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GENERIC DISSIPATION OF ENTANGLEMENT Allan Solomon Open University, UK and Paris University VI, France Received: date / Revised version: date Abstract. We find states for a multi-level system which are stable under a very general model of dissipation, one which is governed simply by generic rate parameters; in general such stable states are not entangled. We exhibit such a state explicitly for a two-qubit system. We then specialize to a more physical model of dissipation, one which is governed by pure dephasing. In such a case it is possible, by choice of the dephasing rates, to have a stable, and limiting, entangled state under the evolution governed by the free hamiltonian and pure decoherence. We exhibit such a choice explicitly which has a stable and limiting two-qubit state of maximum entanglement (Bell state). PACS. 03.67.Bg Entanglement production and manipulation – 03.65.Yz Decoherence; open systems

1 Introduction Entanglement, which according to Schrodinger[1] is the characteristic trait of quantum mechanics, seems to be an increasingly relevant resource in Quantum Information. An insight into the effects of the environment on this property is therefore important. Although many authors have discussed the effect of specific models of dissipation (by which we include both population relaxation and decoherence), it is our intention here to note some results which do not depend on any particular model of dissipation, other than it should satisfy a Liouville-type equation. These conclusions should therefore remain valid for both stochastic and non-stochastic models. We begin by finding those states which remain stable under such a generic model of dissipation. We show that for general dissipation processes involving relaxation (depopulation) and decoherence rates there always exists a stable (i.e. non-dissipating) physical state, which in general will not be entangled. In more specialized models stable entangled physical states are possible. We show that in a decoherence-only system, at least one choice of dephasing parameters, which in fact corresponds to a physical process, leads to a stable entangled state.

2 Generic Model for Dissipation Processes

quantum state, the Schr¨odinger equation1 i

d ρ(t) = [H, ρ(t)] ≡ Hρ(t) − ρ(t)H dt

(1)

where H is the total hamiltonian of the system. The standard form of a general dissipative process in Quantum Mechanics is governed by the Liouville equation obtained by adding a dissipation (super-)operator LD [ρ(t)]: iρ(t) ˙ = [H, ρ(t)] + iLD [ρ(t)].

(2)

In general, uncontrollable interactions of the system with its environment lead to two types of dissipation: phase decoherence (dephasing) and population relaxation. The former occurs when the interaction with the environment destroys the phase correlations between states, which leads to changes in the off-diagonal elements of the density matrix: ρ˙ kn (t) = −i([H, ρ(t)])kn − Γkn ρkn (t)

(3)

where Γkn (for k 6= n) is the dephasing rate between |ki and |ni. The latter happens, for instance, when a quantum particle in state |ni spontaneously emits a photon and moves to another quantum state |ki, which changes the populations according to X [γnk ρkk (t) − γkn ρnn (t)] ρ˙ nn (t) = −i([H, ρ(t)])nn + k6=n

The standard description of a quantum state suitable for our discussion, which will involve open systems, is by means of a density matrix ρ, a positive matrix of trace 1. For a non-dissipative process, the basic equation which determine the evolution of a hamiltonian quantum system may be written in the form of a differential equation for the

(4) where γkn ρnn is the population loss for level |ni due to transitions |ni → |ki, and γnk ρkk is the population gain caused by transitions |ki → |ni. The population relaxation rate γkn is determined by the lifetime of the state 1

we choose units in which ¯ h = 1.

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Allan Solomon: GENERIC DISSIPATION OF ENTANGLEMENT

|ni, and for multiple decay pathways, the relative probability for the transition |ni → |ki. Phase decoherence and population relaxation lead to a dissipation superoperator (represented by an N 2 × N 2 matrix) whose non-zero elements are (LD )[k;n],[k;n] = −Γkn k 6= n (LD )[n;n],[k;k] = +γP k= 6 n nk (LD )[n;n],[n;n] = − n6=k γkn

(5)

where Γkn and γkn are taken to be positive numbers, with Γkn symmetric in its indices. We have here introduced the convenient notation [m; n] = (m − 1)N + n

(6)

The N 2 × N 2 matrix superoperator LD may be thought of as acting on the N 2 -vector r obtained from ρ by r[m;n] ≡ ρmn .

