aa r X i v : . [ qu a n t - ph ] F e b Local transformations of superpositions of entangled states
Iulia Ghiu
Centre for Advanced Quantum Physics, University of Bucharest, PO Box MG-11, R-077125, Bucharest-M˘agurele, Romania
Abstract
Suppose that we have two entangled states | φ i , | ψ i that cannot be converted to any of other two states | φ i , | ψ i by localoperations and classical communication. We analyze the possibility of locally transforming a superposition of | φ i and | ψ i into asuperposition of | φ i and | ψ i . By using the Nielsen’s theorem we find the necessary and su ffi cient conditions for this conversionto be performed. Key words: entanglement transformations, superposition
PACS:
1. Introduction
The entangled states are the main resources in many pro-cesses of quantum processing, such as quantum cryptography,quantum teleportation, quantum telecloning, superdense cod-ing or quantum computation [1]. In order to perform sometasks it is useful to manipulate the entanglement under specialconditions, namely allowing only local operations and classicalcommunication (LOCC). The method of finding the possibletransformations between bipartite entangled states by LOCCwas found by Nielsen [2] and is based on the theory of ma-jorization. Let | Ψ i = P dj = √ α j | j j i and | Φ i = P dj = p β j | j j i be two bipartite states whose Schmidt coe ffi cients are orderedin decreasing order: x ≥ x ≥ ... x d ( x = α and β ). Then | Ψ i → | Φ i by LOCC if and only if α is majorized by β , α ≺ β ,i.e. if for each k = , ..., d [2] k X j = α j ≤ k X j = β j . (1)The Nielsen’s theorem for entanglement manipulation hasmany applications in the process of the catalysis, asymptoticentanglement transformations or for the local distinguishabilityof states.A major interest in entanglement transformations has beenthe catalysis. This enables the conversion between two initiallyinconvertible entangled states assisted by a lent entangled state,which is recovered at the end of the process [3]. The Nielsen’stheorem was recently applied for finding the conditions for astate to be a general catalyst by Song et al. [4]. In a previ-ous work of us [5], we proved that bipartite and tripartite statescannot be used as catalysis states to enable local transforma-tions between the entangled states which belong to the two in-equivalent classes of three-particle states: the GHZ class andthe W class. The inequivalent classes of entangled states haverecently been investigated with a renewed interest: Chattopad-hyay and Sarkar have shown that there is an infinite number of pure entangled states with the same entanglement, but all beingincomparable to each other (i.e. being members of inequiva-lent classes) [6]. Asymptotic entanglement manipulations wereanalysed by Bowen and Datta, who considered di ff erent mea-sures for finding bounds on optimal rates of local entanglementconversions [7].The local distinguishability of orthogonal states was studiedby Horodecki et al. by proposing a method that involves purestates [8]. The key element of their method is the building of astate in a larger system with the help of a superposition insteadof a mixture, which was previously used in the scientific litera-ture. The local indistinguishability is proved by showing that atransformation is impossible under LOCC due to the Nielsen’stheorem. Recently Fan found a general approach for distin-guishing arbitrary bipartite states, i.e. not only entangled statesbut also separable ones, by LOCC [9].Another fundamental problem in quantum information the-ory is the relation between the entanglement of a given stateand the entanglement of its individual terms. The entangle-ment of superposition of states was investigated by Linden etal. , who found upper bounds on the entanglement [10]. Thevon Neumann entropy E ( ψ ) = S (Tr A | ψ ih ψ | ) was employed asa measure of entanglement in this analysis. Suppose that wehave a superposition of two states: | Γ i = α | φ i + β | ψ i . In theparticular case when | φ i and | ψ i are bi-orthogonal states, thefollowing equality holds: E ( Γ ) = | α | E ( φ ) + | β | E ( ψ ) | + h ( | α | ) , (2)where h ( x ) : = − x log x − (1 − x ) log (1 − x ) is the binaryentropy function. Many generalizations of this paper were re-cently given: lower and upper bounds on the entanglement ofsuperposition [11], the entanglement measure used is the con-currence [12, 13], the geometric measure and q-squashed en-tanglement in Ref. [14], multipartite entanglement [15, 16], su-perpositions with more than two components [17]. Despite ofthese generalizations, there are still many unsolved aspects re- Preprint submitted to Physics Letters A November 8, 2018 arding the entanglement of superpositions, one of them beingthe behavior under local manipulations.In this Letter we analyze the following scenario: we startwith two entangled states | φ i , | ψ i , which are inconvertibleto any of other two entangled states | φ i , | ψ i by LOCC. Wewant to investigate the possibility of building a superposition of | φ i and | ψ i that can be locally transformed to a superpositionof | φ i and | ψ i . It turns out that the necessary and su ffi cientconditions to make this conversion realizable involve some in-equalities which have to be satisfied by the Schmidt coe ffi cientsof | φ i , | ψ i , and | φ i .The Letter is organized as follows. In section 2, we derivethe main result, which consists of three propositions that rep-resent the conditions for the local transformation of superposi-tions of entangled states. One example is given in subsection2.2 to illustrate the application of our propositions. Finally, theconclusions are drawn in section 3.
