Concurrence versus purity: Influence of local channels on Bell states of two qubits
Mário Ziman
1,2,3
and Vladimír Bužek
1,3,4
1
Research Center for Quantum Information, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
2
Faculty of Informatics, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic
3
Quniverse, Líščie údolie 116, 841 04 Bratislava, Slovakia
4
Abteilung für Quantenphysik, Universität Ulm, 89069 Ulm, Germany
Received 12 August 2005; published 22 November 2005
We analyze how a maximally entangled state of two qubits e.g., the singlet
s
is affected by the action of
local channels described by completely positive maps E. We analyze the concurrence and the purity of states
E
= E I
s
. Using the concurrence-versus-purity phase diagram we characterize local channels E by their
action on the singlet state
s
. We specify a region of the concurrence-versus-purity diagram that is achievable
from the singlet state via the action of unital channels. We show that even the most general including
nonunital local channels acting just on a single qubit of the original singlet state cannot generate the maxi-
mally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by
local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-
purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other
hand, the depolarization process is represented by a line that forms the upper bound of the region of maps
induced by unital maps.
DOI: 10.1103/PhysRevA.72.052325 PACS numbers: 03.67.Mn, 03.65.Ud, 03.65.Ta
I. INTRODUCTION
From two well-established properties of the
entanglement—namely, from the fact that i interactions
create entanglement and ii entanglement cannot be shared
freely monogamy 1–4—we can conclude that any nonuni-
tary evolution of a single qubit that is entangled with another
qubit is accompanied with a deterioration of the original en-
tanglement between these two qubits.
The aim of this paper is to address the question how local
actions channels affect properties of quantum states of bi-
partite systems. In particular, we will analyze in detail how
the entanglement and the purity of a two-qubit system that
has been originally prepared in a maximally entangled Bell
state depend on the action of a single-qubit channel; i.e., we
assume that one of the qubits of the original Bell pair is
affected by an environment.
We have a twofold task in front of us: First, we will ana-
lyze how local channels affect the entanglement and purity
of the original Bell state. Second, we study how the time
evolution i.e., a one-parametric subset E
t
of the set of all
completely positive maps can be represented as a one-
parametric curve in the concurrence-versus-purity “phase”
diagram. We will focus our attention on Markovian
evolutions—i.e., those one-parametric subsets of channels
for which the semigroup property E
t
E
s
= E
t+s
holds. We will
analyze in detail physical processes such as decoherence,
decay, quantum homogenization, etc., in terms of the
concurrence-versus-purity phase diagram.
Let us first define those quantities that we shall use
through the paper. The purity that measures the degree of
“mixedness” of a state that is described by the density op-
erator will be quantified by the function
P = Tr
2
, 1.1
which equals to unity for pure states and achieves its mini-
mum for maximally mixed state; i.e., for the total mixture
=1/ dI, the purity achieves the minimal value that is equal
to 1/ d.
The entanglement between two quantum systems de-
scribed by a density operator
AB
will be quantified by a
function called the tangle:
= min
=
k
q
k
k
k
q
k
S
2
k
, 1.2
where
k
denotes the projection onto a pure state
k
.
The minimum in Eq. 1.2 is taken over all pure-state
decompositions of the state while the function S
2
k
=21- PTr
B
k
is the so-called linear entropy. The quan-
tity C =
the square root of the tangle is known in
the literature as the concurrence. Wootters 5 has derived a
simple analytic formula for the concurrence of two qubits in
a state :
C = 2 max
j
-
j
j
, 1.3
where
j
are square roots of eigenvalues of the matrix R
=
y
y
*
y
y
and
*
denotes complex conjugation
of the original two-qubit density operator . From these defi-
nitions it is obvious that the entanglement and purity are
closely related quantities and that for two-qubit state they
cannot take arbitrary values.
One of the questions one can ask at this point is, which
two-qubit states are maximally entangled providing that their
purity is fixed and vice versa? This problem has been ad-
dressed in several earlier papers 6–11. In particular, Ish-
izaka and Hiroshima 6 have introduced the so-called maxi-
mally entangled mixed states MEMS’s. These are the states
that for a given value of the purity achieve the maximal
entanglement. In Ref. 7 a slightly more general problem
has been solved. The authors have shown which unitary
PHYSICAL REVIEW A 72, 052325 2005
1050-2947/2005/725/0523259/$23.00 ©2005 The American Physical Society 052325-1