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arXiv:quant-ph/0203037 · Physical Review A

Bounds on entanglement in qudit subsystems

Viv M. Kendon, Karol Życzkowski, William J. Munro

Published Dec 12, 2002• 53 citations
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Abstract

The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterized by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other $N\ensuremath{-}m$ qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than $N/3$ qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller than $N/3$ falls exponentially in the dimension of the Hilbert space. The bulk of pure state Hilbert space thus consists of highly entangled states with multipartite entanglement encompassing at least a third of the qudits in the pure state.

DOI
10.1103/physreva.66.062310

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The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterized by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other N\ensuremath−mN\ensuremath{-}mN\ensuremath−m qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than N/3N/3N/3 qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller than N/3N/3N/3 falls exponentially in the dimension of the Hilbert space. The bulk of pure state Hilbert space thus consists of highly entangled states with multipartite entanglement encompassing at least a third of the qudits in the pure state.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
Quantum entanglement and its paradoxical properties hold the key to an information processing revolution. Much attention has focused recently on the challenging problem of characterizing entanglement. Entanglement for a two qubit system is reasonably well understood; however, the nature and properties of multiple qubit systems are largely unexplored. Motivated by the importance of such systems in quantum computing, we show that typical pure states of N qubits are highly entangled but have decreasing amounts of pairwise entanglement (measured using the Wootter concurrence formula) as N increases. Above six qubits, very few states have any pairwise entanglement and, generally, for a typical pure state of N qubits there is a sharp cut-off where its subsystems of size m become positive partial transpose (i.e. separable or only bound entangled) around N ⪆ 2m + 3, based on numerical analysis up to N = 13.
We show that pure states of multipartite quantum systems are multiseparable (i.e., give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the quantification of multipartite pure-state entanglement are discussed. Further, as an application of the techniques used here, we show that any purification of a bipartite-bound entangled state having a positive partial transpose is tri-inseparable, i.e., has none of its three bipartite partial traces separable.
For a random quantum state on H=Cd⊗CdH=C^d \otimes C^dH=Cd⊗Cd obtained by partial tracing a random pure state on H⊗CsH \otimes C^sH⊗Cs, we consider the whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold s0=s0(d)s_0=s_0(d)s0​=s0​(d) of order roughly d3d^3d3. More precisely, for any a>0a > 0a>0 and for d large enough, such a random state is entangled with very large probability when s<(1−a)s0s < (1-a)s_0s<(1−a)s0​, and separable with very large probability when s>(1+a)s0s > (1+a)s_0s>(1+a)s0​. One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold k0=k0(N)∼N/5k_0 = k_0(N) \sim N/5k0​=k0​(N)∼N/5 such that two subsystems of k particles each typically share entanglement if k>k0k > k_0k>k0​, and typically do not share entanglement if k<k0k < k_0k<k0​. Our methods work also for multipartite systems and for "unbalanced" systems such as Cd⊗Cd′C^{d} \otimes C^{d'}Cd⊗Cd′, d≠d′d \neq d'd=d′. The arguments rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces; some of the auxiliary results may be of reference value. A high-level non-technical overview of the results of this paper and of a related article arXiv:1011.0275 can be found in arXiv:1112.4582.
For a random quantum state on obtained by partial tracing a random pure state on , we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold of order roughly . More precisely, for any and for d large enough, such a random state is entangled with very large probability when , and separable with very large probability when . One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold such that two subsystems of k particles each typically share entanglement if k > k 0 , and typically do not share entanglement if k < k 0 . Our methods also work for multipartite systems and for “unbalanced” systems such as , . The arguments rely on random matrices, classical convexity, high‐dimensional probability, and geometry of Banach spaces; some of the auxiliary results may be of reference value. © 2013 Wiley Periodicals, Inc.