Boundary of the Set of Separable States

Transcription

1 Boundary of the Set of Separale States Mingjun Shi, Jiangfeng Du Laoratory of Quantum Communication and Quantum Computation, Department of Modern Physics, University of Science and Technology of China, Hefei, , P.R.China arxiv:quant-ph/ v1 5 Mar 2001 (Novemer 9, 2018) Typeset using REVTEX shmj@ustc.edu.cn djf@ustc.edu.cn 1

2 Astract The set of all separale quantum states is compact and convex. We focus on the two-quit quanum system and study the oundary of the set. Then we give the criterion to determine whether a separale state is on the oundary. Some straightforward geometrical interpretations for entanglement are ased on the concept and presented susequently. PACS: Bz. I. INTRODUCTION Entanglement and nonlocality are some of the most emlematic concepts emodied in quantum mechanics [1,2]. Recent developments in quantum computation [3,4] and quantum information (e.g. [5 7]) evoke interest in the properties of quantum entanglement. Now we have known that a state of a composite quantum system is called entangled (or inseparale) if it cannot e represented as a convex comination of tensor products of its susystem states[4,8]. In the case of pure states, one has otained a complete understanding, that is, a pure state is entangled if and only if it violates Bell s inequality [9 11]. But the case for mixed states is much more complicated (e.g. [12,13]). Besides the aove definition of entangled state, another qualitative criterion proposed y Peres [14] is more practicale to determine whether a given state is inseparale, although it is sufficient and necessary only for sone simple quantum system, such as 2 2 and 2 3 systems. On the other hand, one is more interested in the quantitative measure of entangled states. Various entanglement measure has een proposed. For example, for two-party system, there are following kinds of measure: the entanglement of formation [4], the relative entropy of entanglement [15,16], and the roustness of entanglement [17], etc. Consider the relative entropy associated with entangled state ρ e of two-quit system. In order to otain the relative entropy of entanglement of ρ e, the crucial point is to find a separale state ρ s such that minimizes the relative entropy S(ρ e ρ s ) defined as 2

3 In general case, to find ρ s is a laorious work. { S(ρ e ρ s ) = tr ρ e ln ρ } e. (1) ρ s Then recall the roustness of entanglement. It is said that given an entangled state ρ e and a separale state ρ s there always exists a minimal real s 0 for which ρ(s) = 1 1+s (ρ e +sρ s ) (2) is separale. Such minimal s, denoted y R(ρ e ρ s ), is called roustness of ρ e relative to ρ s. Then the entanglement emodied in ρ e can e measured y the asolute roustness of ρ e, which is otained y means of finding a separale state ρ s such that R(ρ e ρ s ) is minimal. We can see that oth in relative entropy and in roustness, finding a specific separale state is of essence. Additionally, let s consider another useful tool to study entanglement, i.e., Lewenstein- Sanpera (L-S) decomposition [18]. Lewenstein et.al. have shown that any state of two-quit system can e decomposed as ρ = λρ s +(1 λ)p e, λ [0.1], (3) where ρ s is a separale state, and P e denotes a single pure entangled projector (P e = Ψ e Ψ e ), that is, P e represents a pure entangled state. It is known that there are many different L-S decomposition with varying values of λ and various P e s. Among them is the unique optimal decomposition, the one with the largest λ value in Eq. (3), where ρ = Sρ (opt) s +(1 S)P (opt) e, (4) S = max{λ} is the degree of separaility possessed y ρ. In other words, the entanglement possessed y ρ can e represented as (1 S)E(Ψ e ), where E(Ψ e ) is the entanglement of its pure state 3

4 expressed in terms of the von Neumann entropy of the reduced density matrix of either of its susystems: E(Ψ e ) = tr(ρ A logρ A ) = tr(ρ B logρ B ). (5) Here again, in the sense of maximizing or minimizing some quality, a specific separale state ρ (opt) s appears. From the aove consideration, we show that there are some specific separale states playing an important role in determining the entanglement measure. Then it is meaningful to determine the properties of these separale states. A fact provokes our attention. That is, atleast fortwo-quit system thesetofseparalestates, denotedys, iscompactandconvex [19]. Then oundary of S does exist. We argue in this paper that these special separale states mentioned aove are on the oundary of S. In the following, we give the criterion to determine whether a separale state is on the oundary. Taking this into account, we present more clear interpretations on some concepts concerning quantum entanglement. II. BOUNDARY OF S In this paper, we focus on the two-quit quantum system. The states can e represented in the Hilert space C 2 C 2. So we also call it 2 2 quantum system. We know that all 4 4 Hermitian matrices span a 16-dimensional real Hilert space, denoted y H. All density matrices of states forms a suset of H. We choose Hiler-Schimdt metric, that is, for A H, the norm is defined as A = [tr(a + A)] 1 2 = [tr(a 2 )] 1 2, and for A,B H, the distance of A and B is A B = [ tr(a B) 2] 1 2. (6) For two-quit system,a given state ρ is separale, i.e. ρ S, if and only if the partial transposition of ρ,denoted y ρ T or ρ T a (ρ T = (ρ Ta ) ), is also positive [14,19], or equivalently, the partial time-reversal of ρ, denoted y ρ, is positive. Consider an aritrary separale state ρ s S. Mixing it with the maximal separale state ρ 0 = 1I 4 4 4, we have 4

