International Journal of Quantum Information

We present sufficient criteria for the entanglement of three-qubit states. For some special families of states, the criteria are also necessary for the entanglement. They are formulated as simple sets of inequalities for the mean values of certain observables defined as tensor products of Pauli matrices. The criteria are good indicators of the entanglement in the vicinity of three-qubit GHZ and W states and enjoy the capability of detecting the entangled states with positive partial transpositions. Furthermore, they improve the best known result for the case of W state mixed with the white noise. The efficiency of the criteria is illustrated through several examples.

The flow of quantum information in local dephasing channels is analyzed over short and long times in case the structured reservoir of frequency modes exhibits a spectral gap in the density of modes over low frequencies. The presence of the low-frequency gap with upper cut-off frequency ${\omega }_g$ produces an infinite sequence of long-time intervals over which information backflow appears. Though generally irregular, the time intervals exhibit, under certain conditions, upper bounds: the $n$th episode of information backflow has already started at the instant $\pi (2n-1)∕{\omega }_g$, and already ended at the instant $2\pi n∕{\omega }_g$, for every $n=1,2,\dots$. The intervals become regular over long times, tend to the asymptotic length $\pi ∕{\omega }_g$, as the supremum value, and are described analytically in terms of the structure of the spectral density near the upper cut-off frequency of the spectral gap. Consequently, engineering structured reservoirs of frequency modes with a spectral gap over low frequencies produces in local dephasing channels regular sequences of information backflow and recoherence over long times, along with non-Markovian evolution.

The completeness of some classical statistical mechanical (SM) models is a recent result that has been developed by quantum formalism for the partition functions. In this paper, we consider a 2D classical ${\varphi }^4$ filed theory whose completeness has been proved in [V. Karimipour and M. H. Zarei, Phys. Rev. A 85 (2012) 032316]. We give a new and general systematic proof for the completeness of such a model where, by a few simple steps, we show how the partition function of an arbitrary classical field theory can be derived from a 2D classical ${\varphi }^4$ model. To this end, we start from various classical field theories containing models on arbitrary lattices and also $U(1)$ lattice gauge theories. Then we convert them to a new classical field model on a nonplanar bipartite graph with imaginary kinetic terms. After that, we show that any polynomial function of the field in the corresponding Hamiltonian can approximately be converted to a ${\varphi }^4$ term by adding enough numbers of vertices to the bipartite graph. In the next step, we give a few graphical transformations to convert the final nonplanar graph to a 2D rectangular lattice. We also show that the number of vertices which should be added grows polynomially with the number of vertices in the original model.

The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of $q^2$-ary BCH codes with length $n=q^{2m}+1$ (also called antiprimitive BCH codes in the literature), where $q\ge 4$ is a power of 2 and $m\ge 2$. By a detailed analysis of some useful properties about $q^2$-ary cyclotomic cosets modulo $n$, Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of $q^2$-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.

We consider the estimation of two-qubit interactions when initial states of both qubit can be locally controlled, while the final state of only one qubit can be measured. This amounts to realize a model of quantum channel communication with environment assistance. In such a framework, the unitaries’ parameters space results a tetrahedron in ${ℝ}^3$. On its edges the problem, becoming of single parameter estimation, can be exactly solved and we derive optimal probe states and estimators. Our results show that the possibility of environment assistance is always beneficial, while the usage of entanglement at channels’ input is not.

In $3\times 3$ dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. These rank four states are completely understood. We say that they have rank $(4,4)$ since both a state $\rho$ and its partial transpose ${\rho }^P$ have rank four. The next problem is to understand the extremal PPT states of rank $(5,5)$. We call two states $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-equivalent if they are related by a product transformation. A generic rank $(5,5)$ PPT state $\rho$ is extremal, and both $\rho$ and ${\rho }^P$ have six product vectors in their ranges, and no product vectors in their kernels. The three numbers $\{6,6;0\}$ are $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-invariants that help us classify the state. There is no analytical understanding of such states. We have studied numerically a few types of nongeneric rank five PPT states, in particular, states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-equivalence, where they appear as boundary states on one single five-dimensional face on the set of normalized PPT states. The interior of the face consists of rank $(5,5)$ states with four common product vectors in their kernels, it is a simplex of separable states surrounded by entangled PPT states. We say that a state $\rho$ is $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetric if $\rho$ and ${\rho }^P$ are $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-equivalent, and is genuinely $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetric if it is $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-equivalent to a state $\tau$ with $\tau ={\tau }^P$. Genuine $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetry implies a special form of $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetry. We have produced numerically, by a special method, a random sample of rank $(5,5)$ $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetric states. About 50 of these are of type $\{6,6;0\}$, among those all are extremal and about half are genuinely $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetric. All these genuinely $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetric states can be transformed to have a circulant form. We find however that this is not a generic property of genuinely $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetric states. The remaining $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetric states found in the search have product vectors in their kernels, and they inspired us to study such states without regard to $\mathrm{\text{SL}}\otimes \mathrm{\text{SL}}$-symmetry.

In this paper, entanglement classification shared among the spins of localized fermions in the noninteracting Fermi gas is studied. It is proven that the Fermi gas density matrix is block diagonal on the basis of the projection operators to the irreducible representations of symmetric group $S_n$. Every block of density matrix is in the form of the direct product of a matrix and identity matrix. Then it is useful to study entanglement in every block of density matrix separately. The basis of corresponding Hilbert space are identified from the Schur–Weyl duality theorem. Also, it can be shown that the symmetric part of the density matrix is fully separable. Then it has been shown that the entanglement measure which is introduced in Eltschka et al. [New J. Phys. 10, 043104 (2008)] and Guhne et al. [New J. Phys. 7, 229 (2005)], is zero for the even $n$ qubit Fermi gas density matrix. Then by focusing on three spin reduced density matrix, the entanglement classes have been investigated. In three qubit states there is an entanglement measure which is called 3-tangle. It can be shown that 3-tangle is zero for three qubit density matrix, but the density matrix is not biseparable for all possible values of its parameters and its eigenvectors are in the form of W-states. Then an entanglement witness for detecting non-separable state and an entanglement witness for detecting nonbiseparable states, have been introduced for three qubit density matrix by using convex optimization problem. Finally, the four spin reduced density matrix has been investigated by restricting the density matrix to the irreducible representations of $S_n$. The restricted density matrix to the subspaces of the irreducible representations: $S_{sym}$, $S_{3,1}$ and $S_{2,2}$ are denoted by ${\rho }_{sym}$, ${\rho }_{3,1}$ and ${\rho }_{2,2}$, respectively. It has been shown that some highly entangled classes (by using the results of Miyake [Phys. Rev. A 67, 012108 (2003)] for entanglement classification) do not exist in the blocks of density matrix ${\rho }_{3,1}$ and ${\rho }_{2,2}$, so these classes do not exist in the total Fermi gas density matrix.