Multilevel quantum Otto heat engines with identical particles

A quantum Otto heat engine is studied with multilevel identical particles trapped in one-dimensional box potential as working substance. The symmetrical wave function for Bosons and the anti-symmetrical wave function for Fermions are considered. In two-particle case, we focus on the ratios of \(W^i\) (\(i=B,F\)) to \(W_s\), where \(W^\mathrm{B}\) and \(W^\mathrm{F}\) are the work done by two Bosons and Fermions, respectively, and \(W_s\) is the work output of a single particle under the same conditions. Due to the symmetrical of the wave functions, the ratios are not equal to 2. Three different regimes, low-temperature regime, high-temperature regime, and intermediate-temperature regime, are analyzed, and the effects of energy level number and the differences between the two baths are calculated. In the multiparticle case, we calculate the ratios of \(W^i_M/M\) to \(W_s\), where \(W^i_M/M\) can be seen as the average work done by a single particle in multiparticle heat engine. For other working substances whose energy spectrum has the form of \(E_n\sim n^2\), the results are similar. For the case \(E_n\sim n\), two different conclusions are obtained.     

Efficient parallelizable hashing using small non-compressing primitives

A well-established method of constructing hash functions is to base them on non-compressing primitives, such as one-way functions or permutations. In this work, we present \(S^r\), an \(rn\)-to-\(n\)-bit compression function (for \(r\ge 1\)) making \(2r-1\) calls to \(n\)-to-\(n\)-bit primitives (random functions or permutations). \(S^r\) compresses its inputs at a rate (the amount of message blocks per primitive call) up to almost 1/2, and it outperforms all existing schemes with respect to rate and/or the size of underlying primitives. For instance, instantiated with the \(1600\)-bit permutation of NIST’s SHA-3 hash function standard, it offers about \(800\)-bit security at a rate of almost 1/2, while SHA-3-512 itself achieves only \(512\)-bit security at a rate of about \(1/3\). We prove that \(S^r\) achieves asymptotically optimal collision security against semi-adaptive adversaries up to almost \(2^{n/2}\) queries and that it can be made preimage secure up to \(2^n\) queries using a simple tweak.     

Recursive Computation of Logarithmic Derivatives,Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and Applications

Spheroidal harmonics and modified Bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics (\(P_n^m(x)\), \(Q_n^m(x)\), \(n\ge m\ge 0\), \(x>1\)), the oblate spheroidal harmonics (\(P_n^m(ix)\), \(Q_n^m(ix)\), \(n\ge m\ge 0\), \(x>0\)), and the modified Bessel functions (\(I_n(x)\), \(K_n(x)\), \(n\ge 0\), \(x>0\)) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.     

Good and asymptotically good quantum codes derived from algebraic geometry

In this paper, we construct several new families of quantum codes with good parameters. These new quantum codes are derived from (classical) t-point (\(t\ge 1\)) algebraic geometry (AG) codes by applying the Calderbank–Shor–Steane (CSS) construction. More precisely, we construct two classical AG codes \(C_1\) and \(C_2\) such that \(C_1\subset C_2\), applying after the well-known CSS construction to \(C_1\) and \(C_2\). Many of these new codes have large minimum distances when compared with their code lengths as well as they also have small Singleton defects. As an example, we construct a family \({[[46, 2(t_2 - t_1), d]]}_{25}\) of quantum codes, where \(t_1 , t_2\) are positive integers such that \(1<t_1< t_2 < 23\) and \(d\ge \min \{ 46 - 2t_2 , 2t_1 - 2 \}\), of length \(n=46\), with minimum distance in the range \(2\le d\le 20\), having Singleton defect at most four. Additionally, by applying the CSS construction to sequences of t-point (classical) AG codes constructed in this paper, we generate sequences of asymptotically good quantum codes.     

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.     

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 \(1\le k \le d\).     

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.     

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.     

