Three qubits in a symmetric environment: Dissipatively generated asymptotic entanglement

Annals of Physics 326 (2011) 740–753

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Three qubits in a symmetric environment: Dissipatively generated asymptotic entanglement Fabio Benatti a,b,⇑, Adam Nagy c a b c

Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, 34051 Trieste, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34100 Trieste, Italy Budapest University of Technology and Economics, Muegyetem rkp.3-9, Hungary

a r t i c l e

i n f o

Article history: Received 14 July 2010 Accepted 3 September 2010 Available online 15 September 2010 Keywords: Entanglement Open quantum systems Asymptotic states

a b s t r a c t We study the asymptotic entanglement of three identical qubits under the action of a Markovian open system dynamics that does not distinguish them. We show that by adding a completely depolarized qubit to a special class of two-qubit states, by letting them reach the asymptotic state and by finally eliminating the added qubit, can provide more entanglement than by direct immersion of the two qubits within the same environment. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction A quantum system S interacting weakly with its environment E can be treated as an open quantum system: standard techniques can be applied to obtain a master equation involving the degrees of freedom of S, only. On a long time-scale determined by the weakness of the coupling to the environment, the reduced dynamics consists of a semigroup of trace-preserving completely positive maps on the states of S. These maps are fully consistent transformations of the system S states incorporating the dissipative and noisy effects due to the environment E [1,2]. An irreversible reduced dynamics typically transforms pure states into mixed states, thus spoiling fundamental quantum resources like entanglement [3]. However, the Markovian reduced dynamics resulting from suitably engineered environments may entangle an initial separable state of a bipartite quantum system immersed within it; such an entanglement can even persist asymptotically [4–12]. The properties of the asymptotic states of quantum dynamical semigroups can be studied by looking at the structure of the generator of the reduced dynamics [13–19]: some concrete applications can be found in [9,12,20–22]. ⇑ Corresponding author at: Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, 34051 Trieste, Italy. E-mail address: [email protected] (F. Benatti). 0003-4916/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.09.006

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Most of the results present in the literature concern pairs of open qubits; in the following, we will instead consider three qubits weakly interacting with a common environment such that the resulting master equation affects all possible pairs of qubits in the same way. In this case, the manifold of stationary states can partially be analytically characterized as well as the asymptotics of a particular class of initial states. We shall use the analytically obtained results to construct the following protocol: given two qubits immersed in the just depicted environment, (1) we add a third completely depolarized qubit, (2) leave the three of them to irreversibly evolve until they reach the stationary state, (3) trace away the ancilla. As an indication of the surprises set in store by higher dimensional entanglement with respect to the two-qubit case, we show that, for certain initial two-qubit states, the entanglement obtainable by such a procedure is larger than that achievable by simply letting them reach the stationary state within the bath. 2. Three open qubit model In the following we shall study an open quantum system of three qubits in weak interaction with their environment E. We shall denote by M the algebra of 8  8 matrices x 2 M 8 ðCÞ and by q 2 M the positive matrices of trace 1 that describe the states of S and by SðMÞ the convex set of all states. For sake of simplicity, we shall sometimes write q(x) for the expectation values Tr (qx); further, by x(1), y(2), ð1Þ ð2Þ ð3Þ z(3), respectively q1 ; q2 ; q3 , we will denote the local one-qubit observables x  1  1; 1  y  1; 1  1  z, respectively the one-qubit states q1  1  1; 1  q2  1 and 1  1  q3 . In the weak-coupling limit, the states of an open quantum system evolve in time according to a master equation of the form @ tqt = L[qt], where the generator L incorporates the dissipative and noisy effects due to the environment; the solutions form a semigroup of completely positive maps ct = exp (tL), t P 0 [1,2,9]. In the present case of three qubits, we shall concretely study the following time-evolution equation,

" #   3 3 X X xa ðaÞ 1 n ðbÞ ðaÞ o ðabÞ ðbÞ @ t qt ¼ i r3 ; q t þ C ij rðaÞ rj ri ; qt ¼ L½qt ; i q t rj  2 2 a¼1 a;b¼1

ð1Þ

i;j¼1 ðaÞ

ðabÞ

where ri is the ith Pauli matrix of the ath qubit, and the coefficients C ij the so-called Kossakowski matrix:

0

C ð11Þ B ð12Þ y 0 6 K ¼ @ ðC Þ ðC

ð13Þ y

Þ

C ð12Þ

C ð13Þ

ð22Þ

C

ð23Þ

C

ð33Þ

C ðC

ð23Þ y

Þ

1 C A;

h i3 ðabÞ C ðabÞ ¼ C ij

i;j¼1

form a 9  9 positive matrix,

:

ð2Þ

To the semigroup of completely positive, trace-preserving maps ct = exp (tL) on the states of S, there ^t : M # M, on the matricorresponds the semigroup of completely positive, identity-preserving maps c ces x 2 M, generated by:

"

@ t xt ¼ i

#   3 3 X X xa ðaÞ 1 n ðbÞ ðaÞ o ðabÞ ðaÞ r3 ; xt þ C ij rðbÞ x r  r r ; x ¼b L½xt : t t j i 2 j i 2 a¼1 a;b¼1

ð3Þ

i;j¼1

The analytic solution of (1) can in general be addressed only numerically; we shall instead consider a simpler class of master equations amenable to a partially analytic study. Notice that, by taking the trace of the generator L in (1) with respect to any single qubit, one gets the generator of a master equation relative to the other two qubits; thus, by choosing x1 = x2 = x3 = x and

