using Freudenthal triple systems based on cubic Jordan algebras. For systems with an arbitrary number of fermions and single particle states we propose the Plücker relations as a sufficient and necessary condition of separability.
In this paper, we define the return operator, the cross product operator and the cross Bell bases. Using the cross Bell bases, we give a protocol of teleportation of arbitrary multipartite qubit entanglement, this scheme is a quite natural generalization of the BBCJPW scheme. We find that this teleportation, in fact, is essentially determined by the teleportation of every single unknown qubit state as in the original scheme of BBCJPW. The calculation in detail is given for the case of tripartite qubit.
We examine the problem of exhibiting Bell nonlocality for a two-qudit entangled pure state using a randomly chosen set of mutually unbiased bases (MUBs). Interestingly, even if we employ only two-setting Bell inequalities, we find a significant chance of obtaining a Bell violation if the two parties are individually allowed to measure a sufficient number of MUBs. In particular, for the case of maximally entangled qutrits and ququarts, our numerical estimates indicate that we can obtain near-guaranteed Bell violation by considering only such Bell inequalities. The case of maximally entangled qu…
We investigate the relation between unextendible product bases (UPB) and Bell inequalities found recently in [R. Augusiak et al., Phys. Rev. Lett. 107, 070401 (2011)]. We first review the procedure introduced there that associates to any set of mutually orthogonal product vectors in a many-qubit Hilbert space a Bell inequality. We then show that if a set of mutually orthogonal product vectors can be completed to a full basis, then the associated Bell inequality is trivial, in the sense of not being violated by any nonsignalling correlations. This implies that the relevant Bell inequalities tha…
We numerically investigate entropic Bell inequalities for a pair of entangled qutrits using information-theoretic distances. We show that for this class of inequalities Tsallis entropy is more suitable than Shannon as it reveals non-classicality for a larger set of quantum states. Finally, we find that like probability based inequalities, entropic ones are maximally violated by the non-maximally entangled qutrit state.
Exchange-Symmetrized Qudit Bell Bases and Bell-State Distinguishability
本文针对偶数维量子比特对引入交换对称化的广义Bell基，证明奇数维不存在完整基，并表明在LELM限制下可无歧义区分$2d-1$个Bell态，达到上界，扩展了量子通信应用。
Polarization-entangled photon pair generation based on two scalar scattering processes of the vector four photon scattering has been demonstrated experimentally in high nonlinear microstructure fiber with birefringence. By controlling the pump polarization state, polarization-entangled Bell states can be realized. It is provides a simple way to realize efficient and compact fiber based polarization-entangled photon pair sources.
We study the relationship between Bell states, finite groups and complete sets of bases. We show how to obtain a set of N+1 bases in which Bell states are invariant. They generalize the X, Y and Z qubit bases and are associated to groups of unitary transformations that generalize the sigma operators of Pauli. When the dimension N is a prime power, we derive (in agreement with well-known results) a set of mutually unbiased bases. We show how they can be expressed in terms of the (operations of the) associated finite field of N elements.
A new conditional scheme for generating Bell states of two spatially separated high-Q cavities is reported. Our method is based on the passage of one atom only through the two cavities. A distinctive feature of our treatment is that it incorporates from the very beginning the unavoidable presence of fluctuations in the atom-cavity interaction times. The possibility of successfully implementing our proposal against cavity losses and atomic spontaneous decay is carefully discussed.
We present an alternative definition of quantum entanglement for bipartite system based on Bell inequality and operators' noncommutativity. A state is said to be entangled, if the maximum of CHSH expectation value $F_{\max}$ is obtain by noncommutative measures on each particle of the bipartite system; otherwise, the state is a disentangled state. A uniform measure quantifying the degree of entanglement for any state of the bipartite system is also proposed.
A general matrix approach to study entangled states is presented, based on operator completeness relations. Bases of unitary operators are considered, with focus on irreducible representations of groups. Bell measurements for teleportation are considered, and robustness of teleportation to various kinds of non idealities is shown.
