A Variational Approach to the Quantum Separability Problem (original) (raw)
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Variational approach to the quantum separability problem
Physical Review A
We present the variational separability verifier (VSV), which is a variational quantum algorithm that determines the closest separable state (CSS) of an arbitrary quantum state with respect to the Hilbert-Schmidt distance (HSD). We first assess the performance of the VSV by investigating the convergence of the optimization procedure for Greenberger-Horne-Zeilinger states of up to seven qubits, using both state-vector and shot-based simulations. We also numerically determine the CSS of maximally entangled mixed X states, and subsequently use the results of the algorithm to surmise the analytical form of the aforementioned CSS. Our results indicate that current noisy intermediate-scale quantum devices may be useful in addressing the NP-hard full separability problem using the VSV, due to the shallow quantum circuit imposed by employing the destructive SWAP test to evaluate the HSD. The VSV may also possibly lead to the characterization of multipartite quantum states, once the algorithm is adapted and improved to obtain the closest k-separable state of a multipartite entangled state.
Improved algorithm for quantum separability and entanglement detection
Physical Review A, 2004
Determining whether a quantum state is separable or entangled is a problem of fundamental importance in quantum information science. It has recently been shown that this problem is NPhard. There is a highly inefficient 'basic algorithm' for solving the quantum separability problem which follows from the definition of a separable state. By exploiting specific properties of the set of separable states, we introduce a new classical algorithm that solves the problem significantly faster than the 'basic algorithm', allowing a feasible separability test where none previously existed e.g. in 3-by-3-dimensional systems. Our algorithm also provides a novel tool in the experimental detection of entanglement.
Quantum entanglement: an overview of the separability problem in two quantum bits
Cornell University - arXiv, 2022
The separability problem is one of the basic and emergent problems in present and future quantum information processing. The latter focuses on information and computing based on quantum mechanics and uses quantum bits as its basic information units. In this paper we present an overview of the progress in the separability problem in bipartite systems, more specifically in two quantum bits (qubits) system, from the criterion based on the Bell's inequalities in 1964 to the Li-Qiao criterion and the enhanced entanglement criterion based on the SIC POVMs in 2018.
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions, in which the density matrices are decomposed into various terms representing certain 1-qubit, 2-qubits…, n-qubits measurements. The present method is more general than previous methods for bipartite systems, as it can be used for quantification of entanglement for large n-qubits systems (3 n ≥). We demonstrate the use of the present method by analyzing 3-qubits GHZ states and 3-qubits general Bell-states produced by a certain multiplications of Braid operators, operating on the computational basis of states. Quantum correlations are obtained by measuring all qubits of these systems, while a measurement of a part of these systems gives only classical correlations. Quantification of entanglement, for these systems, is given by the use of HS parameters.
International Journal of Quantum Information, 2016
Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of [Formula: see text]-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the [Formula: see text]-qubit 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 [Formula: see text]-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that [Formula: see text] (MDS) is not separable is that it has an eigenvalue larger than [Formula: see text] for a qubit and a qudit, and larger than [Formula: see t...
A Cone Approach to the Quantum Separability Problem
Arxiv preprint arXiv: …, 2010
Exploiting the cone structure of the set of unnormalized mixed quantum states, we offer an approach to detect separability independently of the dimensions of the subsystems. We show that any mixed quantum state can be decomposed as ρ = (1 − λ)C ρ + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice C ρ = M 1 ⊗ M 2 we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ equals 1. We give a first extension of this result to detect genuine entanglement in multipartite states and show a natural connection between the multipartite separability problem and the classification of pure states under stochastic local operations and classical communication (SLOCC). We argue that this approach is not exhausted with the first simple choices included herein.
Local quantum uncertainty in two-qubit separable states: a case study
Quantum Information Processing, 2015
Recent findings suggest, separable states, which are otherwise of no use in entanglement dependent tasks, can also be used in information processing tasks that depend upon the discord type general non classical correlations. In this work, we explore the nature of uncertainty in separable states as measured by local quantum uncertainty. Particularly in two-qubit system, we find separable X-state which has maximum local quantum uncertainty. Interestingly, this separable state coincides with the separable state, having maximum geometric discord. We also search for the maximum amount of local quantum uncertainty in separable Bell diagonal states. We indicate an interesting connection to the tightness of entropic uncertainty with the state of maximum uncertainty.
Partial scaling transform of multiqubit states as a criterion of separability
Journal of Physics A: Mathematical and General, 2005
The partial scaling transform of the density matrix for multiqubit states is introduced to detect entanglement of the quantum state. The transform contains partial transposition as a special case. The scaling transform corresponds to partial time scaling of subsystem (or partial Planck's constant scaling) which was used to formulate recently separability criterion for continous variables. A measure of entanglement which is a generalization of negativity measure is introduced being based on tomographic probability description of spin states.