Quantum teleportation with close-to-maximal entanglement from a

PHYSICAL REVIEW A, VOLUME 65, 052313

Quantum teleportation with close-to-maximal entanglement from a beam splitter Ngoc-Khanh Tran and Olivier Pfister* Department of Physics, University of Virginia, 382 McCormick Road, Charlottesville, Virginia 22904-4714 共Received 31 July 2001; revised manuscript received 17 December 2001; published 2 May 2002兲 We investigate theoretically quantum teleportation by means of the number-sum and phase-difference variables. We show that a unitary beam-splitter transformation turns two-mode Fock-states into close-to-maximally entangled states, in this case close approximations of the relative-phase eigenstates. These states could be created experimentally using on-demand single-photon sources, but also with any second-quantized bosonic system 共e.g., trapped ions, Bose-Einstein condensates兲. We show that such ‘‘quasi-EPR’’ states can yield near-unity fidelity for the teleportation of coherent states and of ‘‘Schro¨dinger cat’’ states. DOI: 10.1103/PhysRevA.65.052313

PACS number共s兲: 03.67.Hk, 42.50.Dv

I. INTRODUCTION

Quantum teleportation 关1,2兴, ‘‘the disembodied transport of an unknown quantum state from one place to another’’ 关3兴, is an important component of quantum information. It holds great promise for communication between quantum computers, the realization of quantum gates 关4兴 and the implementation of quantum error correction 关5兴. Quantum teleportation is based on maximally entangled states, a purely quantum mechanical feature, initially little noticed outside research on the fundamental issues of quantum theory such as the Einstein-Podolsky-Rosen 共EPR兲 paradox 关6兴. Producing, preserving, and detecting high quality entanglement is an experimental challenge in making reliable quantum teleporters 共and in quantum information in general兲. Initial experiments based on discrete and finite Hilbert spaces have been successful in proving the principle, but hindered by low efficiencies in the production and detection of entangled photons 关7–9兴. The use of continuous quantum variables for teleportation 关2,10–13兴 offers more straightforward detection methods and also the interesting feature of an infinite Hilbert space, much richer in possibilities for encoding quantum information. The first continuous-variable teleportation experiment 关3,10兴 used quadrature-squeezed electromagnetic fields and beam-splitter entanglement, and was based on the experimental realization of the EPR paradox 关14兴 共see also 关15,16兴兲 using continuous quantum optical variables 关17,18兴. Another set of interesting, in part continuous, variables is constituted by the photon number and the phase, which are canonically conjugate in the same sense as energy and time 关19兴. The use of these variables has been proposed to realize the EPR paradox 关17兴 and the corresponding maximally entangled states are therefore usable for teleportation by means of the detection of the photon number difference and phase sum 关12兴, or of the photon number sum and phase difference 关13,20兴. The latter proposal, besides being more practical and connected to the actively explored field of Heisenberglimited interferometry 关21–25兴, has the advantage to be suited to experimental realization with bright squeezed sources, such as the ones demonstrated in Ref. 关15,16兴. In Ref. 关13兴, Cochrane, Milburn, and Munro 共CMM兲 study *Corresponding author. Email address: [email protected] 1050-2947/2002/65共5兲/052313共9兲/$20.00

quantum teleportation using phase-difference eigenstates. In this article, we adopt the definition of Luis and Sa´nchez-Soto 关26兴 of phase-difference eigenstates, generalized to relativephase eigenstates 关27兴, and investigate the significance of such number-phase EPR states in the context of numberphase teleportation. Because the creation of such states in the laboratory is still a challenge for more than two photons 关27兴, we explore the use of approximate EPR states, or quasi-EPR states, such as created when a twin Fock state such as 兩 n 典 a 兩 n 典 b passes through a lossless beam splitter—which would be very simply realizable with on-demand single-photon sources. CMM found that, for such states, the teleportation fidelity is bounded by the classical limit of 50% for an arbitrary coherent state, but reaches 100% for a ‘‘Schro¨dinger cat’’ state proportional to, 兩 ␣ 典 ⫹ 兩 ⫺ ␣ 典 关13兴. This is due, as CMM points out, to the fact that half the quantum amplitudes of this entangled state are null, and does not affect the cat state which presents the same characteristic 共provided that the nonzero amplitudes coincide兲. The gist of this paper is to show that this nulling of half of the EPR amplitudes corresponds to a very narrow set of experimental conditions and disappears as soon as the beam splitter or its input Fock state are not perfectly balanced 共‘‘perfectly’’ referring, for the beam splitter, to the Heisenberg limit兲. As a result, we show that a wider set of quasiEPR states than considered by CMM do yield near-unity fidelity for teleportation of both coherent and cat states. In Sec. II we derive and evaluate close-to-maximally entangled, or quasi-EPR, states that can be created by sending two-mode Fock states through a lossless beam splitter. In Sec. III we describe number-phase teleportation for ideal EPR states, and for quasi-EPR states. We then analyze the fidelity of quasi-EPR based teleportation of coherent and cat states. II. GENERATION OF CLOSE-TO-MAXIMALLY ENTANGLED STATES BY A BEAM SPLITTER A. The Schwinger representation