2. Transformation of superpositions of entangled states
Suppose that we have two bipartite entangled states | φ i and | ψ i with the Schmidt number equal to 2, such that they arebi-orthogonal. Consider other two bipartite, bi-orthogonal en-tangled states with the Schmidt number equal to 2, | φ i and | ψ i , such that the first group of states cannot be transformedto any of the two states of the second group by LOCC: | φ i 6→ | φ i| φ i 6→ | ψ i| ψ i 6→ | ψ i| ψ i 6→ | φ i . (3)Let us define the two superpositions: | Γ i = √ α | φ i + p − α | ψ i ; | Γ i = √ α | φ i + p − α | ψ i . (4)We address the following question: is there α and α such thatthe transformation | Γ i → | Γ i can be performed for arbitrary | φ j i , | ψ j i , j = ,
2? And if the transformation is possible, whatconditions should the coe ffi cients α and α satisfy?It is well known that the entanglement cannot be increasedby LOCC; this means that if | Ψ i → | Φ i , then E ( Ψ ) ≥ E ( Φ ).Chattopadhyay et al. have recently proved that the entangle-ment of two comparable states d × d with d ≥ ff erent(Theorem 2 in [6]), i.e. E ( Ψ ) > E ( Φ ). Accordingly, by employ-ing the equality (2), the necessary condition for performing thetransformation | Γ i → | Γ i reads: h ( α ) + α (cid:2) E ( φ ) − E ( ψ ) (cid:3) + E ( ψ ) < h ( α ) + α (cid:2) E ( φ ) − E ( ψ ) (cid:3) + E ( ψ ) , (5)where the entanglement is given by the von Neumann entropy.Let us analyze the following example. Suppose that the twoinitial bi-orthogonal entangled states are: | φ i = r | i + r | i ; | ψ i = r | i + r | i . (6) Α f Figure 1: Illustration of the inequality (9): f ( α ) < . f ( α ) = h ( α ) − . α and the constant function 0.57017. Thesolution of the inequality is α ∈ (0 , . α ∈ (0 . ,
1) and this isthe condition for E ( Γ ) > E ( Γ ). The two final bi-orthogonal states are: | φ i = r | i + r | i ; | ψ i = r | i + r | i . (7)We can easily check by using the Nielsen’s theorem that theconditions (3) are fulfilled and that the entanglement of the fourstates is: E ( φ ) = . E ( ψ ) = . E ( φ ) = . E ( ψ ) = . . (8)Let us choose α = . We have to determine α such that theinequality (5) is verified: f ( α ) < . , (9)where f ( α ) = h ( α ) − . α . The solution of this in-equality is α ∈ (0 , . α ∈ (0 . ,
1) (see Figure 1)and this is the condition for E ( Γ ) > E ( Γ ).Let us take α = .
