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Physical Review Letters

The Parts Determine the Whole in a Generic Pure Quantum State

Noah Linden, William K. Wootters

Published Dec 20, 2002• 81 citations
Publisher

Abstract

We show that almost every pure state of multiparty quantum systems (each of whose local Hilbert space has the same dimension) is completely determined by the state's reduced density matrices of a fraction of the parties; this fraction is less than about two-thirds of the parties for states of large numbers of parties. In other words, once the reduced states of this fraction of the parties have been specified, there is no further freedom in the state.

DOI
10.1103/physrevlett.89.277906

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•Dec 15, 2011•0 citations
We show that almost every pure state of multiparty quantum systems (each of whose local Hilbert space has the same dimension) is completely determined by the state's reduced density matrices of a fraction of the parties; this fraction is less than about two-thirds of the parties for states of large numbers of parties. In other words, once the reduced states of this fraction of the parties have been specified, there is no further freedom in the state.
It is shown that generic NNN-party pure quantum states (with equidimensional subsystems) are uniquely determined by their reduced states of just over half the parties; in other words, all the information in almost all NNN-party pure states is in the set of reduced states of just over half the parties. For NNN even, the reduced states in fewer than N∕2N∕2N∕2 parties are shown to be an insufficient description of almost all states (similar results hold when NNN is odd). It is noted that real algebraic geometry is a natural framework for any analysis of parts of quantum states: two simple polynomials, a quadratic and a cubic, contain all of their structure. Algorithmic techniques are described which can provide conditions for sets of reduced states to belong to pure or mixed states.
It has been proven by Rana et al. that the nnn-qubit generic Dicke states $|\mathrm{GD}(n,\ensuremath{\ell})\ensuremath{\rangle}$ are uniquely determined, among arbitrary states, by their $(\ensuremath{\ell}+1)$-partite reduced density matrices, and $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$ number of them which have one party common to all are sufficient. We show that among arbitrary states, Dicke-type states are also uniquely determined by their $(\ensuremath{\ell}+1)$-partite reduced density matrices. Here, by Dicke-type states, we mean all multiqubit pure states that are stochastic local operations and classical communication equivalent to the nnn-qubit standard Dicke state $|D(n,\ensuremath{\ell})\ensuremath{\rangle}$. Moreover, we find two kinds of subsets of the $(\ensuremath{\ell}+1)$-partite reduced density matrices set, both of which can determine the Dicke-type states. More importantly, the number of elements in these two subsets is much smaller than $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$.
It is shown that the number of orthogonal density matrices that can be perfectly distinguished locally is bounded above by the total dimension over the average of maximal pure state entanglement in the supports of the density matrices. Thus we establish a direct connection between pure state entanglement and local distinguishability for any se t of orthogonal quantum states, pure or mixed. The result holds for any finite dimensional composite quantum system irrespective of the multipartite structure and dimensions of the individual su bsystems.