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arXiv:quant-ph/0010096 · Physical Review A

Separability properties of tripartite states with U ⊗ U ⊗ U symmetry

Tilo Eggeling, Reinhard F. Werner

Published Mar 21, 2001• 130 citations
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Abstract

We study separability properties in a five-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states that can be written as linear combinations of permutation operators, or equivalently, commute with unitaries of the form $U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}U.$ We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.

DOI
10.1103/physreva.63.042111

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We study separability properties in a five-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states that can be written as linear combinations of permutation operators, or equivalently, commute with unitaries of the form U\ensuremath⨂U\ensuremath⨂U.U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}U.U\ensuremath⨂U\ensuremath⨂U. We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition. Tripartite entanglement is investigated in terms of the relative entropy of tripartite entanglement and of the trace norm.
We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition.
We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding product states have unique decompositions, and show that this is true in most cases if the number of product vectors is sufficiently small. In the three-qubit case, generic five-dimensional spaces give rise to faces of the convex set consisting of all separable states, which are affinely isomorphic to the five-dimensional simplex with six vertices. We construct, as a byproduct, three-qubit entangled positive partial transpose edge states of rank 4 with explicit formulas. This covers those entanglements which cannot be constructed from an unextendible product basis.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
State space structure of tripartite quantum systems is analyzed. In particular, it has been shown that the set of states separable across all the three bipartitions [say, Bint(ABC){\mathcal{B}}^{int}(ABC)Bint(ABC)] is a strict subset of the set of states having positive partial transposition across the three bipartite cuts [say, Pint(ABC){\mathcal{P}}^{int}(ABC)Pint(ABC)] for all the tripartite Hilbert spaces ${\mathbb{C}}_{A}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}_{B}^{{d}_{2}}\ensuremath{\bigotimes}{\mathbb{C}}_{C}^{{d}_{3}}$ with $min{{d}_{1},{d}_{2},{d}_{3}}\ensuremath{\ge}2$. The claim is proved by constructing a state belonging to the set Pint(ABC){\mathcal{P}}^{int}(ABC)Pint(ABC) but not to Bint(ABC){\mathcal{B}}^{int}(ABC)Bint(ABC). For ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}3}$ with $d\ensuremath{\ge}3$, the construction follows from a specific type of multipartite unextendible product bases. However, such a construction is not possible for ${({\mathbb{C}}^{2})}^{\ensuremath{\bigotimes}3}$ since bipartite systems ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ do not allow any unextendible product bases for arbitrary nnn [Phys. Rev. Lett. 82, 5385 (1999)]. For the three-qubit system, we therefore come up with a different construction.
Two important subsets of tripartite pure states, namely {\\em triseparable} states (i.e. states such that the bipartite density matrices obtained by tracing out any party's subsystem are separable) and {\\em triorthogonal} states (i.e. states with a three-party Schmidt-like decomposition) are introduced. It is proved that any triseparable state is triorthogonal and vice-versa. The relevance of this result to quantifying tripartite pure-state entanglement is discussed. It is also proved that any purification of a bipartite bound entangled state with positive partial transpose is {\\em tri-inseparable}. This is a new necessary condition for such bound entanglement.