Teleportation using coupled oscillator states

P.T. Cochrane, G.J. Milburn and W.J. MunroDepartment of Physics, The University of Queensland, QLD 4072, Australia.

(April 14, 2000)

We analyse the variation of teleportation fidelity with resource entanglement for three entangle-ment resources: two mode harmonic oscillator states with fixed total photon number, beam splittergenerated harmonic oscillator states with fixed total photon number and the two-mode squeezedvacuum state. We define corresponding teleportation protocols for each example. The variation ofentanglement in each case is modelled using phase noise.

I. INTRODUCTION

Quantum entanglement plays a central role in theemerging fields of quantum computation [1–5], quantumcryptography [6,7], quantum teleportation [8–13], super-dense coding [14] and quantum communication [15–17].The characterisation of entanglement is a challengingproblem [18–23] and considerable effort has been investedin characterising entanglement in a variety of contexts[24–30].

One such context, quantum teleportation, has playeda crucial role in understanding how entanglement can beused as a resource for communication. The recent ex-perimental demonstrations [31,32] suggest that quantumteleportation could be viewed as an achievable experi-mental technique to quantitatively investigate quantumentanglement. Teleportation is a way of transmitting anunknown quantum state to a distant receiver with farbetter reliability than can be achieved classically. As theentanglement of the enabling resource is degraded, thefidelity of the teleportation protocol is diminished.

In this paper we attempt to make this intuition moreprecise using specific examples from quantum optics.Three entangled resources are considered: two mode har-monic oscillator states with fixed total photon number,beam splitter generated harmonic oscillator states withfixed total photon number and the two-mode squeezedvacuum state [33]. The examples we discuss exhibitquantum correlations between the photon number in eachmode and, simultaneously, between the phase of eachmode. To degrade the entanglement of the resource weconsider phase fluctuations on each mode independently.In the limit of completely random phase we are left withonly the classical intensity (photon number) correlations.The ability of the resources to teleport a state is shownto improve with the energy available in the resource, andtherefore the dimension of the Fock space.

II. ENTANGLEMENT AND TELEPORTATION

Intuitively entanglement refers to correlations betweendistinct subsystems that cannot be achieved in a clas-

sical statistical model. Of course correlations can ex-ist in classical mechanics, but entanglement refers toa distinctly different kind of correlation at the level ofquantum probability amplitudes. The essential differ-ence between quantum and classical correlations can bedescribed in terms of the separability of states [34–40,46].A density operator of two subsystems is separable if it canbe written as the convex sum [40]

ρ =∑A

wAρ′A ⊗ ρ′′A (1)

As an example we consider the density operator for twoharmonic oscillators (a and b) which have correlated en-ergy

ρab =∞∑

n=0

pn|n, n〉〈n, n| (2)

where we use the notation |n, n〉 = |n〉a ⊗ |n〉b. Such astate has perfectly correlated excitation energy in eachmode, a classical correlation, but no entanglement. Onthe other hand the pure state

|Ψ〉ab =∞∑

n=0

√pn|n, n〉 (3)

has the same classical correlation but is not separable.In this form we see that is is possible for a separable andan entangled state to share similar classical correlationsfor some variables. Any state that is not separable isentangled, but it may be difficult to distinguish a separa-ble state from an entangled state. Teleportation howeverwill enable such states to be distinguished.

Consider a communication protocol in which the re-sults of measurements made on a physical system aretransmitted to a distant receiver. The goal of the receiveris to reconstruct the physical state of the source, usingonly local resources, conditioned on the received infor-mation. The communication that takes place is of courseentirely classical. In a teleportation protocol there is oneadditional feature: quantum correlations, entanglement,are first shared between the sending and receiving sta-tion. The degree of entanglement shared by sender and

1

receiver is called the teleportation resource. If there isno shared quantum correlation between the sender andreceiver, the protocol is called classical. The extent andnature of the quantum correlations in the resource deter-mine the fidelity of the protocol. Under ideal conditionsthe unknown state of some physical system at the trans-mitting end can be perfectly recreated in another physicalsystem at the receiving end. If there is no shared entan-glement resource this is impossible. There are many waysin which actual performance can differ from the ideal.In this paper we analyse the change in performance ofteleportation protocols as the entanglement resource isvaried. Our primary objective is to use the fidelity of ateleportation protocol to compare and contrast differentkinds of entangled oscillator states.

In general, teleportation proceeds as follows: thesender, Alice, has a target state, |ψ〉T , she wishes to tele-port to Bob, the receiver (see Figure 1). Alice and Bobeach have access to one part of an entangled bipartitephysical system prepared in the state |ψ〉AB . In this pa-per the bipartite physical system is a two-mode electro-magnetic field. In order to send the state of the target toBob, Alice performs a joint measurement of number sum(N) and phase difference (Φ) on the target and her mode.She then sends the information gained from these mea-surements to Bob via a classical channel. Bob performslocal unitary transformations on the mode in his pos-session according to the information Alice sends to him,thereby attempting to recreate the initial target state.We quantify the quality of the protocol by the modulussquare of the inner product between the target state andthe actual state that Bob reconstructs. This quantity isknown as the fidelity.

j iT

Alice

Bob

N ; �

U

j iout;B

j iAB

Classical Channel

FIG. 1. General teleportation protocol. The entanglement resource |ψ〉quantum state |ψ〉T at |ψ〉out,B.

The fidelity has an important interpretation in quan-tum measurement theory which makes it a particularlyappropriate measure of the quality of a teleportation pro-tocol. Consider for a moment how Bob and Alice wouldestablish that the protocol had succeeded. If two phys-ical systems are in the same pure quantum state, thenthe statistics of all possible measurements made on thosesystems must be the same. In reality of course we can-not measure every physical quantity. The question ofhow many measurements suffice has been the subject ofconsiderable research and forms the basis of quantumstate tomography [41]. To asses how well the telepor-tation protocol has worked we essentially need to knowthe statistical distinguishability of the actual state |ψ〉aachieved and the ideal state |ψ〉i. The answer to thisquestion is known. The statistical distance between twoclassical distributions, p(1)

n , p(2)n is given in terms of the

classical fidelity by

F (p(1), p(2)) =∑

n

√p(1)n , p

(2)n (4)

If we now minimise this over all possible measurementdistributions, we find it is bounded from below by thequantum fidelity [45].