C ð11Þ ¼ C ð22Þ ¼ C ð33Þ ¼ A;

C ð12Þ ¼ C ð21Þ ¼ C ð13Þ ¼ C ð31Þ ¼ C ð23Þ ¼ C ð32Þ ¼ B;

one obtains a highly symmetric generator with Kossakowski matrix

0

A B K ¼ @ By By

B A B

y

B

1

C B A;

A

ð4Þ

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such that any pair of qubits is affected in the same way by the presence of the environment. As apparent from (4), the matrix A governs the dissipative reduced dynamics of each one of the qubits, while B rules the dissipative statistical coupling of pairs of different qubits. However, in the following, we shall further restrict the case to master equations where one and two-qubit terms are the same and choose

0

a

B A ¼ B ¼ @ ib 0

ib 0

1

a

C 0 A;

0

c

c P 0; 0 6 jbj < a:

ð5Þ

pffiffiffi Then, the Kossakowski matrix (2) reads K = 3A  P, where P projects onto the vector ð1; 1; 1Þ= 3 and A P 0. The resulting master equation for the states (Schrödinger time-evolution) and its dual for the system operators (Heisenberg time-evolution) can conveniently be recast as

@ t qt ¼ i

@ t xt ¼ i

x 2

x 2

½S3 ; qt  þ

½S3 ; xt  þ

  1 Aij Si qt Sj  fSj Si ; qt g ¼ L½qt ; 2 i;j¼1

3 X

  1 Aij Sj xt Si  fSj Si ; xt g ¼ b L½xt ; 2 i;j¼1

3 X

ð6Þ

ð7Þ

in terms of global spin operators

S1;2;3 ¼

3 X

ð1Þ ð2Þ ð3Þ rðaÞ 1;2;3 ¼ r1;2;3 þ r1;2;3 þ r1;2;3 :

ð8Þ

a¼1

Remark 1. For three qubits (n = 3), a master equation of the form

 1 n ðaÞ ðaÞ o rj ri ; qt 2 2 a¼1 i;j¼1   n 3 X X 1 n ðbÞ ðaÞ o ðaÞ ðbÞ Bij ri qt rj  rj ri ; qt ; þ 2 a–b¼1 i;j¼1

@ t qt ¼  i

x

½S3 ; qt  þ

n X 3 X



riðaÞ qt rðaÞ j 

Aij

may have direct experimental implication in certain realizations of the driven cavity array proposed in [23]. For n = 2, the above equations have been derived in a physical scenario where the qubits are at a distance from each and immersed in a scalar Bose field in thermal equilibrium [20], while a master equation of the form

@ t qt ¼ i

x 2

½S3 ; qt  þ

3 3 X X

Aij



ðbÞ rðaÞ i q t rj 

a;b¼1 i;j¼1

 1 n ðbÞ ðaÞ o rj ri ; q t ; 2

corresponds to two qubit immersed in an environment described by a thermal, scalar Bose field when the spatial distance among the qubits is negligible [9].

3. Asymptotic states We shall start by briefly reviewing some available results about the stationary states of quantum dynamical semigroups [13–15] and about the tendency to equilibrium of open quantum systems. Let S c ¼ fq 2 SðMÞ : ct ½q ¼ q 8t P 0g denote the set of stationary states of a semigroup of tracepreserving, completely positive maps ct = exp (tL), generated by the master equation @ tqt = L[qt], and ^t ½x ¼ x 8t P 0g the set of operators (n  n matrices) invariant under the identityby Mc ¼ fx 2 M : c ^t ¼ expðt b preserving maps c LÞ generated by the dual time-evolution equation @ t xt ¼ b L½xt . It is convenient to cast the latter equation in diagonal form:

@ t xt ¼ i½H; xt  þ

3  X i¼1

V yi xt V i 

  1 y V i V i ; xt : 2

ð9Þ

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From [15] one knows that, if a full-rank stationary state q1 exists, then ^t ½x ¼ xg is a -subalgebra of M, that is c ^t ½xy  ¼ xy 1. The subset of constant matrices M c ¼ fx 2 M : c ^t ½xy x ¼ xy x for all t P 0. and also c 2. The time-average

1 b E½x ¼ lim T!þ1 T

Z

T

^t ½x dtc

ð10Þ

0

defines a conditional expectation from M onto Mc, that is a completely positive unital map such that

b E½1 ¼ 1;

b E½y1 xy2  ¼ y1 b E½xy2

8y1;2 2 Mc ; 8x 2 M:

ð11Þ

The conditional expectation b E : M # M c has a dual map defined by

Trðq b E½xÞ ¼ TrðE½qxÞ;

8q 2 SðMÞ; x 2 M:

ð12Þ

This is a completely positive, trace-preserving linear map on the state-space SðMÞ such that E½q is a stationary state and E½q ¼ q if q is a stationary state. We are interested in establishing whether, given any initial state q, it goes into an asymptotic state q1 according to

q1 ¼ t!þ1 lim ct ½q ¼ E½q:

