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Physics Letters A 280 (2001) 7–16 www.elsevier.nl/locate/pla

Entangled versus classical quantum states Ph. Blanchard a , L. Jakóbczyk b,∗ , R. Olkiewicz b a Faculty of Physics and BiBoS, University of Bielefeld, Bielefeld, Germany b Institute of Theoretical Physics, University of Wrocław, Pl. M. Borna 9, 50-204 Wrocław, Poland

Received 20 November 2000; accepted 29 December 2000 Communicated by P.R. Holland

Abstract We discuss usefulness of the criterion of fragility with respect to a measurement-like dynamics in the description of entanglement of both pure and statistical states. 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.w; 03.65.Bz; 03.67.a Keywords: Decoherence; “Classical” states; Entanglement

1. Introduction In recent years decoherence has been widely discussed and accepted as the mechanism responsible for the emergence of classicality in a quantum measurement and the absence, in the real world, of Schrödinger-cat-like states [1–4]. The basic idea of decoherence is that classicality is an emergent property induced in quantum open systems by their environments. It is marked by the dynamical transition of the vast majority of pure states of the system to statistical mixtures. In other words, decoherence is a process of continuous interaction between the system and its environment which results in limiting the validity of the superposition principle in the Hilbert space of the system. In consequence, the appearance of environment-induced superselection rules precludes all but a particular subset of states from stable existence. A thorough analysis of the structure of environment-induced superselection rules has been presented in [5,6]. On the other hand, it singles out a subset of preferred states which behave in an effectively classical, predictable manner. Such states are distinguished by their ability to persist despite the environmental monitoring, and therefore they are the ones in which the quantum open system can be observed. They are called “classical”. However, it should be noted that they may not exist at all. In such a case the predictability sieve was introduced [7,8] (see also [9]). It is a procedure which systematically explores states of an open quantum system in order to arrange them and next put on a list, starting with the most predictable ones and ending with those, which are most affected by the environment. Clearly, the states being on the top of the list can be thought of as the most classical ones. By combining predictability with the principle expressing the fact that any superposition of two distinct * Corresponding author. Fax: 0048-71-321-44-54.

E-mail address: [email protected] (L. Jakóbczyk). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 2 6 - 3

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Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

preferred states cannot belong to the same class of stability the notion of “quasi-classical” states was introduced and discussed in [10]. Clearly, the structure of the subset of classical states depends on the induced dynamics for the reduced density matrices which are obtained by tracing out the degrees of freedom of the environment. In general, such a procedure of taking the partial trace leads to a complicated dynamics. However, in the Markovian regime, we may assume that the dynamics is described by the master equation, and so the evolution of the reduced density matrices is given by a dynamical semigroup, i.e., a trace preserving semigroup of completely positive and contractive in the trace norm superoperators. In this Letter we introduce a dynamical semigroup for a bipartite system associated to a “fuzzy” measurement of a continuous family of non-commuting observables. For such a semigroup we determine the class of states which are least affected by the environment and the set of the most unstable states deteriorating quickly to statistical mixtures. It turns out that the most stable states are separable ones, that is they are of the form |ψ ⊗ |φ, whereas the most unstable states corresponds to the most entangled states. This is in full agreement with the idea that one of the criteria for entanglement may be robustness of a state, which was put forward by Gisin and Bechmann-Pasquinucci in [11]. It was suggested there that symmetric (with respect to permutations of the subsystems) maximally entangled states are maximally fragile with respect to noise modelled by a fluctuating Hamiltonian. In the case of measurement-like interaction between the system and its environment considered here we show that the rate of decoherence of a pure state not only determines the set of maximally entangled states but may be also taken as another measure of entanglement. Finally, some remarks about entangled density matrices in the case when the component systems are two-dimensional will be given. In particular, sufficient conditions for entanglement and separability of statistical states are presented.