5 ρ ε = (1 ε)ρ 0 +ερ s, ε [0,1]. (7) Oviously, ρ ε S for each ε. In the geometrical view, ρ ε is on the line segment connecting ρ s and ρ 0, Then consider another form of mixture of ρ s and ρ 0, ρ ε = (1 ε )ρ 0 +ε ρ s ε > 1. (8) At this time, ρ ε is on the extension line from ρ 0 to ρ s. From the theorems in [20], we know that ρ ε can e in the interior of S, on the oundary of S, or outside of S. That depends on the values of ε. So for sufficiently large ε > 1, ρ ε may e outside of S. Let us now otain our main result, which refer to the oundary of S. THEOREM For a given separale state ρ S, if and only if either of the following conditions is satisfied, ρ is on the oundary of S, (i) there exists at least one vanishing eigenvalue for ρ ; (ii) there exists at least one vanishing eigenvalues for ρ. or in the other words, the rank of ρ or ρ is no larger than 3. Proof Note that S H and S is compact and convex and that ρ 0 is in the interior of S. If ρ is on the oundary of S, let us consider the mixture with form of Eq.(8), that is, ρ δ = (1 δ)ρ 0 +δρ, δ > 1. (9) From [20], we know that such ρ δ does not elong to S. Therefore, for a given δ > 1, there are two possiilities for ρ δ. That is, either ρ δ is positive and thus represents an entangled state, or ρ δ is non-positive and does not represent any physical state. In the former case, any point on the line segment from ρ 0 to ρ δ represents a real physical state, and particularly, all the points (except ρ ) on the line segment from ρ to ρ δ are entangled states. Let δ 1, then ρ δ ρ. Correspondingly ρ δ ρ. Under this condition, the distance etween ρ and ρ δ can e aritrarily small. Note that oth ρ and ρ are clearly positive and that ρ δ is also positive. But ρ δ is non-positive. So y connectivity of the space and continuity of δ we conclude that ρ must have vanishing eigenvalue. For the other case in which ρ δ is 5

6 non-positive and does not represent real state, similar analysis shows that ρ has at least one vanishing eigenvalue. To prove the reverse. We first consider a separale state ρ s the eigenvalues of which can e denoted y λ 1 λ 2 λ 3 λ 4 = 0, that is, ρ s has at least one vanishing eigenvalue. We assume that ρ s is in the interior of S. Due to the compactness and convexity of S and with the help of [20], we know that there always exists a ε > 1 such that the mixture ρ ε in the form ρ ε = (1 ε )ρ 0 + ε ρ (ε > 1) is also in the interior of S. The eigenvalues of ρ ε, denoted y λ i, can e easily calculated, that is, λ i = ε λ i + 1 ε, (i = 1,,4). (10) 4 Oviously, for λ 4 = 0, we have λ 4 = 1 ε 4 < 0. Then ρ ε is non-positive and can not elong to S. This result contradicts with the assumption made aove. Hence ρ s can not e in the interior and can only e onthe oundary of S, i.e. ρ s = ρ. Onthe other hand, let s consider a separale state ρ s such that ρ s has one vanishing eigenvalue, i.e. λ 1 λ 2 λ 3 > λ 4 = 0. Also assume that ρ s is in the interior of S. Then ρ ε is the interior point of S for some ε > 1. So ρ ε is necessarily positive. Denote the eigenvalues of ρ ε y λ i,i = 1, 4. Similar analysis will demonstrate that one of the eigenvalues of ρ ε, which is actually λ 4, is negative. It is contrary to the assumption. So ρ s = ρ. III. DISCUSSION Having otained the description of the oundary of S, we now return to think aout some concepts of quantum entanglement. A. Relative Entropy of Entanglement In [15,16], Vedrel et.al. propose the concept of relative entropy of entanglement and give the numerical results for some examples of two-quit entangled states. For instance, given entangled state ρ e as 6

