Parity-Swap State Comparison Amplifier for Schrödinger Cat States
PParity-Swap State Comparison Amplifier for Schr¨odinger Cat States
Gioan Tatsi, Luca Mazzarella and John Jeffers
Department of Physics, University of Strathclyde,John Anderson Building, 107 Rottenrow, Glasgow G4 0NG, U.K. (Dated: March 1, 2021)We propose a postselecting parity-swap amplifier for Schr¨odinger cat states that does not requirethe amplified state to be known a priori. The device is based on a previously-implemented statecomparison amplifier for coherent states. It consumes only Gaussian resource states, which providesan advantage over some cat state amplifiers. It requires simple Geiger-mode photodetectors andworks with high fidelity and approximately twofold gain.
I. INTRODUCTION
Schr¨odinger cat states, superpositions of two coherentstates with coherent amplitudes of the same magnitudebut different phases, have been widely studied for thesignificant role they could play in quantum information[1–5], computation [6, 7] and in fundamental tests [8–11]as resource states. For example Ralph et al. [3] showedthat cat states can be used to implement qubit gates inan all optical quantum computation scheme, where thelogical qubits are encoded in the phase of the coherentstate complex amplitude and Jeong [8] considered thepossibility of testing the Bell inequalities using cat-likestates as resources.The nongaussian character of cat states renders themboth challenging to engineer and fragile. Traditionallythey have been generated using a combination of linearoptics and measurement postselection techniques [12–19].One simple implementation relies on subtracting a pho-ton from a squeezed vacuum state. This generates a stateclose to an odd cat state, but only for small coherent am-plitudes ( α ≤ . II. SCHR ¨ODINGER CAT STATES
Optical coherent states are superpositions of all photonnumbers, | α (cid:105) = e −| α | / ∞ (cid:88) n =0 α n √ n ! | n (cid:105) , (1)where α is a complex number known as the coherentamplitude [32]. The states therefore have photon num-ber probabilities that satisfy a Poisson distribution withmean | α | . Different coherent states are not orthogonaland so cannot be distinguished perfectly and determin-istically by any measurement . They approach orthog-onality if their coherent amplitudes are sufficiently farapart and the states can be distinguished in this limit.Coherent states of low mean photon number naturallydo not have very different amplitudes and their indistin-guishability is the only quantum property that can bemeasured. For this reason coherent states are often de-scribed as the most classical of quantum states.Optical Schr¨odinger cat states are superpositions oftwo coherent states with coherent amplitudes of the samemagnitude but different phases, typically opposite, | α θ (cid:105) = 1 √ θe − | α | ( | + α (cid:105) + e iθ | − α (cid:105) ) . (2)For high values of the mean photon number | α | catstates are superpositions of states that can be dis-tinguished macroscopically. This is in analogy withSchr¨odinger’s original gedanken experiment, which was a r X i v : . [ qu a n t - ph ] F e b introduced [33] in order to advocate the difficulties inapplying quantum mechanics to the macroscopic world.Typically we set θ = { , π } , obtaining the so called even and odd cat states | α θ (cid:105) = 1 √ θe − | α | ( | α (cid:105) + e iθ | − α (cid:105) )= e −| α | √ θe − | α | ∞ (cid:88) n =0 α n + e iθ ( − α ) n √ n ! | n (cid:105) = √ cosh | α | (cid:80) ∞ n =0 α n √ (2 n )! | n (cid:105) θ = 0 √ sinh | α | (cid:80) ∞ n =0 α n +1 √ (2 n +1)! | n + 1 (cid:105) θ = π , (3)so the even cat state | α (cid:105) = | α + (cid:105) contains only even pho-ton numbers and conversely the odd state | α π (cid:105) = | α − (cid:105) only has odd photon numbers. The even and odd statesare mutually orthogonal and, despite the fact that theyare superpostions of almost classical states, the gaps inthe photon number distribution are a signature of non-classicality, whatever the value of | α | . They are also non-gaussian, as attested by the negativity of their Wignerfunction, another widely accepted measure of nonclassi-cality [34]. III. STATE COMPARISON AMPLIFIER
