Bell-diagonal States Research Articles

Entropy-based geometric measures of correlations for bell diagonal states via measurement

Abstract The classification and quantification of quantum correlations are the important issues in quantum information theory. Quantum relative entropy and Rényi relative entropy are both non-negative, and can therefore be regarded as generalized distances between quantum states. We first introduced the method proposed by Brodutch and Modi for constructing measurements of classical and quantum correlations. Subsequently, we use this method to construct measurement-induced geometric classical and quantum correlations based on quantum relative entropy and Rényi− 1 2 relative entropy. For two-qubit Bell diagonal states, the analytic expressions for these geometric classical correlations and quantum correlations are obtained.

Three Dimensional Exploration of the Dynamics of Bell Diagonal States

Three Dimensional Exploration of the Dynamics of Bell Diagonal States

Quantum correlation swapping between Bell-diagonal states

Quantum correlation (QC) swapping is a generalization of quantum entanglement swapping, in which the concerned QCs can be quantum entanglement, QC beyond quantum entanglement, or both of them. In general, QC swapping is different from quantum entanglement swapping. Obviously, the study range of the former is beyond than the latter, and the former can give more interesting and significant applications in quantum information processings. So far, QC swapping has been studied explicitly only for a rather limited set of two-qubit quantum states as the initial states and cases for more general QC swapping with multi-parameter initial states are not known. In this article, we explore the QC swapping case with three-parameter Bell-diagonal (BD) states as the initial states. The familiar QC measure, quantum discord (Ollivier and Zurek 2001 Phys. Rev. Lett. 88 017901), is utilized to quantify all the QCs in the concerned states. It is found that a long-distance shared QC can be acquired through the BD-initial states case of QC swapping. Moreover, some distinct features and characteristics can be exposed.

Performance of entanglement purification including maximally entangled mixed states

Entanglement between distant quantum systems is a critical resource for implementing quantum communication. This property is affected by external agents and can be restored by employing efficient entanglement purification protocols. In this work, we propose an entanglement purification protocol based on two entangling two-qubit operations that replace the usual controlled-NOT (CNOT) gate. These operations arise from a generalised quantum measurement and can be understood as measurement operators in a positive operator-valued measure. Furthermore, two variants of the core protocol are introduced and their performances are studied in terms of the overall success probability of reaching a Bell state and the number of purifiable states. Based on rank-two states, we can obtain analytical expressions for the success probability that we extend and refine using numerical calculations to the case of maximally entangled states. We also consider more general rank-three states to show that our procedure is in general more convenient compared to purification protocols based on Bell diagonal states. Finally, we test the protocols using initial random states. In all cases, we find a better performance and a larger amount of purifiable states using our schemes compared to the CNOT-based protocol.

Mutually unbiased coherence of Bell Diagonal States* *This work is supported by NSFC under No. 11761073.

This study explores the coherence of Bell-diagonal states under two complete sets of mutually unbiased bases (MUBs), focusing on four measures: the l 1 norm, l p norm, Rényi relative α-entropy, and relative entropy. We initially construct a complete set of MUBs in C4 by incorporating two maximally entangled bases unbiased with respect to the autotensor of mutually unbiased bases (AMUBs), subsequently comparing it with the canonical set documented in existing literature. We establish the expressions for Bell-diagonal states under these two complete sets of MUBs and demonstrate that the aggregate of each coherence measure across both sets is equivalent. Additionally, we present the upper bounds for the sums of the l 1 norm, l p norm, and Rényi relative 2-entropy of coherence. Furthermore, we describe the geometric structure associated with the sum of relative entropy coherence, revealing that an increase in the sum of relative entropy of coherence corresponds to an increase in the parameter of the Bell-diagonal states.

