Analytical expression of quantum discord for rank-2 two-qubit states

Quantum correlations characterized by quantum entanglement and quantum discord play important roles in many quantum information processing. We study the relations among the entanglement of formation, concurrence, tangle, linear entropy-based classical correlation and von Neumann entropy-based classical correlation . We present analytical formulae of linear entropy-based classical correlation for arbitrary \(d\otimes 2\) quantum states and von Neumann entropy-based classical correlation for arbitrary \(2\otimes 2\) rank-2 quantum states. From the von Neumann entropy-based classical correlation, we derive an explicit formula of quantum discord for arbitrary rank-2 two-qubit quantum states.     

Geometry of quantum state space and quantum correlations

Quantum state space is endowed with a metric structure, and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical considerations on the quantum state space. In this article, considering the quantum state space being spanned by \(2\times 2\) density matrices, we determine a particular Riemannian metric for a state \(\rho \) and show that if \(\rho \) gets entangled with another quantum state, the negativity of the generated entangled state is, upto a constant factor, equal to square root of that particular Riemannian metric . Our result clearly relates a geometric quantity to a measure of entanglement. Moreover, the result establishes the possibility of understanding quantum correlations through geometric approach.     

Extremal properties of conditional entropy and quantum discord for XXZ,symmetric quantum states

For the XXZ subclass of symmetric two-qubit X states, we study the behavior of quantum conditional entropy \(S_{cond}\) as a function of measurement angle \(\theta \in [0,\pi /2]\). Numerical calculations show that the function \(S_{cond}(\theta )\) for X states can have at most one local extremum in the open interval from zero to \(\pi /2\) (unimodality property). If the extremum is a minimum, the quantum discord displays region with variable (state-dependent) optimal measurement angle \(\theta ^*\). Such \(\theta \)-regions (phases, fractions) are very tiny in the space of X-state parameters. We also discover the cases when the conditional entropy has a local maximum inside the interval \((0,\pi /2)\). It is remarkable that the maxima exist in surprisingly wide regions, and the boundaries for such regions are defined by the same bifurcation conditions as for those with a minimum.     

Protecting tripartite entanglement in non-Markovian environments via quantum partially collapsing measurements

In this paper, the dynamics of tripartite entanglement via \(\pi \)-tangle in independent non-Markovian environments is investigated. The results indicate that the \(\pi \)-tangle vanishes periodically as decoherence time increases with a damping of its revival amplitude due to the memory of the non-Markovian environments. In addition, we present a scheme to protect entanglement of W state from non-Markovian environments by means of the quantum partially collapsing measurements. It is worth mentioning that our scheme is a successful protection for the tripartite quantum system and the effect is better for the larger measurement strength, while the stronger decoherence suppression induces smaller success probability.     

Steering quantum-memory-assisted entropic uncertainty under unital and nonunital noises via filtering operations

In this work, we investigate the dynamic features of the entropic uncertainty for two incompatible measurements under local unital and nonunital channels. Herein, we choose Pauli operators \(\sigma _x \) and \(\sigma _z \) as a pair of observables of interest measuring on particle A, and the uncertainty can be predicted when particle A is entangled with quantum memory B. We explore the dynamics of the uncertainty for the measurement under local unitary (phase-damping) and nonunitary (amplitude-damping) channels, respectively. Remarkably, we derive the entropic uncertainty relation under three different kinds of measurements of Pauli-observable pair under various realistic noisy environments; it has been found that the entropic uncertainty has the same tendency of its evolution during the AD and PD channel when we choose \(\sigma _x \) and \(\sigma _y \) measurement. Besides, we find out that the entropic uncertainty will have an optimal value if one chooses \(\sigma _x \) and \(\sigma _z \) as the measurement incompatibility, comparing with others. Furthermore, in order to reduce the entropic uncertainty in noisy environment, we propose an effective strategy to steer the amount by means of implementing a filtering operation on the particle under the two types of channels, respectively. It turns out that this operation can greatly reduce the entropic uncertainty by modulation of the operation strength. Thus, our investigations might offer an insight into the dynamics and steering of the entropic uncertainty in an open system.     