The fidelity of quantum teleportation protocol is deter-mined by the degree of shared entanglement, the qualityof the measurements made by the sender, the quality ofthe classical communication channels used and finally itdepends on how well Bob can implement the desired uni-tary transformations. In this paper we will discuss only

2

the first of these; the amount of shared entanglement. Inthe original teleportation protocol [8], the bipartite sys-tem was made up of two systems each described by atwo dimensional Hilbert space, that is to say, two qubits,and the shared entangled state was a maximally entan-gled state [42]. In the case of two correlated harmonicoscillators, or two field modes, we cannot define maximalentanglement in quite the same way, as the entropy ofeach component system can be arbitrarily large. In thispaper we define extremal entangled pure states of twofield modes in terms of the total mean photon numberand the total maximum photon number.

In the case of a system with a finite dimensional Hilbertspace, a state of maximum entropy is simply the identityoperator in that Hilbert space. A natural generalisationof this idea to infinite Hilbert spaces would define a max-imum entropy state subject to some constraint, such asmean energy or total energy. These of course define thecanonical ensemble and micro canonical ensemble of sta-tistical mechanics. In the case of entangled pure statesthe Araki-Lieb [43] inequality indicates that the entropyof each component system is equal. As the entropy ofa harmonic oscillator scales with mean energy, this indi-cates that each component subsystem has the same meanenergy. If we maximise the entropy of each subsystemsubject to a constraint on the mean energy, the statemust be a thermal state. The entangled pure two modestate for which the reduced density operator of each modeis thermal is the squeezed vacuum state.

|λ〉 = (1 − λ2)−1/2∞∑

n=0

λn|n, n〉 (5)

The mean photon number in each mode is given byn = λ2/(1 − λ2). If however we constrain the total pho-ton number, N , of each mode we get a very differentdefinition of an entangled our state,

|N〉 =1√

1 +N

N∑n=0

|N − n, n〉 (6)

The entropy of the reduced state of each mode is ln(1+N)while the mean photon number is N/2. We will con-sider n of these entangled pre-states as a teleportationresource. While squeezed vacuum states may be achievedin the laboratory, states with fixed total photon numberhave not been produced, and will not be possible until wehave a reliable N photon source. There are now a coupleof proposals for such sources [44] and it may not be toolong before they are used in teleportation schemes.

Teleportation fidelity for infinite dimensional Hilbertspaces must necessarily vary from unity for an arbitrarytarget state, as the notion of a maximally entangled re-source differs form the finite dimensional case. The tele-portation protocol can also be degraded by unknown in-coherent process that corrupts the purity of the sharedentanglement see Figure 2. Of course in some cases

these incoherent processes may destroy the correlationsentirely, for example by absorbing all the photons in eachmode before Alice and Bob get to use them. In this pa-per, however, we will only consider those decoherenceprocesses that change the purity of the states and leaveunchanged the classical intensity correlations.

j iT

j iout

j iAB

FIG. 2. Using teleportation to analyse entanglement. The decoherence p|ψ〉AB thereby changing the overlap between |ψ〉T and |ψ〉out after the tel

III. IDEAL RESOURCE

In a recent paper by Milburn and Braunstein [33] ateleportation protocol was presented using joint measure-ments of the photon number difference and phase sumon two field modes. This protocol is possible because thenumber difference and phase sum operators commute,thus allowing determination of these quantities simul-taneously and to arbitrary accuracy. The teleportationscheme improves with increasing number of Fock statesin the resource since the resource states tend to simulta-neous eigenstates of number difference and phase sum.

Number sum and phase difference operators also com-mute, implying that if eigenstates of these operators canbe found a teleportation protocol is possible. Such a pro-tocol is discussed below.

Because the number sum and phase difference oper-ators commute we look for simultaneous eigenstates ofthese observables. Consider states of the form

|ψ〉AB =N∑

n=0

dn|N − n〉A|n〉B (7)

which are eigenstates of number sum with eigenvalue N .The labels A and B refer to Alice’s and Bob’s modes re-spectively and the coefficients dn satisfy

∑Nn=0 |dn|2 = 1,

so that the state is normalised. For Eq. (7) to be usedas a resource for teleportation it needs to be entangled;it will be maximally entangled when the dn are all equal.We choose dn = 1√

N+1, giving the resource,

|ψ〉AB =1√N + 1

N∑n=0

|N − n〉A|n〉B. (8)

3

This resource tends towards eigenstates of phase dif-ference as N → ∞. To see this consider the joint phaseprobability density of Eq. (8), which is calculated by pro-jecting onto each mode with the phase state, [47–49]

|φ〉 =∞∑

n=0

einφ|n〉, (9)

and taking the modulus squared, i.e.,

P (φA, φB) = |〈φA|〈φB |ψAB〉|2

=1√N + 1

∣∣∣∣∣N∑

n=0

einφ−

∣∣∣∣∣2

(10)

where φ− = φA −φB . The probability density as a func-tion of N and φ− is shown in Figure 3, and shows that asN gets larger this density becomes sharply peaked aboutφ− = 0 in the interval [−π, π], hence the states of Eq. (8)tend to eigenstates of phase difference with increasing N .

FIG. 3. Joint phase probability density. As N increases the probability density becomes very narrowly peaked about φ = 0.

To decide when N is sufficiently large for the resource(Eq. (8)) to be called a simultaneous eigenstate of phasedifference we need to determine when the probabilitydensity (Eq. (10)) is sufficiently narrow. We make thecriterion that if the full width at half maximum of thedensity is 1% of the domain of φ− ([−π, π]) then N islarge enough. This criterion is met if N > 88.

The state to be teleported can be written in the generalform

|ψ〉T =∞∑

m=0

cm|m〉T , (11)

The subscript T is used to emphasise that this is the“target” mode. We generate the input state to the pro-tocol by taking the tensor product of the target and theresource

|ψ〉TAB = |ψ〉T ⊗ |ψ〉AB

=1√N + 1

∞∑m=0

N∑n=0

cm|m〉T |N − n〉A|n〉B. (12)

Alice’s measurement of the number sum on the targetand her mode projects the input state into the condi-tional state

|ψ(q)〉 =1√N + 1

1√PI(q)

×∑

n

cq−N+n|q −N + n〉T |N − n〉A|n〉B, (13)

where n runs from max(0, N − q) to N , and q is thenumber sum measurement result, and

PI(q) =1

N + 1

∑n

|cq−N+n|2, (14)

is the probability of measuring that result, I merely em-phasises that this probability relates to the idealised re-source Eq. (8). Note that again the summation runs fromn = max(0, N − q) to N . After Alice’s measurement ofthe phase difference the total state is

|ψ(q,φ−)〉 = ei(q−2N)φ−

×∞∑

w,y=0

ei(y−w)φ−δw,q−y|w〉T |y〉A

⊗ 1√N + 1

1√PI(q)

×∑

n

e2inφ−cq−N+n|n〉B, (15)

implying that the state in Bob’s mode is

|ψ〉B =1√N + 1

1√PI(q)

×∑

n

e2inφ−cq−N+n|n〉B. (16)

Using the results q and φ−, and knowledge of the num-ber of Fock states in the resource (N), Bob has sufficientinformation to reproduce the target state. He does thisby amplifying his mode so that |n〉B → |q − N + n〉Band phase shifting it by e−i2nφ− . The unitary amplifi-cation operation is described in [50]. These operationscomplete the protocol and the state Bob finally has inhis possession is

|ψ〉out,B =1√N + 1

1√PI(q)

×∑

n

cq−N+n|q −N + n〉B . (17)

The fidelity of this protocol is

FI =∑

n

|cq−N+n|2. (18)

To see how well the teleportation protocol performs, letus consider some examples.

