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.     

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.     

Generation and protection of steady-state quantum correlations due to quantum channels with memory

We have proposed a scheme of the generation and preservation of two-qubit steady-state quantum correlations through quantum channels where successive uses of the channels are correlated. Different types of noisy channels with memory, such as amplitude damping, phase damping, and depolarizing channels, have been taken into account. Some analytical or numerical results are presented. The effect of channels with memory on dynamics of quantum correlations has been discussed in detail. The results show that steady-state entanglement between two initial qubits whose initial states are prepared in a specific family states without entanglement subject to amplitude damping channel with memory can be generated. The entanglement creation is related to the memory coefficient of channel \(\mu \). The stronger the memory coefficient of channel \( \mu \) is, the more the entanglement creation is, and the earlier the separable state becomes the entangled state. Besides, we compare the dynamics of entanglement with that of quantum discord when a two-qubit system is initially prepared in an entangled state. We show that entanglement dynamics suddenly disappears, while quantum discord dynamics displays only in the asymptotic limit. Furthermore, two-qubit quantum correlations can be preserved at a long time in the limit of \(\mu \rightarrow 1\).     

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.     

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.     

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.     

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.     

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.     

New q-ary quantum MDS codes with distances bigger than $$\frac{ q}{2}$$

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS \([[n,n-2d+2,d]]_q\) codes with minimum distances \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.     

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.     

Pure state ‘really’ informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space \(\mathcal {H}_d\)? The known result is that the number is no less than \(3d-2\). We show that this lower bound is not tight except for \(d=2\) or 4. Then we give an upper bound \(4d-3\). For \(d=2\), many rank-1 POVMs with four elements can determine any pure states in \(\mathcal {H}_2\). For \(d=3\), we show eight is the minimal number by construction. For \(d=4\), the minimal number is in the set of \(\{10,11,12,13\}\). We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in \(\mathcal {H}_4\). For any dimension d, we construct \(d+2k-2\) adaptive rank-1 positive operators for the reconstruction of any unknown pure state in \(\mathcal {H}_d\), where \(1\le k \le d\).     

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 \(\tilde{\chi _2}(\varepsilon )\).     

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.     

Cyclic quantum teleportation

We propose a scheme of cyclic quantum teleportation for three unknown qubits using six-qubit maximally entangled state as the quantum channel. Suppose there are three observers Alice, Bob and Charlie, each of them has been given a quantum system such as a photon or spin-\(\frac{1}{2}\) particle, prepared in state unknown to them. We show how to implement the cyclic quantum teleportation where Alice can transfer her single-qubit state of qubit a to Bob, Bob can transfer his single-qubit state of qubit b to Charlie and Charlie can also transfer his single-qubit state of qubit c to Alice. We can also implement the cyclic quantum teleportation with \(N\geqslant 3\) observers by constructing a 2N-qubit maximally entangled state as the quantum channel. By changing the quantum channel, we can change the direction of teleportation. Therefore, our scheme can realize teleportation in quantum information networks with N observers in different directions, and the security of our scheme is also investigated at the end of the paper.     

Optimal noise functions for location privacy on continuous regions

Users of location-based services are highly vulnerable to privacy risks since they need to disclose, at least partially, their locations to benefit from these services. One possibility to limit these risks is to obfuscate the location of a user by adding random noise drawn from a noise function. In this paper, we require the noise functions to satisfy a generic location privacy notion called \(\ell \)-privacy, which makes the position of the user in a given region \(\mathcal {X}\) relatively indistinguishable from other points in \(\mathcal {X}\). We also aim at minimizing the loss in the service utility due to such obfuscation. While existing optimization frameworks regard the region \(\mathcal {X}\) restrictively as a finite set of points, we consider the more realistic case in which the region is rather continuous with a nonzero area. In this situation, we demonstrate that circular noise functions are enough to satisfy \(\ell \)-privacy on \(\mathcal {X}\) and equivalently on the entire space without any penalty in the utility. Afterward, we describe a large parametric space of noise functions that satisfy \(\ell \)-privacy on \(\mathcal {X}\), and show that this space has always an optimal member, regardless of \(\ell \) and \(\mathcal {X}\). We also investigate the recent notion of \(\epsilon \)-geo-indistinguishability as an instance of \(\ell \)-privacy and prove in this case that with respect to any increasing loss function, the planar Laplace noise function is optimal for any region having a nonzero area.     

