Entanglement sharing via qudit channels: Nonmaximally entangled states may be necessary for one-shot optimal singlet fraction and negativity
aa r X i v : . [ qu a n t - ph ] M a r Entanglement sharing via qudit channels: Nonmaximally entangled states may be necessaryfor one-shot optimal singlet fraction and negativity
Rajarshi Pal ∗ Department of Physics, Indian Institute of Technology Madras, Chennai 600036
Somshubhro Bandyopadhyay † Department of Physics and Center for Astroparticle Physics and Space Science,Bose Institute, EN-80, Sector V, Bidhannagar, Kolkata 700091
We consider the problem of establishing entangled states of optimal singlet fraction and negativity betweentwo remote parties for every use of a noisy quantum channel and trace-preserving LOCC under the assumptionthat the parties do not share prior correlations. We show that for a family of quantum channels in everyfinite dimension d ≥ , one-shot optimal singlet fraction and entanglement negativity are attained only withappropriate nonmaximally entangled states. A consequence of our results is that the ordering of entangledstates in all finite dimensions may not be preserved under trace-preserving LOCC. I. INTRODUCTION
In quantum information theory, entangled states [1, 2] shared between remote parties are considered as resources [2] withinthe paradigm of local operations and classical communication (LOCC) (see for example, [3]). However, any protocol of en-tanglement sharing requires sending quantum systems over quantum channels along with local processing irrespective ofpreshared correlations that may be present between the parties [8–11, 17–20]. It may be noted that recent results [19] stronglysuggest that protocols with prior correlations may not provide any efficiency advantage over the ones without correlations.In this paper we consider the basic protocol between two remote parties, Alice and Bob, who do not share any prior cor-relation. Such a protocol may be described as follows. Alice locally prepares a pure quantum state | ψ i ∈ C d ⊗ C d and sendshalf of it to Bob down a d -dimensional quantum channel Λ . In an ideal scenario where the channel is taken to be noiseless,maximally entangled states are easily established this way. For a noisy channel, which is typically the case, Alice and Bob endup with a mixed state ρ ψ, Λ = ( I ⊗ Λ) ρ ψ where ρ ψ = | ψ i h ψ | or an ensemble for many uses of the channel. Thus in a noisychannel scenario the goal is to establish entangled states that are optimal with respect to some well-defined figure of merit.Entanglement distillation [8–11] provides a solution by converting many copies of ρ ψ, Λ to fewer near-perfect entangled statesthereby requiring many uses of the channel and joint measurements.The present paper considers a one-shot instance of the entanglement sharing problem where the goal is to establish entan-gled states of maximum singlet fraction and entanglement negativity [24] achievable for every single use of the channel (seefor example, [12, 13]). As we will see, the one-shot optimal values of these two quantities are closely related and exhibit similarproperties.The singlet fraction (or maximally entangled fraction) [8–10, 16, 17] of ρ ψ, Λ is given by F ( ρ ψ, Λ ) = max | Φ i h Φ | ρ ψ, Λ | Φ i (I.1)where | Φ i is a maximally entangled state in C d ⊗ C d . The motivation behind choosing singlet fraction as our figure of meritlies in the fact that singlet fraction is an effective measure of usefulness of the state ρ ψ, Λ for quantum information processingtasks, e.g. quantum teleportation [7], superdense coding [4], quantum key distribution [5], and distributed computation [6],which typically require entangled states of very high F , ideally close to unity. It is useful to note that the yield in a distillationprotocol depends on F ( ρ ψ, Λ ) ; in fact, for the distillation protocols to work the singlet fraction of the mixed states must exceeda certain threshold value [8–10].While one may suppose that maximizing F ( ρ ψ, Λ ) given by Eq. (I.1) over all transmitted states | ψ i will yield the desired result,such a supposition may be unfounded. This is because singlet fraction of a state can increase under local trace-preservingoperations (TP-LOCC) [16, 21, 22] which strongly suggests that in a one-shot protocol local post-processing may be requiredto attain the optimal value. Taking this into account, let ρ L ψ, Λ = L ( ρ ψ, Λ ) denote the density matrix under the action of some ∗ Electronic address: [email protected] † Electronic address: [email protected]