(7)

The resulting vector equation is r˙ = Lr = (LH + LD )r

(8)

where LH is the anti-hermitian matrix corresponding to the hamiltonian H. The quantum Liouville equation (2) is very formal and so tells us really very little about the actual dissipation process. For example, the values of the parameters Γkn and γnk are not determined and in general will not lead to a physical process - that is, one under which the state ρ(t) remains a physical state - unless the parameters satisfy various constraints. Such constraints are a result of, for example, a stochastic dissipation theory[2]. Nevertheless, its virtue is that essentially every dissipation process will have to satisfy Eq.(2) and so results derived from its use will have great generality. We proceed to derive three such results.

2.2 Free Hamiltonian and LD We show that the dissipation super-operator LD commutes with a diagonal free hamiltonian super-operator LH0 . We first recall the standard algebraic trick applied in evaluating Liouville equations (see, for example [3]). The correspondence between ρ and r as given in Eq. (7) tells us, after some manipulation of indices, that ˜ r ρ → r ⇒ AρB → A ⊗ B

(10)

using the Kronecker product. Thus the super-operator corresponding to the hamiltonian term in Eq. (2) is ˜ H ⊗I −I ⊗H

(11)

and the corresponding super-operator for the diagonal matrix H0 = diag{h(1) , . . . h(N ) } is (LH0 )[u;v],[r;s] = −i (h(u) − h(v) )δur δvs

(12)

using the notation of Eq. (6) for which (A ⊗ B)[u;v],[r;s] = Aur Bvs .

(13)

Therefore we have [LD , LH0 ][k;n],[r;s] = −i(h(k) −h(n) −h(r) +h(s) )(LD )[k;n],[r;s] . (14) Consulting Eq. (5) we note that all the non-zero elements of LD have vanishing coefficients in Eq. (14); whence [LD , LH0 ] = 0.

2.3 Stable and limiting states of LD + LH0 2.1 Stable states of LD We show that the dissipation superoperator LD of Eq.(5) always possesses a stable eigenvector. Consider the following partial row sums R(α) of LD X R(α) = (LD )[k;k],α (α = 1 . . . N 2 ). (9) k=1...N

Once again referring to Eq. (5) we see that the only nonvanishing contributions are X X (LD )[k;k],[n;n] = γkn k=1...N

n6=k

(n6=k)

and (LD )[n;n],[n;n] = −

X

γkn

n6=k

whose sum is zero. Thus R(α) gives a vanishing row sum for each column α, and LD has vanishing determinant, and thus a zero eigenvalue. An eigenvector corresponding to a zero eigenvalue will therefore be a stable state, which is in fact a physical state (see Subsection 2.4).

Consider again, similarly to Eq.(9), the partial row sums of LD + LH0 R(α) =

X

(LD + LH0 )[k;k],α

(α = 1 . . . N 2 ).

k=1...N

(15) Referring to Eq.(12), we see that the additional terms from the free hamiltonian vanish, and so these partial row sums are again zero. This shows that the result of Subsection(2.1) above are essentially unchanged in the presence of a free hamiltonian H0 , and so LD + LH0 has a zero eigenvalue and possesses a stable eigenvector r0 . The state r0 is also a limiting state as t → ∞. Writing the solution to Eq.(8) in the free hamiltonian case as r(t) = exp (Lt)r(0) = exp (LH0 t) exp (LD t)r(0)

(16)

we note that, since the non-zero eigenvalues of LD are negative, the only remaining state for t → ∞ is the eigenstate r0 (0) of the common zero eigenvalue of LD and LH0 .

Allan Solomon: GENERIC DISSIPATION OF ENTANGLEMENT

2.4 Stable state is separable in general We now consider the entanglement properties of the stable eigenstate of Subsection 2.3. We therefore assume an N -level system, for N ≥ 4. ( The case N = 4 which corresponds to a 2-qubit system is treated in Appendix A. ) Since entanglement is not invariant under a general change of basis, we must specify the basis in which we are working. This is the standard basis of Subsection 2.2, in which the free hamiltonian H0 , which may be considered as the direct product of two free hamiltonians for a bipartite system for example, is diagonal. In this basis a diagonal density matrix ρ is a convex sum of direct products of (pure) states, and so is by definition separable. The stable eigenvector r0 may be obtained very much as the row sums of Subsection(2.1), but now as column sums. It has only N non-zero elements, and they are independent of the Γ ’s and the elements of H0 . The non-vanishing elements of r0 are positive functions of the (relaxation) γ’s only. Just as in the case of the row sums Eq.(9), the elements r0 k are non-vanishing only for k = [n; n]. From Eq.(7) this corresponds to a real diagonal positive matrix ρ0 and therefore is, when appropriately normalized, a physical state and separable. Note that this result corresponds to a generic dissipation, in a sense a worst-case scenario, and in particular applies when there are relaxation terms present. So, with no additional assumptions on the form of the dissipation superoperator LD , the resulting stable state will be separable. In more special cases one might expect to obtain stable entangled states. In particular, since the result holds good a fortiori for physical processes, such as completely positive evolution governed by the stochastic master equation of Lindblad, one may obtain entangled states, stable under physical evolution as we show in the next section.

