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Quantum memory for entangled two-mode squeezed states

K. Jensen,1 W. Wasilewski,1 H. Krauter,1 T. Fernholz,1 B. M. Nielsen,1

A. Serafini,2 M. Owari,3 M. B. Plenio,3 M. M. Wolf,1 and E. S. Polzik11QUANTOP, Danish National Research Foundation Center for Quantum Optics,

Niels Bohr Institute, University of Copenhagen, DK 2100, Denmark2University College London, Department of Physics& Astronomy, Gower Street, London WC1E 6BT3Institut fur Theoretische Physik, Universitat Ulm, Albert-Einstein Allee 11, D-89069 Ulm, Germany

A quantum memory for light is a key element for therealization of future quantum information networks [1–3].Requirements for a good quantum memory are (i) versatil-ity (allowing a wide range of inputs) and (ii) true quantumcoherence (preserving quantum information). Here wedemonstrate such a quantum memory for states possess-ing Einstein-Podolsky-Rosen (EPR) entanglement. Thesemulti-photon states are two-mode squeezed by6.0 dB witha variable orientation of squeezing and displaced by a fewvacuum units. This range encompasses typical input al-phabets for a continuous variable quantum informationprotocol. The memory consists of two cells, one for eachmode, filled with cesium atoms at room temperature witha memory time of about1msec. The preservation of quan-tum coherence is rigorously proven by showing that theexperimental memory fidelity 0.52± 0.02 significantly ex-ceeds the benchmark of0.45 for the best possible classicalmemory for a range of displacements.

A sufficient condition for a memory to be genuinely quan-tum can be formulated via the fidelity between memory in-put and output. If this fidelity exceeds the benchmark de-termined by the best classical device then it can store andpreserve entanglement and hence is a true quantum memory.Classical benchmark fidelities are difficult to calculate, anduntil recently they were known only for coherent states [4]and homogeneously distributed pure states [5]. Storage ofnon-classical correlations between single photons [6–8] anda dual-path superposition state of a photon [9] have beenexperimentally demonstrated with the Raman and electro-magnetically induced transparency (EIT) approaches. Along-side these demonstrations for single photons, quantum mem-ory for multi-photon entangled states is an important and chal-lenging ingredient of quantum information networks.

A particular class of multi-photon entangled states, namelyEPR-type entangled two-mode squeezed states, plays a fun-damental role in quantum information processing with con-tinuous variables (cv) [10, 11]. A quantum memory for entan-gled cv states is valuable for iterative continuous variable en-tanglement distillation [12], continuous variable cluster statequantum computation [13, 14], communication/cryptographyprotocols involving several rounds [15], and quantum illumi-nation [16, 17]. Temporal delay of EPR entangled states hasbeen demonstrated in [18]. Very recently, classical bench-marks for storing a squeezed vacuum state [19] and displacedsqueezed states [20, 21] have been derived. EIT-based mem-ory for squeezed vacuum has been recently reported [22, 23],

albeit with the fidelity below the classical benchmark.Here we report the realization of a quantum memory for

a set of displaced two-mode squeezed states with an uncon-ditionally high fidelity that exceeds the classical benchmark[20].

We store a displaced EPR state of two modes of lighta+and a− with the frequenciesω± = ω0 ± ωL, whereω0 isthe carrier frequency of light. The entanglement conditionisVar(X+ + X−) + Var(P+ − P−) < 2 [24] where canoni-cal quadrature operators obey[X±, P±] = i. For a vacuumstateVar(Xvac) = Var(Pvac) = 1/2. The EPR entangle-ment of thea+ and a− modes is equivalent to simultaneoussqueezing of thecos(ωLt) mode xLc = (X+ + X−)/

√2;

pLc = (P++ P−)/√2 and the correspondingsin(ωLt) mode.

The alphabet of quantum states is generated by displacing thetwo-mode squeezed vacuum state by varying values〈xLc,s〉and 〈pLc,s〉 and by varying the orientation of the squeezedquadrature betweenxL andpL. The displaced squeezed statesare produced (Fig. 1a) using an optical parametric amplifier(OPA) [25] with the bandwidth of8.3MHz and two electro-optical modulators (EOMs)(see the Methods summary for de-tails).