2. Bipartite quantum open system Suppose we have two quantum systems A and B. For simplicity we assume that both are finite-dimensional, i.e., their observables correspond to Hermitian n × n matrices. Therefore, the algebra of the joint system AB equals to Mn2 ×n2 , and statistical states of AB are represented by positive n2 × n2 matrices ρ with tr ρ = 1. If the system AB is closed, it evolves in a unitary way according to the Liouville equation ρ˙ = −i[H, ρ]

(1)

with H = H ∗ ∈ Mn2 ×n2 . However, if the system AB interacts with its environment, then a new term in (1) appears and we obtain so-called master equation ρ˙ = −i[H, ρ] + LD ρ,

(2)

where LD denotes the dissipative part of the corresponding Markov generator L. This part depends explicitly on the type of interaction between the system AB and the environment. In our model we assume that the environmental monitoring is of a measurement-like type with respect to a family of non-commuting projectors. To start with let us recall that the following transformation: Pi ρPi , ρ→ (3)

i

where i P i = I, Pi Pj = δij Pi , describes the change of a statistical state ρ due to the measurement of an observable i ai Pi with a discrete spectrum without recording the result. A dynamical semigroup associated with such a measurement is given by Tt (ρ) = e−κt ρ + 1 − e−κt P (ρ) with the generator ρ˙ = κ P (ρ) − ρ ,

(4)

Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

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where P (ρ) = i Pi ρPi and κ > 0 is the coupling constant. Clearly, this semigroup is only a mathematical idealization of the measurement process as it accomplishes transformation (3) only if t → ∞. Generalizing Eq. (4) to the case of a continuous family of non-commuting projectors we obtain that LD ρ = κ (5) dµ(x) Px ρPx − ρ , where {Px } is a family of projectors such that of projectors of the following type:

dµ(x) Px = I. In our model we assume that we have two families

{PA ⊗ IB } and {IA ⊗ PB }, where PA and PB are one-dimensional projectors associated with states of the system A and B, respectively. Therefore

dµ(n) PA (n) ⊗ IB ρ PA (n) ⊗ IB + dµ(n) IA ⊗ PB (n) ρ(IA ⊗ PB ) − 2ρ , (6) LD ρ = κ CPn−1

CPn−1

where n ∈ CPn−1 , the complex projective space, and n → PA (n) is a tautological map which assigns to a point n ∈ CPn−1 the corresponding one-dimensional projector. By dµ(n) we denote a unique U (n)-invariant measure on CPn−1 normalized in such a way that dµ(n) P (n)A (B) = IA (B) . CPn−1

Roughly speaking, by the operation ρ → (PA ⊗ IB )ρ(PA ⊗ IB ) we check if the system AB described by ρ is in the state PA of the system A and in arbitrary state of the system B. 1 2 In order to compute an explicit form of L D in formula (6) we assume for a while that n = 2. Then CP = S and for a matrix X ∈ M4×4 of the form X = i Ai ⊗ Bi we obtain that dµ(n) (PA ⊗ IB )X(PA ⊗ IB ) = dµ(n) PA (n)Ai PA (n) ⊗ Bi . S2

i

S2

Because, as was shown in [12], 1 dµ(n) PA (n)APA (n) = (A + IA ⊗ tr A), 3 S2

so

2 IA IB LD (X) = κ ⊗ (tr Ai )Bi + (tr Bi )Ai ⊗ − 2Ai ⊗ Bi 3 2 2 i

2 IA 2 IB = κ ⊗ trA (X) + trB ⊗ − 2X = κ[TrA X + TrB X − 2X]. 3 2 2 3

Here trA (trB ) : M4×4 → M2×2 denotes the partial trace with respect to system A (B), respectively, and TrA (TrB ) is the conditional expectation from M4×4 onto a subalgebra of M4×4 isomorphic to IA ⊗ M2×2 (M2×2 ⊗ IB ),

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respectively. It is clear that IA ⊗ trA (X). 2 To simplify further calculations we put κ = 3/2. Then TrA (X) =

LD (X) = TrA (X) + TrB (X) − 2X.