We begin by recalling the definition of the Schwinger representation 关28兴, widely used in quantum optics, of a nondegenerate two-mode field in terms of a fictitious spin. This spin is defined as

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©2002 The American Physical Society

NGOC-KHANH TRAN AND OLIVIER PFISTER

冉冊 冉

a † b⫹b † a

Jx

1 J⫽ J y ⫽ 2 Jz †

J2 ⫽

†

PHYSICAL REVIEW A 65 052313

冊

⫺i 共 a † b⫺b † a 兲 , a † a⫺b † b

冉

†

†

冊

a a⫹b b a a⫹b b ⫹1 , 2 2

共2兲

where a and b are the photon annihilation operators of each mode. The physical meaning of these operators is the following 关26,25兴: J2 represents the total photon number, J z the photon number difference between the two modes, and J x,y are the phase difference, or interference, quadratures. It stems from this that e i ␪ J z is the relative phase shift operator and e i ␪ J x is the rotation carried out by a beam splitter 共homo-/heterodyne measurements兲. The eigenstates of the fictitious spin are the two-mode Fock states 兩 j m 典 z⫽ 兩 n a典 a兩 n b典 b ,

共3兲

and the respective eigenvalues of J 2 and J z are given by j⫽

n a ⫹n b N ⫽ , 2 2

共4兲

n a ⫺n b . m⫽ 2

冉 冊 b1

⫽

冉

␤ cos e i/2( ␣ ⫹ ␥ ) 2

␤ sin e i/2( ␣ ⫺ ␥ ) 2

␤ ⫺sin e ⫺ i/2( ␣ ⫺ ␥ ) 2

␤ cos e ⫺ i/2( ␣ ⫹ ␥ ) 2

冊冉 冊 a0

b0

, 共6兲

where ␣ , ␤ , and ␥ are the Euler angles, corresponds to the rotation operator e ⫺i ␣ J z e ⫺i ␤ J y e ⫺i ␥ J z (ប⫽1). A lossless beam splitter corresponds to the values

␣ ⫽⫺ ␥ ⫽ ␲ /2,

␤ ⫽⫾2 arccos冑R,

共7兲

where R is the reflectivity of the beam splitter 共the transmittivity T is such that R⫹T⫽1).

s kl s k*⬘ l ⫽ ␦ kk ⬘ .

共9兲

An example of the EPR state is the eigenstate, introduced by Luis and Sa´nchez-Soto 关26兴, of the operators of the photonnumber sum and phase difference of two modes a and b. Heeding the point made by Trifonov et al. 关27兴, we will call this state a relative-phase eigenstate rather than a phasedifference eigenstate, thus recalling that the formal definition of a two-mode quantum phase difference operator does not coincide with the 共problematic兲 definition of two singlemode quantum phase operators 关29兴. The relative-phase eigenstate is 兩 ␾ r(N) 典 ⫽

N

1

冑N⫹1

兺

n⫽0

(N)

e in ␾ r 兩 n 典 a 兩 N⫺n 典 b ,

共10兲

where ␾ r(N) ⫽ ␾ 0 ⫹2 ␲ r/(N⫹1) is the phase difference, N is the total photon number, ␾ 0 an arbitrary phase origin, and r苸 关 0,N 兴 . The phase difference is adequately defined with resolution 1/N, i.e. at the Heisenberg limit. In the Schwinger representation, Eq. 共10兲 becomes 兩 ␾ r(2 j) 典 ⫽

共5兲

The Schwinger representation thus makes use of the homomorphism from SU共2兲 onto SO共3兲, which allows one to represent any unitary operation on the two-mode field by a rotation. The general SU共2兲 transformation

a1

兺k s kl s *kl ⬘⫽ ␦ ll ⬘ , 兺l

共1兲

j

1

兺j 冑2 j⫹1 m⫽⫺

(2 j)

e im ␾ r 兩 jm 典 z .