Communication complexity problems (CCPs) are tasks in which separated parties attempt to compute a function whose inputs are distributed among the parties. Their communication is limited so that not all inputs can be sent. We show that broad classes of Bell inequalities can be mapped to CCPs and that a quantum violation of a Bell inequality is a necessary and sufficient condition for an enhancement of the related CCP beyond its classical limitation. However, one can implement CCPs by transmitting a quantum system, encoding no more information than is allowed in the CCP, and extract information…
Cryptanalysis and improvement of the quantum private comparison protocol based on Bell entangled states
本研究发现基于Bell态纠缠交换的量子保密比较协议存在致命漏洞，第三方可借此窃取秘密输入，为此提出使用诱骗光子进行一次窃听检测的改进方案，在修复漏洞的同时降低了Bell态消耗并简化了步骤。
We introduce two multiple qubit controlled-unitary gates with different working principles. We employ these gates and existing quantum gates to propose simple and efficient algorithms that generate multi-term orthonormal entangled Bell-like bases. All algorithms thus far known for constructing entangled bases turn out to be special cases of our method. The Bell-like states in any basis is superposition of $2^m$-terms with equal probabilities (that is, their amplitudes being $\pm 1/\sqrt{2^m}$). The orthogonality of the basis does not permit arbitrary amplitudes. The quantum correlations of the…
The strength of classical correlations is subject to certain constraints, commonly known as Bell inequalities. Violation of these inequalities is the manifestation of nonlocality---displayed, in particular, by quantum mechanics, meaning that quantum mechanics can outperform classical physics at tasks associated with such Bell inequalities. Interestingly, however, there exist situations in which this is not the case. We associate an intriguing class of bound entangled states, constructed from unextendable product bases (UPBs) with a wide family of tasks, for which (i) quantum correlations do no…
The present work is motivated by the question as to what aspect of correlation entailed by the two-qubit state serves as the appropriate quantitative resource for steering. To this end, considering Bell-diagonal states, suitable measures of simultaneous correlations in two and three complementary (mutually unbiased) bases are identified as the relevant resources for quantum steering. Quantitative relations between appropriate measures of quantum steering and the corresponding measures of simultaneous correlations in complementary bases are demonstrated which ensure that for two qubit steerable…
Based on the Bell theorem, it has been believed that a theoretical computation of the Bell correlation requires explicit use of an entangled state. Such a physical superposition of light waves occurs in the downconverter sources used in Bell experiments. However, this physical superposition is eliminated by wave propagation to spatially separated detectors. Bell correlations must therefore result from local waves, and the source boundary conditions of their previously entangled state. In the present model, Bell correlations are computed from disentangled separated waves, boundary conditions of…
Designing generalized elegant Bell inequalities in higher dimensions from a Tsirelson bound
本文提出从Tsirelson界出发，设计更高维度的广义优雅贝尔不等式，首次在三维中推导出具有更大违反能力的贝尔不等式。
We show that bipartite Bell inequalities based on the Einstein-Podolsky-Rosen criterion for elements of reality and derived from the properties of some hyperentangled states allow feasible experimental verifications of the fact that quantum nonlocality grows exponentially with the size of the subsystems, and Bell loophole-free tests with currently available photodetection efficiencies.
Quantum resources can improve communication complexity problems (CCPs) beyond their classical constraints. One quantum approach is to share entanglement and create correlations violating a Bell inequality, which can then assist classical communication. A second approach is to resort solely to the preparation, transmission and measurement of a single quantum system; in other words quantum communication. Here, we show the advantages of the latter over the former in high-dimensional Hilbert space. We focus on a family of CCPs, based on facet Bell inequalities, study the advantage of high-dimensio…
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.