4

1. Example: Target is a number state

Let the target state be a number state,

|ψ〉T = |m〉T , (19)

so the only coefficient available is cm, which is one.

We find that the teleportation fidelity is unity, inde-pendent of the measurement of q, because the only termappearing in the summations of both the fidelity and theprobability is that corresponding to cm. Therefore, thisprotocol works perfectly if the target is a number state.

2. Example: Target is a coherent state

The fidelity as a function of number sum measurement,q, and number of Fock states in the resource minus one,N , for a coherent target state of amplitude, α = 3, isdisplayed in Figure 4.

FIG. 4. Fidelity, F , as a function of number sum measurement, q, and energy in the ideal resource, N , for a coherent stateof amplitude α = 3.

The figure shows that as N increases, the range ofmeasurement outcomes of number sum measurements forwhich the fidelity is high also increases, implying thetechnique is improving with increasing N . Also considerthe probability of finding a given q, which is shown asa function of q and N in Figure 5. This shows that theprobability of obtaining a given number sum is high incorrespondence with the ranges of q for which the fidelityis high. Hence we can say that the protocol works wellas result of this correspondence.

FIG. 5. Probability, P , as a function of number sum measurement, q,state of amplitude α = 3.

We expect this protocol to improve with increasing Nfor two reasons: the approach of the resource states tosimultaneous eigenstates of number sum and phase dif-ference, and an improved resolution of the target state.Firstly, as mentioned in Section III, as N increases the re-source states tend to eigenstates of phase difference andsince they are already eigenstates of number sum theytherefore tend to simultaneous eigenstates of both quan-tities. Because of this the joint measurements of numbersum and phase difference made by Alice become moreaccurate which we would expect to improve the outputof the teleportation. Secondly, the resolution of the tar-get improves with increasing N . What we mean is thatthere are more states available in the resource with whichto “encode” the target before it is teleported. An anal-ogy between this and digital electronics can be made.Consider an analogue to digital converter. Such a deviceconverts (“encodes”) a continuous analogue signal into adigital form in much the same way we do in our telepor-tation protocol. The quality of the digital reproductionof the analogue signal increases as one increases the num-ber of bits one uses to “encode” the analogue signal inthe digital medium. Much the same process occurs herewhere as we add more states to the description of ourresource (i.e. increase N) we are increasing the numberof “bits” used to “encode” the target state. We use thewords “bit” and “encode” very loosely, but the idea iswhat is important. So, in analogy with the analogue todigital converter, as we increase the number of “bits” inour resource we would also expect the quality of output(fidelity) to improve. Overall, this implies that the shapeof the fidelity function in Figure 4 is not surprising, andin fact what we would expect.

IV. BEAM SPLITTER RESOURCE

The resource states discussed in Section III illustratethe protocol well, but do not describe a known physicalinteraction. However the beam splitter interaction can

5

be shown to give a resource with similar properties toEq. (8). The beam splitter interaction is described by[51]

|ψ〉AB = ei π4 (a†b+ab†)|N〉A|N〉B, (20)

where the operators a, a†, b and b† are the usual bosonannihilation and creation operators for modes A and B.N is the number of photons at each port of the beamsplitter.

We can understand these states by considering the bo-son representation of SU(2) algebra;

Jx =12(a†b+ b†a) (21a)

Jy =−i2

(a†b− b†a) (21b)

Jz =12(a†a− b†b) (21c)

J2 = N(N + 1). (21d)

Let us restrict the state space to eigenstates of the num-ber sum such that a†a + b†b = 2N , implying a pseudoangular momentum algebra. The eigenstates of Jz canbe written in the number state basis as

|N,µ〉z = |N + µ〉A|N − µ〉B. (22)

There are 2N + 1 such eigenstates with eigenvalues µ =−2N,−2N + 1, . . . , 0, . . . , 2N − 1, 2N . Consequently, wecan write Eq. (20) in the form

|ψ〉AB = ei π2 Jx |N, 0〉z. (23)

Expanding this in terms of the Jz eigenstates in the Fockbasis we obtain,

|ψ〉AB =N∑

µ=−N

z〈N,µ|ei π2 Jx |N, 0〉z

×|N + µ〉A|N − µ〉B, (24)

where the term z〈N,µ|ei π2 Jx |N, 0〉z is the distribution of

the Jz eigenstates over the range −N ≤ µ ≤ N and canbe obtained from the rotation matrix coefficients (whichare discussed in more detail in Appendix (B)). Eq. (24)is written more simply as

|ψ〉AB =2N∑n=0

dn−N |n〉A|2N − n〉B. (25)

where the dn−N are the rotation matrix coefficients andmaking the change of variables µ = n−N . These statesmay be used as a resource in our number sum, phase dif-ference teleportation protocol because their form is verysimilar to Eq. (7). This protocol proceeds identicallyto that discussed in Section III excepting the additional

knowledge of how the the resource is generated and theinherent properties therein.

