Multicolor multipartite entanglement produced by vector four-wave mixing in a fiber

Multicolor multipartite entanglementproduced by vector four-wave mixing

in a fiber

C. J. McKinstrie, 1 S. J. van Enk,2 M. G. Raymer2 and S. Radic3

1Bell Laboratories, Alcatel–Lucent, Holmdel, New Jersey 077332Oregon Center for Optics and Department of Physics, University of Oregon,

Eugene, Oregon 974033Department of Electrical and Computer Engineering, University of California

at San Diego, La Jolla, California 92093

[email protected]

Abstract: Multipartite entanglement is a resource for quantum com-munication and computation. Vector four-wave mixing (FWM)in a fiber,driven by two strong optical pumps, couples the evolution offour weakoptical sidebands (modes). Depending on the fiber dispersion and pumpfrequencies, the mode frequencies can be similar (separated by less than1 THz) or dissimilar (separated by more than 10 THz). In this report, thediscrete- and continuous-variable entanglement producedby vector FWM isstudied in detail. Formulas are derived for the variances of, and correlationsbetween, the mode quadratures and photon numbers. These formulas andrelated results show that the modes are four-partite entangled.

© 2008 Optical Society of America

OCIS codes:(060.2320) fiber optics, amplifiers and oscillators; (190.4380) nonlinear optics,four-wave mixing; (270.5290) photon statistics; (270.6570) squeezed states.

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1. Introduction

Entanglement is an intrinsically quantum-mechanical property of some quantum states thatdescribe two or more separate systems [1]. These states cannot be described in terms of clas-sical correlations between the systems [2]. Entanglement is a resource for quantum commu-nication and computation [3, 4]. Its use was proposed and demonstrated for dense coding[5, 6, 7, 8, 9, 10], key distribution [11, 12, 13, 14, 15], one-way computation [16, 17, 18, 19],teleportation [20, 21, 22, 23, 24, 25] and teleportation networking [26, 27, 28, 29]. Earlierpapers focused on two-partite discrete-variable protocols, whereas later papers focused on two-partite continuous-variable and multipartite discrete- or continuous-variable protocols. Opti-cal discrete-variable protocols require the detection of individual photons, whereas opticalcontinuous-variable protocols require the measurement ofmode quadratures by balanced ho-modyne detection.

In some of the aforementioned experiments, two-partite (two-product-mode) entanglementis produced by nondegenerate down-conversion in a crystal [35, 36]. In others, a beam split-ter is used to combine the one-product-mode squeezed statesproduced by degenerate down-conversion [37] in separate crystals. Both types of down-conversion are driven by one pump



mode. Three-partite [10, 15, 27, 28, 29, 30] and four-partite [18, 31, 33, 34] entanglementare produced by path-stabilized arrays of beam splitters, which combine the outputs of two ormore sources of one-mode squeezed states or two-mode entangled states. Because these meth-ods are restricted to modes (photons) with the same frequency, spatial multiplexing is requiredto transmit and manipulate the photons. Recently, it was predicted that concurrent difference-and sum-frequency processes in crystals produce entanglement between three or four productmodes with different frequencies [38, 39, 40]. Phase matching two or more concurrent pro-cesses requires the use of periodic poling [41] or birefringence [42], and is only possible forcertain mode frequencies.

Two-product-mode entanglement can also be produced by four-wave mixing (FWM) in afiber [43, 44, 45, 46]. Three different types of FWM are illustrated in Fig. 1. Modulation in-teraction (MI) is the degenerate process in which two photons from the same pump mode aredestroyed and two different product-mode (sideband) photons are created. Phase conjugation(PC) is the nondegenerate process in which two photons from different pumps are destroyedand two different sideband photons are created. Bragg scattering is the nondegenerate processin which a sideband (signal) photon and a pump photon are destroyed, and different sideband(idler) and pump photons are created. MI and PC produce two-frequency entangled states in thesame way that down-conversion produces two-wavevector entangled states, whereas BS com-bines modes with different frequencies in the same way that abeam splitter combines modeswith different wavevectors [47]. Because fibers (and free space) allow frequency multiplexing,it is useful to study the entanglement of modes with different frequencies.

11- 1+ 2- 2 2+

BS

MI

PC

Fig. 1. Frequency diagram for the interaction of two pumps (1and 2) and four sidebands(1± and 2±). Depending on the fiber dispersion and pump frequencies, six different four-wave mixing (FWM) processes can occur, separately or simultaneously. The red, blue andgreen dashed lines denote modulation interaction (MI), phase conjugation (PC) and Braggscattering (BS), respectively.

FWM is driven by nonlinearity and suppressed by dispersion.By tuning the pump frequen-cies judiciously, relative to the zero-dispersion frequency (ZDF) of the fiber, one can controlwhether MI, PC and BS occur separately, or simultaneously [47, 48, 49]. In the latter case,FWM driven by two pumps couples the evolution of four modes with different frequencies, asillustrated in Fig. 1. The classical physics [48, 49] and quantum noise properties [50] of four-mode interactions were studied thoroughly, in the context of classical communication systems.However, the entanglement produced by these interactions was not studied previously, and isthe focus of this report.



The entanglement scheme described herein has several potential advantages over existingschemes. First, it produces four-mode entanglement in one step. (A path-stabilized array ofbeam splitters is not necessary.) Second, the fiber system inwhich it occurs is simple and com-pact. Third, fibers can be manufactured with different ZDFs for different applications. (Photonswith a variety of frequencies can be generated.) Fourth, because the photons are generated in afiber, their transverse structure is suitable for transmission through another fiber. (There is nomode-matching problem.) Fifth, because the generated photons have different frequencies, theycan be transmitted by the same fiber.

2. Four-mode equations

The interaction of two strong, classical pumps (1 and 2) withfour weak, quantum sidebands(1−, 1+, 2− and 2+), is governed by the Hamiltonian

Ha = α(a†1−a1− +a†

1+a1+)+ α(a†1−a†

1+ +a1−a1+)

+ β (a†1−a2−+a1−a†

2−)+ β (a†1+a2+ +a1+a†

2+)

+ β (a†1−a†

2+ +a1−a2+)+ β (a†1+a†

2−+a1+a2−)

+ γ(a†2−a2− +a†

2+a2+)+ γ(a†2−a†

2+ +a2−a2+), (1)

wherea j is the destruction operator of sideband (mode)j, † denotes a Hermitian conjugate,and the nonlinearity coefficientsα = γKP1, β = γK(P1P2)

1/2 andγ = γKP2, whereγK is the Kerrcoefficient of the fiber, andPj is a pump power. Because the phases of modes 1± are measuredrelative to the phase of pump 1, and the phases of modes 2± are measured relative to the phaseof pump 2, the pump phases do not appear explicitly in the Hamiltonian. The terms in the firstline of Eq. (1) model the MI of pump 1, in which 2γ1 → γ1− + γ1+, whereγ j denotes a photonwith frequencyω j . Not only does pump 1 provide nonlinear coupling between modes 1− and1+, it also imposes cross-phase modulation (CPM) on them. The terms in the second line modelthe BS processes in whichγ1− + γ2 → γ1 + γ2− andγ1+ + γ2 → γ1 + γ2+, and the terms in thethird line model the PC processes in whichγ1 + γ2 → γ1− + γ2+ andγ1 + γ2 → γ1+ + γ2−. Bothof these processes are driven by pumps 1 and 2. The terms in thefourth line model the MI ofpump 2, in which 2γ2 → γ2− + γ2+. All six processes are illustrated in Fig. 1.