How the Hawking radiation affect quantum Fisher information of Dirac particles in the background of a Schwarzschild black hole

In this work, the effect of Hawking radiation on the quantum Fisher information (QFI) of Dirac particles is investigated in the background of a Schwarzschild black hole. Interestingly, it has been verified that the QFI with respect to the weight parameter \(\theta \) of a target state is always independent of the Hawking temperature T. This implies that if we encode the information on the weight parameter, then we can affirm that the corresponding accuracy of the parameter estimation will be immune to the Hawking effect. Besides, it reveals that the QFI with respect to the phase parameter \(\phi \) exhibits a decay behavior with the increase in the Hawking temperature T and converges to a nonzero value in the limit of infinite Hawking temperature T. Remarkably, it turns out that the function \(F_\phi \) on \(\theta =\pi \big /4\) symmetry was broken by the influence of the Hawking radiation. Finally, we generalize the case of a three-qubit system to a case of a N-qubit system, i.e., \(|\psi \rangle _{1,2,3,\ldots ,N} =(\cos \theta | 0 \rangle ^{\otimes N}+\sin \theta \mathrm{e}^{i\phi }| 1 \rangle ^{\otimes N})\) and obtain an interesting result: the number of particles in the initial state does not affect the QFI \(F_\theta \), nor the QFI \(F_\phi \). However, with the increasing number of particles located near the event horizon, \(F_\phi \) will be affected by Hawking radiation to a large extent, while \(F_\theta \) is still free from disturbance resulting from the Hawking effects.     

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.     

High-fidelity quantum state preparation using neighboring optimal control

We present an approach to single-shot high-fidelity preparation of an n-qubit state based on neighboring optimal control theory. This represents a new application of the neighboring optimal control formalism which was originally developed to produce single-shot high-fidelity quantum gates. To illustrate the approach, and to provide a proof-of-principle, we use it to prepare the two-qubit Bell state \(|\beta _{01}\rangle = (1/\sqrt{2})\left[ \, |01\rangle + |10\rangle \,\right] \) with an error probability \(\varepsilon \sim 10^{-6}\) (\(10^{-5}\)) for ideal (non-ideal) control. Using standard methods in the literature, these high-fidelity Bell states can be leveraged to fault-tolerantly prepare the logical state \(|\overline{\beta }_{01}\rangle \).     

Decoupled,Unconditionally Stable,Higher Order Discretizations for MHD Flow Simulation

We propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep yet maintains unconditional stability with respect to the time step size, is optimally accurate in space, and behaves like second order in time in practice. The proposed method chooses a parameter \(\theta \in [0,1]\), dependent on the viscosity \(\nu \) and magnetic diffusivity \(\nu _m\), so that the explicit treatment of certain viscous terms does not cause instabilities, and gives temporal accuracy \(O(\Delta t^2 + (1-\theta )|\nu -\nu _m|\Delta t)\). In practice, \(\nu \) and \(\nu _m\) are small, and so the method behaves like second order. When \(\theta =1\), the method reduces to a linearized BDF2 method, but it has been proven by Li and Trenchea that such a method is stable only in the uncommon case of \(\frac{1}{2}< \frac{\nu }{\nu _m} < 2\). For the proposed method, stability and convergence are rigorously proven for appropriately chosen \(\theta \), and several numerical tests are provided that confirm the theory and show the method provides excellent accuracy in cases where usual BDF2 is unstable.     

Compatibility of observables on effect algebras

We compare different notions of simultaneous measurability (compatibility) of observables on lattice \(\sigma \)-effect algebras and more generally, on \(\sigma \)-effect algebras that can be covered by \(\sigma \)-MV-algebras. We prove that every \(\sigma \)-MV-algebra is the range of a \(\sigma \)-additive observable, and we compare the following notions of compatibility of observables: joint measurability, coexistence, joint measurability of binarizations, coexistence of binarizations, smearings of the same observable. We prove that if there is a faithful state on the effect algebra, then any two standard observables that are smearings of the same (sharp) observable admit a generalized joint observable.