ð13Þ

A sufficient condition can be obtained as follows: consider the subset Dc # M of x 2 M such that

b L½xy x  b L½x ¼ 0: L½xy x  xy b From (9) one immediately derives that Mc # Dc and also that x 2 Dc if and only if 3 X ð½x; V i Þy ½x; V i  ¼ 0 () ½x; V i  ¼ 0 8V i : i¼1

Thus, the subset Dc consists of x 2 M which commute with all operators Vi, namely Dc = {Vi}0 where {Vi}0 denotes the so-called commutant of the set {Vi}. The commutant is a subalgebra which need not coincide with the time-invariant -subalgebra Mc. This is however the case if the operators [15] commuting with all Vi also commute with their adjoints and with the Hamiltonian H. Indeed, if  0  0 y fV i g0 ¼ V i ; V yi ; H , then Mc # Dc #  Mc yas (9) 0 yields V i ; V i ; H # M c . 0 ^t ½x ¼ Moreover, the equality fV i g ¼ V i ; V i ; H is also sufficient [15] to guarantee that limt!þ1 c b E½x, for all x 2 M, whence (13) follows by duality. In general, that is for any number of qubits, the time-evolution equation (7) can be written as in (9) by diagonalizing the 2  2 matrix in the upper left corner of A in (5); concretely, in terms of the spin operators Si in (8),

V y1;2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S1  iS2 2ða  bÞ ; 2 

V3 ¼

pffiffiffi c S3 :

ð14Þ

0

If jbj < a and c > 0, fV i g0 ¼ V i ; V yi ; H ¼ fSi g0 so that, according to the above discussion, it follows that Mc = {Si}0 . Therefore, in the concrete cases we are considering, the time-invariant operators coincide with those commuting with all global spin operators Si in (8). In order to establish the asymptotic convergence to stationary states as in (13), we need seek full-rank stationary states: we shall do this in the following for 1, 2 and 3 qubits. 3.1. One qubit For the case of one qubit, a full-rank stationary state of the master equation

@ t qt ¼ i

x 2

½r3 ; qt  þ

3 X i;j¼1

Aij



ri qt rj 

  1 rj ri ; qt ; 2

ð15Þ

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with A = [Aij] as in (5), can be found by considering the corresponding time-evolution equation of the Bloch vector ~ rt in qt ¼ 12 ð1 þ ~ rt  ~ rÞ; that is, ~r_t ¼ 2ðL~rt  ~zÞ, where

0

aþc

B L ¼ @ x=2

x=2 aþc

0

0

0

0

1

0

1

B C ~ z ¼ @ 0 A: 2b

C 0 A; 2a

ð16Þ

z ¼ ð0; 0; b=aÞ and a unique full-rank (jbj < a) stationary state Setting ~ r_t ¼ 0, one finds ~ r 1 ¼ L1~

1 2

b a

q1 ¼ ð1 þ r1 r3 Þ; r1 ¼ :

ð17Þ

Remark 2. Consider two qubits (a = 1, 2 in (8)); one explicitly verifies (see also [9]) that   2 q2 1 ¼ q1  q1 is a full-rank stationary state for (6): L½q1  ¼ 0. The generator of (6) can be extended to the case of n qubits by extending to n the summation index P of single qubit Pauli matrices in (8); further, it can conveniently be recast as L ¼ na;b¼1 Lab where the  n sum is over generators (6) involving only the ath and bth qubit. Let q1 ¼ q1  q1 . . .  q1 ; then, |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 12 n 2 ðn2Þ Lð12Þ ½qn ¼ 0; 1  :¼ ðL11 þ L22 þ L12 þ L21 Þ½q1  ¼ L ½q1   q1

n times

where L12 is the generator in (6) for two qubits and q2 stationary state. This result 1 is a  two-qubit ðabÞ n clearly holds for all pairs (ab), that is L q1 ¼ 0, whence L qn ¼ 0 and qn 1 1 is an n-qubit full-rank stationary state. 3.2. Two qubits  0 As previously observed, fV i g0 ¼ V i ; V yi ; H ¼ fSi g0 independently of the number of qubits. In order to find the commutant {Si}0 for the case of two qubits, we use the Pauli matrices and write

M 3 x ¼ k1 þ

3 X 2 X

ðaÞ

ki

riðaÞ þ

i¼1 a¼1

3 X

kij ri  rj :

i;j¼1

Then, by imposing that [x, Sp] = 0 for all p = 1, 2, 3, {Vi}0 amounts to being the linear span of the identity P 0 matrix 1 and of the symmetric sum T ¼ 3i¼1 ri  ri . It follows that Mc = {Si} is a commutative algebra; 0 00 0 it coincides with its center, Mc ¼ Z ¼ fSi g \ fSi g ¼ M c \ M c and is generated by the two orthogonal projections

P¼

1 ð1  TÞ; 4

Q ¼1P ¼

1 ð3 þ TÞ; 4

ð18Þ

where the first one is 1-dimensional and projects onto the two-qubit singlet state