The two photonic modes are stored in two ensembles of ce-sium atoms contained in paraffin coated glass cells (Fig. 1a)with the ground state coherence time around30msec [3].ω0 isblue detuned by∆ = 855 MHz from theF = 4 → F ′ = 5 ofD2 transition (Fig. 1c). Atoms are placed in a magnetic fieldwhich leads to the precession of the ground state spins withthe Larmor frequencyωL = 2π · 322 kHz. This ensures thatthe atoms efficiently couple to the entangledω± = ω0 ± ωL

modes of light. The two ensembles1(2) are optically pumpedin F = 4,mF = ±4 states, respectively, which leads tothe opposite orientation of their macroscopic spin componentsJx1 = −Jx2 = Jx.

The atomic memory is conveniently described by twosetsc, s of non-local, i.e., joint for the two separate mem-ory cells, canonical atomic operators [26]xAc = (Jrot

y1 −Jroty2 )/

√2Jx, pAc = (Jrot

z1 + Jrotz2 )/

√2Jx, xAs = −(Jrot

z1 −Jrotz2 )/

√2Jx, pAs = (Jrot

y1 + Jroty2 )/

√2Jx where the super-

scriptrot denotes spin operators in a frame rotating atωL. Itcan be shown that the cosine/sine light mode couples only tothe atomicc/s mode, respectively. As a consequence, in theprotocol described below the upper(lower) entangled sidebandmode of light is stored in the1(2) memory cell, respectively.Since the equations describing the interaction are the same,we omit the indicesc, s from now on.

2

Light sent from the sender station to the receiver (memory)station consists of quantumx-polarized modes and a strongy-polarized part which serves as the driving field for interactionwith atoms and as the local oscillator (LO) for the subsequenthomodyne measurement (Fig. 1). The interaction of light anda gas of spin polarized atoms under our experimental condi-tions can be described by the following input-output equations[27]:

x′A =

√

1− κ2

Z2xA + κpL, p′A =

√

1− κ2

Z2pA − κ

Z2xL,

x′L =

√

1− κ2

Z2xL + κpA, p′L =

√

1− κ2

Z2pL − κ

Z2xA,

(1)

where the coupling constantκ is a function of light intensity,density of atoms and interaction time, andZ2 = 6.4 is a func-tion of the detuning only. In the limitκ → Z these equationsdescribe a swap of operators for light and atoms, i.e., a per-fect memory followed by squeezing by a factorZ2. However,in our experiment the swapping time is too long compared tothe atomic decoherence time. We therefore follow another ap-proach which takes a much shorter time.

The sequence of operations of the quantum memory proto-col is shown in Fig. 1b. After the light-atoms interaction dur-ing the storage part of the protocol the output light operatorx′L is measured by the polarization homodyne detection and

the result is fed back onto thep′A with a gaing. This feedbackinto bothc ands modes is achieved by applying two pulses ofthe magnetic field at the frequencyωL to the two cells. Theresultingxfin

A andpfinA for the optimizedg andκ = 1 can befound from Eq. (1)

xfinA =

√

1− 1

Z2xA + pL and pfinA = −xL. (2)

In the absence of decoherence the operatorxL is perfectlymapped on the memory operatorpfinA . The operatorpL isstored inxfin

A with the correct mean value〈pL〉 = 〈xfinA 〉 (since

〈xA〉 = 0) but with an additional noise due toxA.The ability to reproduce the correct mean values of the in-

put state in the memory by adjustingg andκ is a characteristicfeature of our protocol which arguably makes it better suitedfor storage of multi-photon states compared to, e.g., EIT ap-proaches where such an adjustment is not possible.

The additional noise of the initial state of atomsxA is sup-

pressed by the factor√

1− 1Z2 = 0.92 due to the swap in-

teraction. To reduce it further we start the memory protocolwith initializing the atomic memory state in a spin-squeezedstate (SSS) with a squeezedxA. The SSS with variancesVar (xA) = 0.43(3) andVar (pA) = 1.07(5) is generatedas in [28] by the sequence shown as the preparation of initialstate in Fig. 1b following optical pumping of atoms into thestate withVar (xA) = 0.55(4).