Relationship between the field local quadrature and the quantum discord of a photon-added correlated channel under the influence of scattering and phase fluctuation noise

We study quantum correlations and discord in a bipartite continuous variable hybrid system formed by linear combinations of coherent states \(\mathinner {|{\alpha }\rangle }\) and single photon-added coherent states of the form \(\mathinner {|{\psi }\rangle }_{\text {dp(pa)}}= \mathcal {N}/\sqrt{2} (\hat{a}^\dagger \mathinner {|{\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b \pm \hat{b}^\dagger \mathinner {|{\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b)\). We stablish a relationship between the quantum discord with a local observable (the quadrature variance for one subsystem) under the influence of scattering and phase fluctuation noise. For the pure states the quantum correlations are characterized by means of measurement induced disturbance (MID) with simultaneous quadrature measurements. In a scenario where homodyne conditional measurements are available we show that the MID provides an easy way to select optimal phases to obtain information of the maximal correlations in the channels. The quantum correlations of these entangled states with channel losses are quantitatively characterized with the quantum discord (QD) with a displaced qubit projector. We observe that as scattering increases, QD decreases monotonically. At the same time for the state \(\mathinner {|{\psi }\rangle }_{\text {dp}}\), QD is more resistant to high phase fluctuations when the average photon number \(n_0\) is bigger than zero, but if phase fluctuations are low, QD is more resistant if \(n_0=0\). For the dp model with scattering, we obtain an analytical expression of the QD as a function of the observable quadrature variance in a local subsystem. This relation allows us to have a way to obtain the degree of QD in the channel by just measuring a local property observable such as the quadrature variance. For the other model this relation still exists but is explored numerically. This relation is an important result that allows to identify quantum processing capabilities in terms of just local observables.     

Multi-party quantum private comparison protocol base on $$$$-imensional entangle states

In this paper, a novel quantum private comparison protocol with \(l\)-party and \(d\)-dimensional entangled states is proposed. In the protocol, \(l\) participants can sort their secret inputs in size, with the help of a semi-honest third party. However, if every participant wants to know the relation of size among the \(l\) secret inputs, these two-participant protocols have to be executed repeatedly \(\frac{l(l-1)}{2}\) times. Consequently, the proposed protocol needs to be executed one time. Without performing unitary operation on particles, it only need to prepare the initial entanglement states and only need to measure single particles. It is shown that the participants will not leak their private information by security analysis.     

Bidirectional controlled teleportation by using nine-qubit entangled state in noisy environments

A theoretical scheme is proposed to implement bidirectional quantum controlled teleportation (BQCT) by using a nine-qubit entangled state as a quantum channel, where Alice may transmit an arbitrary two-qubit state called qubits \(A_1\) and \(A_2\) to Bob; and at the same time, Bob may also transmit an arbitrary two-qubit state called qubits \(B_1\) and \(B_2\) to Alice via the control of the supervisor Charlie. Based on our channel, we explicitly show how the bidirectional quantum controlled teleportation protocol works. And we show this bidirectional quantum controlled teleportation scheme may be determinate and secure. Taking the amplitude-damping noise and the phase-damping noise as typical noisy channels, we analytically derive the fidelities of the BQCT process and show that the fidelities in these two cases only depend on the amplitude parameter of the initial state and the decoherence noisy rate.     

Controlled quantum secure direct communication by entanglement distillation or generalized measurement

We propose two controlled quantum secure communication schemes by entanglement distillation or generalized measurement. The sender Alice, the receiver Bob and the controllers David and Cliff take part in the whole schemes. The supervisors David and Cliff can control the information transmitted from Alice to Bob by adjusting the local measurement angles \(\theta _4\) and \(\theta _3\). Bob can verify his secret information by classical one-way function after communication. The average amount of information is analyzed and compared for these two methods by MATLAB. The generalized measurement is a better scheme. Our schemes are secure against some well-known attacks because classical encryption and decoy states are used to ensure the security of the classical channel and the quantum channel.     

Optimal quantum thermometry by dephasing

Decoherence often happens in the quantum world. We try to utilize quantum dephasing to build an optimal thermometry. By calculating the Cramér–Rao bound, we prove that the Ramsey measurement is the optimal way to measure the temperature for uncorrelated probe particles. Using the optimal measurement, the metrological equivalence of product and maximally entangled state of initial quantum probes always holds. Contrary to frequency estimation, the optimal temperature estimation can be obtained in the case \(\nu <1\), not \(\nu >1\). For the general Zeno regime (\(\nu =2\)), uncorrelated product states are the optimal choice in typical Ramsey spectroscopy setup. In order to improve the resolution of temperature, one should reduce the characteristic time of dephasing factor \(\gamma (t)\propto t^2\), and the power \(\nu <1\) appears after it. Under the imperfect condition, maximally entangled state can perform better than product state. Finally, we investigate other environmental influence on the measurement precision of temperature. Based on it, we define a new way to measure non-Markovian effect.     

