Simple proof of the quantum benchmark fidelity for continuous-variable quantum devices
aa r X i v : . [ qu a n t - ph ] A p r Simple proof of the quantum benchmark fidelityfor continuous-variable quantum devices
Ryo Namiki
Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan (Dated: April 6, 2011)An experimental success criterion for continuous-variable quantum teleportation and memoriesis to surpass a limit of the average fidelity achieved by the classical measure-and-prepare schemeswith respect to a Gaussian distributed set of coherent states. We present an alternative proof of theclassical limit based on the familiar notions of the state-channel duality and the partial transposition.The present method enables us to produce a quantum-domain criterion associated with a given setof measured fidelities.
In order to realize quantum information processing [1],a central challenge is to establish reliable quantum chan-nels to transmit and storage quantum states faithfully.For a given experimental implementation of a quantumchannel, it is natural to ask whether or not its perfor-mance originates from quantum coherence. This questionis vital to assert the success of an experimental quan-tum teleportation [2] since it transmits quantum statesby consuming quantum entanglement and has to main-tain better fidelity of transmission beyond the classicaltransmission without entanglement [3–5]. In the present,the framework to prove the effect of entanglement canbe applied for a wide class of experiments including theprocesses of quantum memory [6] and quantum key dis-tribution [7]. Associated with the increase of activity inexperimental researches, there has been a growing inter-est in producing more practical and accessible settingsfor the proof of entanglement [8–17].A central notion to demonstrate quantum advantageover the classical processes is to outperform all classicalmeasure-and-prepare (MP) schemes [8–13]. A classicalMP scheme is an entanglement breaking (EB) channelwhich breaks possible entanglement shared between thesystem being subject to the process and any other sys-tem [18]. If a process is incompatible with any EB chan-nel, one can find an entangled state whose inseparabilitysurvives after the entangled subsystem is subject to theprocess. In this case we call the process is in quantumdomain . A natural figure of merit to measure the per-formance of the process is an average of the fidelitiesbetween the ideal output ( target ) states and actual out-put states of the process over a set of input states with acertain prior probability distribution [10, 11]. The classi-cal limit of the average fidelity achieved by the classicalMP schemes is called the quantum benchmark fidelity.Surpassing this fidelity limit is the proof of the entan-glement and basic success criterion of the experiment forimplementing quantum devices [19–21].In quantum optics and continuous-variable quantuminformation processing [22], the coherent state is one ofthe most accessible quantum states, and it is natural totest the device by the input of coherent states. It is theo-retically simple to determine the classical limit assumingthe uniform distribution of coherent states, however, nei- ther testing the input-output relation for every coherentstate nor assuming the displacement covariant propertyfor the real device is feasible. Hence, a Gaussian distri-bution has been employed to observe the performance ona flat distribution over a feasible amount of phase-spacedisplacement [8, 9]. The value of quantum benchmarkfidelity with respect to the Gaussian distributed set ofcoherent states had been conjectured [8], and this con-jecture was proven in [9]. After the rigorous proof [9], theclassical limit fidelity for a class of non-unit-gain tasks isderived in order to deal with highly lossy processes, suchas, a long distance transmission channel and a quantummemory process with a longer storage time [10]. Theproof [9] has also been utilized in the problem of thenon-locality without entanglement [23]. In view of thesegeneral importance, it would be insightful to find a dif-ferent way