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 \(\beta \).     

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 \(|\overline{\beta }_{01}\rangle \).     

On the connection between quantum nonlocality and phase sensitivity of two-mode entangled Fock state superpositions

In two-mode interferometry, for a given total photon number N, entangled Fock state superpositions of the form \((|N-m\rangle _a|m\rangle _b+\mathrm{e}^{i (N-2m)\phi }|m\rangle _a|N-m\rangle _b)/\sqrt{2}\) have been considered for phase estimation. Indeed all such states are maximally mode-entangled and violate a Clauser–Horne–Shimony–Holt (CHSH) inequality. However, they differ in their optimal phase estimation capabilities as given by their quantum Fisher informations. The quantum Fisher information is the largest for the N00N state \((|N\rangle _a|0\rangle _b+\mathrm{e}^{i N\phi }|0\rangle _a|N\rangle _b)/\sqrt{2}\) and decreases for the other states with decreasing photon number difference between the two modes. We ask the question whether for any particular Clauser–Horne (CH) (or CHSH) inequality, the maximal values of the CH (or the CHSH) functional for the states of the above type follow the same trend as their quantum Fisher informations, while also violating the classical bound whenever the states are capable of sub-shot-noise phase estimation, so that the violation can be used to quantify sub-shot-noise sensitivity. We explore CH and CHSH inequalities in a homodyne setup. Our results show that the amount of violation in those nonlocality tests may not be used to quantify sub-shot-noise sensitivity of the above states.     

Two generalized Wigner–Yanase skew information and their uncertainty relations

In this paper, we first define two generalized Wigner–Yanase skew information \(|K_{\rho ,\alpha }|(A)\) and \(|L_{\rho ,\alpha }|(A)\) for any non-Hermitian Hilbert–Schmidt operator A and a density operator \(\rho \) on a Hilbert space H and discuss some properties of them, respectively. We also introduce two related quantities \(|S_{\rho ,\alpha }|(A)\) and \(|T_{\rho ,\alpha }|(A)\). Then, we establish two uncertainty relations in terms of \(|W_{\rho ,\alpha }|(A)\) and \(|\widetilde{W}_{\rho ,\alpha }|(A)\), which read

$$\begin{aligned}&|W_{\rho ,\alpha }|(A)|W_{\rho ,\alpha }|(B)\ge \frac{1}{4}\left| \mathrm {tr}\left( \left[ \frac{\rho ^{\alpha }+\rho ^{1-\alpha }}{2} \right] ^{2}[A,B]^{0}\right) \right| ^{2},\\&\sqrt{|\widetilde{W}_{\rho ,\alpha }|(A)| \widetilde{W}_{\rho ,\alpha }|(B)}\ge \frac{1}{4} \left| \mathrm {tr}\left( \rho ^{2\alpha }[A,B]^{0}\right) \mathrm {tr} \left( \rho ^{2(1-\alpha )}[A,B]^{0}\right) \right| . \end{aligned}$$

     

Quantum Query Complexity of Almost All Functions with Fixed On-set Size

This paper considers the quantum query complexity of almost all functions in the set \({\mathcal{F}}_{N,M}\) of \({N}\)-variable Boolean functions with on-set size \({M (1\le M \le 2^{N}/2)}\), where the on-set size is the number of inputs on which the function is true. The main result is that, for all functions in \({\mathcal{F}}_{N,M}\) except its polynomially small fraction, the quantum query complexity is \({ \Theta\left(\frac{\log{M}}{c + \log{N} - \log\log{M}} + \sqrt{N}\right)}\) for a constant \({c > 0}\). This is quite different from the quantum query complexity of the hardest function in \({\mathcal{F}}_{N,M}\): \({\Theta\left(\sqrt{N\frac{\log{M}}{c + \log{N} - \log\log{M}}} + \sqrt{N}\right)}\). In contrast, almost all functions in \({\mathcal{F}}_{N,M}\) have the same randomized query complexity \({\Theta(N)}\) as the hardest one, up to a constant factor.     

The identity problem of finitely generated bi-ideals

Finitely generated bi-ideals with letters from a selected alphabet A are considered. We solve the equivalence problem for generating systems of bi-ideals, i.e., look for an effective procedure which provides the means of determining if two generating systems \({\langle u_0, . . . , u_{m-1} \rangle}\) and \({\langle v_0, . . . , v_{n-1} \rangle}\) represent equal or different bi-ideals. We offer a method of constructing, for every generating system \({\langle u_0, . . . , u_{m-1} \rangle}\) , an equivalent generating system \({\langle u^{\prime}_{0}, . . . , u^{\prime}_{m-1} \rangle}\) with differing members. We also describe an algorithm for deciding if two generating systems \({\langle u_0, u_1 \rangle}\) and \({\langle v_0, v_1 \rangle}\) are equivalent or not. For a general case, the problem of existence of such an algorithm remains open.     