1 jWi ¼ pffiffiffi ðj0i  j1i  j1i  j0iÞ; 2

ð19Þ

with r3j0i = j0i and r3j1i = j1i.   From Remark 2, q2 1 ¼ q1  q1 is a full-rank stationary state; then, (13) ensures that the asymptotic state q1 corresponding to an initial q is obtained as E½q, by means of (12). In order to construct it, we first construct the conditional expectation b E onto the subalgebra of constant matrices; b E must be such that b E½x ¼ kðxÞP þ lðxÞQ . From the properties (11) of the conditional expectation,

b E½PxP ¼ kðxÞP;

b E½QxQ ¼ lðxÞQ ;

where, with q(x) :¼ Tr (qx),

kðxÞ ¼

TrðPq2 1 PxÞ ; q2 1 ðPÞ

lðxÞ ¼

 Tr Q q2 1 Qx : q2 1 ðQ Þ

F. Benatti, A. Nagy / Annals of Physics 326 (2011) 740–753

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Then, from (19) one gets

q2 1 P ¼

1  r 21 P; 4

b r1 ¼ ; a

ð20Þ

so that, given any initial state q, its asymptotic state q1 is given by (compare with [9])

q1 ¼ E½q ¼

4qðPÞ 4qðQ Þ 4ð1  qðPÞÞ 2 4qðPÞ  1 þ r 21 Pq2 P þ Q q2 q1 þ P: 1 Q ¼ 1  r 21 1 3 þ r21 3 þ r 21 3 þ r 21

ð21Þ

The entanglement content of any two-qubit state q is quantified by the concurrence C(q) [24]: con~ ¼ r2  r2 q r2  r2 and compute the (positive) sider the complex conjugate matrix q*, construct q 2 ~ eigenvalues ki of qq. Then, C(q) = max{0, k1  k2  k3  k4}. For all asymptotic states q1 in (21), one easily calculates

Cðq1 Þ ¼





 1 max 0; 2 4qðPÞ  1  r 21  2ð1  qðPÞÞ 1  r21 : 2 2ð3 þ r 1 Þ

ð22Þ

In [9] the entanglement capability of the environment has been studied by comparing the concurrence of certain initial states with that of their asymptotes; in the following, we shall focus upon the following one-parameter family of initial conditions

qðaÞ ¼ a1 þ ð1  4aÞP; 0 6 a 6 1=3:

ð23Þ

One easily finds that C(q(a)) = max{0, 1  6a}. Furthermore, if

3 þ r 21 ; 0 6 a < aðr1 Þ ¼

6 3  r 21

ð24Þ

where a(r1) is an increasing function of r1: 1/6 6 a(r1) 6 1/3, the corresponding asymptotic states obtained, according to (13), as

q1 ðaÞ ¼ E½qðaÞ ¼

12a 2 3 þ r 21  12a q þ P 3 þ r 21 1 3 þ r 21

ð25Þ

have concurrence

Cðq1 ðaÞÞ ¼

1 3  r 21  3a > 0: 2 3 þ r 21

ð26Þ

Otherwise, namely for a(r1) 6 a, q1(a) is separable. One can then conclude: 1. both q(a) and q1(a) are separable if

1 1 6 aðr 1 Þ 6 a 6 : 6 3

ð27Þ

2. q(a) is separable and q1(a) is entangled if

1 6 a 6 aðr1 Þ: 6

ð28Þ

3. Since a(r1) P 1/6, it follows that, when 0 6 a < 1/6, the initial state q(a) is entangled as well as q1(a); the entanglement difference

DðaÞ :¼ Cðq1 ðaÞÞ  CðqðaÞÞ ¼ 9a

1 þ r 21 1  3 þ r 21 2

ð29Þ

becomes positive (entanglement gain) if

a > a ðr1 Þ ¼

3 þ r 21 ; 18ð1 þ r21 Þ

where a*(r1) is a monotonically decreasing function of r1: 1/6 P a*(r1) P 1/9.

ð30Þ

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3.3. Three qubits As for one and two qubits, in order to fully characterize the set of asymptotic states, one needs the conditional expectation (10); differently from two qubits, in the case of three qubits its complete expression is still escaping us. Indeed, the commutant Mc is not commutative and cannot coincide with its center, M c – Z (see Appendix A); neither does Mc coincide with the commutant of its center ðM c – Z 0 Þ, which is the other case where one would immediately know how to construct the conditional expectation [13,15]. What can be analytically constructed is at least the action of E on certain subsets of initial states. In Appendix A, it is shown that the commutant set {Si}0 = Mc is the linear span of the the 3  3 identity matrix and of the following operators

SðabÞ ¼

3 X

ðbÞ rðaÞ a < b ¼ 2; 3; S ¼ i ri ;

i¼1

3 X

ð3Þ eijk rið1Þ rð2Þ j rk :

ð31Þ

i;j;k¼1

Further, the center Z surely contains the operator

T¼

3 X

Sab ¼ Sð12Þ þ Sð23Þ þ Sð13Þ :

ð32Þ

a
Also, the operators

PðabÞ ¼

1  SðabÞ 2 Mc 4

ð33Þ

are projections such that [P(ab), Si] = 0 for all i = 1, 2, 3. Using (9), given any q 2 SðMÞ, the states

qðabÞ ¼

PðabÞ qPðabÞ

ð34Þ

qðP ðabÞ Þ

are such that

L½qðabÞ  ¼ PðabÞ L½qPðabÞ ) ct ½qðabÞ  ¼ PðabÞ ct ½qP ðabÞ : Moreover, as P

ðabÞ

ct ½qab  ¼ P ðcÞ t

where q

ð35Þ

¼ jWab ihWab j1c projects onto the singlet vector state jWabi of the qubits a and b, then