共11兲

Trifonov et al. introduced the more general state 兩兵␪

(N)

其典⫽

1

N

兺 冑N⫹1 n⫽0

(N)

e i ␪ n 兩 n 典 a 兩 N⫺n 典 b ,

共12兲

which they stated can still be considered a relative-phase eigenstate, however involved or even arbitrary the real set 兵 ␪ (N) n 其 n may be with respect to n 关27兴. Indeed, the whole basis can always be constructed by applying the N⫹1 phase (N) shift operators e i ␾ r J z to 兩 兵 ␪ (N) 其 典 , with 兵 ␪ (N) n 其 n being an initial relative-phase distribution between the two modes a and b. Nonetheless, we will show in Sec. III that successful quantum teleportation demands full initial knowledge of 兵 ␪ (N) n 其n , which is also the relative phase of the entanglement between Alice and Bob, and the number-phase teleportation protocol is not linear in n. becomes very complicated if ␪ (N) n Finally, we recall that maximal entanglement is only attained when N→⬁. C. Generation of EPR and quasi-EPR states

B. Ideal number-phase EPR states

By definition, a maximally entangled two-mode state, or EPR state, is a two-mode quantum state 兩 EPR典 ⫽

s kl 兩 k 典 a 兩 l 典 b , 兺 k,l

共8兲

such that any reduced 共single-mode兲 density matrix of this state Tra,b ( 兩 EPR典具 EPR兩 ) is proportional to the identity matrix, which yields

The experimental realization of relative-phase eigenstates is an arduous problem. Recently, Trifonov et al. reported the experimental realization of a relative-phase eigenstate 共12兲 for N⫽2 关27兴. Their astute method uses a nonbalanced beam splitter to create a two-mode state, all of whose amplitudes have equal modulus. This method is not general in the sense that it cannot work perfectly for N⬎2, as we will see later 共and is immediate from Fig. 2兲. However, the use of a beam splitter to generate EPR or quasi-EPR states stems from quite general arguments indeed.

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PHYSICAL REVIEW A 65 052313

FIG. 1. Modulus of the quantum amplitudes of beam-splitter ( ␤ ) 典 for a relative-phase eigenstate input versus the output 兩 ␾ (20) 0 beam-splitter angle ␤ . N⫽20 photons.

FIG. 2. Modulus of the quantum amplitudes of the beam-splitter output 兩 10 0 ( ␤ ) 典 ⫽ 兩 ⌿ (1) 10 ( ␤ ) 典 versus the beam-splitter angle ␤. N⫽20 photons.

The studies of Heisenberg-limited interferometry 关21–25兴 have led to a thorough understanding of the subtle quantum optical properties of the beam splitter 关30,22–25兴. In particular, a balanced beam splitter swaps the amplitudes and phase properties of the impinging two-mode wave 关31兴, and can also be used to entangle nonclassical optical fields 关3,32,33兴. As is readily seen in the Schwinger representation, a balanced beam splitter corresponds to a ␲ /2 rotation around X and therefore transforms a state from axis Z 共intensity difference兲 to axis Y 共phase difference兲. In the case of an EPR state such as Eq. 共11兲, the phase difference is squeezed and the intensity difference is antisqueezed. Experimentally, this is achievable by sending an intensity-difference-squeezed state through a beam splitter 关15兴. To illustrate this, let us examine the beam-splitter output of the relative phase state 共11兲:

lowest values of m 共this could be viewed as a quantum AbbePorter experiment, with a low-m-pass filter兲. These states are, for N⫽2 j even,

j) j) 兩 ␾ (2 共 ␤ 兲 典 ⫽e ⫺i ␤ J x 兩 ␾ (2 0 0 典

⫽

共13兲

j

1

冑2 j⫹1

兺

m⫽⫺ j

i ⫺m f mj 共 ␤ 兲 兩 jm 典 z ,

共14兲

with where f mj ( ␤ )⫽ 兺 m ⬘ ⫽⫺ j e im ⬘ ( ␾ 0 ⫹ ␲ /2) d mm ⬘ ( ␤ ), j ⫺i ␤ J y 兩 j m ⬘ 典 a rotation matrix element taken d mm ⬘ ( ␤ )⫽ 具 jm 兩 e real by convention and proportional to a Jacobi polynomial 关34兴. This state is displayed for ␾ 0 ⫽0 in Fig. 1. As expected, the result is a narrow state in photon number 共i.e., f mj →0 very fast as 兩 m 兩 → j). j) It is straightforward to see that sending 兩 ␾ (2 0 ( ␲ /2) 典 through another, balanced ( ␤ ⫽⫾ ␲ /2), beam splitter will rej) construct the initial relative phase eigenstate 兩 ␾ (2 0 典 . 共One can see that ␤ → ␲ /2 is necessary to maximal entanglement, since the spread of the state must cover all values of the j) projections of the spin.兲 Now, since 兩 ␾ (2 0 ( ␲ /2) 典 contains but a few nonzero amplitudes, a reasonable method for generating close approximations of EPR states with a balanced beam splitter is to consider ‘‘quantum filtered’’ input states, j) which are derived from 兩 ␾ (2 0 ( ␲ /2) 典 by keeping only the j

(2 j)