3 Pure dephasing So far we have discussed the most general case, when in principle all relaxation and decoherence parameters may be present in the dissipation matrix. This leads to a separable stable state. However, experimentally, the relaxation time T1 for most systems is much longer than the dephasing time T2 so that we may effectively neglect the relaxation rates γ. This leads to a mathematically very simple situation, as the dissipation matrix LD0 is then diagonal. For example, for the two-qubit, 4-level case, LD0 = diag{0, −Γ12 , −Γ13 , −Γ14 , −Γ21 , 0, −Γ23 , −Γ24 , −Γ31 , −Γ32 , 0, −Γ34 , −Γ41 , −Γ42 , −Γ43 , 0}. (17) (Note that in the above Γij = Γji .) In general, the stable states for this dissipation matrix will still be separable. For special choices of the Γ ’s , this need not be the case. However, we are not at liberty to choose the decoherence parameters arbitrarily, since the requirement that the evolution be physical (that is, that ρ(t) remains a physical state) imposes constraints[4] on the values of the Γ ’s. As we shall show in Appendix B, satisfying

3

the constraints imposed by adopting a stochastic process governed by the standard local form[2] of the evolution equations still allows for a stable entangled state. Such a choice is given, for example, by taking Γ14 = Γ41 = 0 in Eq.(17). In Appendix C we show that the evolution of an eigenvector of LD0 corresponding to a zero eigenvalue gives a state with stable maximal entanglement under the combined effect of the free hamiltonian and the resulting dissipation operator.

4 Appendix A: Two-qubit example We consider a four-level (two-qubit) system with free hamiltonian H0 given by 

η1 0 0 0

  0 η2 0 0  0 0 η 0 3 

     

(18)

0 0 0 η4 and for which the associated Liouville operator LH0 is LH0 = −i diag {0, η1 − η2 , η1 − η3 , η1 − η4 , η2 − η1 , 0, η2 − η3 , η2 − η4 , η3 − η1 , η3 − η2 , 0, η3 − η4 , η4 − η1 , η4 − η2 , η4 − η3 , 0}.

(19)

The generic Liouville dissipation LD operator is a rather sparse 16 × 16 matrix (N 2 × N 2 in general) with not more than 4 (in general, N ) non-zero terms in each row or column. Since in the calculation of the zero eigenvector, only the terms corresponding to [n; n] (here 1,6,11,16) occur, we may define a reduced N × N matrix LD red which contains only these terms. 

−γ1 γ12 γ13 γ14

  γ21 −γ2 γ23 γ24  γ  31 γ32 −γ3 γ34

   .  

(20)

γ41 γ42 γ43 −γ4 where for brevity we have put γ1 γ2 γ3 γ4

= = = =

γ21 + γ31 + γ41 γ12 + γ32 + γ42 γ13 + γ23 + γ43 γ14 + γ24 + γ34 .

The eigenvector {f1 (γ), f2 (γ), f3 (γ), f4 (γ)} corresponding to the zero eigenvalue of the reduced matrix Eq.(20), supplemented by the appropriate number of zeros(12), gives a stable state of LD + LH0 . This has the form r0 = {f1 (γ), 0, 0, 0, 0, f2 (γ), 0, 0, 0, 0, f3 (γ), 0, 0, 0, 0, f4 (γ)} (21)

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Allan Solomon: GENERIC DISSIPATION OF ENTANGLEMENT

The explicit forms of the fi are not very illuminating; for example f1 is given by γ12 γ13 γ14 + γ12 γ13 γ24 + γ12 γ13 γ34 + γ12 γ23 γ14 + γ12 γ23 γ24 + γ12 γ23 γ34 + γ12 γ43 γ14 + γ12 γ43 γ24 + γ32 γ13 γ14 + γ32 γ13 γ24 + γ32 γ13 γ34 + γ32 γ43 γ14 + γ42 γ13 γ14 + γ42 γ13 γ34 + γ42 γ23 γ14 + γ42 γ43 γ14 (22) with similar expressions for the other three terms. The vector Eq.(21) corresponds to the density matrix   f1 (γ) 0 0 0 0   0 f2 (γ) 0 ρ0 =  (23) 0 0 f3 (γ) 0  0 0 0 f4 (γ) where the Pfi are assumed to be appropriately normalized so that i (fi ) = 1. This diagonal state is by definition separable.