As previously mentioned, the state comparison ampli-fier (SCAMP) is a nondeterministic amplifier for coherentstates that works with high gain, provides high-qualityoutput and requires only classical resources. The back-bone of the amplifier (Fig. 1) is the mature techniqueof state comparison [35], which has been used in the set-ting of multiple phase encoding quantum receivers [36–38]. In our scheme the coherent state to be amplifiedis drawn uniformly at random from a discrete set, say {| + α (cid:105) , | − α (cid:105)} and is mixed with a guess coherent statedrawn from the set {| + β (cid:105) , | − β (cid:105)} on a beamsplitterwith transmission and reflection coefficients t and r re-spectively. Here we assume that the reflection and trans-mission coefficients are real, so the light incident on thelower input arm picks up a π phase shift upon reflection.The beamsplitter relates the output mode annihilationoperators to those for the input mode as follows (cid:18) ˆ a out ˆ b out (cid:19) = ˆ U BS (cid:18) ˆ a in ˆ b in (cid:19) (4)where ˆ U BS = (cid:18) t − rr t (cid:19) (5)is the beamsplitter transformation matrix with realtransmission and reflection coefficients.Then if the input states chosen are | α (cid:105) and | β (cid:105) thestate after the beamsplitter isˆ U BS | α, β (cid:105) → | t α − r β, t β + r α (cid:105) (6) ✗ |𝛽⟩ |𝛼⟩ 𝜌 𝑜𝑢𝑡 𝑡 ≈ 1 ✓ |0⟩ 𝑡 , 𝑟 𝐷 𝐷 FIG. 1: State Comparison Amplifier comprised of acomparison stage and a subtraction stage.From eq. 6 it is not difficult to see that the amplitudein the mode in the detector arm vanishes if t α = r β , i.ewhen the guess state amplitude is β = t αr . The detectorcannot fire. We choose this amplitude for β and use thenon-firing of the detector as an indication of the success ofthe amplification process and accept the output wheneverit occurs.If, however, the input and guess states are oppositely-phased, say α and − t αr , light leaks into the detectorarm and the detector can fire, in which case the outputstate is rejected. The detector may not fire, however, be-cause the coherent state is not orthogonal to the vacuum, (cid:104) α | (cid:105) = (cid:104)− α | (cid:105) = e − | α | . The detector not firing is there-fore an imperfect indication that the correct guess statehas been chosen. This can limit how well the outputstate mimics to the nominal amplified output. Condi-tioned on the detector not registering an event, the out-put of SCAMP after the comparison state is a mixture ofthe nominal correctly amplified state | gα (cid:105) = | α/r (cid:105) andanother, lower amplitude coherent state, with their re-spective probabilities biased towards the amplified state.We use the quantum fidelity, F, of the output state tothe nominal amplified state as a measure of the outputquality [39]. As one of the states is pure, here the fidelityis given by F =Tr[ | gα (cid:105)(cid:104) gα | ρ out ] (7)= (cid:90) C d ξπ χ gα ( − ξ ) χ out (+ ξ ) (8)where in the second line the fidelity is written in termsof the characteristic functions of the nominal and actualoutput states. The symmetrically ordered characteristicfunction of a general state ρ is defined by χ ρ ( ξ ) = Tr[ ρ ˆ D ( ξ )] (9)in which ξ is a complex parameter, ˆ D ( ξ ) = exp( ξ ˆ a † − ξ ∗ ˆ a )is the displacement operator [40]. The characteristicfunction is in one-to-one correspondence with the den-sity operator as it contains all of the information neces-sary to reconstruct the state. We employ this formalismthroughout the paper leaving calculational details for theAppendix.The fidelity can be increased after the comparisonstage by using another mature technique known as pho-ton subtraction [41], a filtration process that is im-plemented by allowing the output from the compari-son beamsplitter to fall on a second highly transmitting( t ≈
1) beamsplitter, with vacuum input in the secondmode. The reflected mode contains a Geiger mode detec-tor and this time the output is accepted whenever thissecond detector fires, an event signifying that the signalcontained at least one photon. The higher the mean pho-ton number of the input the more likely the detector isto fire, so this filters out states with lower mean photonnumbers. The improvement in the output quality comesat the cost of a lower success probability of the overallscheme. If the comparison beamsplitter is 50:50 the in-correct output of the comparison stage is the vacuum, soall of the incorrect output is filtered by subtraction leav-ing only the amplified state | ± gα (cid:105) = | ± √ α (cid:105) , a perfecttwofold gain. IV. SCAMP FOR SCHR ¨ODINGER CAT STATES
Motivated by the high gain and fidelity of the SCAMPfor sets of individual coherent states, we investigate amodified SCAMP system for the amplification of catstates. In the first subsection we will introduce an up-dated SCAMP scheme for the amplification of cat stateswhich only utilises Gaussian resources, beamsplitters andGeiger-mode detectors. In the second we will present theresults.