兩 ⌿ (1) j 典⫽兩 j 0典z ,

共15兲

j j j 兩 ⌿ (3) j 典 ⫽ 关 f 0 兩 j 0 典 z ⫹ f 1 兩 j 1 典 z ⫹ f ⫺1 兩 j ⫺1 典 z ]/C 1 ,

共16兲

␮ j 2 1/2 and so on, with C ␮ ⫽( 兺 m⫽⫺ ␮ 兩 f m 兩 ) , and are 1 1 兩 ⌿ (2) j 典 ⫽ 关 兩 j 2 典 z ⫹ 兩 j⫺ 2 典 z ]/ 冑2,

j

j

j

共17兲 j

1 1 3 兩 ⌿ (4) j 典 ⫽ 关 f 1 兩 j 2 典 z ⫹ f ⫺ 1 兩 j⫺ 2 典 z ⫹ f 3 兩 j 2 典 z ⫹ f ⫺ 3 兩 j 2

2

2

⫺ 典 z ]/C , 3 2

2

共18兲

3 2

and so on, for N⫽2 j odd. As we will now see, sending these states through a beam splitter gives output states closely resembling EPR states. We call these output states quasi-EPR states. We start with the simplest one 共15兲, which has already been considered by CMM. We denote the general state rotated by a beam splitter by

j

兩 j m ⬘ 共 ␤ 兲 典 ⫽e ⫺i ␤ J x 兩 j m ⬘ 典 z

共19兲

j

⫽

兺

m⫽⫺ j

i m ⬘ ⫺m d mm ⬘ 共 ␤ 兲 兩 j m 典 z . j

共20兲

Figure 2 displays the modulus of the quantum amplitudes of 兩 ⌿ (1) j ( ␤ ) 典 versus ␤ 关see Eq. 共7兲兴 and m. One can see that ␤ → ␲ /2 is still necessary for maximal entanglement. The very value ␤ ⫽ ␲ /2 leads to a problem, however, because every other amplitude of the state is zero, as is well known 关30,25,13兴. This was recalled by CMM when they investigated the use of this state as a teleportation channel and found that, because of this, teleportation fidelity was bounded by 50% 共to the notable exception of Schro¨dinger cat states兲. This situation, however, is changed if one considers an ever-so-slightly imbalanced beam splitter: Fig.

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NGOC-KHANH TRAN AND OLIVIER PFISTER

PHYSICAL REVIEW A 65 052313

FIG. 3. Modulus and phase of the quantum o 兩 ⌿ (1) 共left兲 and amplitudes of 10 (90 ) 典 o 兩 ⌿ (1) (85.5 ) 共right兲. All nonzero amplitudes of 典 10 o 兩 ⌿ (1) 10 (90 ) 典 have a phase equal to ␲ /2.

3 dispays the modulus and phase of the coefficients of (1) 兩 10 0(90o) 典 ⫽ 兩 ⌿ 10 (90o) 典 and 兩 10 0(85.5o) 典 (1) o ⫽ 兩 ⌿ 10 (85.5 ) 典 . (1) Clearly, 兩 ⌿ 10 (85.5o) 典 is closer to an EPR state than (1) o 兩 ⌿ 10 (90 ) 典 : it has the same spread but much more even amplitudes, practically constant for ⫺5⭐m⭐5, and no zeroes at all. The phase distribution is not constant and not simple but this just means that it is a general relative phase state of the form of Eq. 共12兲, which is still a legitimate EPR state in the context we chose for the definition of the relative (1) phase. In fact, 兩 ⌿ 10 (85.5o) 典 is the best quasi-EPR state (1) 兩 ⌿ 10 ( ␤ ) 典 , ᭙ ␤ . In general, we find that the angle ␤ Q that gives the best quasi-EPR state is given by the following phenomenological formula:

␤ Q⫽

冉 冊

␲ 1 1⫺ , 2 N

共21兲

which we have tested, to the best of our numerical capabilities Eq. 共21兲, for N ⫽ 200, 2000, and 20000 photons: the resulting states 兩 ⌿ (1) j ( ␤ Q ) 典 for j⫽100, 1000, and 10000, are plotted in Fig. 4. Note that the size of the low-m flat region does scale proportionally to N. Note that all digits are significant in the beam splitter’s parameters of Fig. 4, which points to an interesting situation. Let us assume that on-demand single-photon sources become a reality 共not an unreasonable asumption兲, which would allow the production of 兩 ⌿ (1) j 典 z in the laboratory. Equation 共21兲 nevertheless poses a serious experimental constraint on the tolerance of the beam-splitter reflectivity R, because the required precision on ␤ , i.e., on R, increases with N if one (1) wants to resolve 兩 ⌿ (1) j ( ␤ Q ) 典 from 兩 ⌿ j ( ␲ /2) 典 and its j inconvenient zeroes. Roughly, ⌬R⬃⌬ ␤ ⬃1/N. Since a beam splitter using state-of-the-art optical coatings and polishing