By applying the (spatial) Heisenberg equations

daj/dz= i[a j ,Ha] (2)

to the Hamiltonian (1), whered/dz denotes a distance derivative, one obtains the four-modeequations

da†1−/dz = −iαa†

1−− iαa1+− iβa†2−− iβa2+, (3)

da1+/dz = iαa†1− + iαa1+ + iβa†

2−+ iβa2+, (4)

da†2−/dz = −iβa†

1−− iβa1+− iγa†2−− iγa2+, (5)

da2+/dz = iβa†1− + iβa1+ + iγa†

2−+ iγa2+. (6)

(The Heisenberg equations describe how the mode operators evolve in time. However, themodes convect at the same speed, so temporal evolution is equivalent to spatial evolution.)Equations (3)–(6) are the standard four-mode equations fororthogonal (perpendicular) pumps[48, 49, 50], with the effects of dispersion neglected.

The weak-dispersion approximation is valid in at least two cases. In the first case, the pumpand sideband frequencies are all near the ZDF of the fiber [51,52], whereas in the second,



the sideband frequenciesω1± andω2± are comparable to the pump frequenciesω1 andω2,respectively [48, 49, 50]. The first configuration, which is illustrated in Fig. 2(a), providesphase-sensitive amplification of a signal polarized at 45◦ to the pumps [53], unimpaired bythe generation of secondary pumps and idlers [51, 52], and has been used to generate cross-polarized photon pairs [54]. The second configuration, which is illustrated in Fig. 2(b), has beenused to generate photons in a polarization-independent manner [55], and to wavelength-convertsignals between the low-loss windows near 1310 and 1550 nm [56]. Most fibers have oneZDF. In such fibers, the range of sideband frequencies for which dispersion can be neglectedis narrow. However, some fibers have two ZDFs [57, 58]. If the pump frequencies are near theZDFs, the range of sideband frequencies for which dispersion can be neglected is broad.

!!

"

1

2

1-

1+

2-

2+

!!

"

1

2

1-

1+

2-

2+

Fig. 2. Polarization diagram for the four-sideband interaction driven by perpendicularpumps. (a) Special case in which the pump-pump frequency difference is twice the pump-sideband difference. (b) General case in which the pump-pump difference is (much) largerthan the pump-sideband difference.

Equations (3)–(6) describe photon generation by MI and PC, and photon exchange by BS.The associated Manley–Rowe–Weiss (MRW) equations [59, 60], which relate the photon num-bers of the modes, are derived in Appendix A.

Despite their complexity, Eqs. (3)–(6) have the simple solutions

a†1−(z) = (1− iαz)a†

1−(0)− iαza1+(0)− iβza†2−(0)− iβza2+(0), (7)

a1+(z) = iαza†1−(0)+ (1+ iαz)a1+(0)+ iβza†

2−(0)+ iβza2+(0), (8)

a†2−(z) = −iβza†

1−(0)− iβza1+(0)+ (1− iγz)a†2−(0)− iγza2+(0), (9)

a2+(z) = iβza†1−(0)+ iβza1+(0)+ iγza†

2−(0)+ (1+ iγz)a2+(0), (10)

which are valid for all distancesz. In the absence of dispersion, the modes grow linearly withdistance, rather than exponentially, because the effects of CPM (wave-number shifts) balancethose of nonlinear coupling (amplification and frequency conversion).

3. Quadrature correlations

Equations (7)–(10) show that each output mode depends on allof the input modes: The outputmodes are correlated. It is common to quantify these correlations in terms of the mode quadra-tures, which can be measured by balanced homodyne detection[61, 62]. For each modej, thequadrature

q j(θ j) = (a†j e

iθ j +a je−iθ j )/21/2, (11)

whereθ j is the phase of the local oscillator used in the detection process. The conjugate quadra-turep j(θ j) = q j(θ j +π/2). These quadratures satisfy the canonical commutation relations. The



quadrature deviationδq j(θ j) = q j(θ j )−〈q j(θ j )〉, (12)

where〈〉 denotes an expectation value. The output deviations dependon the input deviations(quantum fluctuations), but not on the input quadratures (signal amplitudes).

In [63], detailed studies were made of the quantum noise properties of multiple-mode inter-actions characterized by the input–output equations

a j(z) = ∑k[µ jk(z)ak(0)+ ν jk(z)a†k(0)]. (13)

Formulas were derived for the means and variances of the quadratures and photon numbers ofthe modes. Equations (7)–(10) can be written in the form of Eq. (13). For example, ifj = 1−andk = 1−, thenµ = 1+ iαz andν = 0. If j = 1− andk = 1+, thenµ = 0 andν = iαz. Byextending the analysis of [63], one finds that the quadraturecorrelations

〈δq j(θ j)δqk(θk)〉 = ∑l (µ jl e−iθ j + ν∗

jl eiθ j )(µ∗

kleiθk + νkle

−iθk)/2. (14)

The input correlations are real by construction. Because the quadratures commute,〈δq jδqk〉 =〈δqkδq j〉. It follows from these facts and Eq. (14) that the output correlations are also real. Forthe case in whichj = k, Eq. (14) reduces to Eq. (40) of [63]. (For reference, the quadraturesdefined above are larger than those defined in [63] by a factor of 21/2. Both normalizationsappear in the literature [61, 62].)