A. Amplification Scheme
The amplification scheme, as one can see in Fig. 2is an updated SCAMP in which the input state is acat state chosen uniformly at random from the set ofstates {| α + (cid:105) , | α − (cid:105)} while the guess state this time is thesqueezed vacuum | ζ (cid:105) state given by: | ζ (cid:105) = √ sech s ∞ (cid:88) m =0 (cid:112) (2 m !) m ! (cid:18) −
12 exp [ iθ ] tanh s (cid:19) m | m (cid:105) (10)where ζ = e iθ s , 0 ≤ s ≤ ∞ and 0 ≤ θ ≤ π . Note thatthe “guess” state is always | ζ (cid:105) irrespective of the inputstate. This particular choice is motivated by the fact thatthe squeezed vacuum state | ζ (cid:105) , similarly to an even catstate, is a superposition of even photon numbers only,and has a high overlap with | α + (cid:105) , given by ✗ |𝜁⟩|𝛼 ± ⟩ 𝜌 𝑜𝑢𝑡
50: 50 𝑡 ≈ 1 ✓ |0⟩ 𝐷 𝐷 FIG. 2: State Comparison Amplifier for Cat Statescomprising of a comparison stage (50 : 50 beamsplitter)and a subtraction stage. |(cid:104) α + | ζ (cid:105)| = exp (cid:2) − ( α + α ∗ ) tanh s/ (cid:3) cosh s cosh | α | (11)for values of coherent amplitude up to α ≈ | ζ (cid:105) with | α − (cid:105) is zeroas the latter is a state that contains only superpositions ofodd photon numbers in the photon number basis. Oneperhaps would expect that such a mixing of odd andeven photon numbers in a beamsplitter would cause thequality of the output state to deteriorate but, as we showin the appendix, in this case such a mixing process, inconjunction with postselection in the detector arm on nocounts, acts as a quantum channel that evolves the inputstates into squeezed states at the output. In Fig. 3 weplot the squeezing magnitudes that maximise the fidelityof | α + (cid:105) to | ζ (cid:105) for different values of coherent amplitude α and also the fidelity of | ζ (cid:105) with | α + (cid:105) . The squeezingvalues that maximise the fidelity for a given coherentamplitude admit a simple expression when we considerthe coherent amplitude α to be real, s = − sinh − (2 α )2 . (12)The amplification protocol then proceeds as follows1. The input cat state to be amplified and thesqueezed vacuum state are mixed at a 50 : 50 beam-splitter and a Geiger mode detector monitors thepresence or absence of light in one of the output α ( db ) α | s | (a) α (b)FIG. 3: (a) Optimal squeezing in dB as a function ofinput coherent amplitude α , ( s db = − e s ]) . Theinset plot shows the absolute value of the optimalsqueezing in terms of the squeezing parameter.(b) Corresponding maximum fidelity as a function ofinput α .modes. If no event is registered, the output of thiscomparison stage is passed to the next stage.2. The output mode impinges on a second highlytransmissive beamsplitter whose other input is thevacuum state. If the second detector registers aclick the amplification is considered successful andthe output is accepted. Otherwise it is discarded.The condition for successful amplification then can beformally given by P S = P ( D = (cid:55) , D = (cid:88) ) (13)which is the joint probability of the second detectorclicking and the first detector not registering a click. Weshow in the appendix that the comparison stage acts asa quantum channel that evolves input states to squeezedstates. In the case of input cat states the output issqueezed cat states,i.e states of the form ˆ S ( ζ ) | α ± (cid:105) whereˆ S ( ζ ) is the squeezing operator. The photon subtraction stage is needed to amplify the cat states. The details ofthe computation have been left for the appendix.Fig. 3 shows that the amount of squeezing required tooptimise the fidelity between the output and an ideal catstate for a wide range of input cat state sizes is moderate.For example, if | α | = 1 the amount of squeezing requiredis 6 dB - a factor of 4. Even for | α | = 4 a squeezing of 12db ( | s | ≤ .