(7)

Because any matrix from M4×4 may be written in the form i Ai ⊗ Bi so Eq. (7) is true for all M4×4 . It may be generalized for arbitrary n > 2 and so the master Eq. (2) takes form ρ˙ = −i [H, ρ] + TrA (ρ) + TrB (ρ) − 2ρ.

(8)

It is clear that Eq. (8) generates a dynamical semigroup, say Tt , which is also contractive in the operator norm, so-called environment-induced semigroup [6]. Therefore we may arrange all pure states of the system AB with respect to the property how fast they deteriorate to density matrices. Since linear entropy Slin = tr ρ − ρ 2 is a convenient measure of the loss of purity, we take its increase for an initial state P as a basic criterion (see [10]): 1 d Slin (Tt P )t =0 . (9) 2 dt It is worth noting that for finite-dimensional systems the quantity λ always exists and its range is a closed interval [a, b] with a 0. For the semigroup Tt we have λ(P ) =

2 2 λ(P ) 2 − 2 (10) n n for any one-dimensional projector P in Mn2 ×n2 . In order to prove (10) we first calculate the explicit form of λ. Because tr(Tt P ) = 1 for all t 0, so 2−

1 d tr(Tt P )2 t =0 = − tr L(P )P = − tr LD (P )P = 2 − tr(TrA P )P − tr(TrB P )P . 2 dt Because TrA (B) is a conditional expectation, so λ(P ) = −

tr(TrA P )P = tr(TrA P )2 =

1 tr(trA P )2 n

and thus

1 1 2 2 2 tr(trA P ) + tr(trB P ) = 2 1 − tr(trA P ) λ(P ) = 2 − n n

(11)

because, by Lemma 3 in [13], tr(trA P )2 = tr(trB P )2 . Since trA P and trB P may be arbitrary n × n matrices, so 2 tr(trA P )2 + tr(trB P )2 2 n and, hence, inequalities (10) follow. It is not difficult to identify states for which λ takes its maximum and minimum Theorem 1. λ(P ) = λmin λ(P ) = λmax

P = PA ⊗ PB , IB if and only if trA P = and n

if and only if

trB P =

IA . n

Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

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Proof. It is clear that λ(P ) = λmin if and only if trA P and trB P are one-dimensional projectors. Suppose that P = |ψψ|, where |ψ is a unit vector in a Hilbert space HAB of the system AB. Since = trA P is a onedimensional projector, so there exists a unit vector w ∈ HB such that w, (trA P )w = 1. Therefore, for any orthonormal basis {vi } in HAB we have 4