to reach the fundamental benchmark.In this report, we present an alternative proof ofthe quantum benchmark fidelity for continuous-variablequantum devices with respect to the transformation ofGaussian distributed set of coherent states. The proofis based on two well-established notions: the Choi-Jamiolkowski state-channel duality (see, e.g., [24]) andthe partial transpose [25]. The state-channel duality is astandard tool to study the property of quantum channelswhereas the partial transpose plays a central role in thetheory of entanglement. Thanks to these reliable basicswe can directly observe that the problem of the quan-tum benchmark is a type of separability problems on thequantum channel. We also apply the present method togive a quantum-domain criterion associated with a set ofexperimentally measured fidelities.We use a standard notation to denote the coherentstate with the complex amplitude α by | α i and the num-ber state with the photon number n by | n i . The co-herent state is expanded in the number basis as | α i = e −| α | / P ∞ n =0 α n | n i / √ n !. When we work on the statewith two modes, we call the first system A and the secondsystem B .Let us define the average fidelity of a physical pro-cess E for the transformation task on the coherent states {|√ N α i} → { (cid:12)(cid:12) √ ηα (cid:11) } with N, η > F N,η,λ ( E ) := Z p λ ( α ) h√ ηα | E (cid:16) |√ N α ih√
N α | (cid:17) |√ ηα i d α (1)where the prior distribution of a symmetric Gaussianfunction with an inverse width of λ > p λ ( α ) := λπ exp( − λ | α | ) . (2)It reproduces the uniform distribution in the limit λ →
0. In the first proof [9], the unit-gain transformation {| α i} → {| α i} was considered so as to establish a bench-mark for the channel that is expected to retrieve inputstates without disturbance, such as the action of idealquantum teleportation and quantum memory. The factor N was introduced to consider a type of state estimationfrom N -copies of the coherent states | α i ⊗ N in Ref. [23]while the factor η was introduced to consider the effectof loss and amplification in Ref. [10]. Quantum benchmark fidelity.—
The quantum bench-mark fidelity for above transformation task is defined bythe maximum of the fidelity in Eq. (1) with respect to theoptimization of the quantum channel E over EB channelsand shown to be [9, 10, 23]sup E∈ EB F N,η,λ ( E ) = N + λN + λ + η =: F C ( N, η, λ ) (3)where EB stands for the set of EB channels. Since wecan verify the relation F N,η,λ = F Nη , , λη , = F , ηN , λN fromEqs. (1) and (2), it is sufficient to show the relation of Eq.(3) either case of η = 1 [23] or case of N = 1 [10]. In thefollowing we prove Eq. (3) with η = 1. The central ideafor the present proof is to make a connection between thefidelity and a two-mode squeezed state via a sort of thestate-channel duality. Then, the problem turns out to bea problem to find the maximum expectation value of anobservable without entanglement, which can be solved byusing the notion of the partial transpose. Proof.—
Let us consider the following integration withthe parameters s, κ ≥
0, and 0 ≤ ξ < J E ( s, κ, ξ ) := Z d αp s ( α ) h α | A h κα ∗ | B E A ⊗ I B ( | ψ ξ i h ψ ξ | ) | κα ∗ i B | α i A (4)where | ψ ξ i = p − ξ P ∞ n =0 ξ n | n i | n i is the two-modesqueezed state and I represents the identity process. Us-ing the relation h α | | ψ ξ i = p − ξ e − (1 − ξ ) | α | / | ξα ∗ i wecan verify the following identity: J E ( s, κ, ξ ) = s (1 − ξ ) λ F N, ,λ ( E ) (5)where the parameters are connected as λ = s + (1 − ξ ) κ , (6) √ N = κξ. (7) In order to find an upper bound of the fidelity we con-sider an upper bound of J E . If E is a MP scheme, ρ E := E ⊗ I ( | ψ ξ i h ψ ξ | ) is a separable state [18]. Then,there exists a separable state, say ρ E = ρ ⋆ E , correspond-ing to the optimal MP scheme that maximizes J E , i.e., J E ( ρ ⋆ E ) = sup E∈ EB J E . This implies that J E ( ρ ⋆ E ) =sup E∈ EB J E is bounded above by the maximum of J E ( ρ E )when ρ E is optimized over the set of separable states,namely, the following inequality holds,sup E∈ EB J E ( s, κ, ξ ) ≤ max ρ ∈ Sep.
