Self-assisted complete maximally hyperentangled state analysis via the cross-Kerr nonlinearity

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by  Xihan Li

Self-assisted complete maximally hyperentangled state analysis via the cross-Kerr
nonlinearity
Xi-Han Li1,2 , Shohini Ghose2,3
∗
1 Department of Physics, Chongqing University, Chongqing, China
2Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Canada
3 Institute for Quantum Computing, University of Waterloo, Canada
(Dated: January 12, 2016)
We present two complete maximally hyperentangled state analysis protocols for photons entan-
gledinthepolarization andspatial-modedegreesoffreedom. Thefirstprotocolisahyperentangled
Bell state analysis scheme for two photonsand thesecond is a hyperentangled Greenberger-Horne-
Zeilinger (GHZ) state analysis scheme for three photons. In each scheme, a set of mutually or-
6 thogonal hyperentangled basis states are completely and deterministically discriminated with the
1 aidofcross-Kerr nonlinearitiesandlinear optics. Wealso generalize theschemestounambiguously
0 analyze the N-photon hyperentangled GHZ state. Compared with previous protocols, our schemes
2
greatly simplify the discrimination process and reduce the requirements on nonlinearities by using

n the measured spatial-mode state to assist in the analysis of the polarization state. These advan-
a tages make our schemes useful for practical applications in long-distance high capacity quantum
J communication.

