PHYSICAL REVIEW A 90, 012317 (2014)
Determination of W states equivalent under stochastic local operations and classical communication
by their bipartite reduced density matrices with tree form
Xia Wu,
1,2
Guo-Jing Tian,
1
Wei Huang,
1
Qiao-Yan Wen,
1
Su-Juan Qin,
1
and Fei Gao
1,*
1
State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
2
State Key Laboratory of Integrated Services Network, Xidian University, Xian 710071, China
(Received 14 April 2014; published 14 July 2014)
It has been known that, among pure states, N-qubit W states cannot be uniquely determined by their arbitrary
(N − 1) bipartite reduced density matrices. Parashar and Rana proved that among arbitrary states, (N − 1)
bipartite reduced density matrices that the pairs of qubits constitute a star graph or a line graph can uniquely
determine stochastic local operations and classical communication (SLOCC) equivalent W states, and we
generalize this conclusion into tree graph. In this paper, we show that all SLOCC equivalent W states can
be uniquely determined (among pure, mixed states) by their (N − 1) bipartite reduced density matrices, if the
(N − 1) pairs of qubits constitute a tree graph on N vertices, where each pair of qubits represents an edge.
DOI: 10.1103/PhysRevA.90.012317 PACS number(s): 03.67.Mn, 03.65.Ud
I. INTRODUCTION
The understanding of entanglement is very important in
quantum information theory. One of the open questions in
quantum entanglement is to understand the different ways
in which multipartite systems can be entangled. Generally,
local operations and classical communication (LOCC) [1] can
transform one entangled state into another, that is to say LOCC
can change the entanglement properties of the system.
It is natural to define equivalence classes of states generated
by some local equivalence relation. The most general local
equivalence relations are equivalence under stochastic local
operations and classical communication (SLOCC equivalence)
[1–3], and local unitary (LU) equivalence [4]. Two states are
called SLOCC equivalent if they can be obtained from each
other by means of SLOCC. In this paper, we are mainly
concerned with the SLOCC equivalent states of the stan-
dard multiqubit W state which is |W
N
=
1
√
N
(|10,...,0+
|01,...,0+···+|00,...,1).
One property of entanglement is to judge whether a state is
determined by its reduced density matrices (RDMs), and this
is called the “parts and whole” problem. For concreteness, it
means whether lower-order correlations [ the set of K(K
N − 1) parties RDMs] can determine higher-order correlations
(full number of parties, i.e., the whole state). This was first
proposed in [5] by Linden et al., and they proved that two-
particle RDMs can uniquely determine a generic three-qubit
pure state. It was extended to tripartite systems by Diosi in
[6], and Diosi showed that almost all three-party pure states
can be uniquely determined by any two of the two-party
reduced states. Linden and Wootters also proved that for
generic N-qubit pure states, reduced density matrices of less
than about
2N
3
can uniquely specify the full quantum states
in [7]. Furthermore, it was shown in [8] that almost every
N-party pure state (the subsystems are equidimensional) is
determined by its RDMs of just over
N
2
parties. However, there
are no general and effective methods to judge the smallest K
of an arbitrary N -party pure s tate |ψ, where |ψ is uniquely
determined by its K-partite RDMs. Till now, many efforts have
*
gaofei_bupt@hotmail.com
been made by studying some special states, such as the N-qubit
|GHZ
N
state and |W
N
state. Walck and Lyons proved that
|GHZ
N
and its LU equivalent states are not determined by
their (N − 1)-particle RDMs, and they are the only states that
their RDMs cannot determine the whole states [9,10].
For the |W
N
state, Parashar and Rana have got some
interesting results. It was shown in [11] that among arbi-
trary states, N-qubit W -type states (|W
T
N
= a
1
|10,...,0+
a
2
|01,...,0+···+a
N
|00,...,1) are uniquely determined
by their bipartite RDMs, but when only pure states are consid-
ered, (N − 1) of them are sufficient. Furthermore, in [12], the
authors discussed the states which are SLOCC equivalent to
the |W
N
state (denoted by |GW ). They proved that among
arbitrary states, all |GW states are uniquely determined by
their bipartite RDMs ρ
1K
GW
, where K = 2,3,...,N. As we can
see, the index set (pairs of qubits) of ρ
1K
GW
is {12,13,...,1N}
which can be viewed as a mathematical star graph (a graph
is a collection of vertices and a description of which vertices
are connected by an edge [13]). Next, Rana and Parashar also
got that |GW states are also uniquely determined by other
bipartite RDMs, which are ρ
K(K+1)
GW
, K = 1,2,...,N − 1. In
this case, the index set {12,23,...,(N − 1)N } can be viewed
as a line graph. In [12], they also found that (N − 1) is the
optimal number, i.e., |GW cannot be determined by a lesser
number of bipartite RDMs.
Nevertheless, a question was asked in the end of paper [11],
that is whether |GW can be uniquely determined by arbitrary
(N − 1) bipartite RDMs. The answer is no, and the authors
gave a counterexample. Hence, a natural question is what kind
of bipartite RDMs can uniquely determine |GW states? As
we know, among arbitrary states, |GW states are uniquely
determined by their (N − 1) bipartite RDMs whose index is
set corresponding to a star graph or a line graph, and (N −1) is
the optimal number. It is easy to see that N vertices star graphs
and line graphs are parts of tree graphs, and there are (N −d1)
edges in arbitrary tree graphs, similarly. Therefore, it can
be conjectured that |GW states can be uniquely determined
(among pure, mixed states) by their (N − 1) bipartite RDMs
whose index is set corresponding to a tree graph. In this paper,
we devote ourselves to proving that the conjecture is true.
The paper is organized as follows. In Sec. II, we briefly
give the definition of a mathematical graph and its properties.
1050-2947/2014/90(1)/012317(8) 012317-1 ©2014 American Physical Society