TP-LOCC operation L on ρ ψ, Λ . Then, for a fixed transmitted state | ψ i , the maximum achievable singlet fraction is defined as[16] F ∗ ( ρ ψ, Λ ) = max L ∈ TP − LOCC F (cid:0) ρ L ψ, Λ (cid:1) , (I.2)where the maximization is over all TP-LOCC L . Note that, unlike F which can increase under TP-LOCC, F ∗ is a LOCCmonotone [16]. It is important to note that the action of optimal TP-LOCC, say, L ∗ on ρ ψ, Λ results in a density matrix, say ρ ′ ψ, Λ = L ∗ ( ρ ψ, Λ ) . Thus, we can write F ∗ ( ρ ψ, Λ ) = F (cid:0) ρ ′ ψ, Λ (cid:1) . (I.3)The one-shot optimal singlet fraction for the channel Λ is defined as [13] F (Λ) = max | ψ i F ∗ ( ρ ψ, Λ ) , (I.4)where the maximum is taken over all pure state transmissions. Let us now suppose that | ψ opt i is a pure entangled state suchthat (I.4) holds; then, F (Λ) = F ∗ (cid:0) ρ ψ opt , Λ (cid:1) = F (cid:16) ρ ′ ψ opt , Λ (cid:17) . (I.5)The one-shot optimal singlet fraction is related to optimal negativity in the following way. For any two-qudit density matrix σ the following inequality holds [23]: F ∗ ( σ ) ≤ N ( σ ) d (I.6)where N ( σ ) denotes the negativity [24] of the state σ . Now, substituting σ by ρ ψ, Λ in the above inequality and maximizingover all transmitted states | ψ i leads to an upper bound on F (Λ) : F (Λ) ≤ N (cid:0) ρ ψ opt , Λ (cid:1) d ≤ N (Λ) d , (I.7)where N (Λ) = max | ψ i N ( ρ ψ, Λ ) is the optimal negativity.Thus given a quantum channel Λ , the task is to find F (Λ) and N (Λ) and the protocols to achieve these optimal values.Note that, it is quite possible that the optimal values may be attained by sending different pure states. However, the questionthat deserves utmost importance is whether the optimal states are maximally entangled like noiseless channels.To the best of our knowledge, the problem concerning one-shot optimal singlet fraction has been completely solved onlyin the qubit case [13]. In particular, for any qubit channel (which is not entanglement breaking), it was shown that | ψ opt i ,satisfying (I.5) is maximally entangled if and only if the channel is unital, and for any non-unital qubit channel | ψ opt i isnecessarily nonmaximally entangled (for the specific case of amplitude damping channel; see [12]). Further, it was shown thatfor any qubit channel Λ qubit , F (Λ qubit ) can be exactly computed and is given by [13] F (Λ qubit ) = 1 + 2 N (cid:0) ρ Φ + , Λ qubit (cid:1) (I.8)where | Φ + i = √ ( | i + | i ) .In [14, 15], specific examples were given which showed that the ordering of entangled states may change under one-sidedlocal action of a qubit channel and the maximum output entanglement may not be achieved for an input maximally entangledstate [shown for a system of four qubits having configuration (cid:0) C ⊗ C (cid:1) ⊗ (cid:0) C ⊗ C (cid:1) ]. A more systematic way supportingthese observations can be found in [12, 13, 19]. For example, in [13] it was pointed out that for qubit channels, the maximumachievable negativity may not be achieved by sending a maximally entangled state: Using (I.8) and (I.7) we see that N (cid:0) ρ Φ + , Λ qubit (cid:1) ≤ N (cid:0) ρ ψ opt , Λ qubit (cid:1) ≤ N (Λ qubit ) (I.9)Since | ψ opt i is nonmaximally entangled for non-unital channels, the inequality implies that nonmaximally entangled statesalso lead to maximum achievable entanglement negativity; for an amplitude damping channel the inequality (I.9) is strict [13].The question for other nonunital channels, however, remains open.In this paper we extend our previous studies [12, 13] to higher dimensional quantum channels. In particular, we wish toknow whether we can find quantum channels in all higher dimensions d ≥ with properties similar to non-unital qubitchannels. The main results of this paper are the following.• We present a family of quantum channels Ω in every finite dimension d ≥ for which we prove that | ψ opt i is non-maximally entangled. Although we are not able to provide an expression for this optimal state, nonetheless, we obtaina nonmaximally entangled state | ψ ′ i ∈ C d ⊗ C d satisfying the inequality: F (Ω) = F ∗ (cid:0) ρ ψ opt , Ω (cid:1) ≥ F ( ρ ψ ′ , Ω ) > F ∗ ( ρ Φ , Ω ) (I.10)where | ψ ′ i is the eigenvector corresponding to the largest eigenvalue of the density matrix ρ Φ + , ˆΩ with ˆΩ being the dualmap (see the next section for the definition) and | Φ i ∈ C d ⊗ C d being any maximally entangled state. Note that thefirst inequality gives us a lower bound on the one-shot optimal singlet fraction and shows that suitable nonmaximallyentangled states are better than maximally entangled states. Also note that, since F ∗ is a LOCC monotone, the aboveinequality together with the identity (I.5) provides a constructive way to demonstrate that ordering of F ∗ in general isnot preserved under TP-LOCC in all finite dimensions.• Optimal negativity is attained only by appropriate nonmaximally entangled states. Using (I.6), (I.5) and (I.12) it is easyto see that N (Ω) ≥ N (cid:0) ρ ψ opt , Ω (cid:1) > N (cid:0) ρ Φ + , Ω (cid:1) (I.11)where | ψ opt i is nonmaximally entangled. Thus, in all finite dimensions d ≥ we are able to show by explicit constructionthat the maximum output entanglement, as measured by negativity, is not always achieved using a maximally entangledinput state. This, significantly improves upon the previously known examples.We also make the following observation. We find that in higher dimensions an expression analogous to (I.8) does not hold ingeneral. This follows from inequality (see the proof of I.10): F (Ω) > N (cid:0) ρ Φ + , Ω (cid:1) d (I.12)where N (cid:0) ρ Φ + , Ω (cid:1) is the negativity of the density matrix ρ Φ + , Ω . One may argue that there is no convincing reason why oneshould have expected the generalization to hold in the first place; however, the exact formula obtained in [13] prompted us tothink such a generalization, if it holds, would give us a computable formula for one-shot optimal singlet fraction in all finitedimensions. Unfortunately, our optimism turned out to be misplaced. II. RESULTS
A quantum channel Λ is a trace preserving completely positive map characterized by a set of Kraus operators { A i } satisfying P A † i A i = I (see for example, [18]). The dual map ˆΛ , described in terms of the Kraus operators n A † i o , is the adjoint map withrespect to the Hilbert-Schmidt inner product. We say that a channel Λ is unital if its action preserves the Identity: Λ ( I ) = I ,and nonunital if it does not, i.e., Λ ( I ) = I . Moreover, the dual map ˆΛ is trace-preserving, and hence a channel, iff Λ is unital.The one-sided action of a d -dimensional map $ ∈ n Λ , ˆΛ o on a pure state | ψ i ∈ C d ⊗ C d gives rise to a mixed state which canbe conveniently expressed as: ρ ψ, $ = ( I ⊗ $) ρ ψ = X i ( I ⊗ K i ) ρ ψ (cid:16) I ⊗ K † i (cid:17) where the Kraus operators { K i } describe the channel $ and ρ ψ = | ψ i h ψ | is the density matrix corresponding to the pure state | ψ i . We now give two useful lemmas which are applicable to any quantum channel Λ . The first lemma was proved in [14]. Lemma 1.
For a d -dimensional quantum channel Λ , F ∗ ( ρ Φ , Λ ) = F ∗ (cid:0) ρ Φ + , Λ (cid:1) where | Φ + i = √ d P d − i =0 | ii i and | Φ i is anymaximally entangled state in C d ⊗ C d . The proof is simple. Since every maximally entangled state | Φ i ∈ C d ⊗ C d can be written as | Φ i = ( U ⊗ V ) | Φ + i for some U, V ∈ SU ( d ) , using the identity ( I ⊗ V ) | Φ + i = (cid:0) V T ⊗ I (cid:1) | Φ + i we can write | Φ i = ( W ⊗ I ) | Φ + i where W = U V T isalso a unitary operator. Because the channel Λ acts only on the second qudit, we have ρ Φ , Λ = ( W ⊗ I ) ρ Φ + , Λ (cid:0) W † ⊗ I (cid:1) . Thus the density matrices ρ Φ , Λ and ρ Φ + , Λ are connected by a local unitary operator acting on the first system. Because thefirst system never interacts with the channel, this local unitary can always be absorbed in the post-transmission optimal TP-LOCC associated with the state transformations (defined earlier) ρ Φ , Λ → ρ ′ Φ , Λ and ρ Φ + , Λ → ρ ′ Φ + , Λ . Therefore, F ∗ ( ρ Φ , Λ ) = F ∗ (cid:0) ρ Φ + , Λ (cid:1) . Lemma 2.