Before the input state of light undergoes various lossesit is a 6dB squeezed state which we refer to as an ”ini-

tial pure state”. In the photon number representationfor the two modes the state is|Ψ〉 = 0.8|0〉+|0〉− +0.48|1〉+|1〉− + 0.29|2〉+|2〉− + 0.18|3〉+|3〉− + .... Weran the memory protocol for 18 different states definedby {φ = 0, 90 deg; [〈xL〉 ; 〈pL〉] = [0, 3.8, 7.6; 0, 3.8, 7.6]},whereφ is the phase of the squeezing, and〈xL〉 and 〈pL〉are the displacements of the initial state (see Table I). In theabsence of passive (reflection) losses for light and atomicdecoherence we find from Eq. (2) the expected fidelity of0.95, 0.61 for the states squeezed with theφ = 0, φ = 90,respectively, with the mean fidelity of0.78.

In the experiment light is sent to the receiver’s mem-ory station via a transmission channel with the transmis-sion coefficientηloss = .80(4) (which includes the OPAoutput coupling efficiency) resulting in the ”memory inputstate” withVar (xL · cos(φ)− pL · sin(φ)) = 0.20(2) andVar (xL · sin(φ) + pL · cos(φ)) = 1.68(9). The state of lightis further attenuated by the factorηent = .90(1) due to the en-trance (reflection) losses at the windows of the memory cells.Between the interaction and detection light experiences lossesdescribed by the detection efficiencyηdet = .79(2) (see Meth-ods Summary).

Following the storage time of1msec we measure the atomicoperators with the verifying probe pulse in a coherent state.The mean values and variances of the atomic operatorsxfin

A

andpfinA are summarized in Table I (see the Methods summaryfor calibration of the atomic operators). From these valuesandthe loss parameters, we can calculate the noise added duringthe storage process which is not accounted for by transmissionand entrance losses. We find that the memory adds0.47(6)to Var(xfinA ) and0.38(11) to Var(pfinA ), whereas for the idealmemory, according to Eq. (2), we expect the additional noiseto be0.36(5)(due to the finite squeezing of the initial atomicoperatorxA) and0, for the two quadratures respectively. Thisadded noise can be due to atomic decoherence, uncancelednoise from the initial anti-squeezedpA quadrature, and tech-nical noise from the EOMs.

The overlap integrals between the stored states and the ini-tial pure states are given in Table I. The average fidelities cal-culated from the overlap values for square input distributionswith the sizedmax = 0, 3.8 and 7.6 are plotted in Fig. 2. Thechoice of the interaction strengthκ = 1 minimizes the addednoise but leads to the mismatch between the mean values ofthe stored atomic state and of the initial pure state of lightbythe factor

√ηent · ηloss = 0.85. This mismatch is the reason

for the reduction of the experimental fidelity for states withlarger displacements.

The experimental fidelity is compared with the best classi-cal memory fidelity which is calculated [20] from the over-lap of the initial pure state with the state stored in the clas-sical memory positioned in place of the quantum memory.The input distribution used in the calculations is a square{|〈xL〉| , |〈pL〉| ≤ dmax} with all input states with the meanvalues within the square having equal probability and otherstates having probability 0. The squeezing of the input statesis fixed to the experimental value, and all phases of squeezing

3

FIG. 1: Setup and pulse sequence.a. At the sender station two-mode entangled (squeezed) light is generated by the Optical parametric Am-plifier (OPA). A variable displacement of the state is achieved by injecting a coherent input into OPA modulated by electro-optical modulators(EOM). The output of the OPA is shaped by a chopper, and combined on a polarizing beamsplitter with the local oscillator (LO) beam, suchthat the squeezed light is only on during the second probe pulse. A beamshaper and a telescope create an expanded flattop intensity profile. Thelight is then send to the receiver (memory) consisting of twooppositely oriented ensembles of spin-polarized cesium vapour in paraffin coatedcells and a homodyne detector. The detector signal is processed electronically and used as feedback onto the spins obtained via RF magneticfield pulses. b. Pulse sequence for the initiation of the memory, storage, and verification. c. Atomic level structure illustrating interaction ofquantum modes with the memory.

are allowed. The upper bound on the classical benchmark fi-delity is plotted in Fig. 2. The benchmark values have beenobtained by first truncating the Hilbert space to a finite photonnumber and then solving the finite dimensional optimizationemploying semi-definite programming. The truncation of theHilbert space is treated rigorously by upper bounding its effectvia an error bound that is included in the fidelity. As a result,around the valuedmax = 3.5 the theoretical calculated fidelitydoes not decrease further, but remains at a constant level dueto the rapidly increasing truncation error (see the Supplemen-tary Material for the proof of the non-increasing characterofthe dependence of the benchmark ondmax).