We illustrate this variation of the protocol with thepure state form of the resource as given in Eq. (25). Thetotal state is,

|ψ〉TAB = |ψ〉T ⊗ |ψ〉AB

=∞∑

m=0

2N∑n=0

cmdn−N |m〉T |n〉A|2N − n〉B. (26)

The state of the system, conditioned on the number sumresult q, is

|ψ(q)〉 = NBS

min(q,2N)∑n=0

cq−ndn−N

×|q − n〉T |n〉A|2N − n〉B , (27)

where NBS = 1√PBS(q)

and

PBS(q) =min(q,2N)∑

n=0

|cq−n|2|dn−N |2, (28)

is the probability of obtaining the result q, the subscriptdenoting that the probability is for the beam splitter gen-erated resource. After phase difference measurement thetotal state is,

|ψ(q,φ−)〉 =∞∑

w,y=0

ei(y−w−q)φ−δw,q−y|w〉T |y〉A

⊗ NBS

min(q,2N)∑n=0

e2inφ−

×cq−ndn−N |2N − n〉B, (29)

and the state in Bob’s mode is

|ψ〉B = NBS

min(q,2N)∑n=0

e2inφ−

×cq−ndn−N |2N − n〉B . (30)

Applying the amplification |2N − n〉B → |q − n〉B andthe phase shift e−2inφ− , the output state becomes,

|ψ〉out,B = NBS

min(q,2N)∑n=0

cq−ndn−N |q − n〉B , (31)

and the teleportation fidelity is found to be

FBS =1

PBS(q)

∣∣∣∣∣∣min(q,2N)∑

n=0

|cq−n|2dn−N

∣∣∣∣∣∣2

. (32)

6

3. Example: Target is a coherent state

The fidelity function in Figure 6 is almost identical toFigure 4 except that its maximum is one half as opposedto unity. The reason for this is due to every second termof the target is being removed because the every secondterm of the resource is zero. Every second term of theresource is zero because in the case of equal photon num-ber incident onto each port of a 50:50 beam splitter givesrotation matrix coefficients whose every second term iszero. Therefore, only half of the terms in the target arecontributing to the fidelity, so the maximum the fidelitycan be for such a target is one half. As in the ideal case,this resource improves in its ability to teleport a statewith increasing N for exactly the same reasons discussedin Section III

0 50 100 150 200 250

0

50

1000

0.1

0.2

0.3

0.4

0.5

N

q

F

FIG. 6. Fidelity, F , as a function of number sum measurement, q, and energy in the beamsplitter resource, N , for a coherentstate of amplitude α = 3.

The probability function is shown in Figure 7 and hasmuch the same structure as the probability in Figure 5.The “bumps” on the “edge” of the surface are due to thedistribution of the dn−N coefficients not being flat (c.f.the even distribution of Eq. (8)).

0 50 100 150 200 250

0

50

1000

0.05

0.1

0.15

q

N

P

FIG. 7. Probability, P , as a function of number sum measurement, q, and energy in the beamsplitter resource, N , foracoherent state of amplitude α = 3.

4. Example: Target is an even “cat” state

If we have some a priori knowledge about the formof the state we are attempting to teleport, it is possibleto achieve unit fidelity. For instance if we use an even“cat” state, formed from the superposition of two coher-ent states of equal amplitude but opposite sign, [52]

|ψ〉T =|α〉T + | − α〉T√

2 + 2e−2|α|2 , (33)

we find that every second term of the target is zero. Thismeans that even though every second term in the re-source is zero and is removing every second term fromthe target, this has no effect on the resource’s ability toteleport an even “cat” state, as shown by the fidelityfunction in Figure 8.

0 50 100 150 200 250

0

50

1000

0.2

0.4

0.6

0.8

1

N

q

F

FIG. 8. Fidelity, F , as a function of number sum measurement, q, andstate of amplitude α = 3.

This example is interesting since we have seeminglyadded only a small amount of extra information but wehave greatly improved the quality of the protocol. Evensuch incomplete knowledge about the target as its par-ity or form in a given basis can allow one to teleportthe state perfectly with a seemingly imperfect resource.This result implies that it is possible to tailor resourcesfor given applications, so that if only the form of thestate to be teleported is known, then one may be able toconstruct a resource that will teleport the state perfectly,without the resource necessarily being able to teleport anarbitrary state perfectly.

V. DECOHERENCE

We model decoherence by using phase diffusion. Thisretains the classical (number or intensity) correlations inthe system, but alters the quantum (entanglement re-lated) correlations. Phase diffusion is modelled by ap-plying the following phase shift to the resource

7

U(θ) ≡ exp[−i(θaa

†a+ θbb†b)

], (34)

where the combined phase of θa and θb is taken to beGaussian randomly distributed with a zero mean andvariance σ.

P (θ) =1√2πσ

exp(− θ2

2σ

). (35)

Even though a Gaussian distribution is not periodicit can taken to be an approximation to a true periodicdistribution, such as cos2N (θ − θ0), for sufficiently largeN . To see that this is so, consider the probability distri-bution

P (θ) = N cos2N (θ − θ0) (36)

where N is the normalisation. This can be approximatedby

P (θ) ≈ N[1 − (θ − θ0)2

2

]2N

≈ N [1 −N(θ − θ0)2

]≈ N exp

[−N(θ − θ0)2]

= N exp[−1

2(θ − θ0)2

σ

](37)

where σ = 12N . Therefore, a Gaussian distribution can

approximate a true periodic distribution given the con-dition that σ ∼ 1

2N .

A. Squeezed state resource

The phase diffusion model may be illustrated by apply-ing it to the resource discussed by Milburn and Braun-stein [33]. In their paper they describe teleportation us-ing two-mode squeezed vacuum states as an entangle-ment resource for teleportation. The entanglement be-tween resource modes may be altered by changing thesqueezing parameter, λ, and by decohering the resourceusing phase diffusion. The resource for this protocol iswritten in the Fock basis as

|ψ〉AB =√

1 − λ2

∞∑n=0

λn|n〉A|n〉B . (38)

Applying the phase shift U(θ) to the resource gives,

|ψ(θ)〉AB =√

1 − λ2e−i(θaa†a+θbb†b)∞∑

n=0

λn|n〉A|n〉B

=√

1 − λ2

∞∑n=0

e−iθnλn|n〉A|n〉B, (39)

where θ = θa + θb is the Gaussian distributed randomvariable with mean zero and variance σ

We average over all realisations of the phase to givethe resource as a density operator

ρAB = (1 − λ2)∞∑

n,n′=0

λnλn′e−γ(n−n′)2

×|n〉A〈n′| ⊗ |n〉B 〈n′|, (40)

where γ describes the degree of decoherence and is equalto σ/2. The input state is

ρTAB = ρT ⊗ ρAB

= (1 − λ2)∞∑

n,n′=0

cmc∗m′λnλn′

e−γ(n−n′)2

×|m〉T 〈m′| ⊗ |n〉A〈n′| ⊗ |n〉B 〈n′|. (41)

This teleportation takes two paths, one characterisedby measurement of a positive number difference and oneby a negative number difference. We will arrive at two fi-delities, each corresponding to the relevant measurement.