4) is required, which has been experimentallygenerated in a doubly resonant, type I optical parametricamplifier (OPA) operated below threshold [42].
B. Results
Here we present the results of fidelity, gain and proba-bility of success to benchmark the amplifier. We providethe derivation of these results in the Appendix. We haveassumed that the coherent amplitudes of both the out-put states and the nominal states are real, without lossof generality.The performance of the amplifier is benchmarked us-ing the fidelity F between the output state and an idealamplified cat state {| gα + (cid:105) , | gα − (cid:105)} . We note that pho-ton subtraction changes the parity of the cat state [43].To see why that is the case, consider applying the an-nihilation operator, ˆ a , to the even cat state, | α + (cid:105) ∼| + α (cid:105) + | − α (cid:105) ; coherent states are eigenstates of ˆ a satis-fying | α (cid:105) = α | α (cid:105) and thus we getˆ a | α + (cid:105) ∼ ˆ a | + α (cid:105) + ˆ a | − α (cid:105) = α ( | + α (cid:105) − | − α (cid:105) ) (14)The right hand side of the equation is an (un-normalised)odd cat state | α − (cid:105) ∼ | + α (cid:105) − | − α (cid:105) . So for example ifwe start with an even cat state, after photon subtractionwe obtain an odd cat state and this subtraction occursin the limit t ≈
1. The application of ˆ a to states of evenphoton number produces states of odd photon numberand vice versa.In Fig. 4-6 we summarise results for the gain, max-imum fidelity and probability of success for the outputstate after the amplification process. In the following wehave assumed that the dark counts in the detection pro-cess are negligible (they can be made so in pulsed systemsby time filtering around the pulse centre) and we haveconsidered two scenarios where the quantum efficienciesfor the two detectors are both ideal ( η = η = 1) or bothdetectors are 80% efficient( η = η = 0 . t ≈
1; this comes at the expense of the successprobability as in this limit there is hardly any light re-flected to trigger the second detector. One can see that t = t = g = α (a) t = t = g = α (b)FIG. 4: (a) Optimal gain g as a function of the size α ofthe input even cat state. (b) Same plot for an odd catstate. The green line indicates √ α . It is greater than 80 % for input α upto α ≈ . β = 1 .
95 if the transmission coefficient is t = √ .
95 and the detector efficiencies are η = 0 . t = η = t = η = t = η = t = η = α (a) t = η = t = η = t = η = t = η = α (b)FIG. 5: (a) Optimal fidelity as a function of the size α of the input even cat state.(b) Same plot for an odd catstate.the second beamsplitter (higher reflection coefficient) in-creases the probability of success as the probability ofthe second detector clicking depends on the amplitudeof the reflected light that impinges the detector. Forthe same reason the success probability increases withincreasing cat input amplitude. Overall, the amplifierworks with a high fidelity and approximately two-foldintensity gain for a range of input cat state sizes up to α ≈ . β = 1 .
95. Considering, the most realistic implementa-tion where ( t = √ . , η = 0 .
8) and keeping in mindthe practical requirement that input cat states of size α ≥ . β = 1 .
