aa r X i v : . [ qu a n t - ph ] F e b Gaussian Quantum Reading beyond the Standard Quantum Limit
Michele Dall’Arno Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya, 464-8601, Japan (Dated: April 17, 2019)Quantum reading aims at retrieving classical information stored in an optical memory with lowenergy and high accuracy by exploiting the inherently quantum properties of light. We provide anoptimal Gaussian strategy for quantum reading with phase-shift keying encoding that makes use ofsqueezed coherent light and homodyne detectors to largely outperform the Standard Quantum Limit,even in the presence of loss. This strategy, being feasible with current quantum optical technology,represents a viable prototype for a highly efficient and reliable quantum-enhanced optical reader.
I. INTRODUCTION
Digital optical memories store classical information inthe optical properties of a media. They are ubiquitous inmodern applications of classical information processing,such as CDs or DVDs. To retrieve information, opticalreaders illuminate each memory cell with a light probeand measure the outgoing signal. Clearly a tradeoff ex-ists between the energy of the source and the informationit is able to extract. On the one hand, the need for minia-turized and embedded technology push toward the use oflow energetic light sources. On the other hand, reliabil-ity is an unavoidable requirement for modern informationprocessing technology.In the general framework of quantum theory, inher-ently quantum properties of light can be exploited tomaximize the amount of retrieved information per fixedenergy of the light probe. This problem, usually referredto as quantum reading, was recently introduced by Pi-randola [1] and immediately triggered a noticeable in-terest [2–9]. However, most of the experimental as wellas commercial implementations of optical readers makeuse of a suboptimal strategy exploiting coherent lightprobes generated by a laser beam. The maximal amountof information per fixed energy that can be retrievedusing a coherent probe defines the Standard QuantumLimit [10–12] in quantum reading. Since in general Gaus-sian states and measurements are experimentally feasiblewith current optical technology, the question naturallyarises: can Gaussian quantum reading outperform theStandard Quantum Limit, thus providing highly efficientwhile practically feasible quantum reading techniques?We need to be more specific before answering this ques-tion. In applications, the most common ways to encodeclassical information are in the phase or in the amplitudeof the signal. The two methods are usually referred toas phase-shift keying (PSK) and amplitude-shift keying(ASK), respectively. Due to its simplicity, PSK encod-ing has been widely adopted in several ISO and IEEEstandards, such as wireless LAN (wifi), several creditcards, Bluetooth, and satellite communications. Despiteits widespread use, the problem of quantum reading wasmainly addressed in the context of APK encoding [1–5],while only recently the interest drew by quantum readingwith PSK encoding increased. In Ref. [8] the problem of quantum reading of two sig-nals with phase difference π was addressed in the losslessscenario. It was shown that perfect quantum reading canbe achieved by a particular class of entangled coherentstates [13]. Very recently, in Ref. [9] the scenario wasgeneralized to the case of M signals with symmetricallydistributed phases - namely, the relative phase of signal i with respect to a given seed state is 2 πi/M . In particular,it was shown that squeezed coherent states outperformcoherent ones in quantum reading of two signals withphase difference of π in the lossless scenario and with-out any constraint on the measurement applied. Whileof the utmost importance these results leave the afore-mentioned question open: can Gaussian quantum read-ing - namely, quantum reading with Gaussian probes andGaussian measurements [14] - outperform the StandardQuantum Limit in the general lossy scenario?The aim of this work is to affirmatively answer thisquestion in the context of PSK encoding. We provide anoptimal Gaussian strategy exploiting squeezed coherentlight and homodyne detection to perform quantum read-ing for any value of the phase difference between the twosignals and in the presence of loss. A comparison of theoptimal Gaussian strategy with the Standard QuantumLimit shows