4 vi ⊗ w, ψ2 = 1, vi ⊗ w, P vi ⊗ w =

i=1

what implies that ψ = is clear. ✷

4

i=1

i=1 αi vi

⊗ w = v ⊗ w, where v =

4

i=1 αi vi

∈ HA . The second statement of the theorem

It is worth noting that the the most stable states correspond to separable states. On the other hand, the most unstable states which are strongly affected by the environment are the most entangled states. Here we use commonly accepted measure of entanglement of pure states, namely the logarithmic entropy of trA P [14]: S(trA P ) = − tr(trA P ) log(trA P ). Therefore, λ (or λ − 2 + 2/n ) may serve as another measure of entanglement of pure states. If λ(P ) = λmin , then P is separable; if λ(P ) > λmin , then P is entangled; if λ(P ) = λmax , then P is a maximally entangled state. This result seems to be interesting from the quantum computing point of view, where it is essential to maintain quantum coherence in any computing system in order to fully exploit new possibilities opened by quantum mechanics. Since any real computing device cannot be completely separated from its environment, so it is a vital problem to have at least a family of stable states, so-called the noisless quantum code [15], where information could be stored for a sufficiently long time without being effected by errors. However, as we could see, some interactions can make all states unstable. A natural monitoring of a bipartite system described by the master equation (8) leads to the decoherence process for all pure states of the system. What is more, it is the subset of entangled states which is strongly affected by the interaction with the environment. Finally, by the following example we illustrate behaviour of λ for some families of pure states of the coupling of two two-dimensional systems. Example. Let P ∈ M4×4 be a one-dimensional projector. By direct calculations we get that a general form of P for which λ(P ) = 3/2, i.e., trA P = trB P = I/2, is √ √ a 1−a 2 −iθ1 a 1−a 2 −iθ2 a2 a 2 −i(θ1 +θ2 ) e e − e 2 2 2 √ 2 √ 2 2 2 a 1−a 2 −iθ2 a 1−a eiθ1 1−a i(θ1 −θ2 ) 1−a e − e 2 2 2 2 , P = P (a, θ1 , θ2 ) = √ √ 2 a 1−a 2 −iθ1 a 1−a eiθ2 1−a 2 e−i(θ1 −θ2 ) 1−a 2 − e 2 2 2 2 √ √ 1−a 2 iθ2 a 1−a 2 iθ1 a 2 i(θ1 +θ2 ) a2 −2e − 2 e − 2 e 2 where a ∈ [0, 1], θ1 , θ2 ∈ [0, 2π]. In this way we obtain a three-parameter family of maximally entangled states of the two two-dimensional systems. In particular, fixing θ2 = 0, we obtain a two-parameter family of states which contains projectors P1 = P (1, π, 0), P2 = P (1, 0, 0), P3 = P (0, 0, 0), P4 = P (0, π, 0) onto one-dimensional subspaces spanned by so-called Bell vectors e1 , e2 , e3 , e4 :

1 i 1 1 1 1 0 0 0 0 e1 = √ ⊗ ⊗ + − ⊗ ⊗ , e2 = √ , 0 0 1 1 1 1 2 0 2 0

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Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

Fig. 1. S (dotted curve) and λ as the functions of ϕ ∈ [0, 2π ].

i e3 = √ 2

1 0 0 1 ⊗ + ⊗ , 0 1 1 0

1 e4 = √ 2

1 0 0 1 ⊗ − ⊗ . 0 1 1 0

On the other hand, one can find two one-parameter families of projectors 2ϕ 0 0 0 cos 2 0 0 sin2 ϕ sin ϕ 0 0 cos2 ϕ2 0 0 0 2 Q2 (ϕ) = Q1 (ϕ) = , 0 sin ϕ 0 0 0 sin2 ϕ2 0 2 sin ϕ 2

0 0

sin2

ϕ 2

0

0

0

0 0 , 0 0

which are entangled for ϕ ∈ [0, 2π] \ {0, π, 2π} and such that Q1 (π/2) = P1 , Q1 (3π/2) = P2 , Q2 (π/2) = P3 , Q2 (3π/2) = P4 . Obviously, 2ϕ 0 cos 2 . trA Q1 (ϕ) = trA Q2 (ϕ) = trB Q1 (ϕ) = trB Q2 (ϕ) = 0 sin2 ϕ2 Then the measure of entanglement given by the logarithmic entropy is the function of ϕ: ϕ ϕ ϕ ϕ S(ϕ) = − cos2 log cos2 + sin2 log sin2 . 2 2 2 2 On the other hand, λ(ρ) is also a simple function of the parameter ϕ: 1 2 sin ϕ. 2 From the plot of functions λ and S (Fig. 1) we see that λ may be taken as another measure of entanglement of pure states. λ(ϕ) = 1 +

3. Entangled statistical states Having discussed the usefulness of the quantity λ in description of entanglement of pure states we now investigate whether some information about entanglement may be also derived from λ for density matrices. In this section, for simplicity, we restrict our considerations to the case n = 2. First, we have to generalize this measure to

Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

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act on an arbitrary statistical state ρ. It is straightforward λ(ρ) =