For the input distributions arounddmax = 3.8 (a vacuumunit of displacement isd = 1/

√2), the measured fidelity is

higher than the classical bound and thus the memory is a truequantum memory.

Outperforming the classical benchmark means that ourmemory is capable of preserving EPR-entanglement in casewhen one of the two entangled modes is stored while the otheris left propagating. Using experimentally obtained valuesofthe added noise we evaluate the performance of our memory

for the protocol where the upper sidebanda+ mode is storedin one of the memory cells whereas the other EPR modea−is left as a propagating light mode. We find an EPR vari-ance between the stored mode and the propagating mode of1.52 (−1.2dB below the separability criterion) which corre-sponds to the lower bound on the entanglement of formationof ∼ 1/7ebit [29] (see Supplementary Information for detailsof the calculation). Implementing this version of the memorywould require spectral filtering of thea− anda+ modes whichcan be accomplished by a narrow band optical cavity.

In conclusion, we have experimentally demonstrated de-terministic quantum memory for multi-photon non-classicalstates. An obvious way to improve the fidelity is to reduce thereflection losses on the memory cell windows. Other improve-ments involve increasing the initial atomic spin squeezingandreduction of atomic decoherence which should allow for thestorage protocol based on the swap interaction.

4

0 1 2 3 4 5 6 70.35

0.4

0.45

0.5

0.55

0.6

dmax

aver

age

fidel

ity

experimentclassical bound

FIG. 2: Fidelities. The fidelityF calculated from experimental re-sults is shown as circles connected by lines. The theoretical bench-mark values are shown as squares. The horizontal axis is the sizedmax of the input distribution. Note that one vacuum unit of dis-placement corresponds todmax = 1/

√2.

Methods Summary

Verification. By measuring thex′L of the verification

pulse the statistics of the atomic operatorpfinA is obtained. Inanother series we rotatexfin

A into pfinA (and vice versa) using amagneticπ-pulse and obtain the statistics ofxfin

A for the sameinput state of light. Since we assume Gaussian statistics, themean values and the variances ofxfin

A andpfinA are sufficientfor a complete description of the atomic state.

Calibrations. Before performing the storage, we calibratethe interaction strengthκ and the feedback gaing, such thatthe mean values of the light state inside the memory (i.e. af-ter the entrance loss) are transferred faithfully.κ is calibratedby creating a mean value

⟨

p2ndL

⟩

in the second probe pulse.The mean is stored in the atomicx′

A, which is read out afterthe magneticπ-pulse with the third probe pulse. The mea-sured mean of the third pulse is then

⟨

x′3rdL

⟩

= κ2⟨

p2ndL

⟩

from whichκ2is determined. Using similar methods we cancalibrate the electronic feedback gaing.

Generation of the displaced squeezed input states.TheOPA which is pumped by the second harmonic of the masterlaser and generates the entangled squeezed vacuum states isseeded with a fewµW of the master laser light with the car-rier frequencyω0 which is amplitude and phase modulated bytwo electro-optical modulators (EOMs) at a frequency of 322kHz, thus creating coherent states in the± 322 kHz sidebandsaroundω0 (Fig. 1). With such a modulated seed, the outputof the OPA is a displaced two-mode squeezed state. The out-put of the OPA is mixed at a polarizing beamsplitter with thestrong local oscillator (driving) field from the master laser.

Losses. In order to calculate the mean values and vari-ances of the stored state and the input states, we need to knowthe optical losses. We choose to divide the total lossesηtotinto three parts, the channel propagation lossesηloss from theOPA to the front of the memory cells (including the OPA out-put efficiency), the entrance lossηent and the detection losses

Input states Stored states overlap

〈xL〉 〈pL〉 φ⟨

pfinA⟩ ⟨

xfin

A

⟩

Var(

pfinA)

Var(

xfin

A

)