that the former largely outperforms the lat-ter, even in the presence of loss and taking into accountpresent technological limitations in the preparation ofhighly squeezed states. The proposed optimal strategyis suitable for implementation with current quantum op-tical technology, thus representing a viable prototype fora highly efficient and reliable quantum-enhanced opticalreader.The paper is structured as follows. In Sect. II we in-troduce the problem of quantum reading and simplify itunder the assumption that no encoding is done in theamplitude of the signal (e.g. PSK encoding). In Sect. IIIwe provide the Standard Quantum Limit and an opti-mal Gaussian strategy for quantum reading with PSKencoding. In Sect. IV we compare the optimal Gaussianstrategy with the Standard Quantum Limit and demon-strate experimental feasibility of Gaussian quantum read-ing beyond the Standard Quantum Limit. Finally wesummarize our results and propose future developmentsin Sect. V. II. QUANTUM READING
In this Section we formally introduce the quantumreading of optical memories as the problem of determin-ing the tradeoff between energy and probability of errorin the discrimination of quantum channels. Let us firstfix the notation [15].A m -modes quantum optical setup is represented byan Hilbert space H = N mi =1 H i , where H i is the Fockspace representing mode i and a i is the correspondingannihilation operator. In the following we denote with L ( H ) the space of linear operators on H . A quantum state on H is described by a density matrix ρ ∈ L ( H ), namelya positive semidefinite operator satisfying Tr[ ρ ] = 1, andin the following we pictorially represent it with '!& ρ .Up to irrelevant constants its energy is given by E ( ρ ) :=Tr[ N ρ ], where N := P mi =1 a † i a i is the number operator on H . A quantum measurement on H is described by aPOVM Π, namely a map associating to any state ρ ∈L ( H ) a probability distribution p ( j | ρ ) over a set { j } ofoutcomes, and we represent it with "% Π . The mostgeneral transformation from states to states is describedby a quantum channel C : L ( H ) → L ( H ′ ), namely acompletely positive and trace preserving map that werepresent with C .The most general strategy to discriminate a channel C i randomly chosen from a set {C i : L ( H ) → L ( H ) } di =1 with probability p i consists in probing it with a state ρ ∈ L ( H ⊗ K ), where K is an ancillary space, and mea-suring the output state with a d -outcomes [16] POVM Π,namely ρ ?>89 H C i Π =<:; K . (1)The probability of error is given by P e ( ρ, Π) :=1 − P di =1 p i p ( i |C i ( ρ )). Deriving the tradeoff between P e ( ρ, Π) and E ( ρ ) is the aim of quantum reading. Definition 1 (Quantum reading) . The optimal state ρ ∗ and the optimal measurement Π ∗ for quantum reading ofchannels {C i } distributed according to probability { p i } arethose that minimize the error-probability P e ( ρ, Π) whilesatisfying E ( ρ ) ≤ E for given energy threshold E , namely ( ρ ∗ , Π ∗ ) = arg min ( ρ, Π) P e ( ρ, Π) s.t. E ( ρ ) ≤ E. Let us specify the problem more. An m -modes quan-tum optical device [17] is described by a unitary operator U : H → H ′ relating d input optical modes with an-nihilation operators a i ’s on Fock space H i to d outputoptical modes with annihilation operators a ′ i ’s on Fockspace H ′ i . An optical device U is called linear and pas-sive if the a i ’s are linearly related to the a ′ i ’s, namely¯ a ′ = S U ¯ a , where ¯ a = ( a , . . . a d ) and S U is the scatteringmatrix associated to U . A passive device U conserves theenergy, namely for any state ρ one has E ( ρ ) = E ( U ρU † ).A simple example of quantum optical device is the ψ - phase shifter , namely a single mode device represented by the unitary P φ = exp( iφa † a ). Another example is the beamsplitter with transmittivity η , namely a 2-modes de-vice represented by the unitary B η = exp[ θ ( a † b − ab † )],where η = cos θ . For any unitary U we denote with U ( ρ ) := U ρU † the corresponding unitary channel ; wewill consider only linear and passive unitary channels.Loss, affecting any experimental quantum opticalimplementation, can be modeled by a lossy channel E η ( ρ ) := Tr [ B η ( ρ ⊗ | ih | )] with quantum efficiency η , namely a beamsplitter with signal and vacuum as in-puts, and one output mode of which is traced out, orequivalently H E η = H B η | i H "% I .