5, requiring aninput state of size α = 1 . ≈
3% without prior knowledge of the parity ofthe input state. In other words, the amplifier transformsan input cat state to an amplified output cat state of op-posite parity of approximately double the mean photonnumber of the input and works symmetrically on botheven and odd cat states. t = η = t = η = t = η = t = η = α P S (a) t = η = t = η = t = η = t = η = α P S (b)FIG. 6: (a) Corresponding success probability as afunction of the size α of the input even cat state. (b)Same plot for an odd cat state.The fidelity is one indication of the output quality butit does not contain any information about certain fea-tures of the quantum states, such as those contained inthe form of the Wigner function, of which one examplemight be negativity - a signature of nonclassicality. Itis therefore instructive to plot the contour plots of theoutput states after the amplification process and of theequivalent ideal amplified cat states. In Figs. 7, 8 weplot the Wigner functions of the output amplified statewhen the inputs are an even and an odd cat state of size α = 1 and alongside the Wigner functions of the idealamplified cat states. We choose the experimentally feasi-ble parameters of t = √ . , η = 0 . - - - - ( α ) I m ( α ) - (a) - - - - ( α ) I m ( α ) - (b)FIG. 7: (a) Contour plot of the Wigner function of aneven cat state of size α = 1 .
31 (b) Same plot for theoutput state of the swap parity cat scamp given an oddcat state input of size α = 1.ity of its Wigner function, compared to the negativity ofthe Wigner function of the ideal amplified even cat state,is lower (by approximately a factor of 2), but for an eveninput cat state it is comparable. V. THEORETICAL CONSIDERATIONS
To understand how SCAMP can be used to amplifySchr¨odinger cat states we need to consider how photonsubtraction affects a squeezed cat state. A squeezed catstate can be generated by applying the squeezing oper-ator ˆ S ( ζ ) to a cat state | α ± (cid:105) . Photon subtraction thenleads toˆ a ˆ S ( ζ ) | α ± (cid:105) = ˆ S ( ζ ) ˆ S † ( ζ )ˆ a ˆ S ( ζ ) | α ± (cid:105) = ˆ S ( ζ )(cosh s ˆ a | α ± (cid:105) − sinh s ˆ a † | α ± (cid:105) ) , (15)which is a squeezed superposition of a photon subtractedand a photon added cat state. While photon subtrac-tion simply swaps the parity of the cat state, the photon - - - - ( α ) I m ( α ) - - - (a) - - - - ( α ) I m ( α ) - - (b)FIG. 8: (a) Contour plot of the Wigner function of anodd cat state of size α = 1 .
42 (b) Same plot for theoutput state of the swap parity cat scamp given an evencat state input of size α = 1.addition swaps the parity and increases the amplitude ofthe state [44]. One may wonder whether photon additionalone would work but we should note that a photon addedeven cat state is a vacuum removed state. Therefore, thefidelity between an ideal even cat state and the photonadded cat state would be lower than the fidelity betweenthe output state of the cat SCAMP, which produces astate that contains a vacuum component. The combinedeffect of photon addition and photon subtraction, ratherthan state comparison, is the main gain mechanism ofSCAMP for cat states. In principle we could undo thesqueezing by a local unitary operation and we would onlybe left with a superposition of a photon subtracted anda photon added cat state, but it may be challenging toimplement experimentally.The overlap between an ideal cat state and the output of the amplification scheme is (cid:104) β ∓ | ˆ a ˆ S ( ζ ) | α ± (cid:105) = N − ± (cid:104) √ sech se −| β − γ | / − βγ ∗ + β ∗ γ ) / − tanh s ( β ∗ − γ ∗ ) / ( γ − ( β ∗ − γ ∗ ) tanh s )+ 2 √ sech se −| β + γ | / βγ ∗ − β ∗ γ ) / − tanh s ( β ∗ + γ ∗ ) / ( ± γ ± ( β ∗ + γ ∗ ) tanh