By combining Eqs. (7)–(10) with Eq. (14), one finds that the quadrature variances

〈δq21±(θ1±)〉 = [1+2(α2+ β 2)z2]/2, (15)

〈δq22±(θ2±)〉 = [1+2(β 2+ γ2)z2]/2 (16)

and the quadrature correlations

〈δq1−(θ1−)δq1+(θ1+)〉 = αzsin(θ1− + θ1+)− (α2 + β 2)z2cos(θ1− + θ1+), (17)

〈δq1−(θ1−)δq2−(θ2−)〉 = β (α + γ)z2cos(θ1−−θ2−), (18)

〈δq1−(θ1−)δq2+(θ2+)〉 = βzsin(θ1− + θ2+)−β (α + γ)z2cos(θ1− + θ2+), (19)

〈δq1+(θ1+)δq2−(θ2−)〉 = βzsin(θ1+ + θ2−)−β (α + γ)z2cos(θ1+ + θ2−), (20)

〈δq1+(θ1+)δq2+(θ2+)〉 = β (α + γ)z2cos(θ1+ −θ2+), (21)

〈δq2−(θ2−)δq2+(θ2+)〉 = γzsin(θ2− + θ2+)− (β 2+ γ2)z2 cos(θ2− + θ2+). (22)

Although the variances are phase independent, the correlations are phase dependent.For the case in whichP1 ≫ P2 (α ≫ β ≫ γ) and the phases are equal, Eqs. (15) and (17)

reduce to

〈δq21±(θ )〉 ≈ [1+2(αz)2]/2, (23)

〈δq1−(θ )δq1+(θ )〉 ≈ (αz)sin(2θ )− (αz)2cos(2θ ). (24)

These results, which characterize the MI of pump 1, are consistent with the results of [63].Although modes 1+ and 1− are not squeezed separately (their quadrature variances are phase-independent), they are strongly correlated (so superpositions of modes 1+ and 1− have phase-dependent variances). Hence, the MI of pump 1 is a two-mode squeezing interaction [61, 62].Similar formulas characterize the MI of pump 2 (α is replaced byγ).

For the case in whichP1 = P2 (α = β = γ) and the phases are equal, Eqs. (15) and (16) reduceto

〈δq21±(θ )〉 = [1+4(z′)2]/2, (25)

〈δq22±(θ )〉 = [1+4(z′)2]/2, (26)



wherez′ = βz, and Eqs. (17)–(22) reduce to

〈δq1−(θ )δq1+(θ )〉 = z′ sin(2θ )−2(z′)2cos(2θ ), (27)

〈δq1−(θ )δq2−(θ )〉 = 2(z′)2, (28)

〈δq1−(θ )δq2+(θ )〉 = z′ sin(2θ )−2(z′)2cos(2θ ), (29)

〈δq1+(θ )δq2−(θ )〉 = z′ sin(2θ )−2(z′)2cos(2θ ), (30)

〈δq1+(θ )δq2+(θ )〉 = 2(z′)2, (31)

〈δq2−(θ )δq2+(θ )〉 = z′ sin(2θ )−2(z′)2cos(2θ ). (32)

The variances and correlations associated with the MI of pump 1, and the four-mode in-teraction driven by pumps with equal powers, are illustrated in Fig. 3. The input modes have(vacuum-level) variances of 1/2, and are uncorrelated. As distance increases, so also do the vari-ances and correlations. Although the MI and four-mode results are similar, the latter process isnoisier than the former [Eqs. (23), (25) and (26)].

0 1 2 3 4Distance

-10

-5

0

5

10

15

20

Var

ianc

e�

corr

elat

ionHd

BL

0 1 2 3 4Distance

-10

-5

0

5

10

15

20

Var

ianc

e�

corr

elat

ionHd

BL

Fig. 3. Quadrature variances and correlations, normalizedto the input variance 1/2 andmeasured in dB, plotted as functions of distance. (a) MI of pump 1, which involves modes1− and 1+. The solid curve denotes the variance of either mode, whereas the dashed curvedenotes the correlation between the modes [Eqs. (23) and (24)]. The local-oscillator phaseθ = π/2 and the distance parameter isγKP1z. Similar results apply to the interaction be-tween the superposition modesb+ andc+, for which the distance parameter isγK(P1+P2)z[Eqs. (37)–(39)]. (b) Four-mode interaction driven by pumps with equal powers. The solidcurve denotes the variance of any mode, whereas the dashed curve denotes the correlationbetween any pair of modes [Eqs. (25)–(32)]. The phase isπ/2 and the distance parameteris γKPz.

Equations (27)–(32) show that mode 1− has the same correlation properties as mode 2−,and mode 1+ has the same properties as mode 2+. These results prompt consideration of thesum and difference modes

b± = (a1−±a2−)/21/2, (33)

c± = (a1+±a2+)/21/2, (34)

which satisfy the canonical commutation relations. Because the quadrature deviations are linearfunctions of the mode operators and their conjugates, one can deduce the correlations betweenthe superposition modesb± andc± from the correlations between their constituent modes. Forexample,

2〈δqb+δqc±〉 = 〈δq1−δq1+〉± 〈δq1−δq2+〉+ 〈δq2−δq1+〉± 〈δq2−δq2+〉, (35)

2〈δqb−δqc±〉 = 〈δq1−δq1+〉± 〈δq1−δq2+〉− 〈δq2−δq1+〉∓ 〈δq2−δq2+〉. (36)



By proceeding in this way, one finds that

〈δq2b+(θ )〉 = [1+2(2z′)2]/2, (37)

〈δq2c+(θ )〉 = [1+2(2z′)2]/2, (38)

〈δqb+δqc+(θ )〉 = (2z′)sin(2θ )− (2z′)2cos(2θ ). (39)

Equations (37)–(39) are similar to Eqs. (23) and (24), whichdescribe the MI of pump 1 (αz isreplaced by 2z′). Hence, the sum modes participate in a two-mode squeezing interaction. Onealso finds that

〈δq2b−(θ )〉 = 1/2, (40)

〈δq2c−(θ )〉 = 1/2, (41)

〈δqb−(θ )δqc−(θ )〉 = 0. (42)

Furthermore, the correlations between modesb− or c−, and modesb+ or c+, are all zero.These results show that the difference modes are inert. (Their variances and correlations arecharacteristic of vacuum fluctuations.)

4. Superposition modes

For the special case in which the pump powers are equal, one can simplify the four-mode equa-tions by rewriting them in terms of two sum modes, which participate in a two-mode squeezinginteraction, and two difference modes, which do not interact with any other mode (sum or dif-ference). One can also simplify these equations for the general case in which the pump powersare unequal. Define the distance parameterz′ = βzand the normalized HamiltonianH ′

a = Ha/β .Thendaj/dz′ = i[a j ,H ′

a]. Now define the superposition modes

b+ = ε(σa1− +a2−), b− = ε(a1−−σa2−), (43)

c+ = ε(σa1+ +a2+), c− = ε(a1+−σa2+), (44)

where the pump-strength parameterσ = α/β = β/γ and the normalization coefficientε =1/(1+ σ2)1/2. Thenb± andc± satisfy the canonical commutation relations. The+ modes arecoupled to each other, but not the− modes:

b+(z′) = [1+ i(σ +1/σ)z′]b+(0)+ i(σ +1/σ)z′c†+(0), (45)

c†+(z′) = −i(σ +1/σ)z′b†

+(0)+ [1− i(σ +1/σ)z′]c+(0). (46)

In contrast, the− modes are inert:

b−(z′) = b−(0), (47)

c−(z′) = c−(0). (48)

Now let z′′ = (σ + 1/σ)z′. Then Eqs. (45) and (46) describe solutions of the two-mode equa-tions

db+/dz′′ = ib+ + ic†+, (49)

dc†+/dz′′ = −ib+− ic†

+, (50)

which produce two-mode squeezed states. [Had we used definitions (43) and (44) in the pre-vious section, instead of definitions (33) and (34), we wouldhave obtained the variance andcorrelation equations (37)–(39), with 2z′ replaced byz′′.]