For any bipartite operator X ∈ L ( H ⊗ H ), we denotewith Tr [ X ] the partial trace over Hilbert space H .When loss affects m optical modes we write E ¯ η := N i E η i with ¯ η = ( η , . . . η m ). A lossy device U ¯ η is describedby the composition of a unitary channel U and a lossychannel E ¯ η , namely U ¯ η := U ◦ E ¯ η . Analogously, a lossysource ρ ¯ η is described by the preparation of an ideal state ρ followed by a lossy channel E ¯ η , namely ρ ¯ η := E ¯ η ( ρ ), anda lossy measurement Π ¯ η is described by an ideal one Πpreceded by a lossy channel E ¯ η , namely Π ¯ η := E ¯ η ∨ (Π),where C ∨ represents the application of channel C in theHeisenberg picture. In principle lossy channels can beabsorbed in the definition of states and measurements,nevertheless in the following it will be convenient to keepthe contribution of loss in evidence.When each memory cell can be modeled as a lossyoptical device, the quantum reading strategy given byEq. (1) becomes ρ ?>89 E ¯ α H E ¯ β i U i E ¯ γ Π =<:; K ❴ ❴ ❴ ❴ ❴✤✤✤ ✤✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤✤ ✤✤✤❴ ❴ ❴ ❴ ❴ . (2)Here and in the following, the dashed box on the leftsurrounds the lossy preparation, the one in the middle thelossy unknown device, while the one on the right the lossymeasurement. The setup in Eq. (2) can be simplifiedtaking into account the following rule of composition oflossy channels. Lemma 1 (Composition) . Given an Hilbert space H ,the composition of lossy channels E ¯ α : L ( H ) →L ( H ) and E ¯ β : L ( H ) → L ( H ) is a lossy chan-nel E ¯ η : L ( H ) → L ( H ) with efficiency ¯ η =( α β , α β , . . . , α m β m ) , namely E ¯ α ◦ E ¯ β ( ρ ) = E ¯ η ( ρ ) , ∀ ρ ∈ L ( H ) . Proof.
Since E ¯ η = N i E η i , it is sufficient to prove thestatement for a single mode, namely we prove that E α ◦ E β ( ρ ) = E η ( ρ ) with η = αβ . Denote with H and H the ancillary Fock spaces of channels E α and E β re-spectively, namely E α ( ρ ) := Tr [ B α ( ρ ⊗ | ih | )] and E β ( ρ ) := Tr [ B β ( ρ ⊗ | ih | )]. We want to prove thatfor any α and β one has the following equivalence be-tween channels:Tr [( B β ⊗ I )( B α ⊗ I )( ρ ⊗ | ih | )] (3)= Tr [( B η ( ρ ⊗ | ih | )] , namely | i H B α "% I H B β | i H "% I = H B η | i H "% I . (4)By direct computation one obtains [18]( B β ⊗ I )( B α ⊗ I ) = ( B γ † ⊗ I )( B η ⊗ I )( B δ ⊗ I ) , (5)where γ = (1 − α ) / (1 − αβ ) and δ = γβ , namely H B α H B β H = H B η H B γ † B δ H (6)Notice that for clarity the ordering of Fock spaces H and H is exchanged in the two sides of Eq. (6).Upon replacing Eq. (5) into Eq. (3) [or equivalentlyEq. (6) into Eq. (4)] and noticing that B γ † | i = | i ,one gets the statement by unitarily invariance of trace.Lemma 1 allows to absorb channel E ¯ α in Eq. (2) intothe definition of channels E ¯ β and E ¯ γ , so Eq. (2) becomes ρ ?>89 H E ¯ β ′ i U i E ¯ γ ′ Π =<:; K ❴ ❴ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤✤ ✤✤✤❴ ❴ ❴ ❴ ❴ . (7)To further simplify the problem we need to be morespecific about the way information is encoded. Informa-tion can be stored in (i) quantum efficiencies ¯ β ′ i ’s, and(ii) in unitary channels U i ’s. In the former case (i) logi-cal d -it i is encoded into U ◦ E ¯ β ′ i , while in the latter case(ii) logical d -it i is encoded into U i ◦ E ¯ β ′ . Clearly, also acombination of the two encodings is possible. An exam-ple of the former case (i) is ASK encoding , where logical d -it i is encoded into a single-mode lossy optical channel E η i . An example of the latter case (ii) is PSK encod-ing , where logical d -it i is encoded into a lossy φ i -phaseshifter P φ i ◦ E η . In this work we address the problem ofquantum reading in the presence of loss with encoding oftype (ii), and we have the following Lemma. Lemma 2 (Commutation) . Given an Hilbert space H ,for any lossy channel E ¯ η : L ( H ) → L ( H ) such that ¯ η =( η, . . . η ) is a constant vector and for any unitary linearand passive channel U : L ( H ) → L ( H ) , one has that E ¯ η commutes with U , namely U ◦ E ¯ η ( ρ ) = E ¯ η ◦ U ( ρ ) , ∀ ρ ∈ L ( H ) . (8) Proof.