s ) (cid:105) (16)where γ = α cosh s − α ∗ sinh s and N ± is given by N ± =(2 ∓ e − | β | ) (cid:8) | α | (1 ∓ e − | α | )(cosh s + sinh s )+ (2 ± e − | α | )(sinh s − s sinh sRe [ α ] ) (cid:9) (17)The fidelity is then simply F = |(cid:104) β ∓ | ˆ a ˆ S ( ζ ) | α ± (cid:105)| .The state after the comparison stage and upon posts-election on zero clicks, assuming a perfect detector, is asqueezed cat state of a scaled squeezing parameter and ascaled coherent amplitude, both smaller than the equiv-alent quantities of the input states. The squeezing pa-rameter s (cid:48) is given by s (cid:48) = ln (cid:115) cosh s + (1 − r ) sinh s cosh s − (1 − r ) sinh s (18)where s is the squeezing parameter of the input squeezedvacuum state and r the reflection coefficient of thebeamsplitter of the comparison stage.The scaled coherent amplitude, α (cid:48) of the state afterthe comparison stage is given by α (cid:48) = r α cosh s (cid:112) (cosh s ) − (1 − r ) (sinh s ) (19)Then we can plot the ratio of the ideal cat size β , whichmaximises the fidelity to the photon subtracted squeezedcat state, to the input cat size α as a function of theinput cat size α . α FIG. 9: Ideal gain given by subtracting a photon from asqueezed even cat state as a function of input cat size.One can indeed verify that Fig. 4 (a) and Fig. 9 differmarginally due to the fact that photon subtraction canonly be implemented approximately and is limited bythe transmission coefficient of the beamsplitter used toperform the photon subtraction (which is always less thanunity, t = √ .
99 in this case).In this section we have described the theoretical mech-anism for the gain of our device. Physically it is based onmean photon number matching between a resource state(squeezed vacuum) and the input cat state after photonsubtraction conditioning. If the cat state has a signifi-cantly higher photon number than the resource state theoutput will effectively be a photon subtracted cat state.If the resource state has a significantly higher photonnumber than the cat state the output will be a photonsubtracted resource state. Only if they are comparablein mean photon number does the significant state overlapbetween the two give an amplified cat. The approximatephoton number matching of the two states is also thereason for the approximate twofold gain.
VI. OTHER SCHEMES FOR AMPLIFICATIONOF SCHR ¨ODINGER CATS
Figure 9 suggests that we could amplify a Schr¨odingercat state reasonably well by first squeezing the cat stateand then subtracting a photon. The process amounts toa slightly simpler theoretical scheme than ours. How-ever, squeezing low amplitude non-vacuum states in acontrolled fashion is more difficult experimentally, par-ticularly in the pulsed domain. Optical nonlinearitiesare small and require high pump fields with a good modeoverlap with the signal. For these reasons we believe thatsuch a direct-drive scheme is a little less experimentallypractical than ours.The state of the art experimental scheme of [23] whichutilises the cat “breeding” method produces an ampli-fied cat state with amplitude α = 1 .
85 while consumingtwo cat states of amplitude α = 1 .
15 as resources. Theprobability of success is P S = 0 . .
77. Seem-ingly, SCAMP is on par with cat “breeding” schemes interms of fidelity and gain while it offers a lower probabil-ity of success. One has though to take into account thefact that cat “breeding” schemes consume two nongaus-sian resource states, as opposed to just one for SCAMP,that have a low success probability of production cur-rently and therefore offset this advantage. If push-buttonquantum state generation becomes a reality then the in-trinsic advantages of cat “breeding” may be more easilyrealised.