By making an appropriate change of basis, one can convert a two-mode squeezed state intotwo one-mode squeezed states [61, 62]: The degree of correlation (entanglement) between twomodes depends on the basis that defines them [64]. It follows from Eqs. (49) and (50) that thealternative superposition modes

r = (b+ +c+)/21/2, (51)

s = (b+−c+)/21/2 (52)

satisfy the uncoupled one-mode equations

dr/dz′′ = ir + ir †, (53)

ds/dz′′ = is− is†, (54)

which have the solutions

r(z′′) = (1+ iz′′)r(0)+ iz′′r†(0), (55)

s(z′′) = (1+ iz′′)s(0)− iz′′s†(0). (56)

The preceding analysis shows that one can think of a two-modesqueezing interaction astwo separate one-mode squeezing interactions followed by abeam-splitter-like process, whichcombines the two output modes. One can also think of the four-mode interaction as a two-modesqueezing interaction followed by two separate beam-splitter-like processes, in which the twooutput modes are combined with two separate vacuum modes.

5. Photon-number correlations

Not only does the use of superposition modes elucidate the nature of two- and four-mode in-teractions, it also facilitates the derivation of number-state expansions for the state vectors. Byrewriting H ′

a in terms of the first set of superposition modes, which were defined in Eqs. (43)and (44), one (eventually) obtains the transformed Hamiltonian

H ′bc = (σ +1/σ)(b†

+b+ +c†+c+ +b†

+c†+ +b+c+). (57)

Consistent with Eqs. (47) and (48),H ′bc does not depend onb− or c−. The form of Eqs. (2) and

(57) prompts the definitionsz′′ = (σ +1/σ)z′ andH ′′bc = H ′

bc/(σ +1/σ). By rewritingH ′′bc in

terms of the second set of superposition modes, which were defined in Eqs. (51) and (52), oneobtains the alternative transformed Hamiltonian

H ′′rs = r†r +[(r†)2 + r2]/2+s†s− [(s†)2 +s2]/2. (58)

Consistent with Eqs. (53) and (54), ther ands terms inH ′′rs are separate. We denote them byH ′′

randH ′′

s , respectively. In the rest of this section, the superscript′′ will be omitted for simplicity.It follows from Eq. (2) that the aforementioned Hamiltonians are constant operators. Hence, theinput and output states are related by the equation|ψ(z)〉 = exp(iHz)|ψ(0)〉.

Hamiltonians (57) and (58) differ from the standard two- andone-mode Hamiltonians [61,62] because of CPM. One can writeHr = K+ + K− + 2K3 − 1/2, where the operatorsK+ =(r†)2/2,K− = r2/2 andK3 = (r†r +rr †)/4. These operators satisfy the angular-momentum-likecommutation relations[K+,K−] = −2K3 and[K3,K±] = ±K±. By using a standard operator-ordering theorem [61], which is proved in Appendix B, one finds that

exp(iHrz) = exp(γ+K+)exp(γ3K3)exp(γ−K−), (59)



whereγ± = iz/(1− iz), γ3 = −2ln(1− iz) and the phase factor exp(−iz/2) was omitted. If theinput is the one-mode vacuum state|0〉, then the output is the squeezed state

|r〉 =1

(1− iz)1/2

∞

∑n=0

(

iz1− iz

)n [(2n)!]1/2

2nn!|2n〉, (60)

where the basis vectors|2n〉 = (r†)2n|0〉/[(2n)!]1/2. The number-state expansion of|s〉 is simi-lar. [In the numerator of Eq. (60),iz is replaced by−iz.]

One can also writeHbc = K+ +K− +2K3−1, where the operatorsK+ = b†+c†

+, K− = b+c+

andK3 = (b†+b+ + c+c†

+)/2. These operators satisfy the commutation relations stated in theprevious paragraph. By using the same operator-ordering theorem, one finds that exp(iHbcz)can also be written in the form of Eq. (59), whereγ± = iz/(1− iz), γ3 = −2ln(1− iz) and thephase factor exp(−iz) was omitted. If the input is the two-mode vacuum state|0,0〉, then theoutput is the squeezed state

|b+,c+〉 =1

1− iz

∞

∑n=0

(

iz1− iz

)n

|n,n〉, (61)

where the basis vectors|n,n〉 = (b†+c†

+)n|0,0〉/n!. These vectors are consistent with the two-mode MRW equation〈nb+〉= 〈nc+〉, which is a limit of the four-mode MRW equations derivedin Appendix A. By using the inverses of Eqs. (51) and (52), onecan show that formula (61) isequivalent to the direct product of formula (60) and its analog for |s〉.

The properties of the two-mode state described by Eq. (61) are illustrated in Fig. 4. Theprobability that there aren photons in each of superposition modesb+ andc+ is

A(n,z) = z2n/(1+z2)n+1. (62)

This probability distribution (PD) is plotted as a functionof photon number in Fig. 4(a), forshort, intermediate and long distances. As distance increases, so also do the probabilities ofmany-photon states. The related probability that there arek photons in modeb+ andl photonsin modec+ is

Q(k, l ,z) = A(k,z)δ (k, l), (63)

whereδ (k, l) is the Kronecker delta. This joint PD is plotted in Fig. 4(b) for the intermediatedistance. Figure 4(b) illustrates the simple photon-number correlation that exists between thesuperposition modes. These results are similar to the standard results for unstable two-modeinteractions [61, 62], such as MI and PC.

