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

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. Quantifying and characterizing multi-party quantum entanglement is a fundamental problem in the field of quantum information. Roughly speaking, one expects the states that are "most entangled" to be the most valuable resources for carrying out quantum information processing tasks such as quantum communication and quantum teleportation, and to give the most striking philosophical implications in terms of the rejection of local hidden variable theories [1].

Monogamy of quantum correlations in three-qubit pure states

“…Thus, despite belonging to the same SLOCC class D 1,1,1 , the states |GHZ and |WW exhibit contrasting behaviour corresponding to the shareability of correlations amongst subsystems. It is pertinent to point out yet another distinguishing feature of these states [17,18]: the three qubit Wsuperposition state |WW possesses reducible correlations [15,16] while the correlations in the GHZ state are not attributable to its subsystems. The monogamous nature of GHZ state and the polygamous nature of W-superposition state brought out here (in terms of the correlation measure RRQD) thus supports the assertion that the states belonging to the same SLOCC class can exhibit quite dissimilar features -especially with respect to the reducibility and shareability of correlations.…”

Irreducible Multiparty Correlations in Quantum States without Maximal Rank

“…The irreducible n-particle correlation in an n-particle state has been shown to be zero for most n-particle pure states [6], e.g., among n-qubit pure states, the irreducible n-particle correlation is not zero only for the GHZ type pure states [7].…”

“…In particular, we obtain the analytic results for the degrees of irreducible multi-particle correlations for the stabilizer states [8,9,10] and the generalized GHZ states [7].…”

Determination of stabilizer states

“…In [7], the present authors proved that SLOCC equivalent * Email: gaofei bupt@hotmail.com |W n states can be uniquely determined among arbitrary states by their (n − 1) bipartite reduced density matrices whose index set is correspond to a tree graph. For |GHZ n states, it has been shown that there can be only |GHZ n states containing information at the n-particle level [10,11].…”

“…They also shown that the set of just about half the parties contains all the information in almost all n-party pure states [4]. In recent years much effort has been spent on studying quantum states with special forms and properties, such as stochastic local operations and classical communication (SLOCC) equivalent |W n states [5][6][7], Dicke states [8], stabilizer states [9] and generalized |GHZ n states which are LU equivalent to some stabilizer states [10,11]. In [7], the present authors proved that SLOCC equivalent * Email: gaofei bupt@hotmail.com |W n states can be uniquely determined among arbitrary states by their (n − 1) bipartite reduced density matrices whose index set is correspond to a tree graph.…”