Physics Letters A 373 (2009) 3610–3613 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Entanglement transformation between two-qubit mixed states by LOCC ∗ De-Chao Li a,b, a School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316000, China b College of Automation, Northwestern Polytechnical University, Xi’an 710072, China article info abstract Article history: Based on a new set of entanglement monotones of two-qubit pure states, we give sufficient and Received 6 September 2008 necessary conditions that one two-qubit mixed state is transformed into another one by local operations Received in revised form 30 July 2009 and classical communication (LOCC). This result can be viewed as a generalization of Nielsen’s Accepted 30 July 2009 theorem Nielsen (1999) [1]. However, we find that it is more difficult to manipulate the entanglement Availableonline6August2009 transformation between single copy of two-qubit mixed states than to do between single copy of two- Communicated by P.R. Holland qubit pure ones. PACS: © 2009 Elsevier B.V. All rights reserved. 03.67.-a 03.67.Hk 03.65.Ud Keywords: Transformation Mixed state Entanglement 1. Introduction ory [4]. Obviously, for a pure bipartite state the entanglement is completely determined by its Schmidt coefficients according to the Entanglement has important applications in quantum informa- Schmidt decomposition [5]. Therefore, the Von Neumann entropy tion theory [2]. In particular, shared bipartite entanglement is an S(ρ) =−tr(ρ log ρ) has been accepted as a canonical measure of essential resource for processing and transmitting quantum infor- pure bipartite states entanglement. mation. Since most applications of quantum information theory re- However, in practical application people would have to deal quire the maximally entangled state in order to faithfully transmit with mixed states rather than pure ones due to decoherence. quantum information, it is vital necessary to develop the technique Hence, it is extremely important to quantify mixed states entan- of entanglement manipulation by LOCC to produce states with the glement for quantum information processing. This begs the ques- amount of entanglement as great as possible from partially entan- tion: Is majorization a suitable tool for transformation from one gled states [3]. There has been much attention recently concerning mixed state into another one by LOCC yet? If it is not true, what LOCC entanglement manipulation of a single copy of pure states. Nielsen is the condition of ρ −→ σ ?Afunction f : Rd → R is said to provided the necessary and sufficient conditions for pure bipar- be Schur-convex if x ≺ y ⇒ f (x) f (y) [6]. According to this tite states entanglement transformation by LOCC [1]. According to definition the Von Neumann entropy is Schur-concave, that is, Nielsen’s theorem the transformation between pure bipartite states λρ ≺ λσ ⇒ S(ρ) S(σ ). However, the Schumucher noiseless cod- LOCC |φ −→ | ψ can be performed by LOCC if and only if λφ ≺ λψ , ing theorem for quantum information [7] implies that the maxi- where λφ denotes the vector of Schmidt coefficients of trA (|φφ|) mum rate of error-free quantum information transmission is S(ρ) arranged in decreasing order. This result has been extended to qubits per signal. If S(ρ) S(σ ) holds, then the coding theorem LOCC will be violated clearly [8].Sowehavetodevelopothermeasures the case where the probabilistic transformation |φ −→ { p , |ψ } i i to quantify the entanglement of bipartite mixed states. Gour con- can be accomplished iff λ ≺ p λ and therefore majoriza- φ i i ψi sidered an essentially different type of measure of entanglement tion has received renewed attention in quantum information the- for two-qubit√ quantum√ systems [9]. For any two-qubit pure state |ϕ= λ0|00+ λ1|11 the author defined the entanglement x μ for x μ * Correspondence address: School of Mathematics, Physics and Information Sci- as follows: Eμ(|ϕ) = fμ(x) = with ∀0 < μ 1, ence, Zhejiang Ocean University, Zhoushan 316000, China. Tel.: +86 05808180913. 1forx > μ E-mail address: [email protected]. where x = 2min{λ0,λ1}. Based on this definition, Gour gave nec- 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.07.089 D.