ðabÞ

qtðcÞ ;

ð36Þ

is a state of the qubit c. ðcÞ

Proposition 1. The state qt evolves in time according to the master Eq. (6) ðcÞ  E½qðabÞ  ¼ P ðabÞ qðcÞ 1 , where q1 ¼ q1 in (17).

for one qubit and

ðcÞ

Proof. The time-evolution of qt is obtained by tracing over the qubits a and b the expression (36) multiplied by P(ab); by using (35) one gets:

 ðcÞ ðcÞ @ t qt ¼ Trab P ðabÞ L½PðabÞ qt  : By splitting the generator as L ¼

h

L P

ðabÞ

ðcÞ t

q

i

P3

p;q¼1 Lpq ,

one gets

h i ðcÞ ðcÞ ðcÞ ¼ ðLaa þ Lbb þ Lab þ Lba Þ½jWab ihWab jqt þ ðLac þ Lca þ Lcb þ Lbc Þ PðabÞ qt qt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} þP

ðabÞ

h

ðcÞ t

Lcc q

i

LII

:

The first contribution vanishes for it consists of the generator of the master equation (6) for two qubits acting on the projection onto the singlet state; from (21), this state is stationary and the statement follows.

F. Benatti, A. Nagy / Annals of Physics 326 (2011) 740–753

747

Since P(ab) 2 {Si}0 , the trace over the qubits a and b of the second contribution multiplied by P(ab) reads



h i ðcÞ ðcÞ Tr P ðabÞ LII ½PðabÞ qt  ¼ Tr PðabÞ LII 1ðabÞ qt :

 ðaÞ This too; indeed, all Kraus operators contribute with terms of the form Tr P ðabÞ ri or

piece vanishes,  ðbÞ which are both zero as the partial trace of P(ab) is proportional to the 2  2 identity matrix Tr P ðabÞ ri h i ðcÞ ðcÞ and the Pauli matrices are traceless. Therefore, @ t qt ¼ Lcc qt whence the result follows from the fact that Lcc is the generator in (6) for a single qubit which has q1 as full-rank stationary state. ðabÞ One can now fix the action on the projectors PðabÞ ¼ 1S4 and

P¼

3 2 X PðabÞ 3 a
ð37Þ

of the dual map E introduced in (12) which, according to (13), associates to any initial condition the asymptotic states towards which it tends when t ? +1. h 4 Corollary 1. E½P ðabÞ  ¼ 2P ðabÞ qðcÞ 1 and E½P ¼ 3

P3

a
ðabÞ

qðcÞ 1.

Proof. Set q = P(ab) in (36); then, E½P ðabÞ  ¼ limt!þ1 ct ½PðabÞ  ¼ 2P ðabÞ qðcÞ 1 . The second relation follows by the linearity of E. h ðabÞ Remark 3. Notice that while P(ab) 2 Mc and thus b E½P ðabÞ  ¼ P ðabÞ ; P 2 is not an invariant state: h ðabÞ i ðabÞ E P2 – P2 .

The last necessary tool for the applications to be discussed in the next section is the action of the map E on the projection Q ¼ 1  P 2 M c . 8 Proposition 2. E½Q  ¼ 1þr 2

1

2

q3  1r6 1

P3

a
ðabÞ



qðcÞ 1 .

Proof. Since Q 2 Mc, the properties (11) and the algebraic relations (55) applied to b E½x ¼ kðxÞ1þ P3 ðabÞ þ lðxÞS; x 2 M, give a
b E½QxQ ¼ Q b E½xQ ¼ bðxÞQ ;

bðxÞ ¼ kðxÞ þ

3 X

kab ðxÞ:

a;b¼1

 3 Using the time-invariant state q3 ¼ q3 1 ; E q1 1 , one obtains







3 

 3  3 b 3 b Tr Q q3 1 Qx ¼ Tr q1 QxQ ¼ Tr E q1 QxQ ¼ Tr q1 E½QxQ ¼ Tr q1 E½QxQ ¼ bðxÞTrðq3 1 Q Þ:

This gives bðxÞ ¼

TrðQ q3 1 QxÞ Trðq3 1 QÞ

; on the other hand, for all x 2 M,

TrðxE½QÞ ¼ Trð b E½QxQÞ ¼

 TrðQ Þ Tr xQ q3 1 Q : Trðq3 1 QÞ

Then, the result follows using that (see (20))

Pq3 1 ¼

3 3 2 X 1  r 21 X ðcÞ PðabÞ q1  q1  q1 ¼ P ðabÞ q1 ¼ q3 1 P: 3 a
In the next section, we study the following protocol: add a third completely depolarized qubit to a two-qubit initial state q(a) as in (23); let the resulting three qubit state reach equilibirum under the time-evolution governed by (6); eliminate from the asymptotic state the added third qubit.

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F. Benatti, A. Nagy / Annals of Physics 326 (2011) 740–753

We show that the resulting two-qubit state 1. can be entangled when the asymptotic state reached by the two qubits evolving alone would not, that is when a P a(r1); 2. can be more entangled than the initial state, when the asymptotic state of the two qubits evolving alone would not, namely when a 6 a*(r1); 3. the entanglement gain can be larger than D(a) > 0 when a > a*(r1). h 4. Applications We now apply the previous results to the study of the asymptotic entanglement properties of a class of three-qubit states obtained by appending to the two-qubit states (23) a third qubit in the completely depolarized state; we shall thus focus onto initial density matrices of the form

1 a 1  4a ð12Þ q123 ðaÞ ¼ qðaÞ  ¼ 1 þ P ; 0 6 a 6 1=3; 2

2

ð38Þ

2

where, according to the notation of the previous section, P  1 ¼ P ð12Þ . The corresponding asymptotic states are given by the map E : SðMÞ # SðMÞ whose action is given by Corollary 1 and Proposition 2; indeed, writing 1 ¼ P þ Q ,

E½1 ¼

3 X 8 8r2

1  q3 PðabÞ qðcÞ 1 þ 1; 2 2 1 þ r1 3 1 þ r 1 a
123 q123 ðaÞ ¼ 1 ðaÞ ¼ E½q

ð39Þ

3 X 4a 4ar21 ð12Þ ð3Þ 3 q þ PðabÞ qðcÞ q1 : 1 1 þ ð1  4aÞP 1 þ r 21 3ð1 þ r21 Þ a
ð40Þ

According to the last step of the protocol described at the end of the previous section, we trace the asymptotic states q123 1 ðaÞ with respect to the appended qubit: 123 q12 1 ðaÞ ¼ Tr3 ðq1 ðaÞÞ ¼

4a 4ar 2 þ 3ð1  4aÞð1 þ r 21 Þ 2ar2 q1  q1 þ 1 P þ 12  ð1  q1 þ q1  1Þ; 2 2 3ð1 þ r 1 Þ 1 þ r1 3 1 þ r1 ð41Þ

 where P projects onto the two-qubit singlet state. The concurrence C qð12Þ 1 ðaÞ of this two-qubit state 1 can be computed and compared with that of the asymptotic state q (a) in (30); though easy to calculate, the expression of the concurrence is not particularly inspiring and can be found in Appendix B, Eq. (58). The goal is to see whether the addition and final discarding of the added qubit may increase the asymptotic entanglement of q1(a) in (25). We start by considering the case of separable two-qubit state q(a) that cannot get asymptotically entangled by the action of the master equation (6). According to (27), this occurs for 1/6 6 a(r1) 6 a 6 1/3. Consider a third qubit prepared in the totally depolarized state and appended to the qubits 1 and 2 prepared in a state q(a) with a in the above range. According to Appendix B, by tracing the asymptotic 3-qubit state q123 1 ðaÞ over the appended qubit, the qubits 1 and 2 are entangled, that is their concurrence C(q12(a)) > 0, if either 0 6 a 6 a+(r1) where

 3 1 þ r21 pffiffiffiffiffiffiffiffiffiffiffiffi ; aþ ðr1 Þ ¼

 4 3 þ 2r 21 þ 2 dðr 1 Þ

or a ðr1 Þ 6 a 6 13 where

 3 1 þ r21 pffiffiffiffiffiffiffiffiffiffiffiffi ;  4 3 þ 2r 21  2 dðr 1 Þ

a ðr1 Þ ¼

 with dðr1 Þ ¼ ð1  r 21 Þ ð3 þ 2r 21 Þ2  9r 21 P 0. One checks that a+(r1) 6 a(r1); therefore, the first condition is incompatible with (27).

F. Benatti, A. Nagy / Annals of Physics 326 (2011) 740–753

749

0.04 0.03 0.02 0.01 0 0.35

0.3

0.25

1 0.5

0.2

0

a

r



 Fig. 1. C q12 1 ðaÞ when C(q1(a)) = C(q(a)) = 0, r = r1, a = a.

We shall then set r1 so that a ðr1 Þ 6 a 6 13 and let 0.980965 = r* 6 r1 6 1 as calculated in Appendix B, Eq. (64). If a(r1) 6 a(r1), all initial states q(a) with a > a(r1) correspond to asymptotic states q1(a) which are separable, but to reduced asymptotic states q12 1 ðaÞ in (41) that are entangled. The same occurs for a(r1) P a(r1) for initial states q(a) with a > a(r1). Therefore, there are separable states q(a) which do not get asymptotically entangled by direct immersion in the environment described by (6), but do get entangled if a third depolarized qubit is appended to them and then eliminated after reaching stationarity. This phenomenon is numerically confirmed in Fig. 1 where the concurrence of q12 1 ðaÞ is plotted for 0 6 r1 6 1 and 1/5 6 a(r1) 6 a 6 1/3. Remark 4. Admittedly, the range of favorable values of the environment dependent parameter r1 is not so large, as well as the range of separable two-qubit states q(a) that can get entangled by means of the protocol and not by direct immersion in the environment. However, the fact that such a possibility exists is an indication of what might be achievable if one could completely characterize the whole manifold of stationary three-qubit states. Also, instead of tracing away the third qubit, one could perform a less mixing operation on it in such a way that some more entanglement be localized on the remaining two qubits: preliminary results confirm this possibility, but, unfortunately, not to a sufficiently significative extent. Luckily, concerning the second two points listed at the end of Section 3.3, addition of a third completely depolarized qubit and its elimination after reaching the stationary regime, allows for a more substantial improvement on the entanglement that can be gained asymptotically. Let us consider the difference

 D1 ðaÞ :¼ C q12 1 ðaÞ  CðqðaÞÞ

ð42Þ

0.3 0.2 0.1 0 −0.1 1

0.5

0 r

0.2

0.1 a

Fig. 2. D1(a) vs r = r1, 0 6 a = a 6 a*(r1) 6 1/6.