1 d Slin (Tt ρ)t =0 2 dt

and so

1 tr(trA ρ)2 + tr(trB ρ)2 . (12) 2 Because λ is a convex function, so now range of λ is the whole interval [0, 3/2]. Guided by the previous experience we may expect that statistical states with large values of λ are entangled ones. However, in this case the situation is not so simple and only partial information about entanglement may be obtained. To start with we show that the following inequality holds: 1 3 1 1 2 2 λ(ρ) 2 tr ρ 2 − . max tr ρ − , tr ρ − (13) 4 2 2 2 λ(ρ) = 2 tr ρ 2 −

The upper bound is clear since tr(trA ρ)2 + tr(trB ρ)2 1 for any ρ. To prove that λ(ρ) tr ρ 2 − 1/4, what is equivalent to

1 1 tr(trA ρ)2 + tr(trB ρ)2 − , (14) 2 4 notice that both sides are U (2) ⊗ U (2)-invariant. Therefore, without loss of generality we may assume that both trA ρ and trB ρ are diagonal. Hence inequality (14) may be written as tr ρ 2

1 1 (ρ11 + ρ22 )2 + (ρ33 + ρ44 )2 + (ρ11 + ρ33 )2 + (ρ22 + ρ44 )2 , 4 2 where ρii are diagonal elements of matrix ρ. Because tr ρ 2 i ρii2 , it further simplifies to tr ρ 2 +

1 (ρ11 + ρ44 )(ρ22 + ρ33 ). 4 Since i ρii = 1 and ρii 0, so (16) is always true. In a similar way the next lower bound 1 3 tr ρ 2 − 2 2 may be shown. Therefore 1 3 1 2 2 λ(ρ) max tr ρ − , tr ρ − , 4 2 2 λ(ρ)

(15)

(16)

(17)

what ends the proof of inequality (13). Moreover, it should be noted that these bounds are optimal. The comparison of numbers λ(ρ) and tr ρ 2 provides a new sufficient condition for entanglement. Theorem 2. If λ(ρ) > tr ρ 2 , then ρ is entangled. Proof. It follows directly from the property that IA ⊗ trA ρ ρ

and

trB ρ ⊗ IB ρ

for any separable ρ, see [16,17]. ✷ On the other hand, it is well known that any state ρ such that tr ρ 2 1/3 is separable. Therefore, in a twodimensional set describing possible values of tr ρ 2 and λ(ρ) for all density matrices ρ we may distinguish the

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Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

Fig. 2. Range of the function ρ → (tr ρ 2 , λ(ρ)) with A = (1/4, 0), B = (1/3, 1/6), C = (1/2, 1/2), D = (1, 3/2), E = (1, 1), F = (1/2, 1/4), G = (1/3, 1/12).

following regions, see Fig. 2. Points from the interval DE represent pure states with separable states concentrated only in point E. Any point from the triangle CDE (without the interval CE) represents an entangled state. It is worth noting that the boundary CE is optimal, that is for any point in CE there is a separable density matrix representing it. Suppose P1 is a pure state (a one-dimensional projector) in M2×2 and let a P = P1 ⊗ P1⊥ ,

Q = P1⊥ ⊗ P1 ,

ρ = xP + (1 − x)Q,

where x ∈ [0, 1]. Then tr ρ 2 = x 2 + (1 − x)2 ∈ [1/2, 1] nd λ(ρ) = tr ρ 2 . All states represented by points in the triangle ABG are separable. In the pentagon BCEFG we can find both types of density matrices, and the same point may represent simultaneously a separable and entangled state. However, points from the lower boundary of the set in Fig. 2 correspond only to separable states what gives a new sufficient condition for separability. Theorem 3. Suppose a point p = (tr ρ 2 , λ(ρ)) is in the broken line GFE. Then ρ is separable. Proof. First, notice that we may replace ρ by ρ˜ = (U1 ⊗ U2 )ρ U1∗ ⊗ U2∗ , where U1 and U2 are unitary 2 ×2 matrices, in such a way that both trA ρ˜ and trB ρ˜ are diagonal. Clearly, tr ρ˜ 2 = tr ρ and λ(ρ) ˜ = λ(ρ). Let ρ˜ = (aij )4i,j =1 . Then tr ρ˜ 2 =