1. Positive number difference measurement

The state conditioned on the positive number differ-ence, q is

ρ(q)+ =

1 − λ2

P+(q)

∞∑n,n′=0

cn+qc∗n′+qλ

nλn′e−γ(n−n′)2

×|n+ q〉T 〈n′ + q| ⊗ |n〉A〈n′| ⊗ |n〉B 〈n′|, (42)

where

P+(q) = (1 − λ2)∞∑

n=0

|cn+q|2λ2n, (43)

is the probability of finding the result q. After measure-ment of the phase sum the system is in the state

ρ(q,φ+) =∞∑

x,y,x′,z′=0

ei(w+y−x′−z′)φ+δw,q+yδq,x′−z′

×|w〉T 〈x′| ⊗ |y〉A〈z′|

⊗ 1 − λ2

P+(q)

∞∑n,n′=0

e−2i(n−n′)φ+cn+qc∗n′+q

×λnλn′e−γ(n−n′)2 |n〉B 〈n′|, (44)

so the state in Bob’s mode is

ρB =1 − λ2

P+(q)

∞∑n,n′=0

e−2i(n−n′)φ+cn+qc∗n′+q

×λnλn′e−γ(n−n′)2 |n〉B 〈n′|. (45)

After amplification and phase shifting the output stateis

8

ρout,B =1 − λ2

P+(q)

∞∑n,n′=0

cn+qc∗n′+q

×λnλn′e−γ(n−n′)2 |n+ q〉B 〈n′ + q|, (46)

with a corresponding fidelity given by

F+ =1 − λ2

P+(q)

∞∑n,n′=0

|cn+q|2|cn′+q|2λnλn′e−γ(n−n′)2 , (47)

2. Negative number difference measurement

For measurement of negative number difference, q′ =−q, the conditional state is

ρ(q′)− =

1 − λ2

P−(q′)

∞∑m,m′=0

cmc∗m′λm+q′

λm′+q′e−γ(m−m′)2

× |m〉T 〈m′| ⊗ |m+ q′〉A〈m′ + q′|⊗|m+ q′〉B 〈m′ + q′|, (48)

where

P−(q′) = (1 − λ2)∞∑

m=0

|cm|2λ2(m+q′), (49)

is the probability of obtaining the result q′. The fidelityafter teleportation is

F− =1 − λ2

P−(q′)

∞∑m,m′=0

|cm|2|cm′ |2λm+q′λm′+q′

e−γ(m−m′)2 .

(50)

The fidelity as a function of q and γ is shown in Fig-ure 9 and behaves as we would expect, decoherence inthe resource reduces the output quality of the protocolimplying that the entanglement available as a resourcefor teleportation has decreased.

−500

50

0.3

0.2

0.1

00

0.2

0.4

0.6

0.8

q

F

FIG. 9. Fidelity as a function of number difference measurement, q, and degree of decoherence, γ, with a squeezing parametervalue of λ = 0.8 at a two-mode squeezed state resource energy corresponding to N = 100.

If we take γ = 0 and a coherent state of amplitudeα = 6 then we can reproduce the results of Ref. [33].

B. Ideal resource

Applying our decoherence model to Eq. (8) and aver-aging over all realisations of the phase we obtain the totalstate:

ρTAB =1

N + 1

∞∑m,m′=0

N∑n,n′=0

cmc∗m′e−γ(n−n′)2

×|m〉T 〈m′| ⊗ |N − n〉A〈N − n′| ⊗ |n〉B 〈n′| (51)

where γ is the degree of decoherence as before. Aftercompletion of the protocol the fidelity is given by

FI,γ =1

N + 11

PI(q)

∑n,n′

|cq−N+n|2|cq−N+n′ |2e−γ(n−n′)2 .

(52)

where n and n′ run from max(0, N − q) to N and

PI(q) =1

N + 1

∑n

|cq−N+n|2, (53)

is the probability of obtaining the result q as obtained inEq. (14). It is not difficult to show that by letting γ = 0we reproduce the result without noise, Eq. (18).

Let us examine the example of a coherent state, α = 3.The fidelity as a function of γ and q is shown in Fig-ure 10 and shows how decoherence affects the fidelity.As the degree of decoherence (γ) is increased, the fidelitydrops away very quickly. This is because the off-diagonalmatrix elements of ρAB are being “washed out” by the(n− n′)2 term in the exponential. Physically, we are re-ducing the entanglement between the resource modes bymaking measurement of phase more random and wouldexpect the ability of the technique to teleport a state todecrease – Figure 10 shows this effect explicitly.

0

50

100

0

0.1

0.2

0.30

0.2

0.4

0.6

0.8

1

q

F

FIG. 10. Fidelity as a function of number sum measurement, q, and decorresponding to N = 100.

9

C. Beam splitter resource

We add noise to the beam splitter resource state in thesame manner as described in Section VA, obtaining thetotal state,

ρTAB =∞∑

m,m′=0

2N∑n,n′=0

cmc∗m′dn−Nd

∗n′−Ne

−γ(n−n′)2

×|m〉T 〈m′| ⊗ |n〉A〈n′|⊗|2N − n〉B 〈2N − n′|. (54)

After the teleportation protocol, we find that the fidelitywith respect to the initial state is given by

FBS,γ =1

PBS(q)

min(q,2N)∑n,n′=0

|cq−n|2|cq−n′ |2

×dn−Nd∗n′−Ne

−γ(n−n′)2 . (55)

where

PBS(q) =min(q,2N)∑

n=0

|cq−n|2|dn−N |2, (56)

is the probability of obtaining the number sum result q.

FIG. 11. Fidelity as a function of number sum measurement, q, and degree of decoherence, γ, with a beamsplitter resourceenergy corresponding to N = 100.

Again, the fidelity decreases due to decoherence in theresource, except that the fidelity decreases from one halfinstead of one as in Section III.

VI. FULL DECOHERENCE

Full decoherence corresponds to no entanglement be-tween the resource modes and a completely flat phaseprobability distribution. A flat phase probability distri-bution is equivalent to taking the limit γ → ∞ in thefidelities of Section V making the off-diagonal terms inthe density matrix representing the output state ρout,B

zero. Physically, this limit corresponds to retaining thenumber correlations but making a measurement of phasecompletely arbitrary.We take this limit for each of thethree cases discussed in Section V.

A. Idealised resource

A fully decohered resource implies Eq. (52) reduces to

FI,∞ =1

N + 11

PI(q)

N∑n=max(0,N−q)

|cq−N+n|4. (57)

FIG. 12. Fidelity as a function of number sum measurement, q, and ecoherent state target of amplitude α = 3.