cannot give more than ⌬R⬃10⫺6 , N cannot exceed 106 photons in this case. This ‘‘Heisenberg-limit’’-type sensitivity may be quite general, as hints of the same scaling have also been found in the tolerance to error of entanglement operations 共optical pulse times兲 of N trapped ions 关35兴. In fact, by taking a closer look at Fig. 2, one can see that the amplitudes present 1/N-period oscillations versus ␤ . These oscillations are of significant contrast, with the state amplitudes often reaching zero. This, therefore, can pose a problem for experimentally defining 兩 ⌿ (1) j ( ␤ ) 典 as j increases. This problem disappears, however, as soon as one uses a more elaborate input state, such as 兩 ⌿ (2) j 典 共17兲. Such an input state could be obtained using stimulated emission from a single atom, starting from a 兩 ⌿ (1) j 典 z state and having the two beams shine simultaneously and noncollinearly on the excited atom. One will also want to have fast nonradiative decay from the ground state of the transition so as to prevent subsequent absorption, as in laser media. Figure 5 displays the state 兩 ⌿ (2) j (␤)典. Even though the 1/N oscillations are still present, they are significantly attenuated as there are no zero amplitudes, even at ␤ ⫽ ␲ /2. One can therefore use ␤ ⫽ ␲ /2 in this case, which presents the non-negligible advantage of a flat phase distribution (3 ␲ /4,᭙m) for this state 兩 ⌿ (2) j ( ␲ /2) 典 关unlike 兩 ⌿ (1) ( ␤ ) in Fig. 3兴. This is of great importance for teleQ 典 j portation, as we will see in the next section. Finally, it is straightforward and unsurprising to show that the more elaborate 共less low-m-pass filtered兲 reconstructions 兩 ⌿ (3) j 典 共16兲 and 兩 ⌿ (4) 共18兲 give even better results: more even amj 典 plitudes at still constant phase. We will, however, restrain our (2) investigations to the two states 兩 ⌿ (1) j ( ␤ Q ) 典 and 兩 ⌿ j ( ␲ /2) 典 , which are the simplest ones and are also within reasonable reach of foreseeable future technology.

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PHYSICAL REVIEW A 65 052313

FIG. 5. Modulus of the quantum amplitudes of beam-splitter (2) ( ␤ ) 典 versus the beam-splitter angle ␤. N⫽21 photons. output 兩 ⌿ 21/2

making a joint measurement on T and A such that both are projected onto a maximally entangled state. This prevents Alice from obtaining any quantum information about the target state, which as such is destroyed in the process. In turn, said target state is transferred, by ‘‘entanglement transitivity’’ from T to B, i.e., to Bob, who may then reconstruct the exact target state on B using the classically transmitted results of Alice’s measurements 共which contain no quantum information whatsoever兲. The conceptually difficult part is to figure out what measurements should be used by Alice to maximally entangle her two systems A and T. This question was answered by Vaidman in connection with the EPR paradox 关2兴. In the case of number-phase teleportation, use is made of the commuting operators number sum Nˆ T ⫹Nˆ A and 共Hermitian兲 relative phase 关26兴 ⬁

␾ˆ TA ⫽

兩 ⌿ (1) j ( ␤ Q) 典

FIG. 4. Modulus of the quantum amplitudes of for N⫽200, 2000, and 20000 photons 共see Fig. 3 for N⫽20). All states have an almost identical appearance, and are practically indistinguishable from perfect relative-phase eigenstates for ⫺N/5⭐m ⭐N/5, approximately.

N⫹1

兺 兺

N⫽0 r⫽0

兩 ␾ r(N) 典 ␾ r(N) 具 ␾ r(N) 兩 ,

whose measurements project the joint T-A state onto a joint eigenstate of the total number and the relative phase such as Eq. 共10兲. If the same type of entangled state is shared between Alice and Bob, perfect teleportation can in principle be achieved. Let us consider the general case where the initial total state is ⬁

III. NUMBER-PHASE QUANTUM TELEPORTATION

兩 ␺ 典 T 丢 兩 ␾ r(N) 典 AB ⬅

In this section we briefly recall the definition of numberphase teleportation 关13兴 and extend it to general relativephase states. We then consider the use of the quasi-EPR states that were derived above. A. Ideal entanglement resource

Quantum teleportation relies on a maximally entangled state shared by the sender, Alice, and the receiver, Bob. The entanglement concerns two physical systems, A and B, respectively. Alice is in possession of A and also of system T, whose ‘‘target’’ state is the quantum information she needs to transmit. The teleportation process consists, for Alice, in

共22兲

兺

m⫽0

N

cm 兩m典T

兺

n⫽0

(N)

e in ␾ r

冑N⫹1

兩 n 典 A 兩 N⫺n 典 B .