For a d -dimensional quantum channel Λ , F (Λ) ≥ λ max (cid:0) ρ Φ + , Λ (cid:1) where | Φ + i = √ d P d − i =0 | ii i and λ max (cid:0) ρ Φ + , Λ (cid:1) isthe largest eigenvalue of the density matrix ρ Φ + , Λ .Proof. The proof is along the same lines as in the qubit case [13]. We begin by noting that for any | ψ i ∈ C d ⊗ C d , F (Λ) ≥ max ψ F ( ρ ψ, Λ ) = max ψ, Φ h Φ | ρ ψ, Λ | Φ i (II.1)where | Φ i is maximally entangled. Using the relations | Φ i = ( U ⊗ V ) | Φ + i for some U, V ∈ SU ( d ) and ( I ⊗ V ) | Φ + i = (cid:0) V T ⊗ I (cid:1) | Φ + i , it is straightforward to show that F ( ρ ψ, Λ ) = D ψ (cid:12)(cid:12)(cid:12) ρ Φ + , ˆΛ (cid:12)(cid:12)(cid:12) ψ E (II.2)where ˆΛ is the dual channel. From (II.1) and (II.2) we therefore get F (Λ) ≥ λ max (cid:16) ρ Φ + , ˆΛ (cid:17) = λ max (cid:0) ρ Φ + , Λ (cid:1) where we have used λ max (cid:16) ρ Φ + , ˆΛ (cid:17) = λ max (cid:0) ρ Φ + , Λ (cid:1) proved in [13] for any d dimensional channel Λ . Main results
Let us now consider the d -dimensional quantum channel Ω defined by the Kraus operators A i for i = 0 , . . . , d − , A = diag (1 , x , x , . . . , x d − ) ; ( A m ) ij = p − x m δ i δ mj i, j = 0 , . . . d − ∀ m = 1 , . . . , d − (II.3)where < x i < for every i and x i = x j for at least one pair ( i, j ) . That the Kraus operators defined above indeed describea legitimate quantum channel can be seen as follows. First, it is easy to check that (cid:0) A † m A m (cid:1) ik = (cid:0) − x m (cid:1) δ mi δ mk ; A † A = diag (cid:0) , x , x , . . . , x d − (cid:1) (II.4)Clearly the operators A † i A i are positive and moreover, Eqs. (II.4) lead to A † A + d − X m =1 A † m A m = I d × d . (II.5)We now state our result. Theorem 1.
For the d -dimensional quantum channel Ω described above, the following inequalities hold in every finite dimension d ≥ : F (Ω) ≥ F ( ρ ψ ′ , Ω ) > F ∗ ( ρ Φ , Ω ) (II.6) where | ψ ′ i ∈ C d ⊗ C d is a pure state, not maximally entangled, and | Φ i ∈ C d ⊗ C d is any maximally entangled state. The inequalities (II.6) are established through the following results.
Lemma 3.
For any maximally entangled state | Φ i ∈ C d ⊗ C d , λ max (cid:0) ρ Φ + , Ω (cid:1) > N (cid:0) ρ Φ + , Ω (cid:1) d (II.7) for all d ≥ ,Proof. First we obtain λ max (cid:0) ρ Φ + , Ω (cid:1) . The action of the Kraus operators given by (II.3) on | Φ + i are given by: ( I ⊗ A ) (cid:12)(cid:12) Φ + (cid:11) = 1 √ d | i + d − X i =1 x i | i i | i i ! = | φ i , (II.8)and for m = 1 , . . . , d − I ⊗ A m ) (cid:12)(cid:12) Φ + (cid:11) = 1 √ d d − X i =0 | i i A m | i i = 1 √ d d − X i =0 | i i p − x m δ im | i ∵ A m | i i = p − x m δ im | i = 1 √ d p − x m | m i | i = | φ m i . (II.9)Thus, ρ Φ + , Ω = d − X m =0 ( I ⊗ A m ) ρ Φ + (cid:0) I ⊗ A † m (cid:1) = | φ i h φ | + d − X m =1 | φ m i h φ m | . (II.10)As ρ Φ + , Ω is already in the diagonal form, it is straightforward to obtain its largest eigenvalue, λ max (cid:0) ρ Φ + , Ω (cid:1) = 1 d d − X i =1 x i ! . (II.11)Next, we compute negativity N (cid:0) ρ Φ + , Ω (cid:1) . The partial transposed matrix corresponding to ρ Φ + , Ω is given by ρ ΓΦ + , Ω = 1 d | i h | + d − X i =1 x i ( | i i h i | + | i i h i | ) + d − X i,j =1 x i x j | ij i h ji | + d − X i =1 (cid:0) − x i (cid:1) | i i h i | with easily computed eigenvalues, d (multiplicity d ) ; ± x i d , i = 1 , . . . , d − ± x i x j d , i < j i, j = 1 , . . . , d − . As negativity is defined as the absolute value of the sum of the negative eigenvalues [24], we have N (cid:0) ρ Φ + , Ω (cid:1) = 1 d d − X i =1 x i + X ≤ i