FI,∞ as a function of q and N is shown in Figure 12,note that it behaves in much the same way as the noise-less case (Figure 4) where the fidelity stays “high” for anincreasing part of the region of q with increasing N , buthas a significantly lower value.

A feature of interest in Figure 12 is the bump on eachedge of the fidelity distribution. As one scans across qfor a given value of N the fidelity increases, drops fromzero, peaks slightly to remain flat, then peaks again be-fore dropping away to zero. These features may be seenmore clearly in Figure 13 where FI,∞ is plotted againstq for N = 100.

10

0 50 100 1500

0.02

0.04

0.06

0.08

0.1

0.12

q

F

FIG. 13. Slice through fully decohered ideal resource fidelity function at N = 100.

The “bumps” are due to the distribution of the cm co-efficients of the input target state. To understand theorigin of the “bumps” we write the fidelity of Eq. (57)more explicitly,

FI,∞ =

∑Nn=max(0,N−q) |cq−N+n|4∑Nn=max(0,N−q) |cq−N+n|2

. (58)

Eq. (58) shows that we are finding the summation of adistribution with each term raised to the power of fourdivided by the same distribution with each term raised tothe power of two. Note that a coherent state distributionsquared has a larger spread than one raised to the powerof four. The distributions for a coherent state, α = 3are shown in Figure 14. The α2 curve corresponds tothe coefficients of the coherent state squared and the α4

curve corresponds to the coefficients raised to the powerof four.

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

�2

�4

m

c2

m

c4

m

FIG. 14. Comparison of α2 and α4 coefficient distributions. The cm are the values of the coefficients at a given value of theFock number, m.

Let us consider q < N in the range m = [0, 2] whichcorresponds to the coefficients between c0 and c2 inclu-

sive. The summations in Eq. (58) only add up the coef-ficients from c0 to c2 so, the major contribution to thefidelity comes from the α2 curve which corresponds to thedenominator in Eq. (58), hence the fidelity is small. If welet m = 7 the summations add about half of each of thetwo distributions, this corresponds to where the fidelityreaches about half of its maximum value in Figure 13. Aswe increase m further we are adding more of the α4 dis-tribution and the fidelity increases since this correspondsto the numerator in Eq. (58). Now take m = 12 which iswhere the α4 distribution is small again, this correspondsto the maximum of the first “bump” in Figure 13. As wetake larger m, the summation from c0 to cm of the α4

distribution has maximised, but the summation of theα2 distribution has not, so as we take larger m the fi-delity decreases, since the α2 distribution corresponds tothe denominator in Eq. (58). The fidelity then stays ata constant value because both of the summations havereached their respective maxima. This tells us how thefirst “bump” is produced. The second “bump” is pro-duced by exactly the same effect but in reverse order. Atthe m corresponding to the second “bump” we are in theregion where q > N , hence the summation has changedfrom starting at c0 and finishing at cm, to starting atcq−N and finishing at cq. The summations now move asa “window” of length N + 1 over the distributions. Asthe “window” moves across the distributions the contri-bution from the α2 distribution decreases as coefficientsare being dropped from the summation in the denomi-nator, and we see the fidelity increase again, forming thesecond “bump”. As the “window” moves further to theright over the distributions in Figure 14 the fidelity dropsaway to zero as we are no longer adding non-zero termsin the two distributions.

One may notice that the “bump” on the right handside of Figure 13 is lower than that on the left hand side.This is due to the distribution of states in the target. Inthe case that generates Figure 13 we are using a coherentstate of amplitude three. The distribution of a coherentstate of this amplitude is slightly asymmetrical and givesthe slightly lower “bump” to the right we see in Figure 13.If we use a symmetric distribution of coefficients in thetarget state, say a Gaussian of the form,

cn =1√2πσ

e−(n−n0)2

2σ , (59)

then one would expect to see the “bumps” occurring atthe same height. This is exactly what we see in Figure 15,where we have used a Gaussian with n0 arbitrarily set to30 and σ = 3.

11

0 50 100 150 2000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

q

F

FIG. 15. Slice through fully decohered ideal resource fidelity function for a Gaussian distributed target at N = 100.

Because the “bumps” are due to differences in distri-bution width between when we square the terms in thedistribution and when we take the terms in the distri-bution to the power of four, we would expect that if wechoose a flat distribution then we would not see this ef-fect. Take a target state with coefficients (in the Fockbasis) defined by

if n < 57; cn =14

(60a)

if n ≥ 40; cn = 0 (60b)

The distribution of these coefficients squared will havethe same width as that raised to the power of four. FI,∞as a function of q and N for this distribution of coeffi-cients is shown in Figure 16, note that it has no “bumps”as we expect. Hence we conclude that the “bumps” are adirect result of the continuous nature of the distributionof coefficients defining the target state.

0 20 40 60 80 100 120 140 160 180 2000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

q

F

FIG. 16. Fidelity function of a fully decohered ideal resource for a flat target coefficient distribution.

B. Beam splitter resource

The fidelity for a fully decohered resource generatedby the beam splitter interaction reduces from Eq. (55) to

FBS,∞ =∑min(q,2N)

n=0 |cq−n|4|dn−N |2∑min(q,2N)n=0 |cq−n|2|dn−N |2

. (61)

This fidelity as a function of q and N is shown in Fig-ure 17.

FIG. 17. Fidelity, F , as a function of number sum measurement, q, aresource, N , for a coherent state target of amplitude α = 3.

Again, the typical trend of increasing region of highfidelity in q as N increases is shown. Notice that theconstant value of the fidelity function when N is large isapproximately that of the ideal case in Figure 12. Onemay wonder why this is so when previously the beamsplitter version of the protocol could only achieve ap-proximately half that of the ideal version. The reasonfor this is simple and can be explained by consideringEq. (61). In this equation both the numerator and de-nominator are setting half of the terms to zero (due tothe zero terms of the rotation matrix coefficients) mak-ing the ratio almost identical to that in Eq. (57). Hencethe maximum fidelities in the fully decohered limit areapproximately the same for the ideal and beam splittergenerated resources.

C. Squeezed state resource

The fidelities in the completely decohered limit reduceto

F+,∞ =1 − λ2

P+(q)

∞∑n=0

|cn+q|4λ2n (62)

and

F−,∞ =1 − λ2

P−(q′)

∞∑n=0

|cn|4λ2(n+q′). (63)

and give us the surface in Figure 18

12

−500

50 0

0.5

1

0

0.05

0.1

0.15

0.2

q

�

F

FIG. 18. Fidelity, F , as a function of number difference measurement, q, and squeezing parameter, λ for a fully decoheredsqueezed state resource. The target is a coherent state of amplitude α = 3.

