Separability and entanglement of n-qubits and a qubit with a qudit using Hilbert-Schmidt decompositions

TL;DR

This paper uses Hilbert-Schmidt decompositions and eigenvalue criteria to analyze entanglement and separability in multi-qubit and qubit-qudit systems, providing new criteria and insights into their quantum correlations.

Contribution

It introduces a sufficient separability criterion using Hilbert-Schmidt decompositions and SVD, and analyzes entanglement properties of various states, including maximally disordered states and GHZ-diagonal states.

Findings

Eigenvalue conditions indicate non-separability for maximally disordered states.

Peres-Horodecki criterion's effectiveness varies with system parity.

Mixing entangled states with noise affects their entanglement properties.

Abstract

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.