The four-mode formula follows from Eq. (61) and the binomialexpansions of(b†+)n =

εn(σa†1−+a†

2−)n and(c†+)n = εn(σa†

1+ +a†2+)n. The result is

|ψ(z)〉 =1

1− iz

∞

∑n=0

n

∑k=0

n

∑l=0

(

iz1− iz

)n σk+l n! |k, l ,n−k,n− l〉

(1+ σ2)n[k! l ! (n−k)! (n− l)! ]1/2, (64)

where the basis vectors|k, l ,n−k,n− l〉 = (a†1−)k(a†

1+)l (a†2−)n−k(a†

2+)n−l |0,0,0,0〉/[k! l ! (n−k)! (n− l)! ]1/2. These vectors are consistent with the four-mode MRW equations (79)–(81). Ifσ ≫ 1, the only terms that contribute significantly are those forwhich k = n andl = n. In thiscase,

|ψ(z)〉 ≈1

1− iz

∞

∑n=0

(

iz1− iz

)n

|n,n,0,0〉, (65)



0 2 4 6 8Number HnL

-20

-15

-10

-5

0

Pro

babi

lityHd

BL

02

46

b+ 0

2

4

6

c+

-20

-15

-10

-5

P

02

46

b+

Fig. 4. (a) Probability (in dB) that there aren photons in each of modesb+ andc+ [Eq.(62)]. The dashed, dot-dashed and solid lines represent thedistance parametersγK(P1 +P2)z= 0.3, 1.0 and 3.0, respectively. (b) Joint probability distribution (PD) of modesb+andc+ [Eq. (63)] for the intermediate distance. These modes are correlated.

which is the two-mode state produced by the MI of pump 1. A similar result applies to the casein whichσ ≪ 1. (Number states of the form|0,0,n,n〉 are produced by the MI of pump 2.) Forthe intermediate case in whichσ ≈ 1,

|ψ(z)〉 ≈1

1− iz

∞

∑n=0

n

∑k=0

n

∑l=0

(

iz1− iz

)n n! |k, l ,n−k,n− l〉

2n[k! l ! (n−k)! (n− l)! ]1/2. (66)

In this parameter range, the functionsσ +1/σ andσk+l/(1+ σ2)n depend only weakly onσ ,so the pump powers need only be comparable, not equal: Four-mode correlations are robust.

The properties of the four-mode state described by Eq. (66) are illustrated in Figs. 5–7. Thetotal probability that there arem photons in mode 1− is

Pt(m,z) =∞

∑n=m

A(n,z)B(n,m,n−m), (67)

whereA(n,z) was defined in Eq. (62) and

B(n,k, l) = n!/(2nk! l ! ). (68)

B(n,k,n−k) is a binomial distribution, which has the property∑nk=0B(n,k,n−k) = 1. PD (67)

is plotted as a function of photon number in Fig. 5, for short,intermediate and long distances.As distance increases, so also do the probabilities of many-photon states. Although there arequalitative similarities between Figs. 4(a) and 5, there are also quantitative differences.

The conditional probability that there arek photons in mode 1− andl photons in mode 1+,given that there aren photons in each of the superposition modes, is

Qc(k, l) = B(n,k,n−k)B(n, l ,n− l). (69)

This joint PD is plotted in Fig. 6(a) for the intermediate distance. If the MI of pump 1 were tooccur in isolation, the probability would be nonzero only ifk = 4 andl = 4. It is clear from thefigure that the couplings to modes 2− and 2+, which are enabled by pump 2, have significanteffects. Because formula (66) depends symmetrically onl andn− l , the joint PD of modes 1−



0 2 4 6 8Number HnL

-20

-15

-10

-5

0

Pro

babi

lityHd

BL

Fig. 5. Total probability (in dB) that there aren photons in mode 1− [Eq. (67)]. The dashed,dot-dashed and solid lines represent the distance parameters 2γKPz= 0.3, 1.0 and 3.0,respectively. The PDs of modes 1+, 2− and 2+ are identical.

and 2+ is the same as the PD for modes 1− and 1+. The conditional probability that there arek photons in mode 1− andl photons in mode 2−, given that there aren photons in each of thesuperposition modes, is

Rc(k, l) = B(n,k,n−k)δ (n−k, l). (70)

This joint PD is plotted in Fig. 6(b) for the intermediate distance. Modes 1− and 2− are anti-correlated.

01

23

41-0

12

34

1+

-25

-20

-15

-10

P

01

23

41-

01

23

41-0

12

34

2-

-15

-10

-5

P

01

23

41-

Fig. 6. (a) Joint PD (in dB) of modes 1− and 1+ [Eq. (69)] for then= 4 state and the inter-mediate distance-parameter 2γKPz= 1.0. The joint PD of modes 1− and 2+ is identical.(b) Joint PD of modes 1− and 2− [Eq. (70)] for the same state and distance. These modesare anti-correlated.

The total probability that there arek photons in mode 1− andl photons in mode 1+ is

Qt(k, l ,z) =∞

∑n=max(k,l)

A(n,z)B(n,k,n−k)B(n, l ,n− l). (71)

This joint PD is plotted in Fig. 7(a) for the intermediate distance. The total probability thatthere arek photons in mode 1− andl photons in mode 2− is

Rt(k, l ,z) = A(k+ l ,z)B(k+ l ,k, l). (72)



This joint PD is plotted in Fig. 7(b) for the intermediate distance. A comparison of Figs. 4(b)and 7(a) shows that the couplings to modes 2− and 2+ changes significantly the correlationproperties of modes 1− and 1+.

01

23

41-0

12

34

1+

-30

-25

-20

-15

-10

-5

P

01

23

41-

01

23

41-0

12

34

2-

-30

-25

-20

-15

-10

-5

P

01

23

41-

Fig. 7. (a) Joint PD (in dB) of modes 1− and 1+ [Eq. (71)] for the intermediate distance-parameter 2γKPz= 1.0, which should be compared to the PD shown in Fig. 4(b). The jointPD of modes 1− and 2+ is identical. (b) Joint PD of modes 1− and 2− [Eq. (72)] for thesame distance.

6. Entanglement

Consider the pure two-mode state (61). From a mathematical standpoint, the absence of the off-diagonal termscmn|m〉|n〉 prevents|b+,c+〉 from being written as the direct product|b+〉|c+〉.Hence, it is an entangled state. From a physical standpoint,if one measures the number ofphotons in modec+, one determines the number of photons in modeb+: Measuring the secondmode affects the result of a subsequent measurement of the first mode. For any state vector|ψ〉, the associated density operator (matrix)ρ = |ψ〉〈ψ |. The reduced density matrix (RDM)ρb+ = Trc+(ρ), where Tr denotes a trace, characterizes the properties of modeb+, withoutregard to modec+. (The effects of the different states of modec+ are ensemble averaged.) Byfollowing a standard procedure, which is described in Appendix C, one finds that

ρb+(z′′) =∞

∑n=0

A(n,z′′)|n〉〈n|, (73)

whereA(n,z′′) is the probability that there aren photons in modeb+ at the distancez′′ =γK(P1 + P2)z. The formula forρc+ is identical. Because the two-mode state (61) is pure, thefact that both RDMs describe one-mode mixed states confirms that state (61) is entangled. Forthe MI of pump 1, which involves modes 1− and 1+, ρ1− is given by the same formula, withz′′ = γKP1z (because pump 2 is absent): This MI also produces a two-mode entangled state.