共23兲

ˆ TA ˆ T ⫹Nˆ A and of ␾ We assume Alice’s measurements of N yield the respective eigenvalues q and ␾ s(q) . The phase difference measurement should be thought of as a Heisenberglimited interferometric measurement, whose experimental implementation was proposed and numerically modeled, for the states considered here, in Refs. 关23,25兴. The joint TA state is thus left in 兩 ␾ s(q) 典 TA , and the total state after Alice’s measurement is

052313-5

(N)

兩 ␺ M 典 ⫽e iq ␾ r 兩 ␾ s(q) 典 TA 丢 兩 ␺ B 典 B ,

共24兲

NGOC-KHANH TRAN AND OLIVIER PFISTER

PHYSICAL REVIEW A 65 052313

q

兩 ␺ B 典 B ⫽C 共 q 兲

兺 k

e

(N)

⫺ik( ␾ r

(q)

⫹␾s )

c k 兩 k⫹N⫺q 典 B , 共25兲

0

q 兩 c k 兩 2 兴 ⫺1/2. To where k 0 ⫽max关0,q⫺N 兴 and C(q)⫽ 关 兺 k⫽k 0 exactly recover the target state, Bob must then perform, on B, a photon number shift 关13兴 of q⫺N and a phase shift of ␾ r(N) ⫹ ␾ s(q) , i.e.,

兩 ␺ out 典 B ⫽e

(N)

i( ␾ r

(q) ⫹ ␾ s )Nˆ B Pq⫺N 兩 ␺ B 典 B .

(N)

Jz

(N)

␾ˆ TA e i ␾ r

共27兲

Jz

ˆ A )/2 here.兴 ˆ TA . 关J z ⫽(Nˆ T ⫺N instead of ␾ Note that this requirement that the entanglement phase be perfectly known is a very general one and not at all specific to our particular choice of the optical phase variable for the teleportation protocol. This was pointed out by van Enk in Ref. 关36兴. In light of what we have already discussed, it is clear that a state such as 兩 ⌿ (2) j ( ␲ /2) 典 , which has a flat phase distribu(N) in ␾ r i3 ␲ /4 ⫽e , is perfectly suited for this teleportation tion e protocol. It is, however, interesting to investigate the use of the more exotic state 兩 ⌿ (1) j ( ␤ Q ) 典 in this case. The phase distribution is neither flat nor simple 共Fig. 3兲, one phenom(N) 2 enological description of it being e in ␾ r ⫽e in ␲ /2. We are thus in the case of a general relative phase state 兩 兵 ␪ (N) 其 典 given by Eq. 共12兲. If the initial entanglement is given by Eq. 共12兲, 兩 ␺ 典 T 丢 兩 兵 ␪ (N) 其 典 AB ,

共28兲

then the phase difference operator is not given by Eq. 共22兲 but by 关27兴 ⬁

␪ ␾ˆ TA ⫽兺

U 兵B␪ 其 ⫽ e i(⫺1)

兺

e

N⫽0 r⫽0

兩 兵 ␪ (N) 其 典 ␾ r(N) 具 兵 ␪ (N) 其 兩 e

(N) ⫺i ␾ r J z

B. Quasi-EPR resource

We now turn to the use of quasi-EPR states as the entanglement resource, and show how arbitrary coherent and Schro¨dinger cat states can be successfully teleported. Our evaluation of teleportation performance will contain no assessment of decoherence and will be based on the pure-state fidelity F⫽円具 ␺ out 兩 ␺ 典 円2 .

兩 ␺ M 典 ⫽e

N

兩 ␾ s(q) 典 TA 丢 兩 ␺ B 典 B ,

兺 k⫽k

(N)

(N)

(q)

e i ␪ q⫺k ⫺ ␪ k e ⫺ik ␾ s c k 兩 k⫹N⫺q 典 B , 0

兩 QEPR典 AB ⫽

共30兲

q

兩 ␺ B 典 B ⫽C 共 q 兲

共33兲

The entanglement resource is now a quasi-EPR state,

and the postmeasurement total state is (q)

共32兲

, 共29兲

i(q/2) ␾ s

,

which can be interpreted as an intensity-dependent phase shift and might therefore be realizable with a Kerr nonlinearity. It is somewhat puzzling that this additional step is needed if 兩 兵 ␪ (N) 其 典 may indeed be considered as a legitimate relative phase eigenstate, since it is used in the corresponding relative phase measurement of Eq. 共29兲. Our conclusion is that the definition 共29兲 of the general relative-phase operator is not valid in the context of quantum teleportation, possibly because the full EPR nature of the eigenstate of Eq. 共12兲 is used. Indeed, if one only cares about measuring the phase difference between the two modes, any type of initial relative phase distribution 兵 ␪ (N) n 其 n will give the same result, as illustrated in Ref. 关27兴. This obviously ceases to be true when applying in a teleportation operation, where the complete nature of the entangled state must be known. In other words yet, even though the whole relative-phase eigenbasis — and hence the operator — may be generally defined based upon any general state 兩 兵 ␪ (N) 其 典 with arbitrary phase distribution 兵 ␪ (N) n 其 n , it does in fact matter for teleportation that U 兵 ␪ (N) 其 corresponds to a feasible physical measurement, thereby limiting the generality of relative phase states usable for teleportation.

