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arXiv:quant-ph/0403200 · Physical Review A

Three-party pure quantum states are determined by two two-party reduced states

Lajos Diósi

Published Jul 19, 2004• 45 citations
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Abstract

We can uniquely calculate almost all entangled state vectors of tripartite systems $ABC$ if we know the reduced states of any two bipartite subsystems, e.g., of $AB$ and of $BC$. We construct the explicit solution.

DOI
10.1103/physreva.70.010302

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We can uniquely calculate almost all entangled state vectors of tripartite systems ABCABCABC if we know the reduced states of any two bipartite subsystems, e.g., of ABABAB and of BCBCBC. We construct the explicit solution.
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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.
Abstract We investigate whether a generic pure state on a multipartite quantum system can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems’ dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N -partite case, we provide conditions on the subsystems’ size and the nature of the locality constraint, under which random pure states cannot be quasi-locally stabilized generically. Also, allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic, our results offer insights into the extent to which random pure states may arise as unique ground states of frustration-free parent Hamiltonians. We further argue that, to a high probability, pure quantum states sampled from a t -design enjoy the same stabilizability properties of Haar-random ones as long as suitable dimension constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a connection between the tasks of quasi-local state stabilization and unique state reconstruction from local tomographic information, and provide a constructive procedure for determining a generic N -partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties, improving over existing results.
Groverian and geometric entanglement measures of the nnn-party pure state are expressed by the $(n\ensuremath{-}1)$-party reduced state density operator directly. This main theorem derives several important consequences. First, if two pure nnn-qudit states have reduced states of $(n\ensuremath{-}1)$-qudits, which are equivalent under local unitary transformations, then they have equal Groverian and geometric entanglement measures. Second, both measures have an upper bound for pure states. However, this upper bound is reached only for two-qubit systems. Third, it converts effectively the nonlinear eigenvalue problem for the three-qubit Groverian measure into linear eigenvalue equations. Some typical solutions of these linear equations are written explicitly and the features of the general solution are discussed in detail.