Denote with K = N i K i the ancillary Hilbertspace in the definition of lossy channel E ¯ η , namely E ¯ η ( ρ ) = Tr B [ N i B ηi ( ρ ⊗ σ K )] where σ K = N i | ih | i ∈L ( K ) and beamsplitter B ηi acts on Fock spaces H i and K i . Then the statement can be reformulated as H B η U | i K "% I. . . H m B ηm | i K m "% I = H U B η | i K "% I. . . H m B ηm | i K m "% I (9)It follows by direct computation that O i B ηi ( U ⊗ U K ) = ( U ⊗ U K ) O i B ηi , (10)where U K denotes channel U acting on Hilbert space K ,or equivalently H m U B ηm . . . . . . H B η K U K . . . . . . K m = H m B ηm U . . . H B η K U K . . . K m (11)Upon replacing Eq. (10) into Eq. (8) [or equivalentlyEq. (11) into Eq. (9)] the statement follows, namely U ◦ E ¯ η ( ρ ) = Tr K [( U ⊗ I K ) O i B ηi ( ρ ⊗ σ K )] =Tr K [( U ⊗ U K ) O i B ηi ( ρ ⊗ σ K )] =Tr K [ O i B ηi ( U ⊗ U K )( ρ ⊗ σ K )] =Tr K [ O i B ηi ( U ⊗ I K )( ρ ⊗ σ K )] = E ¯ η ◦ U ( ρ ) , where second equality holds due to unitarily invariance oftrace and second-to-last holds since σ K = N i | ih | i .Whenever the hypothesis of Lemma 2 is satisfied,namely losses affecting each mode of unitary channel U are equal - this is true for example when modes are imple-mented by optical-fiber paths of the same length - apply-ing Lemma 2 and Lemma 1 again the quantum readingstrategy in Eq. (7) can be rewritten as ρ ?>89 H U i E ¯ η Π =<:; K ❴ ❴ ❴ ❴ ❴✤✤✤ ✤✤✤❴ ❴ ❴ ❴ ❴ . (12)We proved that when the information is not encoded inthe efficiency of each memory cell, quantum reading oflossy optical devices can be recasted to the discrimina-tion [6, 7, 22–24] of lossless optical devices with a lossydetector. III. GAUSSIAN QUANTUM READING
In this Section we derive the Standard Quantum Limit- namely, the optimal strategy using coherent statesand homodyne measurements - and an optimal Gaussianstrategy - using squeezed coherent states and homodynemeasurements - for quantum reading with PSK encodingfor any value of the phase difference between signals en-coding logical 0 and 1 and in the presence of loss. Let usfirst fix the notation [25–28].A coherent state | α i is obtained by applying the dis-placement operator D ( α ) := e αa † − α ∗ a to the vacuumstate | i , namely | α i := D ( α ) | i . The energy of a co-herent state is given by E ( α ) = | α | . A squeezed coher-ent state [29] | α, ξ i is obtained by subsequently applyingthe squeezing operator S ( ξ ) := e ( ξ ∗ a − ξa † ) and the dis-placement operator D ( α ) to the vacuum state | i , namely | α, ξ i := D ( α ) S ( ξ ) | i . The energy of a squeezed coherentstate is given by E ( α, φ ) = | α | + sinh | ξ | .For any state ρ it is useful to introduce its Wignerfunction W ρ ( x, p ) defined as W ρ ( x, p ) := 1 π Z ∞−∞ h x − y, ψ | ρ | x + y, ψ i e − i py dy, so the Wigner function of squeezed coherent state | e iφ a, e iθ r i is given by W | α,ξ i ( x, p ) = 1 π e − ( x ′ + p ′ ) . where (cid:18) x ′ p ′ (cid:19) = (cid:18) e − r cos θ sin θ − sin θ e r cos θ (cid:19) (cid:18) xp (cid:19) − (cid:18) a cos φa sin φ (cid:19) , and that of coherent state | e iφ a i can be obtained as aparticular case by setting r = 0.The POVM Π ψ describing an homodyne measure-ment [30, 31] along quadrature X ( ψ ) := √ ( e iψ a + e − iψ a † ) associates to state ρ the probability distribution p ( x | ρ, ψ ) given by p ( x | ρ, ψ ) = Z ∞−∞ dpW ρ ( x ′ , p ′ ) , where (cid:18) x ′ p ′ (cid:19) = (cid:18) cos ψ − sin ψ sin ψ cos ψ (cid:19) (cid:18) xp (cid:19) . In the following we denote with p ( x | α, ξ, ψ ) the condi-tional probability distribution of outcome x of homodynemeasurement Π ψ with efficiency η given input squeezedcoherent state | α, ξ i with α = e iφ a and ξ = e iθ r , givenby the Gaussian p ( x | α, ξ, ψ ) := s πσ ( ξ, ψ ) e − ( x − x α,ψ ))2 σ ξ,ψ ) , (13) where x ( α, ψ ) := a cos( ψ − φ ) ,σ ( ξ, ψ ) := s e − r cos ( ψ − θ e r sin ( ψ − θ − η η , while the analogous distribution for coherent state | e iφ a i can be obtained as a particular case by setting r = 0.The term − η η in the definition of σ ( ξ, ψ ) is due to thefact that the conditional probability distribution in thepresence of loss is the convolution of the ideal conditionalprobability distribution with a Gaussian with variance − η η (see Refs. [32–35]).We introduce now the problem of quantum readingwith PSK encoding with a lossy source of squeezed co-herent states E α ( | α, ξ ih α, ξ | ) and lossy homodyne mea-surement E γ ∨ (Π ψ ). With the same notation as in Defi-nition 1, we assume that the (binary) information is en-coded into lossy phase shifters P δ i ◦ E β with i = 0 , p = p = 1 /
2. Since homodyne measurement has infinitelymany outcomes, we introduce a classical postprocessing J (classical wires being denoted with ) outputting bi-nary outcome j , so that j is our guess for the value i ofthe bit encoded into the unknown phase shifter. Thenthe strategy in Eq. (2) becomes | α, ξ i E α E β P δ i E γ *-+, Π ψ J ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ . (14)Clearly, for fixed probe Gaussian quantum reading re-duces to Gaussian state discrimination [36–41].Due to Lemmas 1 and 2 lossy channels can be absorbedin the definition of lossy POVM, namely the strategy inEq. (14) can be rewritten as | α, ξ i P δ i E η *-+, Π ψ J ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ , (15)with η = αβγ . Notice that it is not restrictive to as-sume δ = 0, since a phase shifter P − δ can be reab-sorbed in the definition of input state or homodyne mea-surement, finally leading to the discrimination of phaseshifters P δ = I and P δ = P δ , where δ = − δ δ , with δ ∈ [0 , π ]. Notice that the application of phase shifter P δ to a squeezed coherent state | α, ξ i gives [42] the squeezedcoherent state | e iδ α, e i δ ξ i . Any classical postprocessingof outcome x can be described by a function q ( j | x ) thatevaluates to 1 if one guesses j from outcome x and to 0otherwise, so the probability of error is given by P e = 12 Z dx q (1 | x ) p ( x | α, ξ ψ ) + q (0 | x ) p ( x | e iδ α, e i δ ξ, ψ ) . (16)In the following, we denote with erf( x ) :=2 / √ π R x dt e − t the error function and with Ω( x ) theunit step function that evaluates to 1 if x ≥ Proposition 1 (Standard Quantum Limit) . For any en-ergy threshold E , any efficiency η and any phase δ , theoptimal coherent state | α ∗ i with α ∗ = e iφ ∗ a ∗ and the op-timal homodyne measurement Π ψ ∗ with efficiency η forGaussian quantum reading of phase shifters {I , P δ } withenergy E ( α ) ≤ E are given by φ ∗ = − δ/ , a ∗ = √ E ,and ψ ∗ = π/ . The optimal tradeoff is given by P e = h (cid:16) x ( α ∗ ,ψ ∗ ) σ (0 ,ψ ∗ ) (cid:17)i .Proof. The minimum of P e in Eq. (16) is attained when q ( y | x ) = Ω[( − ) y ( p ( x | α, , ψ ) − p ( x | e iδ α, , ψ ))]and thus P e is given by the overlap between the Gaussians p ( x | α, , ψ ) and p ( x | e iδ α, , ψ ), namely P e = 12 (cid:20) Z A dx (cid:0) p ( x | e iδ α, , ψ ) − p ( x | α, , ψ ) (cid:1)(cid:21) , where A := { x | p ( x | α, , ψ ) ≥ p ( x | e iδ α, , ψ ) } .From Eq. (13) it follows that P e depends on the phases φ and ψ only through the sum ψ − φ , so without loss ofgenerality we fix ψ = ψ ∗ , namely homodyne measure-ment is performed along quadrature P .Notice that for any coherent state | e iφ a i such that E ( e iφ a ) ≤ E , one has that the state | e iφ a ∗ i is suchthat E ( e iφ a ∗ ) = E and P e ( e iφ a ∗ ) ≤ P e ( e iφ a ). In-deed, while σ (0 , ψ ∗ ) does not depend on a , one has that | x ( e i ( δ + φ ) a ∗ , ψ ∗ ) − x ( e iφ a ∗ , ψ ∗ ) | ≥ | x ( e i ( δ + φ ) a, ψ ∗ ) − x ( e iφ a, ψ ∗ ) | . Then the optimal value for parameter a is a ∗ .Notice that for any coherent state | e iφ a ∗ i suchthat E ( e iφ a ∗ ) = E , one has that the state | e iφ ∗ a ∗ i is such that E ( e iψ ∗ a ∗ ) = E and P e ( e iφ ∗ a ∗ ) ≤ P ( e iφ a ). Indeed, while σ (0 , ψ ∗ ) does not depend on a , one has that | x ( e i ( δ + φ ∗ ) a ∗ , ψ ∗ ) − x ( e iφ ∗ a ∗ , ψ ∗ ) | ≥| x ( e i ( δ + φ ) a ∗ , ψ ∗ ) − x ( e iφ a ∗ , ψ ∗ ) | . Then the optimalvalue for phase φ is φ ∗ .We can now introduce the main result of this work,namely an optimal strategy for quantum reading withPSK encoding with squeezed coherent states and homo-dyne measurements. Proposition 2 (Optimal Gaussian quantum reading) . For any energy threshold E , any efficiency η and anyphase δ , the optimal squeezed coherent state | α ∗ , ξ ∗ i with α ∗ = e iφ ∗ a ∗ and ξ ∗ = e iθ ∗ r ∗ and the optimal homo-dyne measurement Π ψ ∗ with efficiency η for Gaussianquantum reading of phase shifters { I, P δ } with energy E ( α ) ≤ E are given by φ ∗ = − δ/ , a ∗ = p E − sinh r ∗ , ψ ∗ = π/ . Whenever θ ∗ = − δ − π Ω( π − δ ) one has r ∗ = 12 ln r(cid:16) E + 1 + − η η (cid:17) − E ( E + 1) cos θ ∗ + cos θ ∗ (cos θ ∗ + 1)(2 E + 1) + − η η , and the optimal tradeoff is given by P e = h (cid:16) x ( α ∗ ,ψ ∗ ) σ ( ξ ∗ ,ψ ∗ ) (cid:17)i .Proof. The minimum of P e in Eq. (16) is attained when q ( y | x ) = Ω[( − ) y ( p ( x | α, ξ, ψ ) − p ( x | e iδ α, e i δ ξ, ψ ))]and thus P e is given by the overlap between the Gaussians p ( x | α, ξ, ψ ) and p ( x | e iδ α, e i δ ξ, ψ ), namely P e = 12 (cid:20) Z A dx (cid:0) p ( x | e iδ α, e i δ ξ, ψ ) − p ( x | α, ξ, ψ ) (cid:1)(cid:21) , (17)where A := { x | p ( x | α, ξ, ψ ) ≥ p ( x | e iδ α, e i δ ξ, ψ ) } .From Eq. (13) it follows that P e depends on the phases φ , θ and ψ only through the sums ψ − φ and ψ − θ , sowithout loss of generality we fix ψ = ψ ∗ , namely homo-dyne measurement is performed along quadrature P .Notice that for any squeezed coherent state | e iφ a, ξ i such that E ( e iφ a, ξ ) ≤ E , one has that the state | e iφ a ( r ) , ξ i with a ( r ) := p E − sinh r is such that E ( e iφ a ( r ) , ξ ) = E and P e ( e iφ a ( r ) , ξ ) ≤ P e ( e iφ a, ξ ).Indeed, while σ ( ξ, ψ ∗ ) does not depend on a , onehas that | x ( e i ( δ + φ ) a ( r ) , ψ ∗ ) − x ( e iφ a ( r ) , ψ ∗ ) | ≥| x ( e i ( δ + φ ) a, ψ ∗ ) − x ( e iφ a, ψ ∗ ) | . Then constraint E ( α, ξ ) ≤ E can be recasted without loss of generalityinto E ( α, ξ ) = E , allowing to eliminate parameter a bywriting a = a ( r ).Notice that for any squeezed coherent state | e iφ a ( r ) , ξ i such that E ( e iφ a ( r ) , ξ ) = E , one has that the state | e iφ ∗ a ( r ) , ξ i is such that E ( e iφ ∗ a ( r ) , ξ ) = E and P e ( e iφ ∗ a ( r ) , ξ ) ≤ P ( e iφ a ( r ) , ξ ). Indeed, while σ ( e iθ r, ψ ∗ )does not depend on φ , one has that | x ( e i ( δ + φ ∗ ) a ( r ) , ψ ∗ ) − x ( e iφ ∗ a ( r ) , ψ ∗ ) | ≥ | x ( e i ( δ + φ ) a ( r ) , ψ ∗ ) − x ( e iφ a, ψ ∗ ) | .Then the optimal value for phase φ is φ ∗ .When θ = θ ∗ , an explicit evaluation of Eq. (17)leads to P e = h x ( e iφ ∗ a ( r ) ,ψ ∗ ) σ ( e iθ ∗ r,ψ ∗ ) ) i , and the opti-mal value for parameter r can be obtained minimizing P e . Since erf( x ) is a monotone increasing function in x ,minimizing P e is equivalent to minimizing x ( e iφ ∗ a ( r ) ,ψ ∗ ) σ ( e iθ ∗ r,ψ ∗ ) .It is lengthy but not difficult to verify that the equation ∂∂r x ( e iφ ∗ a ( r ) ,ψ ∗ ) σ ( e iθ ∗ r,ψ ∗ ) = 0 admits the only solution r = r ∗ and that r ∗ is a minimum, so the statement remainsproved.Notice that for α = α ∗ and ψ = ψ ∗ , for any r the onlychoices of θ such that σ ( ξ, ψ ∗ ) = σ ( e i δ ξ, ψ ∗ ) are θ = − δ and θ = − δ − π , and the choice θ = θ ∗ given by Prop. 2corresponds to the one minimizing σ ( ξ, ψ ∗ ) (see next Sec-tion and Fig. 1). We obtained numerical evidence thatthe choice θ = θ ∗ is optimal whenever (i) δ ≥ π/ E and any efficiency η , or (ii) when δ ≤ π/ E . However, when δ ≤ π/ E the choice θ = θ ∗ is not optimalanymore, and the second statement in Prop. 2 can notbe applied.When δ = π one has θ ∗ = − π , and thus the expres-sion for r ∗ in Prop. 2 is not defined for η = 1. In thiscase the limit η → r ∗ = arcsinh[ E/ √ E + 1]. Notice that this particularexpression for r ∗ when δ = π and η = 1 is optimal alsofor hybrid quantum reading with Gaussian probe and ar-bitrary measurement (see Ref. [9]). IV. OPTIMAL GAUSSIAN STRATEGY VERSUSSTANDARD QUANTUM LIMIT
This Section is devoted to the comparison betweenthe Standard Quantum Limit and the optimal Gaussianstrategy for quantum reading derived in Sect. III.Figure 1 provides a phase-space representation of theWigner function of the states attaining the StandardQuantum Limit (light gray circles) and the optimal Gaus-sian strategy (bold-line ellipses) as given by Props. 1 and2, respectively. The Figure provides an intuitive under-
FIG. 1. Phase-space representation of the Wigner functionof the optimal Gaussian state (bold-line ellipses) | α ∗ , ξ ∗ i forquantum reading of a δ -phase shifter for δ = π, π/ , π/
6, asgiven by Prop. 2. The optimal homodyne measurement isalong quadrature P (horizontal in Figure). The optimal co-herent state (light gray circles) attaining the Standard Quan-tum Limit is also depicted for each value of δ . The optimalstrategy outperforms the Standard Quantum Limit for anyvalue of δ except δ = π/
2, where the two strategies coincide.The maximal advantage over the Standard Quantum Limit isachieved for δ = π and in the regime of small δ . standing of the advantage given by squeezed coherentstates over coherent states. On the one hand, for fixedenergy the more two states are squeezed, the more theirWigner functions get “closer” and thus hardly distin-guishable. On the other hand, when squeezing is per-formed approximately along the quadrature being mea-sured, the Wigner functions become “thinner” as squeez-ing increases. These two phenomena are clearly contrast-ing, but when the optimal tradeoff is taken, a dramaticimprovement in the precision of the discrimination is ex- perienced, as discussed in the next paragraphs. The onlyvalue of the phase δ for which squeezed coherent states donot provide an advantage over coherent states is δ = π/ δ ∼ π and δ ∼
0, since in both cases the Wigner functionof the optimal state ρ ∗ is squeezed approximately alongthe quadrature which is measured - this is rigorously trueonly for δ = π .The advantage of using the δ = π encoding is obvious- it is clearly the choice giving the lower tradeoff betweenenergy and probability of error (see later discussion andFig. 3). Different choices, and in particular the regime δ ∼
0, can be exploited in several applications for tuningthe minimum energy required by a Gaussian reader toretrieve some information. For example, a read-only-oncememory could be implemented by triggering a device toself erase after being exposed to a radiation of energy E , where E is the minimum energy required to read thememory with a given probability of error P e , as given byProp. 2. Notice that the quantum reading problem inthe regime δ ∼ r ∗ givenby Prop. 2 is plotted as a function of E for differentvalues of δ and η . Remarkably, even for very low valuesof the probability of error P e , the corresponding r ∗ iscomparable with experimentally attainable values of thesqueezing parameter, in particular in the regime δ ∼ π .For example, when δ = π , setting the energy E = 4 andthe efficiency of the homodyne measurement η = 0 . P e ∼ . · − with optimalsqueezing parameter given by r ∗ ∼ .