Squashed entanglement and approximate private states

The squashed entanglement is a fundamental entanglement measure in quantum information theory, finding application as an upper bound on the distillable secret key or distillable entanglement of a quantum state or a quantum channel. This paper simplifies proofs that the squashed entanglement is an upper bound on distillable key for finite-dimensional quantum systems and solidifies such proofs for infinite-dimensional quantum systems. More specifically, this paper establishes that the logarithm of the dimension of the key system (call it \(\log _{2}K\)) in an \(\varepsilon \)-approximate private state is bounded from above by the squashed entanglement of that state plus a term that depends only \(\varepsilon \) and \(\log _{2}K\). Importantly, the extra term does not depend on the dimension of the shield systems of the private state. The result holds for the bipartite squashed entanglement, and an extension of this result is established for two different flavors of the multipartite squashed entanglement.     

Entanglement witness criteria of strong-<Emphasis Type="Italic">k</Emphasis>-separability for multipartite quantum states

Let \(H_{1}, H_{2},\ldots ,H_{n}\) be separable complex Hilbert spaces with \(\dim H_{i}\ge 2\) and \(n\ge 2\). Assume that \(\rho \) is a state in \(H=H_1\otimes H_2\otimes \cdots \otimes H_n\). \(\rho \) is called strong-k-separable \((2\le k\le n)\) if \(\rho \) is separable for any k-partite division of H. In this paper, an entanglement witnesses criterion of strong-k-separability is obtained, which says that \(\rho \) is not strong-k-separable if and only if there exist a k-division space \(H_{m_{1}}\otimes \cdots \otimes H_{m_{k}}\) of H, a finite-rank linear elementary operator positive on product states \(\Lambda :\mathcal {B}(H_{m_{2}}\otimes \cdots \otimes H_{m_{k}})\rightarrow \mathcal {B}(H_{m_{1}})\) and a state \(\rho _{0}\in \mathcal {S}(H_{m_{1}}\otimes H_{m_{1}})\), such that \(\mathrm {Tr}(W\rho )<0\), where \(W=(\mathrm{Id}\otimes \Lambda ^{\dagger })\rho _{0}\) is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.     

Indistinguishability of pure orthogonal product states by LOCC

We construct two sets of incomplete and extendible quantum pure orthogonal product states (POPS) in general bipartite high-dimensional quantum systems, which are all indistinguishable by local operations and classical communication. The first set of POPS is composed of two parts which are \(\mathcal {C}^m\otimes \mathcal {C}^{n_1}\) with \(5\le m\le n_1\) and \(\mathcal {C}^m\otimes \mathcal {C}^{n_2}\) with \(5\le m \le n_2\), where \(n_1\) is odd and \(n_2\) is even. The second one is in \(\mathcal {C}^m\otimes \mathcal {C}^n\) \((m, n\ge 4)\). Some subsets of these two sets can be extended into complete sets that local indistinguishability can be decided by noncommutativity which quantifies the quantumness of a quantum ensemble. Our study shows quantum nonlocality without entanglement.     

Bimodal behavior of post-measured entropy and one-way quantum deficit for two-qubit X states

A method for calculating the one-way quantum deficit is developed. It involves a careful study of post-measured entropy shapes. We discovered that in some regions of X-state space the post-measured entropy \(\tilde{S}\) as a function of measurement angle \(\theta \in [0,\pi /2]\) exhibits a bimodal behavior inside the open interval \((0,\pi /2)\), i.e., it has two interior extrema: one minimum and one maximum. Furthermore, cases are found when the interior minimum of such a bimodal function \(\tilde{S}(\theta )\) is less than that one at the endpoint \(\theta =0\) or \(\pi /2\). This leads to the formation of a boundary between the phases of one-way quantum deficit via finite jumps of optimal measured angle from the endpoint to the interior minimum. Phase diagram is built up for a two-parameter family of X states. The subregions with variable optimal measured angle are around 1\(\%\) of the total region, with their relative linear sizes achieving \(17.5\%\), and the fidelity between the states of those subregions can be reduced to \(F=0.968\). In addition, a correction to the one-way deficit due to the interior minimum can achieve \(2.3\%\). Such conditions are favorable to detect the subregions with variable optimal measured angle of one-way quantum deficit in an experiment.     

