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arXiv:quant-ph/0407117 · Physical Review A

Parts of quantum states

Nick S. Jones, Noah Linden

Published Jan 18, 2005• 60 citations
arXivHTMLPDF

Abstract

It is shown that generic $N$-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 $N$-party pure states is in the set of reduced states of just over half the parties. For $N$ even, the reduced states in fewer than $N∕2$ parties are shown to be an insufficient description of almost all states (similar results hold when $N$ 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.

DOI
10.1103/physreva.71.012324

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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.
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.
The generalized nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of nnn qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
Only nnn-qubit Greenberger-Horne-Zeilinger states contain nnn-partite information
The generalized nnn-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of nnn qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the nnn-party level.
How much information (n-1)-party reduced state possesses about n-party pure state quantum correlation? It is shown that any (n-1)-party reduced state uniquely determines the Groverian and Geometric entanglement measures of the original n-party pure state. The theorem has two important consequences. First, if two pure n-qubit states have (n-1)-qubit reduced states equivalent under local unitary(LU) 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.