0. In Refs. [46, 47],the generation of squeezed coherent states with squeezingparameter up to r dB ∼ . dB is reported - namely suchthat r = r dB / (20 log e ) ∼ . δ and η . As the Figure clearly shows, the optimalGaussian strategy for quantum reading largely outper-forms the Standard Quantum Limit, allowing to dramat-ically reduce the probability of error P e for fixed energy E , even in the presence of a realistic amount of loss andwith realistic limitations in the squeezing parameter. Forexample, when δ = π , setting E = 4 and η = 0 . P e ∼ . · − , while the Standard QuantumLimit gives P e ∼ . · − . V. CONCLUSION
In this work we addressed the problem of quantumreading with Gaussian states and homodyne measure-ments. We showed that when no information is encodedin the amplitude of the signal, quantum reading in thepresence of loss can be recasted to the discrimination s i nh (r * ) Ea) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 s i nh (r * ) Eb)
FIG. 2. (Color online) Optimal value sinh ( r ∗ ) of squeezingenergy for Gaussian quantum reading of phase shifters I and P δ as given by Prop. 2 versus the energy threshold E , for δ = π/
180 [Fig. a)] and δ = π [Fig. b)] and for detectionefficiency η = 1 (upper red line in both figures) and η = 0 . r ∼ .
5, namely squeezing energy sinh ( r ) ∼ .
5, isreported [46, 47]. of unitary devices with low energy and high accuracy.We provided the optimal Gaussian strategy for quantumreading with PSK encoding for any value of the phasedifference between the two signals encoding logical 0 and1 and we showed that it dramatically outperforms theStandard Quantum Limit even in the presence of lossand under realistic assumptions on the practically feasi-ble squeezing parameters. The optimal Gaussian strat-egy, consisting in probing a phase shifter with a properlytuned squeezed coherent state and performing an homo-dyne measurement on the output state, is suitable forexperimental implementation with current quantum op-tical technology and represents a proof of principle foran highly efficient and reliable quantum-enhanced opti- P e Ea) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.5 1 1.5 2 P e Eb)
FIG. 3. (Color online) Tradeoff between energy E and prob-ability of error P e in the Gaussian quantum reading of phaseshifters I and P δ , for δ = π/
180 [Fig. a)] and δ = π [Fig.b)], and for detection efficiency η = 1 (lower red line in bothfigures) and η = 0 . δ = π and η = 1,the optimal tradeoff for hybrid quantum reading with Gaus-sian probe and arbitrary measurement (see Ref. [9]) given by P e = (1 − √ − e − E ( E +1) ) is depicted (black dots) in Fig.b). The Figure suggests that the performance of the optimalGaussian strategy is close to that of the hybrid strategy inthe relevant regime of low probability of error. cal reader.A natural generalization of the problem is to allow fordifferent detectors - e.g. not only homodyne, but also het-erodyne or double homodyne - and to investigate whetherthey can further improve the performance of Gaussianquantum reading.Moreover, in Ref. [6] it was proven that the StandardQuantum Limit for quantum reading is attainable with-out an ancillary space - basically, because linear opticaldevices can not create entanglement when their inputsare coherent states. Then, for the purpose of this work -namely, proving that Gaussian quantum reading can out-perform the Standard Quantum Limit - it was sufficientto consider the setup given by Eq. (15).Nevertheless, since linear optical devices can createentanglement when their inputs are squeezed coherentstates [48], a further generalization of the problem is thatof entangled assisted Gaussian quantum reading, wherethe setup in Eq. (15) is replaced by | α , ξ i V H P δ i W E ¯ η *-+, Π J | α , ξ i K *-+, Π ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤✤✤ ✤✤✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤✤✤✤ ✤✤✤✤❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ , and the optimization is performed over squeezed coher-ent states | α i , ξ i i , entangling devices V and W , homodyne measurements Π i , and classical postprocessing J . To un-derstand whether entangled assisted Gaussian quantumreading can further improve the performance of Gaussianquantum reading, one could make use of the numericaltechniques developed in Ref. [6]. ACKNOWLEDGMENTS
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