VII. CONCLUSIONS
In summary, we have presented a theoretical schemefor the amplification of input Schr¨odinger cat statesbased on a nondeterministic amplifier (SCAMP) com-prised of the two mature techniques of state comparisonand photon subtraction. The device both amplifies andswaps the parity of a state chosen from the set of oddand even Schrodinger cat states. Hence we call it theparity-swap cat state comparison amplifier.The implementation of SCAMP for Schr¨odinger catstates would require only a Gaussian resource state(squeezed vacuum), linear optical components andGeiger mode detectors, making it experimentally feasi-ble. The resource state is not randomly chosen, as it isin the standard SCAMP for coherent state amplification.This, coupled with the fact that it is Gaussian gives asignificant advantage over other schemes. The SCAMPitself provides the photon subtraction required to renderthe output nongaussian. The parity-swap cat SCAMPcan work almost symmetrically for both even and oddcat states without prior knowledge of the input state andoffers high fidelity and reasonably high gain for the rangeof input cat state sizes of interest.We have characterised the performance of the amplifiervia gain, fidelity and success probability. The intensitygain is shown to be approximately twofold for both theeven and odd cat states for a wide range of mean inputphoton numbers, reflecting the fact that the gain of astandard SCAMP is approximately two for a 50/50 com-parison beamsplitter. The fidelity of the output with anideal amplified cat state | gα ± (cid:105) depends on the transmis-sion coefficient, t , of the beamsplitter used to performthe photon subtraction stage. The fidelity remains high( F ≥ α up to α ≈ . α = 1 . β = 1 . | α ± i (cid:105) = N ( | α (cid:105) ± i | − α (cid:105) ) (20)do not satisfy this criterion and cannot be amplified inthis fashion. However, such states have never been pro-duced in the laboratory and so are of limited interest. VIII. ACKNOWLEGMENTS
The authors would like to acknowledge financial sup-port from the UK National Quantum Technology Pro-gramme via EPSRC and the Quantum Technology Hubin Quantum Communications (Grant EP/M013472/1).GT would also like to acknowledge EPSRC for a par-tial studentship from the Doctoral Training Grant to theUniversity of Strathclyde.
APPENDIX
In this appendix section we will use the formalism ofcharacteristic functions to sketch the method used to de-rive all of the required results. We will see how the statesand operators can be represented under the formalismand show how one can compute the benchmark metricsconsidered in the main body of the text.
Characteristic Functions
A useful feature of the characteristic function formal-ism is that the trace of operators can be evaluated as anintegral in phase space. More formally, the trace rule fortwo operators ˆ O and ˆ O is given by [45]Tr[ ˆ O ˆ O ] = (cid:90) C d ξπ χ ˆ O ( ξ ) χ ˆ O ( − ξ ) (21)and this formula gives us the fidelity F. States whosecharacteristic function is a Gaussian are known as Gaus-sian states and one such example we have already seen;namely the coherent state | α (cid:105) whose characteristic func-tion is χ α ( ξ ) = exp( ξα ∗ − ξ ∗ α − | ξ | /
2) (22) and the squeezed vacuum state whose characteristic func-tion is χ ζ ( ξ ) = exp (cid:16) − ξ r exp[2 s ] / ξ i exp[ − s ] (cid:17) (23)where ξ r and ξ i are the real and imaginary parts of thecomplex variable ξ .The cat states on the other hand are nongaussian statesas their characteristic functions are given by sums ofGaussians: χ α + ( ξ ) = N (cid:104) exp (cid:0) ξα ∗ − αξ ∗ − | ξ | / (cid:1) + exp (cid:0) − ξα ∗ + αξ ∗ − | ξ | / (cid:1) + exp (cid:0) − ξα ∗ − αξ ∗ − | α | − | ξ | / (cid:1) + exp (cid:0) ξα ∗ + αξ ∗ − | α | − | ξ | / (cid:1) (cid:105) (24) χ α − ( ξ ) = N − (cid:104) exp (cid:0) ξα ∗ − αξ ∗ − | ξ | / (cid:1) + exp (cid:0) − ξα ∗ + αξ ∗ − | ξ | / (cid:1) − exp (cid:0) − ξα ∗ − αξ ∗ − | α | − | ξ | / (cid:1) − exp (cid:0) ξα ∗ + αξ ∗ − | α | − | ξ | / (cid:1) (cid:105) (25)where N ± = ± e − | α | . Performance Benchmark
