aa r X i v : . [ qu a n t - ph ] M a r Converting separable conditions to entanglement breaking conditions
Ryo Namiki
Institute for Quantum Computing and Department of Physics and Astronomy,University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada (Dated: October 19, 2018)We present a general method to derive entanglement breaking (EB) conditions for continuous-variable quantum gates. We start with an arbitrary entanglement witness, and reach an EB condi-tion. The resultant EB condition is applicable not only for quantum channels but also for generalquantum operations, namely, trace-non-increasing class of completely positive maps. We illustrateour method associated with a quantum benchmark based on the input ensemble of Gaussian dis-tributed coherent states. We also exploit our idea for channels acting on finite dimensional systemsand present a Schmidt-number benchmark based on input states of two mutually unbiased basesand measurements of generalized Pauli operators.
An important task for future realization of quantum in-formation technology is to establish a reliable quantumchannel. A powerful tool to estimate an experimentalimplementation of quantum gates is quantum process to-mography. However, it is not always feasible to measurethe input-and-output relations for a set of tomographiccomplete states. Instead of tomographic approach, onemay be interested in probing a basic coherence of quan-tum channels using a small set of feasible input states.The quantum benchmarks provide such a method basedon the context of quantum entanglement [1–7]. A quan-tum benchmark is typically determined by an upperbound of an average fidelity achieved by a class of quan-tum channels called entanglement breaking (EB) [8]. Ifan experimental fidelity surpasses the fidelity bound wecan verify that any classical measure-and-prepare map isunable to simulate the channel. Mathematically, it im-plies the Choi-Jamiolkowsky (CJ) state of the channel isentangled, hence, there exists, at least, one entangled in-put state whose inseparability maintains under the chan-nel action. There have been several works to determinesuch classical fidelities [3–5, 9–11] or other forms of EBlimits [12, 13]. One can also apply the notion of EB lim-its to quantum operations, namely, trace-non-increasingclass of completely positive (CP) maps [14, 15]. In ad-dition to a proof of the inseparability in the physicalprocess, one can demonstrate a more specified type ofchannel’s coherence by quantifying the amount of entan-glement in the CJ state [16–18].Although it has been known that an EB condition ismathematically equivalent to a separable condition, thevarieties of known EB conditions are rather limited com-pared with those of known separable conditions. In fact,one can easily find several systematic methods to producea series of separable conditions [19–21] whereas poten-tial applications of separable conditions to the quantumbenchmark problems have little been mentioned in theliteratures on the separability problems [22, 23].In this report, we present a general method to converta separable condition to an EB condition for continuous-variable quantum channels as a generalization of themethod developed in [13]. Given a formula of entangle-ment witness we compose an EB condition by separately assigning an entangled density operator. After a generalcomposition we illustrate our method associated with aquantum benchmark based on the Gaussian distributedcoherent states [15]. We also exploit our idea for chan-nels acting on finite dimensional systems and present aSchmidt-number benchmark [16] for qudit channels basedon input states of two mutually unbiased bases and mea-surements of generalized Pauli operators.Let