Connecting unextendible maximally entangled base with partial Hadamard matrices

We study the unextendible maximally entangled bases (UMEB) in \(\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}\) and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in \(\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}\), there is a partial Hadamard matrix which cannot be extended to a Hadamard matrix in \(\mathbb {C}^{d}\). As a corollary, any \((d-1)\times d\) partial Hadamard matrix can be extended to a Hadamard matrix, which answers a conjecture about \(d=5\). We obtain that for any d there is a UMEB except for \(d=p\ \text {or}\ 2p\), where \(p\equiv 3\mod 4\) and p is a prime. The existence of different kinds of constructions of UMEBs in \(\mathbb {C}^{nd}\bigotimes \mathbb {C}^{nd}\) for any \(n\in \mathbb {N}\) and \(d=3\times 5 \times 7\) is also discussed.     

Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits

We study the quantum complexity class \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\) of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\), which is an affirmative answer to the question posed by Høyer and ?palek. In sharp contrast to the strict hierarchy of the classical complexity classes: \({\mathsf{NC}^{0} \subsetneq \mathsf{AC}^{0} \subsetneq \mathsf{TC}^{0}}\), our result with Høyer and ?palek’s one implies the collapse of the hierarchy of the corresponding quantum ones: \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}=\mathsf{QAC}^\mathsf{0}_\mathsf{f}=\mathsf{QTC}^\mathsf{0}_\mathsf{f}}\). Then, we show that there exists a constant-depth subquadratic-size quantum circuit for the quantum threshold operation. This allows us to obtain a better bound on the size difference between the \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\) and \({\mathsf{QTC}^\mathsf{0}_\mathsf{f}}\) circuits for implementing the same operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\), there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\) oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a \({\mathsf{QNC}^\mathsf{0}_\mathsf{f}}\) circuit with gates for the quantum Fourier transform.     

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.     

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^d$$

We study mutually unbiased maximally entangled bases (MUMEB’s) in bipartite system \(\mathbb {C}^d\otimes \mathbb {C}^d (d \ge 3)\). We generalize the method to construct MUMEB’s given in Tao et al. (Quantum Inf Process 14:2291–2300, 2015), by using any commutative ring R with d elements and generic character of \((R,+)\) instead of \(\mathbb {Z}_d=\mathbb {Z}/d\mathbb {Z}\). Particularly, if \(d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}\) where \(p_1, \ldots , p_s\) are distinct primes and \(3\le p_1^{a_1}\le \cdots \le p_s^{a_s}\), we present \(p_1^{a_1}-1\) MUMEB’s in \(\mathbb {C}^d\otimes \mathbb {C}^d\) by taking \(R=\mathbb {F}_{p_1^{a_1}}\oplus \cdots \oplus \mathbb {F}_{p_s^{a_s}}\), direct sum of finite fields (Theorem 3.3).     

Experimental estimation of discord in an antiferromagnetic Heisenberg compound $$\hbox {Cu}(\hbox {NO}_{3})_{2}\cdot 2.5\,\hbox {H}_{2}\hbox {O}$$

Temperature-dependent static magnetic susceptibility and heat capacity data were employed to quantify quantum discord in copper nitrate \((\hbox {CN, Cu}(\hbox {NO}_{3})_{2}\cdot 2.5\, \hbox {H}_{2}\hbox {O})\) which is a spin 1/2 antiferromagnetic Heisenberg system. With the help of existing theoretical formulations, quantum discord, mutual information, and purely classical correlation were estimated as a function of temperature using the experimental data. The experimentally quantified correlations estimated from susceptibility and heat capacity data are consistent with each other, and they exhibit a good match with theoretical predictions. Violation of Bell’s inequality was also checked using the static magnetic susceptibility as well as heat capacity data. Quantum discord estimated from magnetic susceptibility as well as heat capacity data is found to be present in the thermal states of the system even when the system is in a separable state.     

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.     

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.     