Bound entangled Bell diagonal states of unequal local dimensions, and their witnesses

Bound entangled Bell diagonal states of unequal local dimensions, and their witnesses

Special features of the Weyl–Heisenberg Bell basis imply unusual entanglement structure of Bell-diagonal states

Abstract Bell states are of crucial importance for entanglement based methods in quantum information science. Typically, a standard construction of a complete orthonormal Bell-basis by Weyl–Heisenberg operators is considered. We show that the group structure of these operators has strong implication on error correction schemes and on the entanglement structure within Bell-diagonal states. In particular, it implies an equivalence between a Pauli channel and a twirl channel. Interestingly, other complete orthonormal Bell-bases do break the equivalence and lead to a completely different entanglement structure, for instance in the share of positive partial transposition (PPT)-entangled states. In detail, we find that the standard Bell basis has the highest observed share on PPT-states and PPT-entangled states compared to other Bell bases. In summary, our findings show that the standard Bell basis construction exploits a very special structure with strong implications to quantum information theoretic protocols if a deviation is considered.

Selective weak measurement reveals super-ergotropy

The concept of ergotropy was previously introduced as the maximum extractable work from a quantum state. Its enhancement, which is induced by quantum correlation via projective measurement, was formulated as the daemonic ergotropy. In this work, we investigate the ergotropy in the presence of quantum correlation via weak measurement because of its elegant effects on the measured system. By considering a bipartite correlated quantum system consisting of main and ancillary systems, we demonstrate that the extractable work by the non-selective weak measurement on the ancilla is always equal to the situation captured by the strong measurement. However, the selective weak measurement interestingly reveals more work than the daemonic ergotropy and the ergotropy of the total system is greater than or equal to the daemonic ergotropy. Moreover, it is shown that for Bell diagonal states, at the cost of losing quantum correlation, the total extractable and thus non-local extractable works can be increased by using measurement. Also, we find that there is no direct relationship between quantum correlation and non-local extractable work for these cases.

Performance comparison of Gilbert’s algorithm and machine learning in classifying Bell-diagonal two-qutrit entanglement

While classifying states as entangled or separable is one of the fundamental tasks in quantum information theory, it is also extremely challenging. This task is highly nontrivial even for relatively simple cases, such as two-qutrit Bell-diagonal states, i.e., mixtures of nine mutually orthogonal maximally entangled states. In this article we apply Gilbert’s algorithm to revise previously obtained results for this class. In particular we use “entanglement cartography” to argue that most states left in [Hiesmayr, B. C. Scientific Reports 11, 19739 (2021)] as unknown to be entangled or separable are most likely indeed separable, or very weakly entangled, beyond any practical relevance. The presented technique can find endless applications in more general cases.

Weak measurement as a tool for characterizing coherence and quantum correlations in bipartite systems

In this article, we study quantum coherence of bipartite state from the perspective of weak measurement, which generalizes the notion of coherence relative to measurement. This is being illustrated by computing coherence for the well-known Bell diagonal and Werner states. Further, we have expanded the notion of coherence to encompass correlations, exploring its connections with other quantifications of nonlocality. We have determined that this coherence measure can be perceived as a generalized version of the Wigner-Yanese skew information measure and several other nonlocal measures. As a natural extension, we introduce uncertainty relations within the context of weak measurements. Additionally, we propose a novel form of variance by considering the weak value of an observable within a comprehensive framework.

The failure of Wigner function negativity to detect quantum dissonance in an open bipartite system

We study the dynamics of Wigner function, quantum discord, entanglement and quantum dissonance for bipartite Bell-diagonal states independently interacting with dephasing reservoirs. We analyze comparatively their behavior under the decoherence effect. Our main goal is to test the ability of Wigner function negativity to measure all kinds of quantum correlations in such system. Particularly, we find that this negativity can only capture entanglement and discord. Of course it succeeded in predicting the behavior of quantum dissonance, but nevertheless, it failed to detect its presence. As a matter of fact, we conclude that Wigner function negativity cannot be used as measure of quantum dissonance in the two qubit open system.