Now consider the pure four-mode state (66). To prove that it is fully four-partite entangled,one must prove that every partition of it is entangled. LetSdenote a subset of the four modesandT denote the complimentary subset. Because the state vector depends symmetrically onk and l , one can interchange the subscripts 1− and 1+, and 2− and 2+, without changingthe properties of a partition. Hence, the independent partitions {S|T} are{1−|1+,2−,2+},{2−|2+,1−,1+}, {1−,1+|2−,2+}, {1−,2−|1+,2+} and{1−,2+|1+,2−}. For the case inwhich the pump powers are equal, the state vector depends symmetrically onk andn−k, andl



andn− l , so one can interchange the subscripts 1− and 2−, or 1+ and 2+, without changingthe properties of the partition. In this case, the second partition is equivalent to the first, andthe fifth is equivalent to the third. Consider the three independent partitions in turn. For thefirst partition, if one measures the photon numbers of modes 1+, 2− and 2+, one determinesl , n andk, and, hence, the photon number of mode 1−. For the third partition, if one measuresthe photon numbers of modes 2− and 2+, one determines the differencek− l and, hence, thedifference between the photon numbers of modes 1− and 1+. For the fourth partition, if onemeasures the photon numbers of modes 1+ and 2+, one determines the sum of the photonnumber of modes 1− and 2−. In each case, the stated measurement onT affects a subsequentmeasurement onS. Hence, state (66) is fully four-partite entangled. To determine the RDMρ1−, one has to trace out modes 1+, 2− and 2+. By following a standard procedure, which isdescribed in Appendix C, one (eventually) finds that

ρ1−(z′′) =∞

∑m=0

Pt(m,z′′)|m〉〈m|, (74)

wherePt(m,z′′) is the total probability that there arem photons in mode 1− at the distancez′′ = 2γKPz. Equation (74) describes a one-mode mixed state, which confirms that state (66) isentangled. The RDMs associated with the other partitions, which are determined in AppendixC, also describe mixed states. These results confirm that state (66) is fully four-partite entan-gled.

The degrees of two-partite entanglement of a pure state are measured by the entropiesE = −Tr(ρSlogρS), whereρS is the RDM associated with the subsetS and logρS is de-fined by its Taylor series. The RDMs (73) and (74) are both written in the diagonal formρ = ∑∞

n=0 pn|n〉〈n|, for which the associated entropyE = −∑∞n=0 pn log(pn). Two-and four-

mode entropies are plotted as functions of distance in Fig. 8. For all distances, the two-modeentropy of modeb+ is higher than the four-mode entropy of 1−. It follows from Eqs. (43) thatmodes 1− andb+ are related by the beam-splitter-like equationa1− = ε(σb+ + b−), wheremodeb− is inert. Combining an entangled state with a vacuum (unentangled) state dilutes theentanglement, so it makes sense that the entropy of mode 1− is lower than that ofb+. However,the entropy of mode 1− is higher when it participates in the four-mode interactionthan when itparticipates in the MI of pump 1, because the nonlinear coupling for the four-mode interactionis stronger than the coupling for the MI (pump 2 is absent).

0 2 4 6 8Distance

0

1

2

3

4

5

Ent

angl

emen

t

Fig. 8. Entanglement (entropy) of mode 1− plotted as a function of the distance parameter2γKPz. The dashed and solid curves denote the two-mode interaction with mode 1+ [Eq.(73)] and the four-mode interaction with 1+, 2− and 2+ [Eq. (74)], respectively. For com-parison, the dot-dashed curve denotes the entropy of modeb+, which interacts with modec+ [also Eq. (73)].



7. Summary

Multipartite entanglement is a resource for quantum communication and computation. Its usehas been demonstrated in dense coding, key distribution, one-way computation and teleporta-tion networking. The standard way to produce such entanglement is to combine several sourcesof one-color, two-product-modeentanglement, which are based on parametric down-conversionin crystals, and combine their outputs using a path-stabilized array of beam splitters. Three-color, pump- and product-mode entanglement is also being studied [65].

Vector four-wave mixing (FWM) in a fiber, driven by two strongpumps, couples the evo-lution of four weak sidebands (modes). This four-mode interaction is a combination of sixtwo-mode interactions (modulation instability, phase conjugation and Bragg scattering), whichoccur simultaneously when dispersion is weak. Two of the mode frequencies are similar to thefrequency of pump 1, and two are similar to the frequency of pump 2. A fiber with one zero-dispersion frequency (ZDF), supports an interaction between modes with similar frequencies(typically separated by about 1 THz), whereas a fiber with twoZDFs supports an interactionbetween modes with dissimilar frequencies (typically separated by more than 10 THz).

In this report, the discrete- and continuous-variable entanglement produced by vector FWMwas studied in detail. Formulas were derived for the variances of, and correlations between,the mode quadratures [Eqs. (25)–(32)]. These formulas showed that the modes are stronglycorrelated, and prompted the reformulation of the interaction in terms of superposition (sumand difference) modes. The sum modes participate in a two-mode interaction [Eqs. (37)–(39)],whereas the difference modes are inert [Eqs. (40)–(42)]. This result allows one to interpretthe four-mode interaction as a two-sum-mode interaction followed by two beam-splitter-likeprocesses, which mix the output (sum) modes with vacuum (difference) modes. The number-state expansion of the state vector was also derived [Eq. (66)]. This formula showed that theinteraction produces four-partite entanglement. It also enabled the derivation of formulas forthe unconditional and conditional photon-number distributions of the modes [Eqs. (67) and(69)–(72)].

In summary, vector FWM in a fiber produces four-color, four-partite entanglement naturally:Multiple sources of two-partite entanglement and path-stabilized arrays of beam splitters are notnecessary. In future work, we will analyze the van Loock–Furasawa inequalities for multipartiteentanglement of Gaussian states [66] to determine the best types of measurements for verifyingand characterizing the entanglement produced in experiments. (Losses cause the output statesto be mixed, not pure.) Further work is also required to determine the practicality of four-colorentanglement in schemes for quantum communication and computation.

Acknowledgments

The research of MR and SR was supported by the National Science Foundation under con-tracts ECS-0621723 and ECS-0406379, respectively.

