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arXiv:1703.10950 · Physical Review A

Almost all four-particle pure states are determined by their two-body marginals

Nikolai Wyderka, Felix Huber, Otfried Gühne

Published Jul 11, 2017• 22 citations
arXivHTMLPDF

Abstract

We show that generic pure states (states drawn according to the Haar measure) of four particles of equal internal dimension are uniquely determined among all other pure states by their two-body marginals. In fact, certain subsets of three of the two-body marginals suffice for the characterization. We also discuss generalizations of the statement to pure states of more particles, showing that these are almost always determined among pure states by three of their $(n\ensuremath{-}2)$-body marginals. Finally, we present special families of symmetric pure four-particle states that share the same two-body marginals and are therefore undetermined. These are four-qubit Dicke states in superposition with generalized GHZ states.

DOI
10.1103/physreva.96.010102

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Physical Review A•Nov 30, 2015•3 citations
We show that generic pure states (states drawn according to the Haar measure) of four particles of equal internal dimension are uniquely determined among all other pure states by their two-body marginals. In fact, certain subsets of three of the two-body marginals suffice for the characterization. We also discuss generalizations of the statement to pure states of more particles, showing that these are almost always determined among pure states by three of their (n\ensuremath−2)(n\ensuremath{-}2)(n\ensuremath−2)-body marginals. Finally, we present special families of symmetric pure four-particle states that share the same two-body marginals and are therefore undetermined. These are four-qubit Dicke states in superposition with generalized GHZ states.
Abstract Determining whether the original global state is uniquely determined by its local marginals is a prerequisite for some efficient tools for characterizing quantum states. In this paper, we show that almost all generic pure states of even N-particle with equal local dimension are uniquely determined among all pure states (UDP) by four of their half-body marginals. Furthermore, we give a graphical description of the marginals for determining genuinely multipartite entangled states, which leads to several lower bounds on the number of required marginals. Finally, we present a construction of N-qudit states obtained from certain combinatorial structures that cannot be UDP by their k-body marginals for some k ⩾ ⌊ N / 2 ⌋ .
The quantum marginal problem asks whether a family of quantum marginals are compatible with a global quantum state. It is of central importance to a wide range of topics in both quantum many body physics and quantum information. Often it can be the case that when a family of quantum marginals are compatible with a global quantum state, that global state is unique. This thesis explores the role of uniqueness in the quantum marginal problem, in particular the distinction between uniqueness among pure states and uniqueness among all states. A particularly important vari- ant of the quantum marginal problem is the symmetric extendibility problem. This thesis makes headway on characterizing the extremal points of a particular class of symmetric extendibility problems, and demonstrates the importance of understand- ing the distinction between being uniquely determined among mixed states and being uniquely determined among pure states in making further progress in this direction.
Determining whether a mixed quantum state is uniquely determined among all states by its k-body marginals (k-UDA) is a fundamental problem in quantum system certification. We develop a range-based approach to this problem by analyzing the structure of the range of the global state. For three-qubit states, we show that states with GHZ-SLOCC-free ranges are 2-UDA at ranks one, three, and four. We derive a necessary and sufficient range criterion for rank-two 2-UDA states and reduce it to a finite quadratic-form test. To cover the remaining range configurations, we formulate an exact range-restricted semidefinite programming criterion and extend it to arbitrary finite-dimensional tripartite states. We also show that every three-qubit state of rank at least five is not 2-UDA, and further extend high-rank obstructions to multipartite systems. For a channel-based multipartite family, we characterize exactly when a state is (n−1)(n-1)(n−1)-UDA and show that lower-order marginals never suffice. Finally, we apply these results to the certification of genuine multipartite entanglement.
We discuss the uniqueness of quantum states compatible with given measurement results for a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it was known that for a ddd-dimensional Hilbert space, there exists a set of $4d\ensuremath{-}5$ observables that uniquely determines any pure state. We show that for case (2), $5d\ensuremath{-}7$ observables suffice to uniquely determine any pure state. Thus, there is a gap between the results for (1) and (2), and we give some examples to illustrate this. Unique determination of a pure state by its reduced density matrices (RDMs), a special case of determination by observables, is also discussed. We improve the best-known bound on local dimensions in which almost all pure states are uniquely determined by their RDMs for case (2). We further discuss circumstances where (1) can imply (2). We use convexity of the numerical range of operators to show that when only two observables are measured, (1) always implies (2). More generally, if there is a compact group of symmetries of the state space which has the span of the observables measured as the set of fixed points, then (1) implies (2). We analyze the possible dimensions for the span of such observables. Our results extend naturally to the case of low-rank quantum states.
The correlations between local measurement results are characteristic for multi-partite quantum systems. Under the condition of local realism, which assumes that properties of physical systems exist independent of being measured, and that space-like separated actions on the system cannot influence each other, these correlations are constrained by Bell-inequalities. Quantum mechanics predicts correlations which are not explainable under these two assumptions. The resulting violation of Bell inequalities was originally studied for pairs of entangled particles (Einstein et al.,1935). Meanwhile these ideas were extended to three-and more partite quantum states in particular with respect to the different kinds of multi-particle entanglement and their implications on the obtainable correlations (Mermin, 1990). Here we present a method to construct characteristic Bell inequalities for particular four-qubit states. These inequalities are a versatile tool, not only to provide evidence for non-locality, but also for the characterization of the states, with respect to detection of genuine multipartite entanglement and to state discrimination. Exemplarily we apply our approach to the symmetric four qubit Dicke-state (Dicke, 1954) with two excitations. (A symmetric N-qubit Dicke state with M excitations is the equal superposition of N-qubit pure product states having M logical 1 's and (N-M) logical 0's.) Experimentally the state was observed with photons as carrier of the qubits, using polarization encoding.
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 $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$.