Geometric discord of tripartite quantum systems

We study the quantification of geometric discord for tripartite quantum systems. Firstly, we obtain the analytic formula of geometric discord for tripartite pure states. It is already known that the geometric discord of pure states reduces to the geometric entanglement in bipartite systems, the results presented here show that this property is no longer true in tripartite systems. Furthermore, we provide an operational meaning for tripartite geometric discord by linking it to quantum state discrimination, that is, we prove that the geometric discord of tripartite states is equal to the minimum error probability to discriminate a set of quantum states with von Neumann measurement. Lastly, we calculate the geometric discord of three-qubit Bell diagonal states and then investigate the dynamic behavior of tripartite geometric discord under local decoherence. It is interesting that the frozen phenomenon exists for geometric discord in this scenario.

Effects of reservoir squeezing on trace-distance correlations and emergence of the pointer basis

We investigate theoretically the dynamics of 1-norm geometric quantum correlations and their classical counterparts in a two-qubit system. Both qubits are initially prepared in Bell-diagonal states and locally coupled to separated thermal squeezed baths or a common squeezed thermal bath via energy-preserving interactions. We then unveil the effects of reservoir squeezing on the abrupt changes in the evolution of geometric correlations. It is found that adequately tuning the squeezing phase can efficiently suppress the dephasing rate and delay the appearance of sudden transitions in geometric correlations. Further, we show that the squeezing phase of the bath renders a different avenue to enhance the finite time interval for frozen quantum correlation. On the other hand, in this context, we show that the squeezing strength of the reservoir exhibits a negative role. In addition, in the common bath case, we observe the steady-state correlations and decoherence-free subspace, which can be governed via squeezing parameters. Moreover, the abrupt change from a decaying regime to a constant nonzero value in classical correlation signals the emergence of a pointer-state basis. We show that the emergence of a pointer-state basis can be delayed by suitably adjusting the bath squeezing parameters. Remarkably, we find the optimal value of the squeezing phase, which introduces maximum retardation in the appearance of a pointer-state basis.

Stationary states of a dissipative two-qubit quantum channel and their applications for quantum machine learning

Entangled-state preparation and preservation are the cornerstones of any quantum information platform. However, the strongest adversaries in quantum information science are unwanted environmental effects such as decoherence and dissipation. Here, we address how to control and harness these unwanted effects that arise from the coupling of a system with its environment, to provide stationary entangled states for quantum machine learning (QML). To do so, we design a dissipative quantum channel, i.e., a two-qubit system interacting with a squeezed vacuum field reservoir, and study the output state of the channel by solving the corresponding master equation, especially, in the small squeezing regime. We show that the time-dependent output state of the channel is the so-called two-qubit X-states that generalize many families of entangled two-qubit states. Also, by considering a general Bell diagonal state as the initial state of the system, we reveal that this dissipative channel generates two well-known classes of entangled mixed state and Werner-like states in the steady-state regime. Moreover, this channel provides an efficient way to determine whether a given initial state results in a stationary entangled state or not. Finally, we examine the potential application of the designed two-qubit channel for QML. In this line, we propose a general theoretical scheme for quantum neural networks (QNNs) implemented with the variational quantum circuits, which encode data in continuous variables (CVs) of the two-qubit states. The linear (also referred to as affine) and nonlinear (activation function) transformations are enacted in the QNN using the stationary states of the two-qubit channel and measurement process, respectively. Finally, we proceed to test our proposed model by solving some supervised binary classification tasks. Integrating the non-unitary transformation and parallel-processed neural computing on such a two-qubit channel establishes the requirements for a meaningful QNN. Such a CV-QNN model with sufficient layers may execute any algorithm implementable on a universal CV quantum computer.