A: Manley–Rowe–Weiss equations

Let n j = a†j a j be the photon-number operator of mode (sideband)j. (In the main text,n was

used to denote a photon number.) Then Eqs. (3)–(6) imply that

dzn1− = iα(a†1−a†

1+−a1−a1+)+ iβ (a†1−a2−−a1−a†

2−)

+ iβ (a†1−a†

2+−a1−a2+), (75)

dzn1+ = iα(a†1−a†

1+−a1−a1+)+ iβ (a†1+a†

2−−a1+a2−)

+ iβ (a†1+a2+−a1+a†

2+), (76)



dzn2− = iγ(a†2−a†

2+−a2−a1+)+ iβ (a1−a†2−−a†

1−a2−)

+ iβ (a†1+a†

2−−a1+a2−), (77)

dzn2+ = iγ(a†2−a†

2+−a2−a2+)+ iβ (a†1−a†

2+−a1−a2+)

+ iβ (a1+a†2+−a†

1+a2+). (78)

By combining Eqs. (75)–(78), one obtains the Manley–Rowe–Weiss equations

dz(n1−−n1+) = dz(n2+−n2−), (79)

dz(n1− +n2−) = dz(n1+ +n2+), (80)

dz(n1−−n2+) = dz(n1+−n2−). (81)

Equations (79)–(81) are not independent. They are different ways of expressing the sameconstraint. The terms on the left side of Eq. (79) pertain to the MI of pump 1, in which2γ1 → γ1− + γ1+, whereas the terms on the right side pertain to the MI of pump 2, in which2γ2 → γ2−+ γ2+. If these processes were to occur in isolation,〈n1−〉−〈n1+〉 and〈n2−〉−〈n2+〉would be constants, because sideband photons are produced in pairs. The left side of Eq.(80) pertains to the BS process in whichγ1− + γ2 → γ1 + γ2−, whereas the right side pertainsto the process in whichγ1+ + γ2 → γ1 + γ2+. If these processes were to occur in isolation,〈n1−〉+ 〈n2−〉 and〈n1+〉+ 〈n2+〉 would be constants, because photons are exchanged betweenthe sidebands. The left side of Eq. (81) pertains to the PC process in whichγ1+γ2 → γ1−+γ2+,whereas the right side pertains to the process in whichγ1 + γ2 → γ1+ + γ2−. If these processeswere to occur in isolation,〈n1−〉− 〈n2+〉 and〈n1+〉− 〈n2−〉 would be constants, because side-band photons are produced in pairs. In the presence of four-sideband coupling, the aforemen-tioned combinations of photon numbers are not constant. They evolve, subject to constraints(79)–(81).

B: Operator-ordering theorem

The main results of this report, Eqs. (60), (61) and (64), were obtained by the use of anoperator-ordering theorem (OOT). Although such theorems are common in the quantum-opticsliterature [61], they are not common in the optical-communications literature. Consequently, inthis appendix the OOT (59) will be proved from first principles.

The proof of this OOT relies on the Baker–Campbell–Hausdorff (BCH) lemma

exp(a)bexp(−a) =∞

∑n=0

[a,b]n/n!, (82)

where a and b are operators, and thenth-order commutator[a,b]n is defined recursively:[a,b]0 = b, [a,b]1 = [a,b] and[a,b]n = [a, [a,b]n−1]. By expanding the exponentials on the leftside of Eq. (82) in Taylor series, one finds that

exp(a)bexp(−a) =∞

∑n=0

n

∑m=0

amb(−a)n−m

m!(n−n)!. (83)

Equations (82) and (83) are equivalent if and only if the commutator

[a,b]n =n

∑m=0

n!amb(−a)n−m

m!(n−m)!. (84)



It is easy to verify that Eq. (84) is valid forn= 0 andn= 1 (the first non-trivial order). Supposethat it is valid for some ordern≥ 1. Then the next-order commutator

a[a,b]n− [a,b]na =n

∑m=0

n![am+1b(−a)n−m−amb(−a)n−ma]

m!(n−m)!, (85)

=n+1

∑m=1

n!amb(−a)n+1−m

(m−1)!(n+1−m)!+

n

∑m=0

n!amb(−a)n+1−m

m!(n−m)!. (86)

There are three cases to consider. First, form= 0 the second series on the right side of Eq. (86)contributes the terma0b(−a)n+1 = (n+ 1)!a0b(−a)n+1/[0!(n+ 1)!]. Second, form = n+ 1the first series contributes the terman+1b(−a)0 = (n+1)!an+1b(−a)0/[0!(n+1)!]. Third, for1≤ m≤ n, the combined contribution is

n!amb(−a)n+1−m

(m−1)!(n−m)!

[

1n+1−m

+1m

]

=(n+1)!amb(−a)n+1−m

m!(n+1−m)!. (87)

In each case, themth-term in the expansion of the commutator[a,b]n+1 has the form requiredby Eq. (84). This result proves the BCH lemma (82).

Equation (59) provides a normally-ordered formula for the Schrodinger evolution-operatorexp(iHz), whereH is a Hamiltonian andz is a distance variable. In this reportH = K+ +2K3 +K−, where the operatorsK± andK3 satisfy the commutation relations[K+,K−] = −2K3

and [K3,K±] = ±K±. (Formulas for these operators were stated in the main text.) Define thefunction

F(z) = exp[i(K+ +2K3+K−)z]. (88)

Because theK-operators form a closed set under commutation, one can rewrite Eq. (88) in thenormally-ordered form

F(z) = exp[ip(z)K+]exp[iq(z)K3]exp[ir (z)K−], (89)

wherep, q andr are functions ofz (to be determined). It follows from Eq. (88) that

F ′ = i(K+ +2K3 +K−)F, (90)

whereF ′ = dF/dz. Likewise, it follows from Eq. (89) that

F ′ = (ip′K+ + iq′eipK+K3e−ipK+ + ir ′eipK+eiqK3K−e−iqK3e−ipK+)F. (91)

By using lemma (82) and the aforementioned commutation relations, one finds that

eipK+K3e−ipK+ = K3− ipK+, (92)

eiqK3K−e−iqK3 = K−e−iq, (93)

eipK+K−e−ipK+ = K−−2ipK3− p2K+. (94)

By using these results to simplify Eq. (91), and equating thecoefficients ofK+, K3 andK− inEqs. (90) and (91), one obtains the differential equations

p′− ipq′− p2(r ′e−iq) = 1, (95)

q′−2ip(r ′e−iq) = 2, (96)

r ′e−iq = 1, (97)



respectively. For the boundary (initial) conditionsp(0) = 0,q(0) = 0 andr(0) = 0, the solutionsof Eqs. (95)–(97) are

ip(z) = iz/(1− iz), (98)

iq(z) = −2log(1− iz), (99)

ir (z) = iz/(1− iz). (100)

Equations (98)–(100) are consistent with the formulas forγ± andγ3 stated after Eq. (59).