ρ be a density operator and write an expectationvalue of an operator ˆ O as Tr[ ˆ Oρ ] = h ˆ O i ρ . A general formof separable conditions can be written by a function ofexpectation values for a set of operators { ˆ O i } i =1 , , ··· ,N as F ( h ˆ O i ρ , h ˆ O i ρ , · · · , h ˆ O N i ρ ) ≥ . (1)A special case is based on an operator called the entan-glement witness ˆ W that satisfiesTr( ˆ W ρ s ) = h ˆ W i ρ s ≥ ρ s = P p i ( | a i i h a i | ) A ⊗ ( | b i i h b i | ) B .It implies that ρ is entangled when h ˆ W i ρ < W . We can readily extend our methodfor the general form in Eq. (1). This form includes non-linear terms of expectation values and is often referredto as the non-linear witness .We consider a two-mode system AB described bybosonic field operators satisfying the commutation rela-tions [ a, a † ] = [ b, b † ] = 1. Let us suppose ˆ W is expressedin the anti-normal order regarding to the field operators { b, b † } for the second system B such asˆ W ( a, a † , b, b † ) = X n,m W ( n,m ) ( a, a † ) b n ( b † ) m . (3)Then, we can rewrite it asˆ W = X W n,m ( a, a † ) b n B ( b † ) m = X W ( n,m ) ( a, a † ) Z ( α ∗ ) n α m | α ∗ i h α ∗ | d απ = Z ˆ W ( a, a † , α ∗ , α ) | α ∗ i h α ∗ | d απ , (4)where we used the closure relation for coherent states R | α i h α | d α/π = B . Here, we ex-press the closure with α ∗ , the complex conjugate of α ,for a notation convention. Equation (4) impliesTr( ˆ W ρ ) =Tr A (cid:20)Z ˆ W ( a, a † , α ∗ , α ) h α ∗ | ρ | α ∗ i B d απ (cid:21) , (5)where Tr A denotes partial trace over system A .Let ψ = ψ AB be an entangled density operator of thetwo-mode field AB . We define an ensemble of states { p α , ϕ α } on a one-mode system as p α := Tr [ A ⊗ ( | α ∗ i h α ∗ | ) B ψ AB ] ,ϕ α := h α ∗ | ψ AB | α ∗ i B /p α . (6)Note that ϕ α is a type of the relative states of | α ∗ i re-garding ψ AB and p α is a probability density satisfying R p α d α/π = 1.Let us consider the local action of a physical map E forthe state ψ , J = E A ⊗ I B ( ψ ) (7)where E is a CP map acting on system A and I is theidentity map. When E is a trace-decreasing operation, wecan formally normalize J as a density operator by J/P s with P s := Tr[ J ] = Z p α Tr[ E ( ϕ α )] d α/π, (8)where we use the relations in Eq. (6). Note that we have P s = 1 for the trace-preserving maps. Substituting ρ = J/P s into Eq. (5) we can writeTr( ˆ W ρ ) = 1 P s Tr (cid:20)Z ˆ W ( a, a † , α ∗ , α ) p α E ( ϕ α ) d απ (cid:21) . (9)Here, system B is traced out and Tr( ˆ W ρ ) is representedby the mean values of operators on system A over chan-nel’s outputs E ( ϕ α ) subjected to the input state { ϕ α } .Let us suppose that E is an EB map, i.e., E ( ρ ) = P i Tr[ M i ρ ] σ i with M i ≥ P i M i ≤
1, and a set ofdensity operators { σ i } . Then, ρ becomes a separabledensity operator and Tr( ˆ W ρ ) has to fulfills the separablecondition of Eq. (2). Therefore, we obtain the followingEB condition:1 P s Tr (cid:20)Z ˆ W ( a, a † , α ∗ , α ) p α E ( ϕ α ) d απ (cid:21) ≥ . (10)In this manner one can compose an EB condition from aseparable condition by separately assigning an entangledstate ψ . To be concrete, the inequality of Eq. (10) is anecessary condition for entanglement breaking, and anyviolation of this inequality implies that the map E cannotbe an EB map. For a non-linear witness in the form of Eq. (1), we sim-ply assign an operators ˆ W i for each of ˆ O i and express itsexpectation value as in Eq. (9) by repeating the proce-dure above. Then, we can generally convert separableconditions in the form of Eq. (1) into EB conditions byreplacing the relevant expectation values as follows: h ˆ O i i ρ → P s Tr (cid:20)Z ˆ W i ( a, a † , α ∗ , α ) p α E ( ϕ α ) d απ (cid:21) . (11)Note