Throughput maximization for speed scaling with agreeable deadlines

We study the following energy-efficient scheduling problem. We are given a set of n jobs which have to be scheduled by a single processor whose speed can be varied dynamically. Each job \(J_j\) is characterized by a processing requirement (work) \(p_j\), a release date \(r_j\), and a deadline \(d_j\). We are also given a budget of energy E which must not be exceeded and our objective is to maximize the throughput (i.e., the number of jobs which are completed on time). We show that the problem can be solved optimally via dynamic programming in \(O(n^4 \log n \log P)\) time when all jobs have the same release date, where P is the sum of the processing requirements of the jobs. For the more general case with agreeable deadlines where the jobs can be ordered so that, for every \(i < j\), it holds that \(r_i \le r_j\) and \(d_i \le d_j\), we propose an optimal dynamic programming algorithm which runs in \(O(n^6 \log n \log P)\) time. In addition, we consider the weighted case where every job \(J_j\) is also associated with a weight \(w_j\) and we are interested in maximizing the weighted throughput (i.e., the total weight of the jobs which are completed on time). For this case, we show that the problem becomes \(\mathcal{NP}\)-hard in the ordinary sense even when all jobs have the same release date and we propose a pseudo-polynomial time algorithm for agreeable instances.     

No nexiste nce of n-qubit u nexte ndible product bases of size $$2^ n-5$$2 n-5

It is known that the n-qubit system has no unextendible product bases (UPBs) of cardinality \(2^n-1\), \(2^n-2\) and \(2^n-3\). On the other hand, the n-qubit UPBs of cardinality \(2^n-4\) exist for all \(n\ge 3\). We prove that they do not exist for cardinality \(2^n-5\).     

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.     

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.     

A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

A new weak Galerkin (WG) finite element method is developed and analyzed for solving second order elliptic problems with low regularity solutions in the Sobolev space \(W^{2,p}(\Omega )\) with \(p\in (1,2)\). A WG stabilizer was introduced by Wang and Ye (Math Comput 83:2101–2126, 2014) for a simpler variational formulation, and it has been commonly used since then in the WG literature. In this work, for the purpose of dealing with low regularity solutions, we propose to generalize the stabilizer of Wang and Ye by introducing a positive relaxation index to the mesh size h. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along the interior element edges. When the norm index \(p\in (1,2]\), we strictly derive that the WG error in energy norm has an optimal convergence order \(O(h^{l+1-\frac{1}{p}-\frac{p}{4}})\) by taking the relaxed factor \(\beta =1+\frac{2}{p}-\frac{p}{2}\), and it also has an optimal convergence order \(O(h^{l+2-\frac{2}{p}})\) in \(L^2\) norm when the solution \(u\in W^{l+1,p}\) with \(p\in [1,1+\frac{2}{p}-\frac{p}{2}]\) and \(l\ge 1\). It is recovered for \(p=2\) that with the choice of \(\beta =1\), error estimates in the energy and \(L^2\) norms are optimal for the source term in the sobolev space \(L^2\). Weak variational forms of the WG method give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented without requiring a sufficiently large penalty factor as in the usual discontinuous Galerkin methods. In addition, numerical results illustrate that the proposed WG method with an over-relaxed factor \(\beta (\ge 1)\) converges at optimal algebraic rates for several low regularity elliptic problems.     

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

Cost-efficient design of a quantum multiplier–accumulator unit

This paper proposes a cost-efficient quantum multiplier–accumulator unit. The paper also presents a fast multiplication algorithm and designs a novel quantum multiplier device based on the proposed algorithm with the optimum time complexity as multiplier is the major device of a multiplier–accumulator unit. We show that the proposed multiplication technique has time complexity \(O((3 {\hbox {log}}_{2}n)+1)\), whereas the best known existing technique has \(O(n{\hbox {log}}_{2} n)\), where n is the number of qubits. In addition, our design proposes three new quantum circuits: a circuit representing a quantum full-adder, a circuit known as quantum ANDing circuit, which performs the ANDing operation and a circuit presenting quantum accumulator. Moreover, the proposed quantum multiplier–accumulator unit is the first ever quantum multiplier–accumulator circuit in the literature till now, which has reduced garbage outputs and ancillary inputs to a great extent. The comparative study shows that the proposed quantum multiplier performs better than the existing multipliers in terms of depth, quantum gates, delays, area and power with the increasing number of qubits. Moreover, we design the proposed quantum multiplier–accumulator unit, which performs better than the existing ones in terms of hardware and delay complexities, e.g., the proposed (\(n\times n\))—qubit quantum multiplier–accumulator unit requires \(O(n^{2})\) hardware and \(O({\hbox {log}}_{2}n)\) delay complexities, whereas the best known existing quantum multiplier–accumulator unit requires \(O(n^{3})\) hardware and \(O((n-1)^{2} +1+n)\) delay complexities. In addition, the proposed design achieves an improvement of 13.04, 60.08 and 27.2% for \(4\times 4\), 7.87, 51.8 and 27.1% for \(8\times 8\), 4.24, 52.14 and 27% for \(16\times 16\), 2.19, 52.15 and 27.26% for \(32 \times 32\) and 0.78, 52.18 and 27.28% for \(128 \times 128\)-qubit multiplications over the best known existing approach in terms of number of quantum gates, ancillary inputs and garbage outputs, respectively. Moreover, on average, the proposed design gains an improvement of 5.62% in terms of area and power consumptions over the best known existing approach.     