C: Reduced density operators

The two-mode state vector in Eq. (61) can be written in the compact form

|ψ〉 =∞

∑n=0

an|n〉b|n〉c, (101)

wherean(z) = (iz)n/(1− iz)n+1, andb andc are abbreviations forb+ andc+, respectively. Theassociated density operator (matrix)ρ = |ψ〉〈ψ |. By combining this definition with Eq. (101),one finds that

ρ =∞

∑n=0

∞

∑n′=0

ana∗n′ |n〉b|n〉c〈n′|b〈n

′|c. (102)

The reduced density matrix (RDM)ρb = Trc(ρ) = ∑∞n′′=0〈n

′′|cρ |n′′〉c characterizes the proper-ties of modeb, without regard to modec. When one calculates this partial trace, one encountersthe summations∑∞

n′=0 ∑∞n′′=0〈n

′′|n〉c〈n′|n′′〉c = ∑∞n′=0 ∑∞

n′′=0〈n′|n′′〉c〈n′′|n〉c = ∑∞

n′=0 δn′n, whereδi j is the Kronecker delta. By applying this result to Eq. (102),one obtains the RDM

ρb =∞

∑n=0

|an|2|n〉b〈n|b. (103)

Equation (103) is equivalent to Eq. (73). It describes a mixed state, because there is no couplingbetween the eigenstates associated with different photon numbersn. For reference, the preced-ing analysis shows that the identity Tr(|φ〉〈φ ′|) = 〈φ ′|φ〉 is valid even when|φ〉 and |φ ′〉 areparts of a higher-dimensional DM and the operation is a partial trace.

The four-mode state vector in Eq. (66) can be written in the compact form

|ψ〉 =∞

∑n=0

n

∑k=0

n

∑l=0

anbnkbnl|k〉1|l〉2|n−k〉3|n− l〉4, (104)

wherebnk = [n!/2nk!(n− k)!]1/2 and modes 1−, 1+, 2− and 2+ were relabeled 1, 2, 3 and4, respectively. This state vector depends symmetrically on k andn−k, andl andn− l . Inter-changing either pair of indices causes the same set of terms to appear in reverse order. Hence,one can interchange the mode subscripts 1 and 3, and 2 and 4, separately. The state vector alsodepends symmetrically onk andl , so one can interchange the subscripts 1 and 2, and 3 and 4,simultaneously. The associated DM

ρ =∞

∑n=0

n

∑k=0

n

∑l=0

∞

∑n′=0

n′

∑k′=0

n′

∑l ′=0

anbnkbnl(an′bn′k′bn′l ′)∗

× |k〉1|l〉2|n−k〉3|n− l〉4〈k′|1〈l

′|2〈n′−k′|3〈n

′− l ′|4. (105)

Define the RDMsρi jk = Trl (ρ), ρi j = Trkl(ρ) andρi = Tr jkl (ρ). Then it follows from theaforementioned symmetries that there are only two distincttwice-reduced DMs,ρ24 andρ34,



and there is only one distinct thrice-reduced DM,ρ4. These RDMs are all reductions ofρ234=Tr1(ρ). By rewriting the first set of summations in Eq. (105) as∑∞

k=0 ∑∞n=k ∑n

l=0, rewriting thesecond set in similar way and tracing out mode 1, one finds that

ρ234 =∞

∑k=0

∞

∑n=k

n

∑l=0

∞

∑n′=k

n′

∑l ′=0

anbnkbnl(an′bn′kbn′l ′)∗

× |l〉2|n−k〉3|n− l〉4〈l′|2〈n

′−k|3〈n′− l ′|4. (106)

The RDM ρ24 = Tr3(ρ234). By rewriting thek-, n- and n′-summations in Eq. (106) as

∑∞n=0 ∑∞

n′=0 ∑k+k=0, wherek+ = min(n,n′), and tracing out mode 3, one finds that

ρ24 =∞

∑n=0

n

∑k=0

n

∑l=0

n

∑l ′=0

|an|2|bnk|

2bnlb∗nl′ |l〉2|n− l〉4〈l

′|2〈n− l ′|4. (107)

The binomial identity∑nk=0 |bnk|

2 = 1 allows one to rewrite Eq. (107) in the simpler form

ρ24 =∞

∑n=0

n

∑l=0

n

∑l ′=0

|an|2bnlb

∗nl′ |l〉2|n− l〉4〈l

′|2〈n− l ′|4. (108)

Let Vi j be the product vector-space of modesi and j. Then Eq. (108) shows thatρ24 projectssubspaces ofV24, in which the total photon number of modes 2 and 4 isn, onto themselves:There is no coupling between the subspaces associated with different values ofn. Hence,ρ24

describes a mixed state.The RDM ρ4 = Tr2(ρ24). By rewriting the summations in Eq. (108) as∑∞

l=0 ∑∞l ′=0 ∑∞

n=n− ,wheren− = max(l , l ′), and tracing out mode 2, one finds that

ρ4 =∞

∑l=0

∞

∑n=l

|an|2|bnl|

2|n− l〉4〈n− l |4, (109)

which also describes a mixed state. By rewriting the summations in Eq. (109) as∑∞n−l=0 ∑∞

n=n−l ,and interchangingl andn− l , one obtains the alternative equation

ρ4 =∞

∑l=0

∞

∑n=l

|an|2|bnl|

2|l〉4〈l |4, (110)

which is equivalent to Eq. (74).The RDM ρ34 = Tr2(ρ234). By rewriting thek-, n- and n′-summations in Eq. (106) as

∑∞n=0 ∑∞

n′=0 ∑k+k=0, wherek+ = min(n,n′), rewriting the l - and n- summations as∑∞

l=0 ∑∞n=l ,

rewriting thel ′- andn′- summations in a similar way, and tracing out mode 2, one findsthat

ρ34 =∞

∑l=0

∞

∑n=l

∞

∑n′=l

k+

∑k=0

(anbnkbnl)(an′bn′kbn′l )∗|n−k〉3|n− l〉4〈n

′−k|3〈n′− l |4. (111)

Equation (111) shows thatρ34 projects subspaces ofV34, in which the photon numbers of modes3 and 4 differ byd = k− l , onto themselves: There is no coupling between the subspacesassociated with different values ofd. Hence,ρ34 describes a mixed state.

The RDM ρ4 = Tr3(ρ34). By rewriting the l -, n- and n′-summations in Eq. (111) as

∑∞n=0 ∑∞

n′=0 ∑l+l=0, wherel+ = min(n,n′), and tracing out mode 3, one finds that

ρ4 =∞

∑n=0

n

∑l=0

n

∑k=0

|an|2|bnk|

2|bnl|2|n− l〉4〈n− l |4. (112)

By using the binomial identity, interchangingl andn− l , and rewriting then- andl -summationsas∑∞

l=0 ∑∞n=l , one can rewrite Eq. (112) in the form of Eq. (110). Hence, theRDM ρ4 does not

depend on the order in which the reductions are made.



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