7 ρ e = λ Φ + Φ + +(1 λ) 01 01, λ (0,1), (11) where Φ + = 1 2 ( ). The separale state which minimize the relative entropy S(ρ e ρ s ) (cf. Eq.(1)) is said to e ρ s = λ ( 1 λ 2 2 ) λ 2 ( 1 λ ) ( H.C.) 2 ( + 1 λ ) λ λ ( 1 λ ) (12) 2 2 Straightforward computation shows that ρ s has a vanishing eigenvalue. Then ρ s is on the oundary of S. This result also holds in the other examples mentioned in [15,16]. So we can say, at least for these specific cases, the relative entropy of entanglement can e achieved on the oundary of the set of separale states. It is possile to generalize this result. B. Roustness of Entanglement Recall that the roustness of entanglement proposed y Vidal et al.[17]. The geometrical interpretation has een otained in [21], in which the oundary of S is not yet explicitly defined. The discussion in this Letter can e supplementary to [21]. We can say that for a given entangled state ρ e of two-quit system, if s in Eq.(2) is minimal among all separale states ρ s, i.e. s is the asolute roustness of ρ e, then ρ s and ρ(s) in Eq.(2) are necessarily on the oundary of S. C. Lewenstein-Sanpera Decomposition Now let us give our interpretation for L-S decomposition. Recall the form of L-S decomposition represented y Eq.(3). Using Hilert-Schmidt metric defined y Eq.(6), we can express the λ in Eq. (3) as λ = ρ P e ρ s P e. (13) As shown in Fig.1, λ can e regarded as the ratio of the length of line segment AB to that of AC. With P e (point A in Fig. 1) and ρ (point B) fixed, λ reaches the maximal when ρ s 7

8 is on the oundary of S, i.e., the length of AC is minimal. In the geometrical view, this result is straightforward. So we can always rewrite Eq. (5) as ρ = ηρ +(1 η)p e, λ [0,1], (14) where ρ is on the oundary and η is the maximal λ associated with P e. Oviously,for Eq.(4) ρ (opt) s as S = max{η}. is on the oundary of S and can e written as ρ (opt) while S can e expressed In [22], Englert et al. have erred to what is involved in L-S decomposition, that is, if ρ = Sρ (opt) +(1 S)P e (opt) istheoptimaldecomposition, then(a)thestate(1+ε) 1 (ρ (opt) +εp e (opt) ) is non-separale for ε > 0; and () the state ρ (opt) +( 1 S 1)(P(opt) e P e ) is either non-positive or non-separale for each P e P(opt) e. It is said in [16] that their verification is rather complicated even in seemingly simple cases. But in our view, it is easy to understand. Now we give our explanation as follows. For conclusion (a), since the optimal decomposition of ρ has een otained, ρ (opt) necessarily on the oundary of S. Thus the convexity of S guarantees that all points on the line segment AD (cf Fig. 1) can not elong to S and without choice represent entangled states. These states are expressed as the convex sum of P (opt) e (1+ε) 1 (ρ (opt) is and ρ (opt), i.e., +εp e (opt) ),ε > 0. In addition, this conclusion also holds without the optimal condition. That is, for any decomposition determined y Eq. (14), conclusion (a) is valid. For conclusion (), let or equivalently, ρ x = ρ (opt) + 1 S S (P(opt) e P e ), (15) Sρ x +(1 S)P e = Sρ(opt) +(1 S)P (opt) e. (16) The right-handed of Eq. (16) is just the optimal decomposition of ρ. We already know that S is the largest η of all possile decomposition (cf Eq.(14)). If the mixture on the left-hand of Eq.(16) does represent ρ, the weight of ρ x must e less than S. In other words, the 8

9 left-hand of Eq.(16) can not e a real L-S decomposition aout ρ. Hence ρ x / S. Theore, either ρ x can not e positive, or can represent a non-separale state. The conclusion () can e demonstrated in Fig. 2. Of course the optimal condition is necessary in this case. So far we have discussed the oundary of the set of all separale states. Using this concept, we give straightforward explanations for some entanglement measure of the mixed states, mainly for L-S decomposition. We feel that introducing oundary may e helpful for etter understanding of entanglement. Acknowledgement 1 This project is supported y the National Nature Science Foundation of China ( and ) and the Science Foundation of USTC for Young Scientists. 9

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11 [19] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A223, 1 (1996). [20] T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970). [21] J. F. Du, M. J. Shi, X.Y. Zhou and R. D. Han, Phys. Lett. A267, 244 (2000). [22] B. Englert and N. Metwally, e-print archive: quant-ph/ Figure Caption Fig. 1 P e (point A) denotes an entangled pure state. ρ (point B) represents a given non-separale mixed state. S is the set of all separale states. ρ s (point C) is in the interior of S and ρ (point D) is on the oundary of S. Fig. 2 P (opt) e and P e are two different entangled pure states. P (opt) e and ρ (opt) construct the optimal L-S decomposition of ρ. ρ x is outside of set S. P e and ρ x does not construct the L-S decomposition of ρ. 11

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