85. One can compute the Schmidt coef-ficients of | Γ i and | Γ i , respectively, and these are (we writethem in a decreasing order): | Γ i : 108200 ; 64200 ; 16200 ; 12200 ; | Γ i : 119200 ; 51200 ; 18200 ; 12200 . (10)Since these Schmidt coe ffi cients do not satisfy the majorizationinequalities (1), it means that the transformation cannot be per-formed by LOCC.We know that the decreasing of entanglement E ( Γ ) > E ( Γ )is not equivalent with the possibility of conversion between bi-partite entangled states [2]. This example shows that there are α and α , which satisfy the inequality (5), i.e. E ( Γ ) > E ( Γ ),and at the same time the conversion between these two states isnot realizable by LOCC.2 .1. The main result The bi-orthogonal states | φ j i and | ψ j i with the Schmidtnumber equal to 2 have the following general expressions: | φ i = p ξ | i + p − ξ | i , ξ >
12 ; | ψ i = √ η | i + p − η | i , η >
12 ; | φ i = p ξ | i + p − ξ | i , ξ >
12 ; | ψ i = √ η | i + p − η | i , η > . (11)In addition we assume that ξ j > η j , (12) j = ,
2. This is not a restriction, since if (12) is not satisfied,we can convert | i ↔ | i and | i ↔ | i by LOCC and obtainthe condition (12). Also we impose that the conditions (3) areverified and, due to the Nielsen’s theorem, these are equivalentto η > ξ . Accordingly, due to the inequalities (12), we get:12 < η < ξ < η < ξ < . (13)We show that the conversion | Γ i → | Γ i is possible forarbitrary | φ j i , | ψ j i by proving the following: • if ξ ∈ " ξ η − ξ + η , ! , then α ∈ " η − ξ + η , ξ ξ ( Proposition 1 ); • if ξ ∈ " − η − ξ − η , ξ η − ξ + η ! , then α ∈ " ξ , η − ξ + η ! ( Proposi-tion 2 ); • if ξ ∈ , − η − ξ − η ! , then α ∈ " ξ , − η − ξ − η ( Proposition 3 ).In addition we must have α ≥ α ξ ξ , which means that the firstmajorization inequality is the necessary and su ffi cient conditionto enable the transformation. Proposition 1.
Let | φ i , | ψ i , | φ i , | ψ i be the states given by (11), beingcharacterized by ξ j , η j satisfying (13). If ξ ∈ " ξ η − ξ + η , ! , thenfor α ∈ " η − ξ + η , ξ ξ and α > we have the following: √ α | φ i + p − α | ψ i → √ α | φ i + p − α | ψ i by LOCC i f and only i f α α ≤ ξ ξ . (14) Proof.
Let | Γ j i , j = , α ≥ η − ξ + η > , therefore the largest Schmidtcoe ffi cient of | Γ i is α ξ .The inequality α ≥ η − ξ + η is equivalent to α (1 − ξ ) >η (1 − α ). Hence the state | Γ i| Γ i = p α ξ | i + p α (1 − ξ ) | i + p (1 − α ) η | i + p (1 − α )(1 − η ) | i is written with the Schmidt coe ffi cients in the decreasing order: α ξ > α (1 − ξ ) > (1 − α ) η > (1 − α )(1 − η ) . (15)By using the condition α > , we obtain that α ξ is thelargest Schmidt coe ffi cient of the state | Γ i .Therefore, the conditions of the hypothesis give us a cer-tain order of the Schmidt coe ffi cients of | Γ i and the largestSchmidt coe ffi cient of | Γ i . We will prove in the following thenecessity and su ffi ciency of the equivalence (14). ′ ⇒ ′ The necessity: If the transformation | Γ i → | Γ i is possible, then the majorization inequalities (1) are satisfied.The first inequality reads α ξ ≤ α ξ , which is the conclusion. ′ ⇐ ′ The su ffi ciency: We know that α ξ ≤ α ξ . (16)This is the first majorization inequality of the Nielsen’s theo-rem. In addition, we have to prove that the other two inequal-ities (1) are satisfied. Before proceeding we have to determinethe order of the Schmidt coe ffi cients of | Γ i .Let us observe α ≥ α ξ ξ ≥ ξ η − ξ + η ξ > η − ξ + η , (17)where the last inequality is given by (A.1) and is demonstratedin the Appendix A. We have used the fact that the four pa-rameters satisfy (13). The condition given in the hypothesis ξ ≥ ξ η − ξ + η is required in order to have α ≤
1. The inequal-ity (17) reads: α (1 − ξ ) > η (1 − α ) . Therefore we have thefollowing ordered Schmidt coe ffi cients of | Γ i : α ξ > α (1 − ξ ) > (1 − α ) η > (1 − α )(1 − η ) . (18)Since we have α ≤ α ξ ξ < α , we get α ξ + α (1 − ξ ) < α ξ + α (1 − ξ ) . (19)Now we use the inequality (B.1), which is proved in the Ap-pendix B:(1 − α )(1 − η ) > (1 − α )(1 − η ) (20)or equivalently α ξ + α (1 − ξ ) + (1 − α ) η < α ξ + α (1 − ξ ) + (1 − α ) η . (21)The inequalities (16), (19), and (21) represent the majorizationinequalities (1) required by the Nielsen’s theorem, hence thetransformation | Γ i → | Γ i can be performed by LOCC. Proposition 2.