Tr [ ρM ] , where Sep. represents the set of separable states and M := Z p s ( α ) | α i h α | ⊗ | κα ∗ i h κα ∗ | d α. Note that the maximum over separable states can beachieved by a product state and that the optimizationover product states is equivalent to the optimizationover their partial transpose. Hence, for any ρ ∈ Sep. ,we can verify Tr[ ρM ] ≤ max φ,ϕ Tr M | φ i h φ | ⊗ | ϕ i h ϕ | =max φ,ϕ Tr M Γ[ | φ i h φ |⊗| ϕ i h ϕ | ] = max ψ,ϕ TrΓ[ M ] | φ i h φ |⊗| ϕ i h ϕ | where Γ[ · ] denotes the partial transposition map.This impliessup E∈ EB J E ( s, κ, ξ ) ≤ max ρ ∈ Sep. Tr ρ Γ[ M ] ≤ k Γ[ M ] k (8)where the last inequality comes from the fact that themaximum over separable states is no larger than the max-imum over all physical states and k · k := max h u | u i =1 h u | ·| u i denotes the maximum eigenvalue. Since the trans-pose of the coherent state with respect to the num-ber basis acts as a phase conjugation, by taking thereplacement | κα ∗ i h κα ∗ | → | κα i h κα | on M we haveΓ[ M ] = R p s ( α ) | α i h α | ⊗ | κα i h κα | d α . By using thebeam-splitter transformation ˆ V |√ κ α i | i = | α i | κα i we can write k Γ[ M ] k = k ˆ V † Γ[ M ] ˆ V k = (cid:13)(cid:13)(cid:13)(cid:13)Z p s ( α ) | p κ α ih p κ α | ⊗ | ih | d α (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) κ s (cid:19) ⊗ | i h | (cid:13)(cid:13)(cid:13)(cid:13) = ss + 1 + κ (9)where T (¯ n ) := 11 + ¯ n ∞ X n =0 (cid:18) ¯ n n (cid:19) n | n ih n | is the thermal state with the mean photon number ¯ n .Equations (8) and (9) lead tosup E∈ EB J E ( s, κ, ξ ) ≤ ss + 1 + κ . Using this relation and Eqs. (5), (6) and (7), we havesup E∈ EB F N, ,λ ( E ) ≤ λ (1 − ξ ) 1 N + λ + 1 . (10)From the condition s ≥ λ − ξ ≤ N + λ. (11)From Eqs. (10) and (11), we obtain the upper boundsup E∈ EB F N, ,λ ( E ) ≤ N + λN + λ + 1 = F C ( N, , λ ) . (12)This bound can be achieved by the EB channel E EB ( ρ ) := π R h α | ρ | α i (cid:12)(cid:12)(cid:12) √ NαN + λ E D √ NαN + λ (cid:12)(cid:12)(cid:12) d α . We thus havesup E∈ EB F N, ,λ ( E ) ≥ F N, ,λ ( E EB ) = F C ( N, , λ ). Thisconcludes Eq. (3) with η = 1. (cid:4) It is well-known that the inseparability of two-modeGaussian states can be characterized by using the stan-dard form of the covariant matrices of the Gaussian statesunder the local Gaussian unitary operators [26]. Simi-larly, one-mode Gaussian channels can be described bya pair of 2-by-2 matrices that determines the transfor-mation of the covariant matrices and are classified into afew standard forms under the suitable unitary operationsbefore-and-after the channel [27]. Two of the standardforms are relevant to quantum domain channels. In bothforms one can find a proper set of the parameters (
N, η, λ )so that the classical limit fidelity is surpassed if the givenchannel is in quantum domain [10]. In this sense, theoutput-target fidelity with the Gaussian distributed setof coherent states is capable of detecting any one-modeGaussian channels in quantum domain.In experiments we usually obtain a finite set of mea-sured fidelities. The set of data is not enough to directlycalculate the integration in Eq. (1), and F ( E ) is esti-mated by using additional assumptions. It is better ifone can check a quantum domain criterion directly asso-ciated with the set of measured fidelities without addi-tional assumptions. In the following we present a generaltheorem to produce a quantum domain criterion associ-ated with a given set of measured fidelities. The proofof this theorem is essentially the same as above proof. Itis remarkable that the criterion can be generated by asimple calculation of a maximum eigenvalue. In-situ generation of a quantum-domain condition.—