Let | ψ ′ i be the eigenvector corresponding to the eigenvalue λ max (cid:16) ρ Φ + , ˆΩ (cid:17) . Then, λ max (cid:0) ρ Φ + , Ω (cid:1) = F ( ρ ψ ′ , Ω ) . More-over, | ψ ′ i is not maximally entangled.Proof. From Eq. (II.2) we know that for any pure state | ψ i ∈ C d ⊗ C d , F ( ρ ψ, Ω ) = D ψ (cid:12)(cid:12)(cid:12) ρ Φ + , ˆΩ (cid:12)(cid:12)(cid:12) ψ E . As | ψ ′ i is the eigenvectorcorresponding to the eigenvalue λ max (cid:16) ρ Φ + , ˆΩ (cid:17) , this means, λ max (cid:16) ρ Φ + , ˆΩ (cid:17) = D ψ ′ (cid:12)(cid:12)(cid:12) ρ Φ + , ˆΩ (cid:12)(cid:12)(cid:12) ψ ′ E = F ( ρ ψ ′ , Ω ) . Using the identity λ max (cid:16) ρ Φ + , ˆΛ (cid:17) = λ max (cid:0) ρ Φ + , Λ (cid:1) [13] for any quantum channel Λ , we therefore have λ max (cid:0) ρ Φ + , Ω (cid:1) = F ( ρ ψ ′ , Ω ) . On the other hand we have already shown that λ max (cid:0) ρ Φ + , Ω (cid:1) > F ∗ ( ρ Φ , Ω ) . Therefore, F ( ρ ψ ′ , Ω ) > F ∗ ( ρ Φ , Ω ) for any maxi-mally entangled state | Φ i from which we conclude that | ψ ′ i is not a maximally entangled state.Inequalities (II.13) and Lemma 4 conclude the proof of the theorem. III. CONCLUSIONS
For any given d -dimensional quantum channel Λ with d ≥ , its one-shot optimal singlet fraction F (Λ) defines the maxi-mum singlet fraction achievable for entangled states established between two remote observers for every use of the channel.Recall that F (Λ) = F ∗ (cid:0) ρ ψ opt , Λ (cid:1) = F (cid:16) ρ ′ ψ opt , Λ (cid:17) . (III.1)Thus, F (Λ) quantifies how useful a channel Λ is either for direct applications for quantum information processing tasks, e.g.teleportation [17] or for entanglement distillation where the yield depends upon the singlet fraction of the noisy states.For qubit channels F (Λ) can be exactly computed and the relevant questions have been satisfactorily answered before[13]. The results, however, point towards two counter-intuitive features. Foremost among them is that | ψ opt i is nonmaxi-mally entangled if and only if the channel is nonunital. And the next is, for nonunital qubit channels maximum achievableentanglement negativity using a maximally entangled state cannot be more than what is attained by sending | ψ opt i . In fact,for an amplitude damping channel (a nonunital channel) it was further shown that optimal negativity is obtained only by anonmaximally entangled state.Motivated by the above results we wanted to understand how well the results and observations made for qubit channelshold in higher dimensions. We presented a family of qudit channels Ω in all finite dimensions d ≥ for which we provedproperties similar to nonunital qubit channels. In particular, we proved that one-shot optimal singlet fraction and negativityare attained only using appropriate nonmaximally entangled states. However, we also find that a generalized version of theformula that allows us to compute the optimal singlet fraction exactly for qubit channels does not hold in general in higherdimensions.While a lot of results had been obtained characterizing quantum channels, we believe that much less is understood whenit comes to characterizing quantum channels through the notions of optimal singlet fraction and entanglement measures.In higher dimensions almost every interesting question is left open, and probably a good way to address them is to solvethe questions for specific channels of interest e.g. a depolarizing channel. Such results can provide us with useful insights.Another paradigm within which where we can ask similar questions is entanglement distribution in the presence of presharedcorrelations. Acknowledgments