0

750

F. Benatti, A. Nagy / Annals of Physics 326 (2011) 740–753

0.4 0.3 0.2 0.1 0 0 0.5 1 0.1

0.2

0.3

0.4

Fig. 3. D2(a) vs r = r1, a*(r1) 6 a = a 6 1/3.

in the range a < a*(r1). For these values of a, no entanglement gain can be achieved by letting the two open qubits evolve towards their stationary state; that is,

DðaÞ ¼ Cðq1 ðaÞÞ  CðqðaÞÞ 6 0: However, by adding a completely depolarized qubit and eliminating it after reaching the stationary state, one may get D1(a) > 0 as shown in Fig. 2 which exhibits the range of parameters r1 (depending on the environment) and a (labeling the initial state) for which this is possible. Next, consider the difference

D2 ðaÞ :¼ Cðq12 1 ðaÞÞ  Cðq1 ðaÞÞ

ð43Þ

in the range a > a*(r1) where two qubits present an entanglement gain, D(a) > 0. Such an entanglement gain may be increased by adding a third depolarized qubit as shown in Fig. 3. 5. Conclusions We have studied the asymptotic states of a Lindblad master equation describing the reduced dynamics of three qubits weakly coupled to an environment that affects in the same way any pair of qubits. By applying standard algebraic techniques, we could control the asymptotic states of a particular family of initial three-qubit states of which one is completely depolarized. We showed that, after eliminating the latter from the asymptotic state, the remaining two qubits may show more entanglement than the asymptotic two-qubit state achievable by direct immersion within such an environment. This phenomenon can be regarded as an asymptotic manifestation of the richer structure of irreversible entanglement generation in higher dimensional discrete systems that was observed at short times in [10]. Appendix A P ðaÞ Given the operators Si ¼ 3a¼1 ri , i = 1, 2, 3, the commutant set Mc = {Si}0 is found by expanding a generic x 2 M by means of tensor products of the Pauli matrices:

x ¼ k0 1 þ

3 X a¼1;i¼1

ð1Þ

kai

rðaÞ i þ

3 X a
ð2Þ

ðaÞ

kai;bj ri

rðbÞ j þ

3 X

ð3Þ

ð1Þ

kijk ri

ð3Þ rð2Þ j rk

ð44Þ

i;j;k¼1

and then imposing [x, Si] = 0 for all i = 1, 2, 3. By using the Pauli algebraic relations one finds the following equalities 3 X

ð1Þ

ka‘

e‘pi ¼ 0 8 i; p ¼ 1; 2; 3;

ð45Þ

‘¼1 3  X ð2Þ ð2Þ kai;b‘ e‘pj þ ka‘;bj e‘pi ¼ 0 8 a < b ¼ 2; 3; i; j; p ¼ 1; 2; 3; ‘¼1

ð46Þ

751

F. Benatti, A. Nagy / Annals of Physics 326 (2011) 740–753 3 X

 ð3Þ ð3Þ ð3Þ kij‘ e‘pk þ ki‘k e‘pj þ k‘jk e‘pi ¼ 0 8 i; j; k; p ¼ 1; 2; 3;

ð47Þ

‘¼1 ð1Þ

ð2Þ

ð2Þ

ð3Þ

whence kai ¼ 0 for all a; i ¼ 1; 2; 3; kai;bi ¼ kaj;bj for all a < b = 2, 3 and i, j = 1, 2, 3, while kijk ¼ keijk . It thus follows that the commutant set is fSi g0 ¼ f1; SðabÞ ; Sg; a; b ¼ 1; 2; 3, namely the linear span of 1 and

SðabÞ ¼

3 X

3 X

ðbÞ rðaÞ a < b ¼ 2; 3; S ¼ i ri

i¼1

eijk rið1Þ rjð2Þ rð3Þ k :

ð48Þ

i;j;k¼1

Unlike for two qubits, the commutant set is not commutative; indeed, with a, b, c different indices,

½SðabÞ ; SðacÞ  ¼ 2ieabc SðbcÞ ;

fSðabÞ ; SðacÞ g ¼ 2SðbcÞ ;

½SðabÞ ; S ¼ 4iðSðbcÞ  SðacÞ Þ;

ð49Þ

a < b;

ð50Þ

whence

T¼

3 X

Sab ¼ Sð12Þ þ Sð23Þ þ Sð13Þ ) ½T; SðabÞ  ¼ ½T; S ¼ 0;

ð51Þ

a
so that T belongs to the center Z ¼ fSi g0 \ fSi g00 ¼ M c \ M 0c . Other useful algebraic relations are as follows