Let us write a set of input states {| ψ i i} , a set of targetstates {| ψ ′ i i} , and a prior probability distribution { p i } with P i p i = 1. We can show that the following theoremholds: A process E is in quantum domain if¯ F [ E ; p i ; ψ i → ψ ′ i ] > d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i p i | ψ ′ i i h ψ ′ i | ⊗ | ψ i i h ψ i | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (13)where the average fidelity is given by¯ F [ E ; p i ; ψ i → ψ ′ i ] := X i p i h ψ ′ i | E ( | ψ i i h ψ i | ) | ψ ′ i i , and d is the dimension of the Hilbert space spanned bythe set of input states {| ψ i i} . Note that the experimentdetermines the set of the fidelities {h ψ ′ i | E ( | ψ i i h ψ i | ) | ψ ′ i i} whereas the choice of the probability distribution { p i } isarbitrary. Proof.—
Let {| u k i} k =0 , , , ··· ,d − be an orthonormalbasis of the d -dimensional Hilbert space. We define themaximally entangled state of a two- d level system by | Φ d i := P d − k =0 | u k i | u k i / √ d . We also define the com-plex conjugation of the d -dimensional state by | ψ ∗ i := P k h ψ | u k i | u k i = √ d h ψ | | Φ d i . Then, we can write¯ F [ E ; p i ; ψ i → ψ ′ i ]= d X i p i h ψ ′ i | A h ψ ∗ i | B E A ⊗ I B ( | Φ d i h Φ d | ) | ψ ∗ i i B | ψ ′ i i A = d Tr[
M ρ E ] (14)where we write M = P i p i ( | ψ ′ i i h ψ ′ i | ) A ⊗ ( | ψ ∗ i i h ψ ∗ i | ) B and ρ E = E A ⊗ I B ( | Φ d i h Φ d | ). The state ρ E is the stan-dard Choi-Jamiolkowski isomorphism. In the continuous-variable case, we have used a two-mode squeezed stateinstead of an unnormalizable maximally entangled state[24].If the process E is a MP scheme, ρ E = E A ⊗ I B ( | Φ d i h Φ d | ) belongs to the set of separable states [18].Hence, the maximum of the average fidelity over all MPschemes is bounded above by the maximum of the finalexpression of Eq. (14) achieved by the optimization ofthe state ρ E over separable states. This impliesmax E∈ EB ¯ F [ E ; p i ; ψ i → ψ ′ i ] ≤ d max ρ ∈ Sep.
Tr[
M ρ ] . (15)Since the optimization over separable states can be con-verted into the optimization over their partial transpose,we have max ρ ∈ Sep.
Tr[
M ρ ] = max ρ ∈ Sep.
Tr[Γ[ M ] ρ ] (16)where Γ stands for the partial transposition map again.Since the maximum over separable states is boundedabove by the maximum over all physical states, we havemax ρ ∈ Sep.
Tr[Γ[ M ] ρ ] ≤ max ρ Tr[Γ[ M ] ρ ] = k Γ[ M ] k . (17)When we choose the partial transposition of the secondsystem with respect to the basis {| u k i} , we haveΓ[ M ] = X i p i | ψ ′ i i h ψ ′ i | ⊗ | ψ i i h ψ i | . (18)Concatenating Eqs. (15)-(17) and (18) we can see thatthe maximum fidelity over all MP schemes is boundedabove by right hand side of Eq. (13). Hence, if a quantumchannel provides the fidelity higher than this limit, it isincompatible with any classical MP scheme. (cid:4) Consequently, if Ineqs. (15) and (17) are tight we canimmediately obtain the classical limit just by the calcu-lation of the maximal eigenvalue of the operator in righthand side of Eq. (13). This is the case for the followingexample.