F

0.0 0.0 0 -0.06 0.25 0.52(2) 1.99(3) 0.62

0.0 3.8 -0.06 3.19 0.60

3.8 0.0 -3.47 -0.42 0.57

3.8 3.8 -3.39 2.89 0.49

0.0 0.0 90 -0.07 0.06 1.95(6) 0.73(1) 0.55

0.0 3.8 -0.06 3.14 0.42

3.8 0.0 -3.22 0.48 0.46

3.8 3.8 -3.21 3.59 0.50

0.0 7.6 0 -0.03 6.30 0.55(2) 2.01(4) 0.49

7.6 0.0 -6.83 -0.46 0.37

3.8 7.6 -3.20 6.07 0.35

7.6 3.8 -6.54 2.80 0.22

7.6 7.6 -6.40 6.03 0.15

0.0 7.6 90 -0.08 6.24 2.12(8) 0.78(3) 0.18

7.6 0.0 -6.37 0.59 0.35

3.8 7.6 -3.13 6.75 0.32

7.6 3.8 -6.38 3.79 0.43

7.6 7.6 -6.36 6.72 0.27

TABLE I: The three first columns display the mean displacementsand the squeezing phase (φ = 0 corresponds toxL being squeezed)of the initial pure light states. The next four columns display themean values and variances of the atomic states after the storage. Thelast column displays the overlap between the initial pure light statesand the stored atomic states. Vacuum state variances are0.5. Theuncertainties on the variances are calculated as the standard deviationof the variances within each subgroup of the input states.

ηdet, such thatηtot=ηloss · ηent · ηdet (all the η’s are inten-sity transmission coefficients). From the measurement of thequadratures of the squeezed light (with variances0.29(1) and1.34(6), we find the total lossesηtot = 0.567(35). We mea-sure the transmission through the cells of 0.817(20), the trans-mission through the optics after the cells of 0.889(10) and es-timate the efficiency of the photodiodes to be 0.98(1). As-signing one half of the losses through the cells to the entrancelosses and another half to the detection losses we findηent =√0.817 = 0.90(1), ηdet =

√0.817 · 0.889 · 0.98 = 0.79(2)

andηloss = ηtot/ (ηentηdet) = 0.80(4).

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ACKNOWLEDGMENTS

This work was supported by EU projects QESSENSE,COMPAS and COQUIT.

AUTHOR CONTRIBUTIONS

Experimental group: K. J., W. W., H. K., T. F., B. M. N.and E. S. P. Calculation of the classical benchmark: M. O., M.B. P., A. S. and M. M. W.

6

SUPPLEMENTARY INFORMATION

Memory added noise

As discussed in the main text, in the experiment the finalatomic states acquire extra noise from losses and decoher-ence not included in the basic theory. This extra noise isaccounted for in two steps. In the first step the initial purelight state propagates through the transmission and memoryentrance lossesηlossηent (modelled by a beamsplitter) whichadd vacuum noise to the state of light. In the second step extranoise is added in the process of performing a unity-gain stor-age of the light state. We model this extra added noise withtwo operatorsSx andSp with zero mean values. The finalatomic state is now (see also Eq. (2) in the main text)

xfinA =

√

1− 1

Z2xA +GppureL +

√

1−G2pvacL + Sx,

pfinA =−GxpureL −

√

1−G2xvacL + Sp, (3)

where we defined the gain of the memoryG =√ηlossηent =

0.85, which equals the ratio of the mean values of the storedatomic state and the intial pure state (this gain should not bemistaken with the electronic feedback gain).xvac

L andpvacL arevacuum operators with the zero mean and the variance 1/2.Taking into account thatZ2 = 6.4 and the initial atomic noiseis Var (xA) = 0.43 we can calculate the expected variancesof the final atomic state from the above equation.

Var(

xfinA

)

=

(

1− 1

Z2

)

Var (xA) +G2Var (ppureL )

+(

1−G2) 1

2+ Var (Sx) ,

Var(

pfinA)

=G2Var (xpureL ) +

(

1−G2) 1

2+ Var (Sp) . (4)

The variances ofxpureL andppureL equal1/ (2s) or s/2 depend-

ing on the phase of the squeezing, wheres = 4 in the exper-iment. By inserting these experimental parameters and thevariances of the final atomic state, we can calculate the addednoise. The results are given in Table II for the two cases wherewe stored squeezed vacuum.

Stored state Added noise

φ Var(

xfin

A

)

Var(

pfinA)

Var (Sx) Var (Sp)

0 2.02 0.52 0.08 0.29

90 0.72 1.90 0.13 0.32

TABLE II: The table shows the measured final atomic state variancesVar

(

xfin

A

)

andVar(

pfinA)

and the added extra noiseVar (Sx) andVar (Sp).

