Operator Entanglement of Two-Qubit Joint Unitary Operations Revisited: Schmidt Number Approach

The operator entanglement of two-qubit joint unitary operations is revisited. The Schmidt number, an important attribute of a two-qubit unitary operation, may have connection with the entanglement measure of the unitary operator. We find that the entanglement measure of a two-qubit unitary operators is classified by the Schmidt number of the unitary operators. We also discuss the exact relation between the operator entanglement and the parameters of the unitary operator.     

Schmidt Number Entanglement Measure for Multipartite k-nonseparable States

In this paper, an entanglement measure for multipartite quantum states with respect to k-partition was introduced, which is called Schmidt number entanglement measure for multipartite k-nonseparable states, it is simply denoted by k-ME SN. We show that this measure is well-defined, i.e., it satisfies some basic properties as an entanglement measure. In addition, we give a super bound and lower bound of k-ME SN for multipartite pure states according to the definition of joint k-Schmidt number with respect to k-partition. Furthermore, we give some examples to show that Schmidt number entanglement measure can quantify the strength of entanglement for multipartite quantum states.

     

Schmidt modes and entanglement

Motivated by their frequent use in quantum mechanical studies of entanglement, we give a brief overview of Schmidt modes and Schmidt decompositions of two-particle wave functions. We discuss methods of their derivation and include a little-known approach used in the original work by E. Schmidt [Math. Annalen, 63 (1906), 433]. This employs the bipartite wave function itself rather than the more complicated two-party reduced density matrix. As an illustration, Schmidt modes for two-photon polarisation qutrits are derived in a general form. The derivation is accompanied by a series of simple examples with special choices of parameters. Relationships between Schmidt modes, polarisation Stokes vectors and entanglement are also discussed.     

Persistent entanglement in arrays of interacting particles

We study the entanglement properties of a class of N-qubit quantum states that are generated in arrays of qubits with an Ising-type interaction. These states contain a large amount of entanglement as given by their Schmidt measure. They also have a high persistency of entanglement which means that approximately N/2 qubits have to be measured to disentangle the state. These states can be regarded as an entanglement resource since one can generate a family of other multiparticle entangled states such as the generalized Greenberger-Horne-Zeilinger states of 相似文献   

Quantum Phase Transitions,Entanglement Spectrum,and Schmidt Gap in Bond-Alternative S=1 Antiferromagnetic Chain

Bipartite entanglement, entanglement spectrum, and Schmidt gap in S=1 bond-alternative antiferromagnetic Heisenberg chain are investigated by the infinite time-evolving block decimation （iTEBD） method. The quantum phase transition （QPT） from the singlet-dimer phase to the Haldane phase can be detected by the singular behavior of bipartite entanglement, the sudden change of the entanglement spectrum, and the completely vanishing of the Schmidt gap. The critical point is determined to be around rc ~- 0.587, and the second-order character of the QPT is verified. Doubly degenerate entanglement spectra of both even and odd bonds are observed in the Haldane phase, by which one can distinguish the Haldane phase from the singlet-dimer phase easily. Nearest-neighbor antiferromagnetic correlations and next-nearest-neighbor ferromagnetic correlations are found in the whole parameter region. At the critical massless point, although exponentially decaying antiferromagnetie correlation is observed, it approaches to a constant value finally. Therefore, long-range correlations exist and the correlation length becomes divergent at the critical point.     

Double- and multi-photon pair production and electron-positron entanglement

We investigate entanglement of electrons and positrons produced via absorption by a vacuum of two or several photons from two external electromagnetic waves. The waves are modelled by finite-length focused pulses. Structures of the arising electron and positron wave packets are investigated in the momentum and coordinate representations. Conditional and unconditional widths of these wave packets, as well as the Schmidt number K are found, and they are used to evaluate the degree of entanglement. The conditions are found when entanglement is large. It is shown that the highest entanglement can be reached at nonrelativistic energies of electrons and positrons. Possibilities of observing the entanglement effects in experiments on pair production are discussed.     

Analysis and interpretation of high transverse entanglement in optical parametric down conversion

Quantum entanglement associated with transverse wave vectors of down conversion photons is investigated based on the Schmidt decomposition method. We show that transverse entanglement involves two variables: orbital angular momentum and transverse frequency. We show that in the monochromatic limit high values of entanglement are closely controlled by a single parameter resulting from the competition between (transverse) momentum conservation and longitudinal phase matching. We examine the features of the Schmidt eigenmodes, and indicate how entanglement can be enhanced by suitable mode selection methods.     

Impact of quantum–classical correspondence on entanglement enhancement by single-mode squeezing

Quantum entanglement between two field modes can be achieved through the collective squeezing of the two respective modes. If single-mode squeezing is performed prior to such a two-mode squeezing, an enhancement of entanglement production can happen. Interestingly, the occurrence of this enhancement can be implicitly linked to the local classical dynamical behavior via the paradigm of quantum–classical correspondence. In particular, the entanglement generated through quantum chaos is found to be hardly enhanced by prior squeezing, since it is bounded by the saturation value of the maximally entangled Schmidt state with fixed energy. These results illustrate that entanglement enhancement via initial squeezing can serve as a useful indicator of quantum chaotic behaviour.     