The surface we see in Figure 18 has the same form –except for being diminished in fidelity – as the squeezedstate fidelity function over λ and q for no damping (Fig-ure 19). Unfortunately, we have yet to come up withan adequate explanation as to why the top of the peakdecreases as λ increases. Nevertheless, it is not difficultto show that the position of the peak in q is determinedby the target state. Here we use a coherent state of am-plitude, α = 3, which implies a peak value at q = 9,corresponding identically to Figure 18.

−500

50 0

0.5

1

0

0.2

0.4

0.6

0.8

1

q

�

F

FIG. 19. Fidelity, F , as a function of number difference measurement, q, and squeezing parameter, λ, for a squeezed stateresource without decoherence. The target is a coherent state of amplitude α = 3.

VII. CONCLUSIONS

We have shown that a teleportation scheme involvingcoupled oscillator states using number sum and phase dif-ference measurements is possible, given sufficiently largenumbers of Fock states in the resource. The ability ofthe scheme to reliably teleport a state was shown to im-prove as the number of Fock state in the resource in-creases. In the case of the beam splitter generated re-source this physically means more photons incident on

the beam splitter ports. We have illustrated the effectsof decoherence (in the form of phase diffusion) in threeentanglement resources (ideal, beam splitter generatedand squeezed state) on the fidelity of teleportation andhave related this qualitatively to the change in entan-glement of the resource. We have also shown that theability of the resources to teleport a state increases withincrease in the Fock space dimension.

ACKNOWLEDGMENTS

PTC acknowledges the financial support of the Cen-tre for Laser Science and the University of QueenslandPostgraduate Research Scholarship.

[1] D. P. DiVincenzo, Science 270, 255 (1995).[2] L. K. Grover, Phys. Rev. Lett. 79, 325 (1997).[3] A. Ekert and R. Jozsa, quant-ph/9803072 (unpublished).[4] R. Jozsa, quant-ph/9707034 (unpublished).[5] V. Vedral and M. B. Plenio, quant-ph/9802065 (unpub-

lished).[6] C. H. Bennett, G. Brassard, and N. D. Mermin, Phys.

Rev. Lett. 68, 557 (1992).[7] J. Kempe, Phys. Rev. A 60, 910 (1999).[8] C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).[9] C. H. Bennett et al., Phys. Rev. Lett. 76, 722 (1996).

[10] S. L. Braunstein, G. M. D’Ariano, G. J. Milburn, andM. F. Sacchi, quant-ph/9908036 (unpublished).

[11] M. B. Plenio and V. Vedral, quant-ph/9804075 (unpub-lished).

[12] S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, quant-ph/9910030 (unpublished).

[13] S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80,869 (1998).

[14] S. L. Braunstein and H. J. Kimble, quant-ph/9910010(unpublished).

[15] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V.Thapliyal, Phys. Rev. Lett. 83, 3081 (1999).

[16] B. Schumacher, Phys. Rev. A 54, 2614 (1996).[17] B. Schumacher and M. A. Nielsen, Phys. Rev. A 54, 2629

(1996).[18] S. Hill and W. K. Wooters, Phys. Rev. Lett. 78, 5022

(1997).[19] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K.

Wooters, Phys. Rev. A 54, 3824 (1996).[20] W. K. Wooters, quant-ph/9709029 (unpublished).[21] D. P. DiVincenzo et al., quant-ph/9803033 (unpub-

lished).[22] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight,

Phys. Rev. Lett. 78, 2275 (1997).[23] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619

(1998).

13

[24] D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 3566(1999).

[25] D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 1455(1999).

[26] P. Horodecki, M. Horodecki, and R. Horodecki, Phys.Rev. Lett. 82, 1056 (1999).

[27] V. Buzek et al., Phys. Rev. A 55, 3327 (1997).[28] W. Dur and J. I. Cirac, quant-ph/0002028 (unpublished).[29] M. Murao, M. B. Plenio, and V. Vedral, Phys. Rev. A

61, (2000).[30] V. Vedral, quant-ph/9908047 (unpublished).[31] A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A.

Fuchs, H. J. Kimble and E. S. Polzik, Science, 282, 706(1998).

[32] D. Boschi et al., Phys. Rev. Lett. 80, 1121 (1998).[33] G. J. Milburn and S. L. Braunstein, Phys. Rev. A 60,

937 (1999).[34] P. Rungta et al., quant-ph/0001075 (unpublished).[35] P. Deuar, W. J. Munro, and K. Nemoto, quant-

ph/0002002 (unpublished).[36] A. O. Pittenger and M. H. Rubin, quant-ph/0001110 (un-

published).[37] W. Dur, J. I. Cirac, M. Lewenstein, and D. Bruß, quant-

ph/9910022 (unpublished).[38] O. Rudolph, quant-ph/0002026 (unpublished).[39] P. Horodecki, quant-ph/9703004 (unpublished).[40] A. Peres, Phys. Rev. Lett. 77, 1413 (1996).[41] M. Raymer, M. Beck and D. McAllister, Phys. Rev. Lett.

72, 1137 (1994); R. Laflamme, E. Knill, W. H. Zurek,P. Catasti and S. V. S. Mariappan, Phil Trans. R. Soc.Lond. A 356, 1941 (1998).

[42] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, W. K.Wooters, Phys. Rev. A 54, 3824 (1996).

[43] A. Wherl, Rev. Mod. Phys. 50, 221 (1978).[44] A. Imamoglu and Y. Yamamoto, Phys. Rev. Lett. 72,

210 (1994); C. L. Foden, V. I. Talyanskii, G. J. Milburn,M. L. Leadbeater and M. Pepper, to appear Phys. RevB (2000); B. Lounis and W. E. Moerner, Nature ??????(2000).

[45] M. Nielsen and I. Chuang, Quantum Computation andQuantum Information, (Cambridge University Press,Cambridge 2000).

[46] C. M. Caves and G. J. Milburn, quant-ph/9910001 (un-published).

[47] D. F. Walls and G. J. Milburn, Quantum Optics(Springer-Verlag, Berlin, 1995).

[48] L. Susskind and J. Glogower, Physics 1, 49 (1964).[49] S. L. Braunstein, C. M. Caves, and G. J. Milburn, Annals

of Physics 247, 135 (1996).[50] H. M. Wiseman, Ph.D. thesis, The University of Queens-

land, 1994.[51] B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75,

2944 (1995).[52] P. T. Cochrane, G. J. Milburn, and W. J. Munro, Phys.