ðSðabÞ Þ2 ¼ 3  2SðabÞ ;

a; b ¼ 1; 2; 3;

S2 ¼ 2ð3  TÞ:

ð52Þ

From the first relations it follows that

1  SðabÞ 2 M c ¼ fSi g0 ¼ f1; SðabÞ ; Sg; 4   3 2 X 1 1 P¼ 1  T 2 M c \ M0c PðabÞ ¼ 3 a
PðabÞ ¼

ð53Þ ð54Þ

are two-dimensional, respectively four-dimensional projections. In particular, P(ab) is the tensor product of the projection onto the singlet state of the qubits a and b with the identity matrix for the qubit c. Furthermore, the projection Q ¼ 1  P 2 M c \ M 0c fulfils

QSðabÞ ¼ Q

8a < b;

QS ¼ 0:

ð55Þ 0

Other projections commuting with Mc, that is in the commutant M c are thus all sub-projections q 6 Q for which qQ = Q = Qq, whence

qSðabÞ ¼ qQSðabÞ ¼ qQ ¼ q;

qS ¼ qQS ¼ 0:

However, unless q = Q, these projections q cannot belong also to Mc; this is proved by writing P q ¼ k1 þ 3a
0

q12 1 ðaÞ ¼

0

0

1 B B 0 B 1 þ r 21 @ 0

y

u

u

y

0C C C; 0A

0

0

x

0

0

1

xþ

ð56Þ

752

F. Benatti, A. Nagy / Annals of Physics 326 (2011) 740–753

where

a

ð1 r 1 Þð3ð1 r1 Þ þ 2r21 Þ; 3

 3 1 þ r21  4að3 þ 2r 21 Þ u¼ : 6

x ¼

y¼

3ð1 þ r 21 Þ  2að3 þ 5r21 Þ ; 6

ð57Þ ð58Þ

The concurrence of such a state is

  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 





 1 3 1 þ r 2  4a 3 þ 2r 2  2a 1  r 2 9 þ 9r 2 þ 4r4

 C q12 ð a Þ ¼ : max 0; 1 1 1 1 1 1 3 1 þ r 21

More explicitly, set dðr1 Þ ¼ 1 



12 1ð

C q

r 21

  ð3 þ 2r21 Þ2  9r21 P 0; then,

ð59Þ

 pffiffiffiffiffiffiffiffiffiffiffiffi





2a 2 3 þ 2r 21  dðr 1 Þ 3 1 þ r 21 1

 p ffiffiffiffiffiffiffiffiffiffiffiffi ; P aÞ ¼ a > a ðr Þ ¼  1 if

  1 3 3 1 þ r 21 4 3 þ 2r 21  2 dðr1 Þ 

ð60Þ pffiffiffiffiffiffiffiffiffiffiffiffi



 2a 2ð3 þ 2r21 Þ þ dðr 1 Þ

 C q12 1 ðaÞ ¼ 1  3 1 þ r 21

 3 1 þ r 21 pffiffiffiffiffiffiffiffiffiffiffiffi : if 0 6 a < aþ ðr1 Þ ¼

 4 3 þ 2r 21 þ 2 dðr1 Þ ð61Þ

The lower bound a(r1) is a decreasing function,

3 1 ¼ a ð1Þ 6 a ðr1 Þ 6 ¼ a ð0Þ; 10 2

ð62Þ

while the upper bound a+(r1) monotonically increases,

1 3 ¼ aþ ð0Þ 6 aþ ðr 1 Þ 6 ¼ aþ ð1Þ: 6 10

ð63Þ

While a+(r1) is always in the permitted range 0 6 a 6 13, it turns out that

a ðr1 Þ 6

1 3

if 0:980965 ¼ r  6 r 1 6 1;

ð64Þ

where r* is such that a-(r*) = 1/3. References [1] R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications, Lect. Notes Phys., vol. 286, Springer-Verlag, Berlin, 1987. [2] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2002. [3] D. Bruss, G. Leuchs, Lectures on Quantum Information, Wiley-Vch, 2007. [4] D. Braun, Phys. Rev. Lett. 89 (2002) 277901. [5] A. Beige et al. J. Mod. Opt. 47 (2000) 2583. [6] L. Jakobczyk, J. Phys. A 35 (2002) 6383. [7] L. Jakobczyk, J. Phys. B 43 (2010) 015502. [8] F. Benatti, R. Floreanini, M. Piani, Phys. Rev. Lett. 91 (2003) 070402. [9] F. Benatti, R. Floreanini, Int. J. Mod. Phys. B 19 (2005) 3063. [10] F. Benatti, A.M. Liguori, A. Nagy, J. Math. Phys. 49 (2008) 042103. [11] A. Isar, Open Sys. Inf. Dynam. 16 (2009) 205. [12] B. Kraus, H.P. Büchler, S. Diehl, et al. Phys. Rev. A 78 (2008) 042307. [13] A. Frigerio, Lett. Math. Phys. 2 (1977) 79. [14] H. Spohn, Lett. Math. Phys. 2 (1977) 33. [15] A. Frigerio, Comm. Math. Phys. 63 (1978) 269. [16] F. Fagnola, R. Rebolledo, Lect. Notes Math. 1882 (2006) 161.

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