that the obtained EB condition depends on thechoice of the entanglement ψ which determines the stateensemble { p α , ϕ α } owing to Eq. (6). Accordingly, a dif-ferent choice of ψ could constitute a different EB condi-tion even the original separable condition is the same.Let us illustrate our method associated with a familiarcase of the fidelity-based quantum benchmark [4, 5, 11,15]. In experiments of quantum optics, coherent statesare commonly available as a state of laser light. It isthus feasible to probe an experimental quantum gate byan input of coherent states. We will consider an inputensemble of the Gaussian distributed coherent states [24].This ensemble can be associated with the case that ψ isa two-mode squeezed state. In fact, by substituting thetwo-mode squeezed state | ψ ξ i = p − ξ P ∞ n =0 ξ n | n i | n i with ξ ∈ (0 ,
1) into Eqs. (6), we obtain the ensemble ofGaussian distributed coherent states, p α = (1 − ξ ) e − (1 − ξ ) | α | ,ϕ α = | ξα i h ξα | . (12)Let X ≥ u, v ) be a pair of real number thatfulfills u + v = 1 and u = 0. Let us define an witnessoperatorˆ W :=
11 + X − π Z e − X | α | | vα i h vα | ⊗ | uα ∗ i h uα ∗ | d α, (13)such that h ˆ W i ≥ W is already expanded inthe local coherent states similar to the form in Eq. (4)it is no need to consider the operator ordering. FromEqs. (9), (12), and (13) we can writeTr[ ˆ W J ] = 11 + X − πP s u (cid:18) λ + Xξ u (cid:19) × Z e − λ | α | h√ ηα | E ( | α i h α | ) |√ ηα i d α, (14)where λ = ξ − ( Xu − +(1 − ξ )) and η := v / ( ξu ) . Usingthe condition of Eq. (10) and taking the limit X → P s − u λπ Z e − λ | α | h√ ηα | E ( | α i h α | ) |√ ηα i d α ≥ , (15)where u = (1 + λ + η ) / (1 + λ ). This corresponds to thefidelity-based quantum benchmark for general CP maps[15]. In Ref. [15], its derivation is based on the dualityof semidefinite programing. For quantum channels (thetrace-preserving class of CP maps; P s = 1), one can findother derivations in Refs. [4, 5, 11].Note that there is a wide interest in formulating separa-ble conditions based on the moments of canonical quadra-ture variables [19, 26–28]. The moments of quadraturevariables can be directly observed by homodyne measure-ments in experiments. Among all, second-order condi-tions have been widely used as a feasible method for en-tanglement detection. It is well-known that the sum con-dition [26] and the product condition [28] are sufficientfor witnessing two-mode Gaussian entanglement. By ap-plying our method we can translate them into the EBconditions with the input ensemble of the Gaussian dis-tributed coherent states in [13] (Corollary 1 and Propo-sition, respectively), which are sufficient to witness one-mode Gaussian channels in the quantum domain, namely,one-mode Gaussian channels being nonmember of theEB class. Further, the formalism developed in Ref. [13]would be usable as a quantitative quantum benchmarkbecause it can be related to entanglement of formationon the CJ state (See Ref. [29]). Similar statements couldhold for the fidelity-based approach. In fact, the entan-glement witness of Eq. (13) is known to be sufficient forwitnessing two-mode Gaussian entanglement [25] and thefidelity-based EB condition is also sufficient for detectingone-mode Gaussian channels in the quantum domain [5].However, its connection to a meaningful entanglementmeasure is left open.In the rest of this report, we discuss the case of thephysical process acting of a finite dimensional system.The key mechanism to introduce the ensemble of inputstates { p α , ϕ α } in Eq. (6) is the coherent-state expressionof system B in Eq. (4). Analogously, we can introduce astate ensemble by decomposing the witness operator witha set of hermitian operators ˆ h on