Bell’s inequality and entanglement in qubits
Received: May
Bell's inequality and entanglement in qubits
Po-Yao Chang 0 1 2 5 6
Su-Kuan Chu 0 1 2 3 6
Chen-Te Ma 0 1 2 4 6
0 College Park , Maryland, 20742 , U.S.A
1 NIST/University of Maryland , USA
2 Piscataway , New Jersey, 08854 , U.S.A
3 Joint Quantum Institute and Joint Center for Quantum Information and Computer Science
4 Department of Physics and Center for Theoretical Sciences, National Taiwan University
5 Center for Materials Theory, Rutgers University
6 Taipei 10617 , Taiwan, R.O.C
We propose an alternative evaluation of quantum entanglement by measuring the maximum violation of the Bell's inequality without information of the reduced density matrix of a system. This proposal is demonstrated by bridging the maximum violation of the Bell's inequality and a concurrence of a pure state in an n-qubit system, in which one subsystem only contains one qubit and the state is a linear combination of two product states. We apply this relation to the ground states of four qubits in the Wen-Plaquette model and show that they are maximally entangled. A topological entanglement entropy of the Wen-Plaquette model could be obtained by relating the upper bound of the maximum violation of the Bell's inequality to the generalized concurrence of a pure state with respect to di erent bipartitions.
Topological Field Theories; Topological States of Matter
1 Introduction 2 3 4
Applications to the Wen-Plaquette model
Maximum violation of the six-qubit state
being the density matrix of a Hilbert space H = HA
formed experimentally by the observation of the violation of the Bell's inequality [2]. The
original theorem, proposed by John S. Bell [
3
], states that correlations between the
outcomes of di erent measurements of two separated particles must satisfy the inequality under
local realism. The violation of the constraints (the Bell's inequality) indicates the quantum
e ect of correlations or \entangledness" in quantum systems, which could be presented in
two-qubit systems theoretically [
4
]. Although the violation of the Bell's inequality may not
reveal the general structure of entanglement of a quantum state, the relation between the
entanglement, measured in terms of the concurrence [5], and the violation of the Bell's
inequality was shown in two-qubit systems [6, 7]. The generalization for higher-qubit systems
is still unclear.
In this letter, we discuss relations between the maximum violation of the Bell's
inequality of an n-qubit Bell's operator [8] and the concurrence of a pure state when the
i-th qubit operators in the Bell's operator are n
, where n is a unit vector and
are
Pauli matrices. One crucial point is that quantum entanglement needs information of the
reduced density matrix of a system, but the Bell's inequality does not. At rst glance,
this suggests that a quantitative entanglement measurement by the Bell's inequality is
di cult. Thus, bridging the maximum violation of the Bell's inequality and measures of
{ 1 {
quantum entanglement provides a huge application of an entanglement measure without a
bipartition to detect entanglement quantities.
There are various n-qubit systems exhibiting topological properties such as the toric
code model [9] and the Wen-Plaquette model [10]. One of the topological signature is
that the total quantum dimension of quasi-particles could be detected from the universal
term in the entanglement entropy [11, 12], i.e., topological entanglement entropy [13, 14].
This motivates us to apply our theorem to the Wen-Plaquette model. We
nd that the
upper bound of the maximum violation of the Bell's inequality in the Wen-Plaquette model
indicates that the ground state is maximally entangled. The use of the maximally entangled
property for a six-qubit state in the Wen-Plaquette model could be related to the topological
entanglement entropy via the maximum violation of the Bell's inequality.
2
Entanglement and maximum violation
A Bell's operator of n qubits is de ned iteratively as Bn [8]: Bn = Bn 1
12 An + A0n
+
A0n ; where An = an
and A0n = a0
n
are the operators in the n-th
qubit with an and a0n being unit vectors and
= ( x; y; z) being the Pauli matrices. The
operators 12 Bn 1 and 12 Bn0 1 act on the rest of the qubits. Notice that we choose 12 B1 = b
with b and b0 being unit vectors. It is known that for an n-qubit system,
the upper bound of the expectation value of the Bell's operator Tr( Bn)
to the violation of the Bell-CHSH inequality [2].
2 n+21 [8] leads
For a given density matrix , the maximum expectation value of a Bell's operator
is referred to as the maximum violation of the Bell's inequality. Here we demonstrate a
relation between the maximum violation of the Bell's inequality and a concurrence of a
pure state (an entanglement quantity) in an n-qubit system when the all i-th operators in
the Bell's operator are Ai and A0i for 2
i < n:
B~n = B1
A2
A3
An 2
An 1
An + A0n
+B10
A02
A03
A0n 2
A0n 1
An
A0n :
(2.1)
1
2
1
2
To proceed our derivati (...truncated)