Example.—
Let us consider the uniform set of inputstates over the d -dimensional Hilbert space and trans-formation task of a unitary map by setting the targetstate | ψ ′ i = U | ψ i for any input | ψ i . In this case it iswell-known that the classical limit fidelity is given by [3–5, 17, 28]¯ F ( d ) c := max E∈ EB Z dψ h ψ | U † E ( | ψ i h ψ | ) U | ψ i = max E∈ EB Z dψ h ψ | E ( | ψ i h ψ | ) | ψ i = 2 d + 1 . where R dψ denotes the Haar measure and the secondequation comes from the fact that the total action ofan EB channel followed by a unitary map can be de-scribed by a single EB channel. Hence, it is sufficientto consider the case that the task is the identity trans-formation, i.e., | ψ ′ i = | ψ i . For the uniform ensembleof input states, the state of Eq. (18) becomes the so-called Werner state [29, 30], and is decomposed intoΓ[ M ] = R dψ | ψ i h ψ | ⊗ | ψ i h ψ | = ( f ) / [ d ( d + 1)],where f := P i,j | u i i h u j | ⊗ | u j i h u i | is the flip operator.Hence, we have k Γ[ M ] k = d ( d +1) , and obtain the in-equality max E∈ EB ¯ F ≤ d k Γ[ M ] k = d +1 = ¯ F ( d ) c throughEqs. (15), (16), and (17). The inequality is saturated bythe EB channel E EB ( ρ ) = P j U | u j i h u j | ρ | u j i h u j | U † . This can be confirmed by the following equations:¯ F = R dψ h ψ | U † E EB ( | ψ i h ψ | ) U | ψ i = Tr[ P j | u j i h u j | ⊗| u j i h u j | ( d | Φ d i h Φ d | )] / [ d ( d + 1)] = d +1 where weused the relation R dψ | ψ i h ψ | ⊗ | ψ ∗ i h ψ ∗ | = ( d | Φ d i h Φ d | ) / [ d ( d + 1)] in the second line (see, e.g., [30]).Hence we obtain the tight classical limit. In the previousapproaches [3–5, 17, 28], the problem is treated as a typeof state estimation in Refs. [3, 4] and is also connected toa limit of optimal cloning in [28] whereas it is addressedas separability problems in Refs. [5, 17]. Our approachis somehow close to the approach of Ref. [5] in the sensethat the maximally entangled state plays a central role.In conclusion, we have presented an alternative proof ofthe quantum benchmark fidelity with respect to a Gaus-sian distributed set of coherent states. The main ideaof proof is to use a sort of the state-channel duality toassociate the average fidelity to the two-mode squeezedstate. Then, the partial transpose is utilized to make thebound on the fidelity as a separability problem. Basedon this method we have also presented a general theoremto produce a quantum-domain criterion associated with aset of measured fidelities. The theorem can be utilized ina wide class of experiments. The present method wouldbe useful to further comprehend the property of quantumchannels.R.N. acknowledges support from JSPS. [1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information , (Cambridge University Press,Cambridge, 2000).[2] C. Bennett et al. , Phys. Rev. Lett. 70, 1895 (1993).[3] S. Popescu, Phys. Rev. Lett. 72, 797 (1994).[4] S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259(1995).[5] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.Rev. A 60, 1888 (1999).[6] A.I. Lvovsky, B.C. Sanders, and W. Tittel, Nature Pho-tonics 3, 706 (2009); K. Hammerer, A.S. Sorensen, andE.S. Polzik, Rev. Mod. Phys.
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