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Register a new userSeparability and entanglement of n-qubits and a qubit with a qudit using Hilbert-Schmidt decompositions
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Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, with emphasis on maximally disordered subsystems (MDS) (i.e., density matrices rho(MDS) for which tracing over any subsystem gives the unit density matrix). A sufficient condition that rho(MDS) is not separable is that it has an eigenvalue larger than 1/d for a qubit and a qudit, and larger than 1/2^(n-1) for n-qubits system. The PTU transformation does not change the eigenvalues of the n-qubits MDS density matrices for odd n. Thus the Peres-Horodecki criterion does not give any information about entanglement of these density matrices, but this criterion is useful for indicating inseparability for even n. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the W state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8 GHZ density matrices with probabilities p(i), is analyzed as function of these probabilities. In some cases we show that the Peres-Horodecki criterion is both sufficient and necessary.
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The use of entanglement witness (EW) for non-full separability and the Bell operator for non-local hidden-variables (LHV) model are analyzed by relating them to the Hilbert-Schmidt (HS) decomposition of n-qubits states and these methods are applied explicitly to some 3 and 4 qubits states. EW for non-full separability (fs) is given by fs parameter minus operator G where the choice of G in the HS decomposition leads to the fs parameter and to the condition for non-separability by using criterions which are different from those used for genuine entanglement. We analyze especially entangled states with probability p mixed with white noise with probability 1-p and find the critical value p(crit.) above which the system is not fully separable. As the choice of EW might not be optimal we add to the analysis of EW explicit construction of fully separable density matrix and find the critical value p below which the system is fully separable. If the two values for p coincide we conclude that this parameter gives the optimal result. Such optimal results are obtained in the present work for some 3 and 4 qubits entangled states mixed with white noise. The use of partial-transpose (PT ) (say relative to qubit A) gives also p value above which the system is not fully separable. The use of EW gives better results (or at least equal) than those obtained by PT .
The entanglement witness is an important and experimentally applicable tool for entanglement detection. In this paper, we provide a nonlinear improvement of any entanglement witness for $2otimes d$ quantum systems. Compared with any existing entanglement witness, the improved separability criterion only needs two more measurements on local observables. Detailed examples are employed to illustrate the efficiency of the nonlinear improvement for general, optimal and non-decomposable entanglement witnesses.
The practically useful criteria of separable states $rho=sum_{k}w_{k}rho_{k}$ in $d=2times2$ are discussed. The equality $G({bf a},{bf b})= 4[langle psi|P({bf a})otimes P({bf b})|psirangle-langle psi|P({bf a})otimes{bf 1}|psiranglelangle psi|{bf 1}otimes P({bf b})|psirangle]=0$ for any two projection operators $P({bf a})$ and $P({bf b})$ provides a necessary and sufficient separability criterion in the case of a separable pure state $rho=|psiranglelanglepsi|$. We propose the separability criteria of mixed states, which are given by ${rm Tr}rho{{bf a}cdot {bf sigma}otimes {bf b}cdot {bf sigma}}=(1/3)Ccosvarphi$ for two spin $1/2$ systems and $4{rm Tr}rho {P({bf a})otimes P({bf b})}=1+(1/2)Ccos2varphi$ for two photon systems, respectively, after taking a geometrical angular average of ${bf a}$ and ${bf b}$ with fixed $cosvarphi={bf a}cdot{bf b}$. Here $-1leq Cleq 1$, and the difference in the numerical coefficients $1/2$ and $1/3$ arises from the different rotational properties of the spinor and the transverse photon. If one instead takes an average over the states in the $d=2$ Hilbert space, the criterion for two photon systems is replaced by $4{rm Tr}rho {P({bf a})otimes P({bf b})}=1+(1/3)Ccos2varphi$. Those separability criteria are shown to be very efficient using the existing experimental data of Aspect et al. in 1981 and Sakai et al. in 2006. When the Werner state is applied to two photon systems, it is shown that the Hilbert space average can judge its inseparability but not the geometrical angular average.
The generation and control of quantum correlations in high-dimensional systems is a major challenge in the present landscape of quantum technologies. Achieving such non-classical high-dimensional resources will potentially unlock enhanced capabilities for quantum cryptography, communication and computation. We propose a protocol that is able to attain entangled states of $d$-dimensional systems through a quantum-walk-based {it transfer & accumulate} mechanism involving coin and walker degrees of freedom. The choice of investigating quantum walks is motivated by their generality and versatility, complemented by their successful implementation in several physical systems. Hence, given the cross-cutting role of quantum walks across quantum information, our protocol potentially represents a versatile general tool to control high-dimensional entanglement generation in various experimental platforms. In particular, we illustrate a possible photonic implementation where the information is encoded in the orbital angular momentum and polarization degrees of freedom of single photons.
We analyze entanglement and nonlocal properties of the convex set of symmetric $N$-qubits states which are diagonal in the Dicke basis. First, we demonstrate that within this set, positivity of partial transposition (PPT) is necessary and sufficient for separability --- which has also been reported recently in https://doi.org/10.1103/PhysRevA.94.060101 {Phys. Rev. A textbf{94}, 060101(R) (2016)}. Further, we show which states among the entangled DS are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens new directions to address entanglement and non-locality on higher dimensional symmetric states, where presently very few results are available.
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