system B as followsˆ W = X l w ( l ) A ⊗ ˆ h ( l ) B = X l w ( l ) A ⊗ X j h j ( l ) | j ( l ) ih j ( l ) | B , (16)where { h ( l ) j , (cid:12)(cid:12) j ( l ) (cid:11) } represents the spectral decompositionof ˆ h ( l ) . This implies the set of input states similarly toEq. (6) as p j,l := Tr h A ⊗ ( | j ( l ) ih j ( l ) | ) B ψ AB i ϕ j,l := h j ( l ) | ψ AB | j ( l ) i B /p j,l . (17)Therefore, instead of Eq. (10), we obtain an EB conditionin the following form:1 P s X j,l p i,l h ( l ) j Tr h ˆ w ( l ) E ( ϕ j,l ) i ≥ , (18) where we define P s = P j,l Tr[ p j,l E ( ϕ j,l )]. Note that anexample of the decomposition in Eq. (16) can be obtainedby choosing a Hilbert-Schmidt orthonormal basis on sub-system B .Finally, using this framework we will derive a Schmidtnumber benchmark [16] for quantum operations actingon a d -dimensional (qudit) system. The Schmidt numberbenchmark of class k + 1 ( k ∈ [1 , d − k or less than k . This class ofquantum channels is called k -partial EB channels [30–32],and k = 1 represents the class of EB channels.Let us consider a Schmidt number-( k + 1) witnesse fortwo d -dimension system given in Ref. [33], g k,d − h ˆ Z A ˆ Z † B + ˆ Z † A ˆ Z B + ˆ X A ˆ X B + ˆ X † A ˆ X † B i ≥ , (19)where g k,d = [( d − k ) cos ω + ( d + k )] /d , and ˆ Z := P d − j =0 e iωj | j ih j | and ˆ X := P d − j =0 | j + 1 ih j | , are the gener-alized Pauli operators. Here, we assumed a fixed Z -basis {| i , | i , · · · , | d − i} with modulo- d conditions | j + d i = | j i and ω := 2 π/d . By expanding ˆ Z and ˆ X respectively in Z -basis {| j i} and X -basis {| ¯ j i} , which is defined through | l i := ˆ Z l (cid:16) √ d P d − j =0 | j i (cid:17) = ˆ Z l | i , we can see that the op-erators on system B in Eq. (19) can be expressed by theprojections onto the mutually unbiased bases, {| j i , | ¯ j i} .Using this expansion and J = E A ⊗ I B ( | Φ d i h Φ d | ) /P s with | Φ d i = d − / P d − j =0 | j i | j i with Eqs. (17) and (18) we ob-tain the following necessary condition for k -partial EBmaps: P s g k,d − d − X j =0 Tr (cid:2) ( ˆ Ze − iωj + ˆ Z † e iωj ) E ( | j i h j | )+ ( ˆ Xe − iωj + ˆ X † e iωj ) E ( (cid:12)(cid:12) − j (cid:11) (cid:10) − j (cid:12)(cid:12) ) (cid:3) /d ≥ . (20)Hence, a violation of this condition implies a quantitativequantum benchmark for the Schmidt class k + 1, namely,any Kraus representation of E has, at least, one Krausoperator whose rank is k + 1 or higher. An experimentaltest would be executed by input states of two mutuallyunbiased bases and projections to these bases similarlyto the result in Ref. [16]. Note that we can readily extendthe result in Ref. [16] for quantum operations acting onqudit states by using the normalized state J = E A ⊗ I B ( | Φ d i h Φ d | ) /P s .In summary, we have presented a method to con-vert separable conditions to EB conditions for bosonicsingle-mode channels. Given an entanglement witnesswe can generate an EB condition by separately assigningan entangled state that determines the ensemble of in-put states. By considering a normalization of this statethe resultant EB condition becomes applicable to gen-eral quantum operations, namely, trace-non-increasingclass of CP maps. As an example we present a differ-ent derivation of the fidelity-based quantum benchmarkin Ref. [15] starting from a separable condition given inRef. [25]. Although we focus on single-mode operations,our method can be straightforwardly extended for multi-mode bosonic quantum channels/operations. We havealso developed a similar framework for quantum opera-tions acting on finite dimension systems and presented aSchmidt number benchmark for quantum operations. Acknowledgments
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