The Hardy test of nonlocality can be seen as a particular case of the Bell tests based on the Clauser-Horne (CH) inequality. Here we stress this connection when we analyze the relation between the CH-inequality violation, its threshold detection efficiency, and the measurement settings adopted in the test. It is well known that the threshold efficiencies decrease when one considers partially entangled states and that the use of these states, unfortunately, generates a reduction in the CH violation. Nevertheless, these quantities are both dependent on the measurement settings considered, and in…
Alternative Bell's states and teleportation
本文提出两量子比特纠缠态的替代正交基（部分依赖连续参数），给出量子电路与代码，并研究基于这些态的量子隐形传态及其电路实现。
We present a setup for quantum cryptography based on photon pairs in energy-time Bell states and show its feasability in a laboratory experiment. Our scheme combines the advantages of using photon pairs instead of faint laser pulses and the possibility to preserve energy-time entanglement over long distances. Moreover, using 4-dimensional energy-time states, no fast random change of bases is required in our setup : Nature itself decides whether to measure in the energy or in the time base.
We study the Bell nonlocality of high dimensional quantum systems based on quantum entanglement. A quantitative relationship between the maximal expectation value B of Bell operators and the quantum entanglement concurrence C is obtained for even dimension pure states, with the upper and lower bounds of B governed by C.
In this paper, a multi-party quantum private comparison (MQPC) scheme is suggested based on entanglement swapping of Bell entangled states within d-level quantum system, which can accomplish the equality comparison of secret binary sequences from n users via one execution of scheme. Detailed security analysis shows that both the outside attack and the participant attack are ineffective. The suggested scheme needn't establish a private key among n users beforehand through the quantum key distribution (QKD) method to encrypt the secret binary sequences. Compared with previous MQPC scheme based o…
We propose a method of the Bell bases decomposition to teleport an arbitrary unknown N-qubit state through a nonmaximally entangled quantum channel, and give the universal decomposition matrix of N-qubit. Using the decomposition matrix, we can easily obtain the collapsed state at the receiver site. The inverse matrix that the decomposition matrix is just the transformation matrix that the receiver can manipulate. The decomposition matrix is a function of the parameters of the quantum channel. After defining the sub-matrix of the quantum channel, we find that the decomposition matrix is a tenso…
The construction of Bell inequalities based on Platonic and Archimedean solids (Quantum 4 (2020), 293) is generalized to the case of orbits generated by the action of some finite groups. A number of examples with considerable violation of Bell inequalities is presented.
We numerically demonstrate that "mode-entangled states" based on the transverse modes of classical optical fields in multimode waveguides violate Bell's inequality. Numerically simulating the correlation measurement scheme of Bell's inequality, we obtain the normalized correlation functions of the intensity fluctuations for the two entangled classical fields. By using the correlation functions, the maximum violations of Bell's inequality are obtained. This implies that the two classical fields in the mode-entangled states, although spatially separated, present a nonlocal correlation.
When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations.These Bell states are in one to one correspondence with the N^2 elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function.Thanks to the properties of general…
Pump-Free Microwave-Optical Bell Pair Generation for Teleportation-Based Quantum Transduction
针对量子转导中光学泵浦的复杂性，提出无泵浦微波-光学贝尔对源，利用自旋或原子系统生成损耗鲁棒的预示贝尔对，实现千赫兹速率和高保真度，用于基于隐形传态的转导。
Recently, Liu W et al. proposed a two-party quantum private comparison (QPC) protocol using entanglement swapping of Bell entangled state (Commun. Theor. Phys. 57(2012)583-588). Subsequently, Liu W J et al. pointed out that in Liu W et al.'s protocol, the TP can extract the two users' secret inputs without being detected by launching the Bell-basis measurement attack, and suggested the corresponding improvement to mend this loophole (Commun. Theor. Phys. 62(2014)210-214). In this paper, we first point out the information leakage problem toward TP existing in both of the above two protocols, an…
All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.
Coherent states in minimal-length Quantum Mechanics: inequivalent characterizations and emergent squeezing
研究最小长度量子力学中一维谐振子的相干态，发现标准表征（湮灭算符本征态、位移真空态、最小不确定波包）不再等价，并揭示相空间轨迹变形和内在压缩机制。
We find that multidimensional determinants "hyperdeterminants", related to entanglement measures (the so-called concurrence or 3-tangle for the 2 or 3 qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the …
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