4 i=1

aii2 +

i=j

|aij |2

Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

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and

2 1 tr(trA ρ) ˜ 2 + tr(trB ρ) ˜ 2 = aii + (a11 + a44 )(a22 + a33 ). 2 4

i=1

Suppose first that p ∈ GF. Then λ(ρ) ˜ = tr ρ˜ 2 − 1/4 and hence

1 1 ˜ 2 + tr(trB ρ) ˜ 2 = tr ρ˜ 2 + . tr(trA ρ) 2 4 It further implies that 4 i=1

1 aii2 + (a11 + a44 )(1 − a11 − a44 ) = tr ρ˜ 2 + . 4

Since (a11 + a44 )(1 − a11 − a44 ) 1/4 for any a11 + a44 ∈ [0, 1], so 4

aii2 = tr ρ˜ 2 ,

i=1

that is, ρ˜ is diagonal and thus separable. Hence ρ is separable, too. Suppose now that p ∈ FE. Then λ(ρ) ˜ = 3/2 tr ρ˜ 2 − 1/2 and so

1 1 1 tr(trA ρ) ˜ 2 + tr(trB ρ) ˜ 2 = tr ρ˜ 2 + . 2 2 2 Hence 4

aii2 + (a11 + a44 )(1 − a11 − a44 ) =

i=1

1 1 tr ρ˜ 2 + . 2 2

Let us denote x = a11 + a44 . Clearly, 0 x 1. We show that

1 1 2 x(1 − x) c, where c = 1 − tr ρ˜ ∈ 0, . 2 2 Suppose, on the contrary, that x(1 − x) > c, that is, √ √ 1 1 − 4c 1 1 − 4c x∈ − , + . 2 2 2 2 The most general form of diagonal elements aii is as follows: a11 = a,

a22 = b,

a33 = 1 − x − b,

a44 = x − a,

where a ∈ [0, x] and b ∈ [0, 1 − x]. Hence 4

aii2 = 2a 2 − 2xa + 2b2 − 2(1 − x)b + x 2 + (1 − x)2 x 2 + (1 − x)2 = 1 − 2x(1 − x)

i=1

and so 4 i=1

aii2 + x(1 − x) 1 − x(1 − x) < 1 − c =

1 1 tr ρ˜ 2 + , 2 2

(18)

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Ph. Blanchard et al. / Physics Letters A 280 (2001) 7–16

a contradiction. Therefore (a11 + a44 )(1 − a11 − a44 )

1 1 − tr ρ˜ 2 . 2

(19)

By (18) and (19) 2 1 1 1 tr ρ˜ 2 + 1 − tr ρ˜ 2 + aii . 2 2 2 4

4

i=1

tr ρ˜ 2 .

4

Hence Because i=1 aii2 tr ρ˜ 2 , so we have equality what implies that ρ˜ is diagonal, hence separable. Thus ρ is also a separable state. ✷ 2 i=1 aii

By the above analysis we can see that the measurement-like interaction between a bipartite system and its environment described by the master equation (8) makes all statistical states, except for a totally mixed state I/4, unstable. However, the degree of stability is not uniform. For fixed tr ρ 2 states with the smallest possible value of λ are separable. In the case tr ρ 2 = 1, i.e., for pure states, the dependence of λ on entanglement is even more transparent, and the value of λ provides a new measure of entanglement. This idea may be also applied to a larger number of systems coupled together selecting a subset of the most unstable states, and providing in this way an analytical criterion for maximal entanglement, which will be reported in a future paper.

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