The variances ofSx andSp are quite small (less than onevacuum unit), and can be attributed to atomic decoherence andnoise of the anti-squeezed light quadrature which can feed intothe atomicpfinA due to imperfect electronic feedback.

Local/non-local operators and cos/sine vs sideband picture

Below we will describe the quantum memory protocol inthe language of local atomic operators and the upper and lowersideband modes instead of using the non-local atomic opera-tors and the cosine and sine light modes. The local atomicoperators and the non-local operators are connected by

xA1 =1√2(xAc + pAs) , pA1 =

1√2(pAc − xAs) ,

xA2 =1√2(xAc − pAs) , pA2 =

1√2(pAc + xAs) , (5)

where the subscripts 1 and 2 refers to the first and secondatomic ensemble, respectively. We can also find the relationbetween the upper and lower sideband modes of light and itscosine and sine mode

xLc =1√2(x+ + x−) , pLc =

1√2(p+ + p−) ,

xLs =1√2(p− − p+) , pLs =

1√2(x+ − x−) . (6)

We write the input-output equations for the quantum memoryas

xfinAc =GppureLc +Ox, pfinAc =−Gxpure

Lc +Op,

xfinAs =GppureLs +Ox, pfinAs =−Gxpure

Ls +Op, (7)

with the definitions

Ox =

√

1− 1

Z2xA +

√

1−G2pvacL + Sx

Op =√

1−G2xvacL + Sp. (8)

UsingSx ≈ 0.1 andSp ≈ 0.3, we findVar (Ox) = 0.44 andVar (Op) = 0.60. Equation (7) can be re-written by insertingEqs. (5) and (6), and we find

xfinA1 =Gppure+ +

Ox +Op√2

, pfinA1 =−Gxpure+ +

Op −Ox√2

,

xfinA2 =Gppure− +

Ox −Op√2

, pfinA2 =−Gxpure− +

Op +Ox√2

.

(9)

We see that the upper sideband is stored in the first atomicensemble and the lower sideband is stored in the second en-semble.

Storage of one part of an EPR-entangled pair

In the experiment, the sender prepares and sends a two-mode entangled state to the receiver, who then stores it inhis quantum memory. Although the memory is a true quan-tum memory as proven by its ability to outperform the classi-cal benchmark, the noise added in the storage process to both

7

EPR modes leads to a separable state of the two memory cells.Note that for input states with the squeezing directionφ = 0displaced up to3.8 (Table I), we are very close to having twodisplaced entangled atomic ensembles after the storage, sincethe parameterE describing EPR-entanglement

E ≡Var(

xfinA1 − xfin

A2

)

+Var(

pfinA1 + pfinA2

)

=2 ·[

Var(

pfinAc

)

+Var(

pfinAs

)]

=2 · (0.52 + 0.52) = 2.08, (10)

is only slightly above 2.

Since outperforming the classical benchmark is sufficientto prove that the memory is capable of storing entanglement,we should be able to think of a modification of the experimentwhich can do exactly that. One example of such an experi-ment is the protocol, where Alice sends only one mode of theEPR-entangled pair to Bob for storage, and the other mode issent to a third person, Charlie. In this case only one of thetwo entangled modes gets distorted by the memory. After thestorage, Bob and Charlie perform a joint measurement to testwhether there is entanglement between Bob’s stored atomicstate and Charlie’s light state. In practice, Alice would haveto separate the upper and lower sidebands, which could forinstance be done using a cavity which transmits one sidebandand reflects the other. The initial entanglement of the upperand lower sidebands is characterized by

E ≡Var(

xpure+ − xpure

−)

+Var(

ppure+ − ppure−)

= 2/s.(11)

Alice then sends the EPR-entangled upper sideband (togetherwith a lower sideband in the vacuum state) to Bob who storesthe upper sideband in the first ensemble (and the vacuum inthe lower sideband in the second ensemble). After the storageBob and Charlie share the entanglement, since

E ≡Var(

xpure− + pfinA1

)

+Var(

ppure− − xfinA1

)

=1

2s(1 +G)

2+

s

2(1−G)

2+Var (Ox) +

+Var (Op) = 1.51 < 2, (12)

which has been calculated using the experimentally obtainedparameters of our memory from Table II, Eq. (9) and utilizingthe fact thatVar

(

xpure+ + xpure

−)

,Var(

ppure+ + ppure−)

= s.We conclude, based on the experimental performance of ourquantum memory for the storage of both modes of the twomode entangled states, that if we instead stored only one partof the EPR-pair, one of the atomic memories would be entan-gled with the other part of the EPR-pair after the storage.