Entanglement and quantum correlation measures for quantum multipartite mixed states

Entanglement, and quantum correlation, are precious resources for quantum technologies implementation based on quantum information science, such as quantum communication, quantum computing, and quantum interferometry. Nevertheless, to our best knowledge, a directly or numerically computable measure for the entanglement of multipartite mixed states is still lacking. In this work, (i) we derive a measure of the degree of quantum correlation for mixed multipartite states. The latter possesses a closed-form expression valid in the general case unlike, to our best knowledge, all other known measures of quantum correlation. (ii) We further propose an entanglement measure, derived from this quantum correlation measure using a novel regularization procedure for the density matrix. Therefore, a comparison of the proposed measures, of quantum correlation and entanglement, allows one to distinguish between quantum correlation detached from entanglement and the one induced by entanglement and, hence, to identify separable but non-classical states. We have tested our quantum correlation and entanglement measures, on states well-known in literature: a general Bell diagonal state and the Werner states, which are easily tractable with our regularization procedure, and we have verified the accordance between our measures and the expected results for these states. Finally, we validate the two measures in two cases of multipartite states. The first is a generalization to three qubits of the Werner state, the second is a one-parameter three qubits mixed state interpolating between a bi-separable state and a genuine multipartite state, passing through a fully separable state.

Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts

We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension d ge 3, a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with special algebraic and geometric structures and applications in quantum information processing tasks in different dimensions. Extending analytical and numerical methods and results for Bell diagonal qutrits (d=3), we successfully classify more than 75% of representative Bell diagonal PPT states for d=4. Via those representative states we are able to estimate the volumes of separable and bound entangled states among PPT ququarts (d=4). We find that at least 75.7% of all PPT states are separable, 1.7% bound entangled and for 22.6% it remains unclear whether they are separable or bound entangled. Comparing the structure of bound entangled states and their detectors, we find considerable differences in the detection capabilities for different dimensions and relate those to differences of the Euclidean geometry for qutrits (d=3) and ququarts (d=4). Finally, using a detailed visual analysis of the set of separable and bound entangled Bell diagonal states in both dimensions, qualitative observations are made that allow to better distinguish bound entangled from separable states.

Complementary relations between l p norm coherence and mixedness of quantum states

As an important resource, quantum coherence is one of the fundamental features of the quantum world. It is essential to understand the relationship between the resource and the mixedness of an open quantum system. We present complementary relations between the l p norm of coherence and the normalized linear entropy mixedness. We then study the geometry of the complementary relations for Bell-diagonal states, as well as the evolutions of Werner and isotropic states under bit flip, phase flip, bit-phase flip, depolarizing, generalized amplitude damping and amplitude damping channels. Our results include some existing conclusions as special cases.

Critical temperature for the existence of entanglement in the Bell-diagonal states

We study systems formed by two-qubits in the Bell-diagonal states interacting in equilibrium with a thermal reservoir. We report some geometric relations between the entanglement and the system temperature, and calculate the minimum temperature above which the entanglement of the two-qubit system vanishes.

Separability Criteria Based on the Weyl Operators.

Entanglement as a vital resource for information processing can be described by special properties of the quantum state. Using the well-known Weyl basis we propose a new Bloch decomposition of the quantum state and study its separability problem. This decomposition enables us to find an alternative characterization of the separability based on the correlation matrix. We show that the criterion is effective in detecting entanglement for the isotropic states, Bell-diagonal states and some PPT entangled states. We also use the Weyl operators to construct an detecting operator for quantum teleportation.

Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits

With a probability of success of 95% we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres–Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be separable, free entangled or bound entangled. Leveraging a geometrical representation of these states in Euclidean space, novel methods are presented that allow the classification of separable and bound entangled Bell diagonal states in an efficient way. Moreover, the classification allows the precise determination of relative volumes of the classes of separable, free and bound entangled states. In detail, out of all Bell diagonal PPT states 81.0% pm 0.1% are determined to be separable while 13.9 pm 0.1% are bound entangled and only 5.1 pm 0.1% remain unclassified. Moreover, our applied criteria are compared for their effectiveness and relation as detectors of bound entanglement, which reveals that not a single criterion is capable to detect all bound entangled states.