N⫹1 (N) i␾r Jz

q ( ␲ /2)N ˆ2 B

共26兲

共In the particular case where Alice’s measurement results are q⫽N and ␾ s(q) ⫽⫺ ␾ r(N) , Bob does not have to do anything.兲 One can see from this that the phase distribution of the initial entanglement resource has to be corrected for, along with Alice’s measurement result. If this distribution is unknown or too complicated to correct, teleportation will fail. This correction can also be made by Alice, by simply shifting her phase operator, i.e., using e ⫺i ␾ r

(N) into a ‘‘flat-phase’’ state for which ␪ (N) n ⫽ ␾ 0 ⫽ cst, ᭙n. This transformation may be single-mode and applied by Bob, B as U 兵 ␪ (N) 其 兩 ␺ B 典 B ⫽ 兩 ␺ out 典 B , or two-mode and applied by AlA ˆ TA U A†(N) . In the aforementioned ice, by measuring U 兵 ␪ (N) 其 ␾ 兵␪ 其 (1) case of 兩 ⌿ j ( ␤ Q ) 典 , one possibility is

共31兲

which shows that Bob needs more than a mere phase shift to properly reconstruct 兩 ␺ 典 : with 兵 ␪ (N) n 其 n fully known, he needs the unitary transformation U 兵 ␪ (N) 其 that transforms 兩 兵 ␪ (N) 其 典

兺

n⫽0

s Nn 兩 n 典 A 兩 N⫺n 典 B ,

共34兲

N 兩 s Nn 兩 2 ⫽1. As announced before, we only treat the where 兺 n⫽0 (2) cases of 兩 ⌿ (1) j ( ␤ Q ) 典 and 兩 ⌿ j ( ␲ /2) 典 . Their respective decompositions in terms of Eq. 共34兲 are found using Eqs. 共3兲– 共5兲 and 共19兲, and their amplitudes are, respectively,

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N/2 s Nn ⫽i ⫺n⫹(N/2) d n⫺(N/2), 0共 ␤ Q 兲 ,

共35兲

QUANTUM TELEPORTATION WITH CLOSE-TO-MAXIMAL . . .

FIG. 6. Teleportation fidelity of an ␣ ⫽3 coherent state, using (2) ( ␤ ) 典 versus the beam-splitter angle ␤ the quasi-EPR resource 兩 ⌿ 21/2 and the number-sum measurement result q. N⫽21 photons.

s Nn ⫽

1

冑2

冋

N/2 i ⫺n⫹(N/2) d n⫺(N/2), (1/2)

N/2 ⫺id n⫺(N/2), ⫺1/2

冉 冊册 ␲ 2

冉冊 ␲ 2

共36兲

.

The postmeasurement total state is 兩 ␺ M 典 ⫽ 兩 ␾ s(q) 典 TA 丢 兩 ␺ B 典 B ,

PHYSICAL REVIEW A 65 052313

FIG. 8. Teleportation fidelity of an ␣ ⫽3 coherent state, using the quasi-EPR resource 兩 ⌿ (1) 10 ( ␤ ) 典 , versus the beam-splitter angle ␤ and the number-sum measurement result q. N⫽20 photons.

Before we plot F(q) for the two quasi-EPR states, we must address the question of the dependence of the fidelity on the measured value q of the number sum: this implies conditional teleportation even for an ideal relative-phase state (s Nn ⫽const ᭙N,n), which should not be. Equation 共40兲 has an upper bound, which can be found by using the Schwartz inequality:

冏兺

共37兲

q

k⫽k 0

q

兩 ␺ B 典 B ⫽C 共 q 兲

兺 k⫽k

q

N 兩 c k 兩 2 s q⫺k 兩2⭐

q

兺

兩 c k兩

k⫽k 0

2

兺

k⫽k 0

兩 c k兩

2

冏

N 兩 s q⫺k

2

.

共41兲

(q)

N e ⫺ik ␾ s c k s q⫺k 兩 k⫹N⫺q 典 B , 共38兲 0

The fidelity is thus bounded by

q where C(q)⫽( 兺 k⫽k 兩 c k 兩 2 兩 s Nn 兩 2 ) ⫺1/2. This yields

q

0

F共 q 兲⭐

q

兩 ␺ out 典 B ⫽C 共 q 兲

兺

k⫽k 0

N c k s q⫺k 兩k典B ,

冏兺

F共 q 兲⫽

k⫽k 0

冏

兺

k⫽k 0

兩 c k兩

2

2

.

共42兲

The physical meaning of this Schwartz inequality thus is the following: if k 0 ⫽0 共i.e., q⭐N, which is automatic for N →⬁) and q is very large compared to the spread of the target state 共denoted by k max , such that c k⬎k max ⫽0兲, then the inequality 共42兲 becomes

N 兩 c k 兩 2 s q⫺k

q

兩 c k兩 2.