Monotonicity of classical fidelity benchmark

The following aims at proving that the classical benchmarkis a non-increasing function ofdmax. That is, the broader

0 1 2 3 4 5 6 70.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

dmax

aver

age

fidel

ity

experimentcalculated benchmarkmonotonic benchmark

FIG. 3: Fidelities. The fidelityF calculated from experimental re-sults is shown as circles connected by lines. The calculatedbench-mark values are shown as squares and the monothonic benchmarkvalues are shown as diamonds. The horizontal axis is the sizedmax

of the input distribution. Note that one vacuum unit of displacementcorresponds todmax = 1/

√2.

is the a priori distribution in phase space, the more difficultit gets for a classical scheme to achieve a certain fidelity.Here ”classical scheme” refers either to entanglement break-ing quantum channels or to channels whose Choi matrix has apositive partial transpose.

This supplement justifies the horizontal asymptote for thebenchmark in Fig. 2 of the main text. Without this a rigoroustreatment of the truncation errors (as in ref. Owari et al in themain text) would for largedmax lead to an increasing boundfor the classical benchmark (see Fig. 3).

Recall that our benchmark is based on the average fidelity

F (T ) :=

∫

R2

dξq(ξ)

∫

[0,2π)

dθ

2πtr[

T(

Nλ(ρθ,ξ))

ρθ,ξ]

, (13)

whereρθ,ξ are the initial pure squeezed coherent states withmeanξ and orientationθ, Nλ is an attenuation channel whichdecreases the intensity by a factorλ, T is the channel corre-sponding to a hypothetical classical memory andq(ξ) is theprobability density distribution of the input alphabet in phasespace. The benchmark is then given by

F := supT

F (T ),

where the supremum runs over allT ’s corresponding to clas-sically possible schemes.

We want to compare different values ofF corresponding todifferent distributionsq, so we regardF as a function ofq andwriteFq. Let us define

T ′(ρ) :=

∫

R2

dη p(η)W †ηT

(

W√ληρW

†√λη

)

Wη, (14)

whereWη is the Weyl operator which displaces byη in phasespace andp(η) is some probability density distribution. NotethatT ′ characterizes a classical, albeit coarse-grained, scheme

8

if T does. In this case

Fq ≥∫

R2

dξ q(ξ)

∫

[0,2π)

dθ

2πtr[

T ′(Nλ(ρθ,ξ))

ρθ,ξ]

(15)

=

∫

dθ

2πdξ dη p(η)q(ξ)tr

[

T(

Nλ(ρθ,ξ+η))

ρθ,ξ+η

]

=

∫

R2

dξ q′(ξ)

∫

[0,2π)

dθ

2πtr[

T(

Nλ(ρθ,ξ))

ρθ,ξ]

(16)

= Fq′ (T ), (17)

where we used thatW√ληNλ(ρ)W

†√λη

= Nλ

(

WηρW†η

)

and

we introduced the convolution

q′(ξ) :=

∫

R2

dη q(ξ − η)p(η).

ChoosingT to be the optimal classical scheme forq′ we ob-

tain that

Fq ≥ Fq′ , (18)

under the assumption thatq′ is obtained fromq by convolu-tion with another probability densityp. If q and q′, for in-stance, were two Gaussians, then Eq. (18) holds wheneverq′

is broader thanq sincep can then be chosen to be a Gaussianwhose variance is the difference between those ofq′ andq.

Similar holds if, as in the experimental setup,q is a flat-topdistributions on a square(−d, d]× (−d, d]. Taking the convo-lution with a discrete distributionp(η) = 1

4

∑4i=1 δ(η − ηi),

where theηi’s are the four corners of the square[−d, d] ×[−d, d], leads to a flat-top distributionq′ on the square(−2d, 2d] × (−2d, 2d]. This justifies to use the benchmarkcomputed fordmax = 3.8 again fordmax = 7.6 as done inFig. 2 of the main text.