Let | φ i , | ψ i , | φ i , | ψ i be the states given by (11), beingcharacterized by ξ j , η j satisfying (13). If ξ ∈ " − η − ξ − η , ξ η − ξ + η ! ,then for α ∈ " ξ , η − ξ + η ! and α > we have the following: √ α | φ i + p − α | ψ i → √ α | φ i + p − α | ψ i by LOCC i f and only i f α α ≤ ξ ξ . (22)3 roof. Since α ∈ " ξ , η − ξ + η ! with ξ ≥ − η − ξ − η , we obtainthe following ordered Schmidt coe ffi cients of | Γ i : α ξ > (1 − α ) η > α (1 − ξ ) > (1 − α )(1 − η ) . (23)By using the condition α > , we obtain that α ξ is the largestSchmidt coe ffi cient of the state | Γ i . ′ ⇒ ′ The necessity is obvious. ′ ⇐ ′ The su ffi ciency: We use the following inequalities α ≥ α ξ ξ > α ≥ ξ > η − ξ + η , (24)where the last inequality is demonstrated in the Appendix A(A.2). Hence the ordered Schmidt coe ffi cients of | Γ i are: α ξ > α (1 − ξ ) > (1 − α ) η > (1 − α )(1 − η ) . Thefirst majorization inequality is verified. Further we start with ξ (1 − ξ ) > η (1 − ξ ) which can be written as ξ ξ > ξ − η + η ξ . (25)By using the condition α ≥ ξ , we obtain α α ≥ ξ ξ > ξ − η + η α . (26)This is equivalent to α = α ξ + α (1 − ξ ) > α ξ + η (1 − α ) . (27)With the help of the inequality (B.1) given in the Appendix B,we find that the third majorization inequality is satisfied: α ξ + (1 − α ) η + α (1 − ξ ) < α ξ + α (1 − ξ ) + (1 − α ) η . (28)The inequalities α ξ ≤ α ξ , (27), and (28) are the majoriza-tion inequalities and this leads to the fact that the conversioncan be realized by LOCC. Proposition 3.
Let | φ i , | ψ i , | φ i , | ψ i be the states given by (11), beingcharacterized by ξ j , η j satisfying (13). If ξ ∈ , − η − ξ − η ! , thenfor α ∈ " ξ , − η − ξ − η and α > we have the following: √ α | φ i + p − α | ψ i → √ α | φ i + p − α | ψ i by LOCC i f and only i f α α ≤ ξ ξ . (29) Proof.
Since α ∈ " ξ , − η − ξ − η ! with ξ < − η − ξ − η , we have: α ξ > (1 − α ) η > (1 − α )(1 − η ) > α (1 − ξ ) . (30)By using the condition α > , we obtain that α ξ is the largestSchmidt coe ffi cient of the state | Γ i . ′ ⇒ ′ The necessity is obvious. ′ ⇐ ′ The su ffi ciency: We have α > α ≥ ξ > η − ξ + η , wherethe last inequality is demonstrated in the Appendix A (A.2).Hence the ordered Schmidt coe ffi cients of | Γ i are: α ξ >α (1 − ξ ) > (1 − α ) η > (1 − α )(1 − η ) . The first majorization inequality is verified. Then we have α ξ + α (1 − ξ ) > α ξ + η (1 − α ) and this is the second majorization inequality.Further we start from the inequality ξ − ξ − η > − ξ and dueto the fact that α ≥ ξ we obtain ξ ξ > α − − ξ − η . On the otherhand we have α α ≥ ξ ξ > α − − ξ − η , (31)which leads to α (1 − ξ ) > (1 − η )(1 − α ) . This inequality isequivalent to α ξ + (1 − α ) η + (1 − η )(1 − α ) < α ξ + α (1 − ξ ) + η (1 − α ) (32)and represents the third majorization inequality. Hence the localtransformation is possible. Let us apply our result for performing the transformation be-tween the states defined at the beginning of Section 2, namelythe states of Eqs. (6) and (7): ξ = , η = , ξ = , and η = . Firstly we compute ξ η − ξ + η = . − η − ξ − η = . . We see that ξ ∈ − η − ξ − η , ξ η − ξ + η ! , therefore we apply theProposition 2. We have η − ξ + η = .