5 Appendix B: Physical evolution Completely positive evolution of the system is guaranteed by the Lindblad form of the dissipation superoperator LD

6 Appendix C: Stable Bell state We now consider stability under the evolution described in Appendix B. We therefore choose Γ14 = Γ41 = 0 in the dissipation matrix Eq.(17). An eigenvector r0 of Eq(17) of the eigenvalue 0 in this case is given by r0 = {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1}.

(29)

The evolution of this vector under the combined free hamiltonian Eq.(19) and the pure dephasing matrix Eq.(17) is, as in Eq.(16) r0 (t) = exp (L0 t)r0 (0) = exp (LH0 t) exp (LD0 t)r0 (0) = {1, 0, 0, e−it(η1 −η4 ) , 0, 0, 0, 0, 0, 0, 0, 0, e−it(η4 −η1 ) , 0, 0, 1}.

(30)

This stable vector corresponds to the Bell state √ θ = t (η1 − η4 ) . (31) |ψB i = (1/ 2)(|0, 0i + eiθ |1, 1i) This state has maximal entanglement (of formation), equal to 1, evaluated as the Von Neumann entropy of the partial trace.

2

LD [ρ(t)] =

N ª 1 X© [Vs ρ(t), Vs† ] + [Vs , ρ(t)Vs† ] 2 s=1

(24)

where the matrices Vs are arbitrary. The standard basis for N × N matrices is given by (Eij )mn = δim δjn

(i, j, m, n = 1 . . . N )

(25)

Relabelling using our notation Eq. (6) we choose V[i;j] = a[i;j] E[i;j] .

(26)

In the pure dephasing case, comparison of Eq.(24) with Eq.(3) gives 1 (|a[i;i] |2 + |a[j;j] |2 ) (i, j = 1 . . . N ). (27) 2 For N = 4 this gives 6 pure dephasing parameters. Choosing Γ14 (= Γ41 ) = 0 gives the following parametrization of the remaining non-zero Γ s Γij =

1 |a6 |2 2 1 Γ13 = |a11 |2 2 1 Γ23 = (|a6 |2 + |a11 |2 ) 2 1 Γ24 = |a6 |2 2 1 Γ34 = |a11 |2 . (28) 2 This leads to the constrained values Γ24 = Γ12 Γ34 = Γ13 and Γ23 = Γ12 + Γ 13, thus leaving essentially two free dephasing parameters Γ12 and Γ13 . This pure dephasing system is tightly constrained, but satisfies complete positivity of evolution. In Appendix C we show that it admits a stable maximally entangled state. Γ12 =

7 Conclusions In this paper we first considered a very general scenario, which we termed a generic model, in which dissipation is governed only by population relaxation and decoherence rates, for which no assumptions are made. In general such a model is too unrestricted to always result in a physical dissipation process; a stochastic evolution governed by, for example, Lindblad’s form of the local equations, will be included, but there will be additional constraints on the dissipation terms. In spite of this generality, it is possible to show that there always exists a stable state under the action of a free hamiltonian (i.e. diagonal in the chosen basis) plus the dissipation matrix. We explicitly exhibited such a state in terms of the population relaxation rates (only on which it depends). This state will be separable in general if there are no restrictions on the dissipation parameters. We then considered the more restricted case of pure dephasing. Again, for a completely general scenario, the stable state will be separable. However, for specific choices of the dephasing parameters, a stable entangled state may result. We explicitly exhibited such a stable state of maximal entanglement (Bell state) by choosing one of the dephasing parameters equal to zero. This choice is an allowable one under the condition that the evolution be completely positive; e.g. governed by the Lindblad form; however, in this case such a choice results on constraints on the other dephasing parameters.

8 Acknowledgement The author benefitted from conversations with Dr Sonia Schirmer.

Allan Solomon: GENERIC DISSIPATION OF ENTANGLEMENT

References 1. E. Schr¨ odinger, Proceedings of the Cambridge Philosophical Society 31,(1935)555. 2. G. Lindblad, Comm. Math. Phys. 48, (1976) 119 G. Lindblad, Comm. Math. Phys. 40, (1975) 147 V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, (1976) 821 . 3. T. F. Havel, J. Math. Phys. 44,(2003)534 4. S.G. Schirmer and A.I. Solomon, Phys. Rev. A 70,(2004) 022107.

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