Typically the performance of amplifiers can be charac-terised by the fidelity, F, and the probability of success, P s of the amplifier. The fidelity metric quantifies thecloseness of two quantum states. In this case it quantifiesthe closeness of the output state after the amplificationto an ideal output state. If the ideal output state is purethen F has a simple form given byF = Tr[ ρ ρ ] (26)where ρ , ρ are some general single mode states and atleast one of them is pure. In terms of characteristic func-tions, as we have already seen, the fidelity is given byF = (cid:90) d ξπ χ ( ξ ) χ ( − ξ ) (27)The probability of success metric on the other hand de-pends on the working details of the amplifier. For theSCAMP the probability of success is defined as the jointprobability of the second detector clicking given that thefirst detector did not register a click, P s = P ( D = (cid:55) , D = (cid:88) ). The probability for a detector on Geigermode to not register a click is given by the Kelley-Kleinerformula [46] P (cid:55) = Tr[ ρ ˆ π (cid:55) ] = Tr[ ρ : exp( − η ˆ a † ˆ a ) :] (28)0where η is the quantum efficiency of the detector andthe colons indicate normal ordering of the operators. Interms of the characteristic function the no counts opera-tor is given by χ ˆ π (cid:55) ( ξ ) = 1 η exp( − − η η | ξ | ) (29)and the probability of success is given by P (cid:55) = (cid:90) d ξπ χ ( ξ ) χ ˆ π (cid:55) ( ξ ) (30)The probability of success if a detector clicks on the otherhand is give by P (cid:88) = Tr[ ρ ˆ π (cid:88) ] = Tr[ ρ : − exp( − η ˆ a † ˆ a ) :] (31)The click operator, ˆ π (cid:88) , in the characteristic function for-malism is given by x ˆ π (cid:88) ( ξ ) = πδ (2) ( ξ ) − x ˆ π (cid:55) ( ξ ) (32)and the probability of success is given by P (cid:88) = (cid:90) d ξπ χ ( ξ ) χ ˆ π (cid:88) ( ξ ) (33) Output of Comparison Stage
Now that we have introduced the formalism of thecharacteristic functions we can have a close look at whathappens when a coherent state | α (cid:105) and a squeezed vac-uum | ζ (cid:105) are incident on a beamsplitter and one of thetwo output modes is projected onto the vacuum state.The input to the beamsplitter is given by χ in ( ξ , ξ ) = χ α ( ξ ) χ ζ ( ξ ) (34)The output (global) state after the beamsplitter can befound by transforming the arguments of the characteris-tic function of the input as follows χ out ( ξ , ξ ) = χ α ( t ξ + r ξ ) χ ζ ( t ξ − r ξ ) (35)Postselection then on zero clicks, assuming a perfect de-tector, leads to χ ( ξ ) = 1 πP (cid:55) (cid:90) χ out ( ξ , ξ ) χ ˆ π (cid:55) ( ξ ) d ξ = exp (cid:34) − x { cosh s + (1 − r ) sinh s } s − (1 − r ) sinh s ) − y { cosh s − (1 − r ) sinh s } s − (1 − r ) sinh s ) − ir xIm [ α ] cosh s (cosh s + (1 − r ) sinh s )cosh s − (1 − r ) sinh s + 2 ir yRe [ α ] cosh s (cosh s − (1 − r ) sinh s )cosh s − (1 − r ) sinh s (cid:35) (36) where P (cid:55) is given by P (cid:55) = 1 π (cid:90) (cid:90) χ out ( ξ , ξ ) χ ˆ π (cid:55) ( ξ ) d ξ d ξ = (cid:115) e − s + r sinh s )( e s − r sinh s ) ∗ exp (cid:34) − e s (1 − r ) sinh s Re[ α ] e s − r sinh s − e − s (1 − r ) sinh s Im[ α ] e − s + r sinh s (cid:35) (37)The characteristic function of a squeezed coherentstate, ˆ S ( ζ ) | α (cid:105) is given by χ ζ,α ( ξ ) = exp (cid:0) − x e s / − y e − s / − ie s x Im[ α ] + 2 ie − s Re[ α ] (cid:1) (38)where ξ = x + iy .If we compare Eq. 36 to Eq.38 we can see that theyboth represent the same state with the transformations s → s (cid:48) , and α → α (cid:48) given by s (cid:48) = ln (cid:115) cosh s + (1 − r ) sinh s cosh s − (1 − r ) sinh s (39) α (cid:48) = r α cosh s (cid:112) (cosh s ) − (1 − r ) (sinh s ) (40)Then we can think of the comparison stage as a quantumchannel that transforms a coherent state to a squeezedcoherent state of squeezing parameter s (cid:48) and scaled co-herent amplitude α (cid:48) . 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