89, which means that α ∈ [0 . , . α = . We must have α ≥ α ξ ξ = . . (33)By defining α = .
98 we know that the conversion is possi-ble by LOCC. Indeed one can verify that the ordered Schmidtcoe ffi cients of the two superpositions are: | Γ i : 6751000 ; 2001000 ; 751000 ; 501000 ; | Γ i : 6861000 ; 2941000 ; 121000 ; 81000 (34)and that they satisfy the majorization inequalities.
3. Conclusions
In the present Letter, we have derived three propositionswhich represent the necessary and su ffi cient conditions to en-able the transformations of superpositions of entangled statesby LOCC. By applying the Nielsen’s theorem we have shownthat the two coe ffi cients of superpositions α and α dependon some inequalities which involve the Schmidt coe ffi cients ofonly three states | φ i , | ψ i , and | φ i . The analysis reported inthis Letter could lead to a deeper understanding of the behaviorof entanglement under LOCC and may be relevant in the futurework on entanglement manipulations.4 cknowledgement This work was supported by the Romanian Ministry of Edu-cation and Research through Grant IDEI-995 / A. The proof of the inequality (17)
In this appendix we will prove the following inequality,which is used for proving Proposition 1. If < η < ξ <η < ξ < , then ξ η − ξ + η > ξ η − ξ + η (A.1) Proof.
Let us observe that ( ξ − η )(1 − ξ ) > , which isequivalent to ξ > η − ξ + η . (A.2)We have η > ξ > ξ η − ξ + η from which we obtain η > ξ η (1 + η ) η (1 − ξ + η ) + ξ η . (A.3)On the other hand we know that ξ > η , which together withEq. (A.3) leads to: ξ η (1 − ξ + η ) > ξ η (1 − ξ + η ) . (A.4) B. The proof of the inequality (20)
Here we will prove a second inequality, namely: If < η <ξ < η < ξ < , and α ξ ≤ α ξ , then the following inequal-ity holds:(1 − α )(1 − η ) > (1 − α )(1 − η ) . (B.1) Proof.
The inequality α (1 − η )( ξ − ξ ) > − η )( α ξ − ξ ) + (1 − η ) ξ (1 − α ) > . Furtherwe get(1 − η )( α ξ − ξ ) + (1 − η ) ξ (1 − α ) > . (B.2)This last inequality is equivalent to α ξ ξ > α (1 − η ) + η − η − η . (B.3)From the hypothesis we have α ≥ α ξ ξ , therefore we obtain α > α (1 − η ) + η − η − η , (B.4)which is equivalent to (1 − α )(1 − η ) > (1 − α )(1 − η ) . References [1] M. A. Nielsen and I. L. Chuang,
Quantum Computation and QuantumInformation (Cambridge University Press, U. K., 2000)[2] M. A. Nielsen, Phys. Rev. Lett. , 436 (1999)[3] D. Jonathan and M. B. Plenio, Phys. Rev. Lett. , 3566 (1999)[4] W. Song, Y. Huang, N. L. Liu, and Z. B. Chen, J. Phys. A: Math. Gen. ,785 (2007)[5] I. Ghiu, M. Bourennane, and A. Karlsson, Phys. Lett. A , 12 (2001)[6] I. Chattopadhyay and D. Sarkar, Phys. Rev. A , 050305(R) (2008)[7] G. Bowen and N. Datta, IEEE Trans. Inf. Theory , 3677 (2008)[8] M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Phys. Rev. Lett. , 047902 (2003)[9] H. Fan, Phys. Rev. A , 014305 (2007)[10] N. Linden, S. Popescu, and J. A. Smolin, Phys. Rev. Lett. , 100502(2006)[11] G. Gour, Phys. Rev. A , 052320 (2007)[12] J. Niset and N. J. Cerf, Phys. Rev. A , 042328 (2007)[13] Y. C. Ou and H. Fan, Phys. Rev. A , 022320 (2007)[14] W. Song, N. L. Liu, and Z. B. Chen, Phys. Rev. A , 054303 (2007)[15] C. S. Yu, X. X. Yi, and H. S. Song, Eur. Phys. J. D , 273 (2008)[16] D. Cavalcanti, M. O. Terra Cunha, and A. Ac´ın, Phys. Rev. A , 042329(2007)[17] Y. Xiang, S. J. Xiong, and F. Y. Hong, Eur. Phys. J. D , 257 (2008), 257 (2008)