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arXiv:1505.06261

Entanglement of Four-Qubit Rank-222 Mixed States

Eylee Jung, DaeKil Park

Published May 23, 2015• 0 citations
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Abstract

It is known that there are three maximally entangled states $\ket{Φ_1} = (\ket{0000} + \ket{1111}) / \sqrt{2}$, $\ket{Φ_2} = (\sqrt{2} \ket{1111} + \ket{1000} + \ket{0100} + \ket{0010} + \ket{0001}) / \sqrt{6}$, and $\ket{Φ_3} = (\ket{1111} + \ket{1100} + \ket{0010} + \ket{0001}) / 2$ in four-qubit system. It is also known that there are three independent measures ${\cal F}^{(4)}_j \hspace{.2cm} (j=1,2,3)$ for true four-way quantum entanglement in the same system. In this paper we compute ${\cal F}^{(4)}_j$ and their corresponding linear monotones ${\cal G}^{(4)}_j$ for three rank-two mixed states $ρ_j = p \ket{Φ_j}\bra{Φ_j} + (1 - p) \ket{\mbox{W}_4} \bra{\mbox{W}_4}$, where $\ket{\mbox{W}_4} = (\ket{0111} + \ket{1011} + \ket{1101} + \ket{1110}) / 2$. We discuss the possible applications of our results briefly.

DOI
10.48550/arxiv.1505.06261

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We introduce two tangle-based four-party entanglement measures t1{t}_{1}t1​ and t2{t}_{2}t2​, and two negativity-based measures n1{n}_{1}n1​ and n2{n}_{2}n2​, which are derived from the monogamy relations. These measures are computed for three four-qubit maximally entangled and WWW states explicitly. We also compute these measures for the rank-2 mixture ${\ensuremath{\rho}}_{4}=p|{\text{GHZ}}_{4}\ensuremath{\rangle}\ensuremath{\langle}{\text{GHZ}}_{4}|+(1\ensuremath{-}p)|{W}_{4}\ensuremath{\rangle}\ensuremath{\langle}{W}_{4}|$ by finding the corresponding optimal decompositions. It turns out that ${t}_{1}({\ensuremath{\rho}}_{4})$ is trivial and the corresponding optimal decomposition is equal to the spectral decomposition. Probably, this triviality is a sign of the fact that the corresponding monogamy inequality is not sufficiently tight. We fail to compute ${t}_{2}({\ensuremath{\rho}}_{4})$ due to the difficulty in the calculation of the residual entanglement. The negativity-based measures ${n}_{1}({\ensuremath{\rho}}_{4})$ and ${n}_{2}({\ensuremath{\rho}}_{4})$ are explicitly computed and the corresponding optimal decompositions are also derived explicitly.
The existence of a maximally entangled pure state is a cornerstone result of entanglement theory that has paramount consequences in quantum information theory. A natural generalization of this property is to consider whether a notion of maximal entanglement is possible among all states with the same spectrum (where the aforementioned case of pure states corresponds to the particular choice in which the spectrum is a delta distribution, i.e., rank-1 states). Despite positive evidence in the past that such a notion might exist at least in the case of two-qubit states, it was recently shown in [Phys. Rev. Lett. 133, 050202 (2024)] that the answer to the above question is negative. This reference proved this for particular choices of the spectrum in the case of rank-2 two-qubit density matrices. While this settles the problem in general, it still leaves open whether there are other choices of the spectrum outside the case of pure states where a maximally entangled state for a fixed spectrum might exist. In this work we extend this impossibility result to all rank-2 and rank-3 two-qubit states as well as for a large class of eigenvalue distributions in the case where the rank equals four.
There has recently been much discussion regarding entanglement transformations in terms of local filtering operations and whether the optimal entanglement for an arbitrary two-qubit state could be realized. We introduce an experimentally realizable scheme for manipulating the entanglement of an arbitrary state of two polarization-entangled qubits. This scheme is then used to provide some perspective to the mathematical concepts inherent in this field with respect to a laboratory environment. Specifically, we look at how to extract enhanced entanglement from systems with a fixed rank, and, in the case where the rank of the density operator for the state can be reduced, show how the state can be made arbitrarily close to a maximally entangled pure state. In this context we also discuss bounds on entanglement in mixed states.
We have studied mixed states in the system of three qubits with the property that all their partial transposes are positive; these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. In random numerical searches, we find entangled PPT states with a large variety of rank combinations. For ranks equal to five or higher, we find both extremal and nonextremal PPT states of nearly every rank combination, with the restriction that the square sum of the four ranks of an extremal PPT state can be at most 193. We have studied especially the rank-four entangled PPT states, which are found to have rank four for every partial transpose. These states are all extremal because of the previously known result that every PPT state of rank three or less is separable. We find two distinct classes of rank-4444 entangled PPT states, identified by a real valued quadratic expression invariant under local SL(2,C)\text{SL}(2,\mathbb{C})SL(2,C) transformations, mathematically equivalent to Lorentz transformations. This quadratic Lorentz invariant is nonzero for one class of states (type I in our terminology) and zero for the other class (type II). The previously known states based on unextendible product bases are a nongeneric subclass of the type-I states. We present analytical constructions of states of both types, general enough to reproduce all the rank-4444 PPT states we have found numerically. We can not exclude the possibility that there exist nongeneric rank-four PPT states that we do not find in our random numerical searches.
We present a measure of quantum entanglement which is capable of quantifying the degree of entanglement of a multi-partite quantum system. This measure, which is based on a generalization of the Schmidt rank of a pure state, is defined on the full state space and is shown to be an entanglement monotone, that is, it cannot increase under local quantum operations with classical communication and under mixing. For a large class of mixed states this measure of entanglement can be calculated exactly, and it provides a detailed classification of mixed states. We also present a connection to the best separable approximation of an entangled state of a two-qubit system.