-C. Li / Physics Letters A 373 (2009) 3610–3613 3611 essary and sufficient conditions for entanglement transformation Definition 2.2. The entanglement monotones of the mixed state ρ between two probability distribution of two-qubit pure states by is defined as the average entanglement of the pure states of the LOCC. However, this result cannot be viewed as the entanglement decomposition, minimized over all decompositions of ρ: transformation between two-qubit mixed states by LOCC. For ex- + + − − + + = ∀ ∈ ] ample, let = 1 |Φ Φ |+ 1 |Φ Φ | and =|Φ Φ | with Ek(ρ) min pi Ek(ψi), k (0, 1 . (3) ρ 2 2 σ {p ,|ψ } ± | ±| i i |Φ = 00√ 11 . It is well known that ρ cannot be transformed 2 to σ by LOCC, although they still meet the condition mentioned Definition 2.3. (See [11].) The concurrence of two-qubit mixed in Ref. [9]. In this Letter we first define a new set of entangle- state ρ is defined as ment monotones of two-qubit pure states. The necessary and suf- = ficient conditions for entanglement transformation between two- C(ρ) min pi C(ψi). (4) {pi ,|ψi } qubit mixed states by LOCC is then represented. Following the Wootters’ results [11], we can obtain that C(ρ) = 2. The condition for entanglement transformation between max{0,λ1 − λ2 − λ3 − λ4}, where the λi s are the square roots of ∗ two-qubit mixed states the eigenvalues of ρ(σ2 ⊗ σ2)ρ (σ2 ⊗ σ2) in decreasing order. Fur- thermore, for any mixed state ρ there always exists a pure-state We first briefly summarize some basic concepts and results ensemble with at most four states such that the following equa- that are needed for further treatment. An ensemble of pure states, tion holds: { | } 4 which is usually represented bypi, ψi , is characterized by a fi- = = nite set of positive numbers pi ( i pi 1) and a corresponding set C(ρ) pi C(ψi). (5) of normalized vectors |ψi of the Hilbert space H [10]. The density i=1 operator ρ (a trace one, semi-definite positive operator) associated By Eq. (5),wehave to {pi , |ψi} is defined as: Proposition 2.4. ρ = pi|ψiψi|, pi = 1, pi 0. (1) i=1 i=1 1 − 1 − C 2(ρ) Ek(ρ) = ∧ 1. (6) 2k Following the ideas presented in Ref. [9],wefirstdiscusstheen- { ;| } −→LOCC tanglement transformation between two ensembles pi ψi Proof. Note that there exists an ensemble {pi ;|ψi} such that √ {q ;|φ }. Suppose Alice and Bob share a pure state |ψ,andthen − − 4 2 j j 4 1− 1−C 2(ρ) 1 1 ( = pi C(ψi )) C = p C .So ∧ 1 = i 1 ∧ perform a quantum operation ε which outputs the pure states (ρ) i=1 √i (ψi ) 2k 2k − − 2 |ψi (i = 1, 2,...,n) with probability pi . We then consider that 4 1 1 C (ψi ) 4 1 = pi ∧ 1 = pi E(ψi ), where we use the Alice and Bob obtain an ensemble {p , |ψ }. Alice and Bob per- i 1 2k √ i i i − − 2 convex of function f x = 1 1 x ∧ 1 in the second inequality. form again quantum operations ε which output the pure states √( ) 2k i |φ with conditional probability p for all the possible out- 1− 1−C 2(ρ) j ji This means that ∧ 1 is a lower bound of p E(ψ ). | | = | | 2k √ i i i come states, that is, ε ( ψi ψi ) j p ji φ j φ j . Thus, the trans- i 1− 1−C 2(ρ) = Hence, we get E(ρ) ∧ 1. formation ε (ε1, ε2,...,εn), defined between the ensembles 2k √ { | } { | } | = − ∧ pi, ψi and q j, φ j , outputs the statesφ j with probability On the other hand, i pi C(ψ√i ) 2k i pi (1 E(ψi))E(ψi ) 1. = | | = | | = − ∧ q j i pi p ji, that is,ε ( i pi ψi ψi ) i, j pi p ji φ j φ j with Define a function g(x) 2k (1 x)x 1. It is easy to see = = that g(x) is concave and monotonically increasing on [0, 1 ].We q j i pi p ji, where j p ji 1. 2 | | = − ∧ √ We write√ an arbitrary two-qubit pure state ψ as ψ therefore have pi C(ψi ) 2k (1 i pi E(ψi ))( ipi E(ψi )) 1. x|00+ 1 − x|11 with x 1 according to Schmidt decompo- 2 Since g is monotonically increasing, min{pi ,|ψi } pi C(ψi ) sition. Firstly, we define a distinct measure of entanglement of { | } = f (min√ pi , ψi pi E(ψi )) f (E(ρ)) holds. Hence E(ρ) two-qubit pure state as follows: 1− 1−C 2(ρ) ∧ 2 2k 1. Definition 2.1. Let x be the minimal Schmidt coefficients of Remark 2. An ensemble that achieves the minimum in Eq. (4) is | | | trA ( ψ ψ ). We define entanglement of two-qubit pure state ψ referred to an optimal one in this Letter. as: Next we consider the entanglement transformation between = x ∧ ∀ ∈ ] Ek(ψ) 1, k (0, 1 , (2) two-qubit mixed states by LOCC. Let {p , |ψ } and {q ;|φ } be k i i j j two optimal ensembles of two-qubit mixed states ρ and σ ,re- where ∧ is the min operator. Especially, E0(ψ) = limk→0 Ek(ψ) = 1. spectively. We say −→LOCC { ;| } −→LOCC { ;| } Remark 1. It is easy to see that Ek(ψ) mentioned above is indeed Definition 2.5.