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arXiv:0707.4428 · Physical Review Letters

Only n -Qubit Greenberger-Horne-Zeilinger States Are Undetermined by Their Reduced Density Matrices

Scott N. Walck, David W. Lyons

Published Feb 4, 2008• 37 citations
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Abstract

The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of n qubits that are not uniquely determined among pure states by their reduced density matrices of n-1 qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the n-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.

DOI
10.1103/physrevlett.100.050501

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The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of n qubits that are not uniquely determined among pure states by their reduced density matrices of n-1 qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the n-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.
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.
Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so-called Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecision, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the Greenberger–Horne–Zeilinger state by stochastic local operations and classical communication, and discuss generalizations of our results to an arbitrary number of qubits.
We show that not all four-party pure states are Greenberger-Horne-Zeilinger (GHZ) reducible (i.e., can be generated reversibly from a combination of two-, three-, and four-party maximally entangled states by local quantum operations and classical communication asymptotically). We also present some properties of the relative entropy of entanglement for those three-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.
Nonproduct nnn-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized nnn-qubit Greenberger-Horne-Zeilinger states and their local unitary equivalents, for $n\ensuremath{\geqslant}3$, $n\ensuremath{\ne}4$. We characterize the Lie algebra of the stabilizer subgroup for these states. For n=4n=4n=4, there is an additional maximal stabilizer subalgebra, not local unitary equivalent to the former. We give a canonical form for states with this stabilizer as well.
It is proved that all states of three spin-12\frac{1}{2}21​ particles exhibiting an ``all versus nothing'' contradiction between quantum mechanics and the local realism of Einstein, Podolsky, and Rosen are exactly the Greenberger-Horne-Zeilinger (GHZ) states and the states obtained from them by local unitary transformations. The proof is obtained by showing that there are at most four elements (except for a different sign) in a set of mutually commuting nonlocal spin observables in the three-qubit system and using the certain algebraic properties that Pauli's matrices satisfy. We show that only does such a set of four nonlocal spin observables present a Greenberger-Horne-Zeilinger-Mermin-like argument. This also reveals the equivalence between the GHZ theorem and maximal violation of the Bell inequality.