Entanglement cost in practical scenarios

We quantify the one-shot entanglement cost of an arbitrary bipartite state, that is, the minimum number of singlets needed by two distant parties to create a single copy of the state up to a finite accuracy, by using local operations and classical communication only. This analysis, in contrast to the traditional one, pertains to scenarios of practical relevance, in which resources are finite and transformations can be achieved only approximately. Moreover, it unveils a fundamental relation between two well-known entanglement measures, namely, the Schmidt number and the entanglement of formation. Using this relation, we are able to recover the usual expression of the entanglement cost as a special case.     

Local Channels Preserving Maximal Entanglement or Schmidt Number

It is well know that entanglement is invariant under local unitary operations. In this paper we show that a local channel preserves maximal entanglement state (MES) or preserves pure states with Schmidt number r (r is an arbitrarily fixed integer) if and only if it is a local unitary operation. That is, the only local channel that leaves entanglement invariant is the local unitary operation.     

The schmidt number of pure continuous-variable bipartite entangled states and the method of its calculation

The Schmidt number—the measure of entanglement of pure states of a continuous-variable bipartite system—is analytically calculated for a simple model of photon-atom scattering.     

Exact formula of the distribution of Schmidt eigenvalues for dynamical formation of entanglement in quantum chaos

The exact formula of the one-level distribution of the Schmidt eigenvalues is obtained for dynamical formation of entanglement in quantum chaos. The formula is based on the random matrix theory of the fixed-trace ensemble, and is derived using the theory of the holonomic system of differential equations. We confirm that the formula describes the universality of the distribution of the Schmidt eigenvalues in quantum chaos.     

High Intensity Generation of Entangled Photons in a Two‐Mode Electromagnetic Field

At present, the sources of entangled photons have a low rate of photon generation. This limitation is a key component of quantum informatics for the realization of such functions as linear quantum computation and quantum teleportation. In this paper, we propose a method for high intensity generation of entangled photons in a two‐mode electromagnetic field. On the basis of exact solutions of the Schrödinger equation, when electrons interact in an atom with a strong two‐mode electromagnetic field, it is shown that there may be large quantum entanglement between photons. The quantum entanglement is analyzed on the basis of the Schmidt parameter. It is shown that the Schmidt parameter can reach very high values depending on the choice of characteristics of the two‐mode fields. We find the Wigner function for the considered case. Violation of Bell's inequalities for continuous variables is demonstrated.     

Entanglement cost of implementing controlled-unitary operations

We investigate the minimum entanglement cost of the deterministic implementation of two-qubit controlled-unitary operations using local operations and classical communication (LOCC). We show that any such operation can be implemented by a three-turn LOCC protocol, which requires at least 1 ebit of entanglement when the resource is given by a bipartite entangled state with Schmidt number 2. Our result implies that there is a gap between the minimum entanglement cost and the entangling power of controlled-unitary operations. This gap arises due to the requirement of implementing the operations while oblivious to the identity of the inputs.     

Random quantum correlations and density operator distributions

Randomly correlated ensembles of two quantum systems are investigated, including average entanglement entropies and probability distributions of Schmidt decomposition coefficients. Maximal correlation is guaranteed in the limit as one system becomes infinite dimensional. The reduced density operator distributions are compared with distributions induced via the Bures and Hilbert-Schmidt metrics.     

各向异性光子晶体中二能级原子和自发辐射场间的纠缠

应用Von Neumann熵和Schmid tnumber K两种纠缠度量讨论了各向异性光子晶体中二能级原子和自发辐射场间纠缠度的演化性质.研究发现,原子-光场纠缠度的演化与原子上能级和光子晶体能带带边的相对位置有关,当原子上能级处于光子晶体禁带内,原子-光场纠缠度将保持稳定,当原子上能级处于光子晶体能带中,原子-光场纠缠度先增大后衰减到零.纠缠度的大小还与原子的初态有关.可以通过控制原子的初态和原子上能级与带边的相对位置来控制原子-光场纠缠度的演化特性.     

Exotic entanglement scaling of Heisenberg antiferromagnet on honeycomb lattice

The scaling behaviors of entanglement entropy (EE) against dimension cut-off of density matrix renormalization group (DMRG) in an anisotropic Heisenberg model on honeycomb lattice are investigated. In the gapped dimer phase, the entanglement spectrum (ES) exhibits large gaps and the EE shows an unexpected linear scaling before convergence. In contrast in the gapless Néel phase, the ES decays in a much smoother way, and the EE scales logarithmically. Our calculations show that the linear scaling in the dimer phase originates from one dominant Schmidt number plus n (nearly) degenerate Schmidt numbers that are much smaller than the dominant one. The non-trivial entanglement-scaling properties of the dimer and Néel phases could potentially be used for their detections.     

Multi-mode Einstein-Podolsky-Rosen Entangled State Representation and Its Applications

Our primary purpose of this work is to explicitly construct the general multipartite Einstein-Podolsky-Rosen (EPR) entangled state in multi-mode Fock space for a system with different masses of particles, which makes up a new quantum mechanical representation owing to completeness relation and orthogonal property. Its entanglement can be seen more clearly by analyzing its standard Schmidt decomposition. In addition, some applications of the multipartite entanglement are proposed including deriving the generalized Wigner operator and squeezing operator.