Improved constructions for quantum maximum distance separable codes

In this work, we further improve the distance of the quantum maximum distance separable (MDS) codes of length \(n=\frac{q^2+1}{10}\). This yields new families of quantum MDS codes. We also construct a family of new quantum MDS codes with parameters \([[\frac{q^2-1}{3}, \frac{q^2-1}{3}-2d+2, d]]_{q}\), where \(q=2^m\), \(2\le d\le \frac{q-1}{3}\) if \(3\mid (q+2)\), and \(2\le d\le \frac{2q-1}{3}\) if \(3\mid (q+1)\). Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

Non-determinism in Gödel’s System <Emphasis Type="Italic">T</Emphasis>

The calculus T^? is a successor-free version of Gödel’s T. It is well known that a number of important complexity classes, like e.g. the classes logspace, \(\textsc{p}\), \(\textsc{linspace}\), \(\textsc{etime}\) and \(\textsc{pspace}\), are captured by natural fragments of T^? and related calculi. We introduce the calculus T^∽, which is a non-deterministic variant of T^?, and compare the computational power of T^∽ and T^?. First, we provide a denotational semantics for T^∽ and prove this semantics to be adequate. Furthermore, we prove that \(\textsc{linspace}\subseteq \mathcal {G}^{\backsim }_{0} \subseteq \textsc{linspace}\) and \(\textsc{etime}\subseteq \mathcal {G}^{\backsim }_{1} \subseteq \textsc{pspace}\) where \(\mathcal {G}^{\backsim }_{0}\) and \(\mathcal {G}^{\backsim }_{1}\) are classes of problems decidable by certain fragments of T^∽. (It is proved elsewhere that the corresponding fragments of T^? equal respectively \(\textsc{linspace}\) and \(\textsc{etime}\).) Finally, we show a way to interpret T^∽ in T^?.     

Two families of BCH codes and new quantum codes

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths \(n=\frac{q^{2m}-1}{q^2-1}\) and \(n=\frac{q^{2m}-1}{q+1}\) (\(m\ge 3)\) over the finite field \(\mathbf {F}_{q^2}\) are studied. The maximum designed distances \(\delta ^\mathrm{new}_\mathrm{max}\) of these dual-containing BCH codes are determined by a careful analysis of properties of the cyclotomic cosets. NNS BCH codes which achieve these maximum designed distances are presented, and a sequence of nested NNS BCH codes that contain these BCH codes with maximum designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these NNS BCH codes. The new quantum codes presented here include many classes of good quantum codes, which have parameters better than those constructed from narrow-sense BCH codes, negacyclic and constacyclic BCH codes in the literature.     

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

Unextendible maximally entangled bases in $${\mathbb {C}}^{pd}\otimes {\mathbb {C}}^{qd}$$

The construction of unextendible maximally entangled bases is tightly related to quantum information processing like local state discrimination. We put forward two constructions of UMEBs in \({\mathbb {C}}^{pd}\otimes {\mathbb {C}}^{qd}\)(\(p\le q\)) based on the constructions of UMEBs in \({\mathbb {C}}^{d}\otimes {\mathbb {C}}^{d}\) and in \({\mathbb {C}}^{p}\otimes {\mathbb {C}}^{q}\), which generalizes the results in Guo (Phys Rev A 94:052302, 2016) by two approaches. Two different 48-member UMEBs in \({\mathbb {C}}^{6}\otimes {\mathbb {C}}^{9}\) have been constructed in detail.     

A Stable Fast Time-Stepping Method for Fractional Integral and Derivative Operators

A unified fast time-stepping method for both fractional integral and derivative operators is proposed. The fractional operator is decomposed into a local part with memory length \(\varDelta T\) and a history part, where the local part is approximated by the direct convolution method and the history part is approximated by a fast memory-saving method. The fast method has \(O(n_0+\sum _{\ell }^L{q}_{\alpha }(N_{\ell }))\) active memory and \(O(n_0n_T+ (n_T-n_0)\sum _{\ell }^L{q}_{\alpha }(N_{\ell }))\) operations, where \(L=\log (n_T-n_0)\), \(n_0={\varDelta T}/\tau ,n_T=T/\tau \), \(\tau \) is the stepsize, T is the final time, and \({q}_{\alpha }{(N_{\ell })}\) is the number of quadrature points used in the truncated Laguerre–Gauss (LG) quadrature. The error bound of the present fast method is analyzed. It is shown that the error from the truncated LG quadrature is independent of the stepsize, and can be made arbitrarily small by choosing suitable parameters that are given explicitly. Numerical examples are presented to verify the effectiveness of the current fast method.