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.     

The Rate of Convergence to the Limit of the Probability of Encountering an Accidental Similarity in the Presence of Counter Examples

This paper refines the main result of [1], where the limit \( - {e^{ - a}} - a{e^{ - a}}\left[ {1 - {e^{ - c\sqrt a }}} \right]\) was proved for the probability of encountering an accidental similarity between two parent examples without \(m = c\sqrt n \) counter examples if each parent example and counter example is described by a series of \(\sqrt n \) independent Bernoulli trials with success probability \(p = \sqrt {a/n} \). In this paper, we prove that the rate of convergence to the limit is proportional to \({n^{\frac{1}{2}}}\).     

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.     

Construction of quantum caps in projective space <Emphasis Type="Italic">PG</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, 4) and quantum codes of distance 4

Constructions of quantum caps in projective space PG(r, 4) by recursive methods and computer search are discussed. For each even n satisfying \(n\ge 282\) and each odd z satisfying \(z\ge 275\), a quantum n-cap and a quantum z-cap in \(PG(k-1, 4)\) with suitable k are constructed, and \([[n,n-2k,4]]\) and \([[z,z-2k,4]]\) quantum codes are derived from the constructed quantum n-cap and z-cap, respectively. For \(n\ge 282\) and \(n\ne 286\), 756 and 5040, or \(z\ge 275\), the results on the sizes of quantum caps and quantum codes are new, and all the obtained quantum codes are optimal codes according to the quantum Hamming bound. While constructing quantum caps, we also obtain many large caps in PG(r, 4) for \(r\ge 11\). These results concerning large caps provide improved lower bounds on the maximal sizes of caps in PG(r, 4) for \(r\ge 11\).     

The panpositionable panconnectedness of crossed cubes

In many parallel and distributed multiprocessor systems, the processors are connected based on different types of interconnection networks. The topological structure of an interconnection network is typically modeled as a graph. Among the many kinds of network topologies, the crossed cube is one of the most popular. In this paper, we investigate the panpositionable panconnectedness problem with respect to the crossed cube. A graph G is r-panpositionably panconnected if for any three distinct vertices x, y, z of G and for any integer \(l_1\) satisfying \(r \le l_1 \le |V(G)| - r - 1\), there exists a path \(P = [x, P_1, y, P_2, z]\) in G such that (i) \(P_1\) joins x and y with \(l(P_1) = l_1\) and (ii) \(P_2\) joins y and z with \(l(P_2) = l_2\) for any integer \(l_2\) satisfying \(r \le l_2 \le |V(G)| - l_1 - 1\), where |V(G)| is the total number of vertices in G and \(l(P_1)\) (respectively, \(l(P_2)\)) is the length of path \(P_1\) (respectively, \(P_2\)). By mathematical induction, we demonstrate that the n-dimensional crossed cube \(CQ_n\) is n-panpositionably panconnected. This result indicates that the path embedding of joining x and z with a mediate vertex y in \(CQ_n\) is extremely flexible. Moreover, applying our result, crossed cube problems such as panpositionable pancyclicity, panpositionably Hamiltonian connectedness, and panpositionable Hamiltonicity can be solved.