Rev. A 59, 2631 (1999).[53] L. C. Biedenharn and J. D. Louck, Angular Momentum

in Quantum Physics—Theory and Application (Addison-Wesley, Reading, MA, 1981).

APPENDIX A: OPERATORS

1. Number projectors

We construct the number sum projector from consid-eration of the total number state

ρ =∞∑

n,m=0

|n〉A⊗〈n||m〉B 〈m| (A1)

The action of the number sum operator on this state gives

N+ρ =∞∑

n,m=0

(n+m)|n〉〈n| ⊗ |m〉〈m| (A2)

letting n = k+l2 and m = k−l

2 we find that after measure-ment of the number sum k the conditional state Eq. (A2)becomes

ρ(k) =∞∑

k=0

k

×k∑

l=−k

∣∣∣∣ k + l

2

⟩⟨k + l

2

∣∣∣∣ ⊗∣∣∣∣ k − l

2

⟩⟨k − l

2

∣∣∣∣ (A3)

letting 2n = l + k gives

ρ(k)∞∑

k=0

kk∑

n=0

|n〉〈n| ⊗ |k − n〉〈k − n|. (A4)

So, after a trivial change of variables, for the number summeasurement result q the projector is

N+ =q∑

p=0

|q − p〉|p〉〈p|〈q − p|. (A5)

To find the number difference projectors we follow asimilar procedure to above; acting the number differenceoperator on the state Eq. (A1) gives

N−ρ =∞∑

n,m=0

(n−m)|n〉〈n| ⊗ |m〉〈m|. (A6)

Letting n−m = k and n+m = l we find that the stateconditional on the measurement result k is

ρ(k) =∞∑

k=−∞k

×∞∑

l=|k|

∣∣∣∣ l + k

2

⟩ ⟨l + k

2

∣∣∣∣ ⊗∣∣∣∣ l− k

2

⟩ ⟨l − k

2

∣∣∣∣ (A7)

We will arrive at two number difference projectors, onecorresponding to a positive number difference result andone to a negative number difference result. Choosing

14

k ≥ 0 will give us the positive number difference resultand simplifies Eq. (A7) to

ρ(k) =∞∑

k=0

k

×∞∑

l=k

∣∣∣∣ l + k

2

⟩ ⟨l + k

2

∣∣∣∣ ⊗∣∣∣∣ l − k

2

⟩ ⟨l − k

2

∣∣∣∣ , (A8)

letting 2n = l − k gives the conditional state as

ρ(k) =∞∑

k=0

k

∞∑n=0

|n+ k〉〈n+ k| ⊗ |n〉〈n|. (A9)

Making a change of variables gives the number differenceprojector for the positive number difference result q,

N (+)− =

∞∑p=0

|p+ q〉1〈p+ q| ⊗ |p〉2〈p|, (A10)

To find the projector for a negative number differenceresult we choose k < 0 and set k′ = −k in Eq. (A7) whichsimplifies to

ρ(k′) =∞∑

k′=0

k′

×∞∑

l=k′

∣∣∣∣ l − k′

2

⟩ ⟨l − k′

2

∣∣∣∣ ⊗∣∣∣∣ l+ k′

2

⟩ ⟨l + k′

2

∣∣∣∣ (A11)

Letting 2n = l − k simplifies this to

ρ(k′) =∞∑

k′=0

k′∞∑

n=0

|n〉〈n| ⊗ |n+ k′〉〈n+ k′|, (A12)

so, after a change of variables, the projector for a nega-tive number difference measurement q′ = −q is

N (−)− =

∞∑p=0

|p〉1〈p| ⊗ |p+ q′〉2〈p+ q′| (A13)

2. Phase projectors

The phase difference and the phase sum projection op-erators are derived by integrating the total phase opera-tor over the relevant variable.

The total phase operator is

|φ1〉1〈φ1| ⊗ |φ2〉2〈φ2| =∞∑

w,x,y,z

ei[(w−x+y−z)φ++(x−w+y−z)φ−]

×|w〉1〈x| ⊗ |y〉2〈z|. (A14)

The phase difference projector is obtained by integrat-ing Eq. (A14) over φ+ in the region [0, 2π],

|φ−〉〈φ−| =∞∑

w,x,y,z=0

ei(x−w+y−z)φ−δw−x,z−y

×|w〉1〈x| ⊗ |y〉2〈z|. (A15)

and the phase sum projector is obtained similarly

|φ+〉〈φ+| =∞∑

w,x,y,z=0

ei(w−x+y−z)φ+δx−w,z−y

×|w〉1〈x| ⊗ |y〉2〈z|. (A16)

APPENDIX B: ROTATION MATRIXCOEFFICIENTS

The distribution of states in the resource is describedby the rotation matrix coefficients defined by

djm′,m(β) = z〈j,m′|e−iβJx |j,m〉z . (B1)

It is easier to calculate these matrix elements using Jy sowe rotate about Jz .

z〈j,m′|e−iβJx |j,m〉z= z〈j,m′|e−i π

2 Jze−iβJyei π2 Jz |j,m〉z

= z〈j,m′|e−i π2 m′

e−iβJyei π2 m|j,m〉z

= e−i π2 (m′−m)

z〈j,m′|e−iβJy |j,m〉z= e−i π

2 (m′−m)Djm′,m(β) (B2)

where

Djm′,m(β) = z〈j,m′|e−iβJy |j,m〉z , (B3)

which are discussed in detail in [53]. These rotation ma-trix coefficients may be written in the more explicit form

Djm′,m(β) = [(j +m′)!(j −m′)!(j +m)!(j −m)!]

12

×∑

s

(−1)m′−m+s(cos β

2

)2j+m−m′−2s (sin β

2

)m′−m+2s

(j +m− s)!s!(m′ −m+ s)!(j −m′ − s)!,

(B4)

where s ranges over all integer values where the factorialsare non-negative.

For equal photon number incident on each port of a50:50 beam splitter we have m = 0, j = N and β = π

2 .The distribution of the coefficients defined in Eq. (B1) isshown in Figure 20.

15

−40 −30 −20 −10 0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

m0

dj

m0;m(�)

FIG. 20. Distribution of djm′,m(β) coefficients fo

Note that every second term is zero, this is the rea-son why continuous target states will have a maximumfidelity of a half using the beam splitter generated re-source.

One finds that as N increases the region in the mid-dle of the distribution becomes flatter, thereby approach-ing the entanglement properties of the flat distributiondn = 1/

√N + 1, discussed in Section III.

16