共39兲

and the fidelity is q

兺

k⫽k 0

共40兲

k max

N 兩 s q⫺k 兩2

F共 q 兲⭐

FIG. 7. Teleportation fidelity of an ␣ ⫽3 cat state, using the (2) quasi-EPR resource 兩 ⌿ 21/2 ( ␤ ) 典 , versus the beam-splitter angle ␤ and the number-sum measurement result q. N⫽21 photons.

兺

k⫽0

兩 c k 兩 2 ⫽1,

共43兲

FIG. 9. Teleportation fidelity of an ␣ ⫽3 cat state, using the quasi-EPR resource 兩 ⌿ (1) 10 ( ␤ ) 典 , versus the beam-splitter angle ␤ and the number-sum measurement result q. N⫽20 photons.

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NGOC-KHANH TRAN AND OLIVIER PFISTER

PHYSICAL REVIEW A 65 052313

As mentioned earlier, we considered two different target states, a simple coherent state 兩 ␣ 典 T and a ‘‘Schro¨dinger cat’’ state, the macroscopic, definitely nonclassical superposition ⬁

C 共 兩 ␣ 典 T ⫹ 兩 ⫺ ␣ 典 T )⫽C

兺 n⫽0

关 1⫹ 共 ⫺1 兲 n 兴

␣n

冑n!

兩 n 典 T , 共45兲

2

FIG. 10. Cross section of Fig. 8 for q⫽19. The peak value is 99.3% at ␤ ⫽ ␤ Q ⫽85.5o (0.5o steps兲. Note that ␤ ⫽90o gives the classical fidelity value F⫽49.8%, consistently with the results of CMM.

which corresponds to the fidelity of the ideal EPR resource. All of this is in fact ensured by maximal entanglement N →⬁, and a finite k max . When this is fulfilled, the probability that q⬍k max becomes negligible and the teleportation becomes unconditional. Otherwise, one can have severe fidelity limitations, as in the worst case N⬍q⬍k max : q⬍k max

F共 q 兲⭐

兺

k⫽q⫺N⬎0

兩 c k 兩 2 ⬍1,

共44兲

where C⫽(2⫹2e ⫺2 兩 ␣ 兩 ) ⫺1/2. We calculated the teleportion fidelity F(q, ␤ ) for both these states with ␣ ⫽3, using the (1) two quasi-EPR states 兩 ⌿ (2) j ( ␤ ) 典 and 兩 ⌿ j ( ␤ ) 典 关assuming the initial-entanglement-phase correction 共32兲 is successfully applied by Bob兴 for the respective entanglement resources. Results are plotted in Figs. 6–9. Once again, the limited spread in q of the high fidelity region is only due to the limitations of our computation. Had we been able to perform the simulations with a larger N, this effect would have disappeared. These teleportation protocols are unconditional. What is essential is that, in all cases, we obtain F⯝100%. This occurs for ␤ ⫽ ␲ /2 in Figs. 6 and 7 and for ␤ ⫽ ␤ Q 共see Fig. 10兲 in Figs. 8 and 9, thereby confirming our analysis that quasi-EPR states can be efficient teleportation resources. IV. CONCLUSION

where the sum over the target state probability becomes truncated. Besides being lower, the fidelity will also depend strongly on the value of q 共conditional teleportation兲. In our computations, N is limited to a few tens of photons (1) ( ␤ Q) 典 as in Fig. 4 is a lighter 关calculating the state 兩 ⌿ 10000 (2) computational load than that of calculating 兩 ⌿ 100 ( ␲ /2) 典 兴, making it difficult to approach N→⬁. Therefore, the fidelity displayed in our figures has a remaining dependence on the value of q and the illusion of conditionality. The arguments above should have, however, convinced the reader that such is not the true physical situation for number-phase teleportation, which can be truly unconditional. As far as our numerical results are concerned, it is simple to see that, for a coherent target state 兩 ␣ 典 ( ␣ real兲, the high-fidelity region is given by q苸 关 k max ,N⫺k min 兴 where NⰇk max , i.e., q苸 关 ␣ 2 ⫹ ␣ ,N ⫺ ␣ 2⫹ ␣ 兴 .

We have demonstrated that efficient quantum teleportation is indeed possible with close-to-maximally entangled states. We chose the simplest quasi-EPR states, which would be straightforward to generate should on-demand singlephoton sources become an experimental reality. Note, however, that the same formalism applies equally well to secondquantized bosons in general, which opens possibilities for the use of trapped ions or Bose-Einstein condensates. We also investigated the use of generalized relative-phase states for quantum teleportation and showed that these states lead to serious complications if the basis ‘‘generator’’ 兩 兵 ␪ 其 典 has a more complicated phase distribution than a simple phase offset. This therefore raises a question about the precise physical significance of the generalized relative-phase operator 共29兲 of Ref. 关27兴.

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