New asymmetric quantum codes over $$F_{q}$$

Two families of new asymmetric quantum codes are constructed in this paper. The first family is the asymmetric quantum codes with length \(n=q^{m}-1\) over \(F_{q}\), where \(q\ge 5\) is a prime power. The second one is the asymmetric quantum codes with length \(n=3^{m}-1\). These asymmetric quantum codes are derived from the CSS construction and pairs of nested BCH codes. Moreover, let the defining set \(T_{1}=T_{2}^{-q}\), then the real Z-distance of our asymmetric quantum codes are much larger than \(\delta _\mathrm{max}+1\), where \(\delta _\mathrm{max}\) is the maximal designed distance of dual-containing narrow-sense BCH code, and the parameters presented here have better than the ones available in the literature.     

Tripartite and bipartite quantum correlations in the XXZ spin chain with three-site interaction

Tripartite and bipartite quantum correlations in the three-qubit XXZ Heisenberg spin chain with two types of three-site interactions and an external magnetic field are investigated. We show that the increase in XZY ? YZX interaction can enhance the robustness of both tripartite and bipartite correlations, whereas the increase in XZX \(+\) YZY interaction could improve the robustness of tripartite quantum correlations, but diminish the robustness of bipartite quantum correlations. Tripartite measurement-induced disturbance is the most robust against temperature, and bipartite entanglement is the most fragile. Tripartite entanglement is even more robust than bipartite quantum discord when XZX \(+\) YZY or XZY ? YZX interaction is relatively large. The cooperative effect of XZX \(+\) YZY and XZY ? YZX interaction could induce bipartite entanglement even at high temperature. The cooperative effect of XZX \(+\) YZY and XZY ? YZX interaction is the most optimal to improve the robustness of all quantum correlations when the magnetic field is negative. When the magnetic field is positive, the effective of XZY ? YZX interaction alone is more ideal to preserve different quantum correlations.     

Master Lovas–Andai and equivalent formulas verifying the $$\frac{8}{33}$$ two-qubit Hilbert–Schmidt separability probability and companion rational-valued conjectures

We begin by investigating relationships between two forms of Hilbert–Schmidt two-rebit and two-qubit “separability functions”—those recently advanced by Lovas and Andai (J Phys A Math Theor 50(29):295303, 2017), and those earlier presented by Slater (J Phys A 40(47):14279, 2007). In the Lovas–Andai framework, the independent variable \(\varepsilon \in [0,1]\) is the ratio \(\sigma (V)\) of the singular values of the \(2 \times 2\) matrix \(V=D_2^{1/2} D_1^{-1/2}\) formed from the two \(2 \times 2\) diagonal blocks (\(D_1, D_2\)) of a \(4 \times 4\) density matrix \(D= \left||\rho _{ij}\right||\). In the Slater setting, the independent variable \(\mu \) is the diagonal-entry ratio \(\sqrt{\frac{\rho _{11} \rho _ {44}}{\rho _ {22} \rho _ {33}}}\)—with, of central importance, \(\mu =\varepsilon \) or \(\mu =\frac{1}{\varepsilon }\) when both \(D_1\) and \(D_2\) are themselves diagonal. Lovas and Andai established that their two-rebit “separability function” \(\tilde{\chi }_1 (\varepsilon )\) (\(\approx \varepsilon \)) yields the previously conjectured Hilbert–Schmidt separability probability of \(\frac{29}{64}\). We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we newly find its two-qubit, two-quater[nionic]-bit and “two-octo[nionic]-bit” counterparts, \(\tilde{\chi _2}(\varepsilon ) =\frac{1}{3} \varepsilon ^2 \left( 4-\varepsilon ^2\right) \), \(\tilde{\chi _4}(\varepsilon ) =\frac{1}{35} \varepsilon ^4 \left( 15 \varepsilon ^4-64 \varepsilon ^2+84\right) \) and \(\tilde{\chi _8} (\varepsilon )= \frac{1}{1287}\varepsilon ^8 \left( 1155 \varepsilon ^8-7680 \varepsilon ^6+20160 \varepsilon ^4-25088 \varepsilon ^2+12740\right) \). These immediately lead to predictions of Hilbert–Schmidt separability/PPT-probabilities of \(\frac{8}{33}\), \(\frac{26}{323}\) and \(\frac{44482}{4091349}\), in full agreement with those of the “concise formula” (Slater in J Phys A 46:445302, 2013), and, additionally, of a “specialized induced measure” formula. Then, we find a Lovas–Andai “master formula,” \(\tilde{\chi _d}(\varepsilon )= \frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left( -\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right) }{\Gamma \left( \frac{d}{2}+1\right) ^2}\), encompassing both even and odd values of d. Remarkably, we are able to obtain the \(\tilde{\chi _d}(\varepsilon )\) formulas, \(d=1,2,4\), applicable to full (9-, 15-, 27-) dimensional sets of density matrices, by analyzing (6-, 9, 15-) dimensional sets, with not only diagonal \(D_1\) and \(D_2\), but also an additional pair of nullified entries. Nullification of a further pair still leads to X-matrices, for which a distinctly different, simple Dyson-index phenomenon is noted. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence. A two-qubit separability probability of \(1-\frac{256}{27 \pi ^2}\) is obtained based on the operator monotone function \(\sqrt{x}\), with the use of \(\tilde{\chi _2}(\varepsilon )\).     

Entangled photon-added coherent states

We study the degree of entanglement of arbitrary superpositions of m, n photon-added coherent states (PACS) \(\mathinner {|{\psi }\rangle } \propto u \mathinner {|{{\alpha },m}\rangle }\mathinner {|{{\beta },n }\rangle }+ v \mathinner {|{{\beta },n}\rangle }\mathinner {|{{\alpha },m}\rangle }\) using the concurrence and obtain the general conditions for maximal entanglement. We show that photon addition process can be identified as an entanglement enhancer operation for superpositions of coherent states (SCS). Specifically for the known bipartite positive SCS: \(\mathinner {|{\psi }\rangle } \propto \mathinner {|{\alpha }\rangle }_a\mathinner {|{-\alpha }\rangle }_b + \mathinner {|{-\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b \) whose entanglement tends to zero for \(\alpha \rightarrow 0\), can be maximal if al least one photon is added in a subsystem. A full family of maximally entangled PACS is also presented. We also analyzed the decoherence effects in the entangled PACS induced by a simple depolarizing channel . We find that robustness against depolarization is increased by adding photons to the coherent states of the superposition. We obtain the dependence of the critical depolarization \(p_{\text {crit}}\) for null entanglement as a function of \(m,n, \alpha \) and \(\beta \).     

Quantum coherence of two-qubit over quantum channels with memory

Using the axiomatic definition of the quantum coherence measure, such as the \(l_{1}\) norm and the relative entropy, we study the phenomena of two-qubit system quantum coherence through quantum channels where successive uses of the channels are memory. Different types of noisy channels with memory, such as amplitude damping, phase damping, and depolarizing channels effect on quantum coherence have been discussed in detail. The results show that quantum channels with memory can efficiently protect coherence from noisy channels. Particularly, as channels with perfect memory, quantum coherence is unaffected by the phase damping as well as depolarizing channels. Besides, we also investigate the cohering and decohering power of quantum channels with memory.     

Probing quantum coherence,uncertainty, steerability of quantum coherence and quantum phase transition in the spin model

In this paper, we study the relation among quantum coherence, uncertainty, steerability of quantum coherence based on skew information and quantum phase transition in the spin model by employing quantum renormalization-group method. Interestingly, the results show that the value of the local quantum uncertainty is equal to the local quantum coherence corresponding to local observable \(\sigma _z\) in XXZ model, and unlikely in XY model, local quantum uncertainty is minimal optimization of the local quantum coherence over local observable \(\sigma _x\) and this proposition can be generalized to a multipartite system. Therefore, one can directly achieve quantum correlation measured by local quantum uncertainty and coherence by choosing different local observables \(\sigma _x\), \(\sigma _z\), corresponding to the XY model and XXZ model separately. Meanwhile, steerability of quantum coherence in XY and XXZ model is investigated systematically, and our results reveal that no matter what times the QRG iterations are carried out, the quantum coherence of the state of subsystem cannot be steerable, which can also be suitable for block–block steerability of local quantum coherence in both XY and XXZ models. On the other hand, we have illustrated that the quantum coherence and uncertainty measure can efficiently detect the quantum critical points associated with quantum phase transitions after several iterations of the renormalization. Moreover, the nonanalytic and scaling behaviors of steerability of local quantum coherence have been also taken into consideration.