8

PACS numbers: 03.67.Hk, 03.67.Dd, 03.65. Ud mode degrees of freedom. Our schemes take advantage
]
h ofinformationaboutthespatial-modestatetoefficiently
p analyze the polarization state of the photons, thus sig-
-
t I. INTRODUCTION nificantly improving on past protocols.
n
Although a set of mutually orthogonal basis states
a
u canintheorybecompletelydiscriminated,completeBell
Quantumentanglementplaysacrucialroleinquantum
q state analysis (BSA) and GHZ state analysis (GSA) of
information processing. It is a key resource for quantum
[ photonscannotberealizedbylinearopticsalone,without
communication tasks such as quantum key distribution
1 [1, 2], dense coding [3, 4], teleportation [5], secret shar- resorting to ancillaries [12–15]. It has been shown that
v the four Bell states for two photons can only be classi-
ing [6–8], quantum secure direct communication [9–11]
9 fiedintothreegroupsandtheoptimalsuccessprobability
andothers. Amongthemanydifferenttypesofentangled
2 of state analysis is 50% using only linear optics. How-
states,theBellstatesfortwoqubitsandtheGreenberger-
0
ever, the success probability can be improved by several
2 Horne-Zeilinger(GHZ)statesforthreeormorequbitsare
means, for example, the use of auxiliary entanglement
0 the most popular resourcesin quantuminformation pro-
inanotherdegreeoffreedom(DOF)[16–20],viaassistant
. cessing. These states have two terms in their simplest
1 states [21, 22] and through a nonlinear interaction such
0 form and are maximally entangled bases for the Hilbert
as the cross-Kerrnonlinearity [23].
6 space. The analysis of Bell states and GHZ states not
1 only has fundamental significance in quantum informa- Whereas traditional entanglement involves particles
: tion theory but is also of practicaluse in quantum infor- that are only entangled in one degree of freedom, hyper-
v
i mationprocessing. Thenumberofthesebasisstatesthat entanglementinvolvesparticlessimultaneouslyentangled
X can be distinguished usually determines the capacity of in more than one degree of freedom, and has attracted
r quantum communication schemes. Thus much effort has much attention in recent years. There are several DOFs
a
been made in the past to perform state analysis, with ofa photonsuch aspolarization,spatialmode, time-bin,
complete discrimination being the ultimate goal. Gen- frequency,etc,thatcanbeusedtoconstructhyperentan-
erally speaking, Bell states and GHZ states can be de- gled states. Hyperentanglement has the appealing fea-
scribedbytwokinds ofinformation: bitinformationand turethateachphotoncarriesinformationintwoormore
phase information. Bit information describes the pari- DOFs andthe DOFs canbe manipulatedindependently.
tiesbetweenanytwoparticlesandthephaseinformation This can improve both the security and the channel ca-
denotes the relative phase between the two terms. Un- pacity of quantum communications [24]. Recently, ex-
ambiguous state discrimination can be accomplished by perimental preparation of hyperentanglement has been
obtaining both kinds of information. In this paper, we reported [25–27]. Hyperentanglement has many applica-
present efficient and practical schemes for complete, un- tions in quantum information processing, such as com-
ambiguous state discrimination of N-photon states that plete Bell-state analysis [16–20], hyper-parallelquantum
aremaximallyentangledinbothpolarizationandspatial- computing [28, 29], deterministic entanglement purifica-
tionprotocols[30–33]andquantumrepeaters[34]. There
has also been interesting recent progress in hyperentan-
glement concentration and hyperentanglement purifica-
Emailaddress: [email protected] tion [35–43].
∗
2
To date, the most popular hyperentangled states of ourscheme, 64three-photonhyperentangledGHZ states
photons are those entangled in polarization and spatial- can be unambiguously discriminated. By maintaining
mode DOFs,since the manipulationtechniquesforthese the spatial-mode coherence in the first step and using it
two DOFs is mature. In high-capacityquantumcommu- to assist in the discrimination of polarization states, our
nication schemes that utilize hyperentangled channels, schemesgreatlysimplify the processandsignificantlyre-
hyperentangled Bell state analysis (HBSA) and hyper- duce the required nonlinearities compared with previous
entangled GHZ state analysis (HGSA) are key steps re- protocols. We also generalize the scheme to a complete
quired to read the information. Hyperentangled state and deterministic analysis of N-photon hyperentangled
discrimination needs to confirm the bit and phase infor- GHZ states. A detailed discussion and summary is pro-
mation for both DOFs, which is more challenging than vided in the last section.
traditional entangled state analysis. Considering both
DOFs together, there are 16 hyperentangled Bell states
intotalfortwophotons. Ithasbeenshownthatthese16 II. COMPLETE HYPERENTANGLED BELL
states can be classified into only 7 groups via linear op- STATE ANALYSIS
tics[44,45]andthuscannotbecompletelydistinguished.
In 2010, Sheng et al. proposed the first complete HBSA Thetwo-photonhyperentangledBellstatecanbewrit-
scheme using the cross-Kerr nonlinearity [46]. In their ten as
scheme,threequantumnondemolitiondetectors(QNDs)
Υ = Θ Ξ . (1)
constructed using cross-Kerr nonlinearities are used to AB P AB S AB
| i | i ⊗| i
read out the bit and phase information of the spatial-
HereAandB denotethetwophotonsandthesubscripts
mode state and the bit information of the polarization
P and S represent the polarization and spatial-mode
state respectively. Then the phase information of the
DOF, respectively. Θ can be one of the following
P AB
polarizationDOF is obtained by measurement in the di- | i
four Bell states in the polarization DOF,
agonalbasisofthepolarizationstate. Withthese4bitsof
information, 16 polarization-momentum hyperentangled 1
Φ = (HH VV ) , (2)
± P AB
Bell states can be unambiguously discriminated. Later, | i √2 | i±| i
Xia et al. presented an efficient HGSA protocol using a 1
Ψ = (HV VH ) . (3)
similar principle [47]. In these two schemes, the spatial- ± P AB
| i √2 | i±| i
mode state is analyzed in the first step. Although the
QNDspreservethephotons,thecoherenceofthespatial- H and V indicate the horizontal and the vertical po-
| i | i
mode state is destroyed. If preserved, it can be useful larizations, respectively. The spatial-mode state ΞS AB
| i
for the discrimination of the polarization state. Unlike is one of the four Bell states in the spatial modes
complete BSA and GSA schemes which resort to several
1
auxiliary tools such as additional entanglement, ancil- Φ± S = (a1b1 a2b2 )AB, (4)
| i √2 | i±| i
lary states or nonlinear interactions, the main resource
1
forcompleteHBSAandHGSAisthenonlinearity. Com- Ψ± S = (a1b2 a2b1 )AB. (5)
plete HBSA schemes were also realized with the help of | i √2 | i±| i
quantum-dotspinsinopticalmicrocavitiesandnitrogen-
vacancy centers in resonators [48–50]. Recently, Liu et Herea1(b1)anda2(b2)arethetwopossiblespatialmodes
of photon A(B). Taking into account the two DOFs to-
al. proposed a complete nondestructive analysis of the
gether, there are 16 hyperentangled Bell states and our
two-photonsix-qubit hyperentangledBell states assisted
task is to distinguish them completely.
by cross-Kerr nonlinearity, in which the photons are en-
Before we describe our scheme, we introduce the basic
tangled simultaneously in the polarization and two lon-
principle of the cross-Kerr nonlinearities which play a
gitudinal momentum DOFs [51].
centralroleinourspatial-modestatediscrimination. The
In this paper we first present a simplified complete Hamiltonian describing the interaction between a signal
HBSA scheme which deterministically distinguishes 16 state ψ sandaprobecoherentstate α pinthenonlinear
| i | i
hyperentangled Bell states of two photons. The bit and medium can be written as
phase information of the spatial-mode state is read by
H =h¯χa a a a . (6)
two QNDs constructed with the cross-Kerr nonlinear- †s s †p p
ity [52]. The key point is that neither the photons nor
Here a (a )and a (a ) arethe creationand annihilation
†s †p s p
the spatial-mode state will be destroyed by the QNDs.
operations for the signal (probe) state, respectively. χ
With the help of the preserved spatial-mode entangle-
is the coupling strength of the nonlinearity and depends
ment, the bit and phase information of the polariza-
on the material. After the interaction with the signal
tion state can be deduced simultaneously by two single-
state inthe medium, the coherentstate picksup a phase
photon Bell state measurements (SPBSMs). We then
shift which is proportional to the photon number N of
describeacompleteHGSAprotocolforthethree-photon
the signal state,
hyperentangled GHZ state, in which three assistant co-
herent states and three SPBSMs are required. With α αeiNθ . (7)
p
| i →| i
3
Here θ = χt and t is the interaction time. By measur- as
ing the phase shift via the X-quadrature measurement,
1
the number of photons can be read out without destroy- Φ± S α1 = (a1b1 a2b2 )α1
ing the photons. We can choose X-quadrature measure- | i | i √2 | i±| i | i
ments that do not distinguish phase shifts differing in 1
sign“ ”. Thisfeaturepreservesthecoherenceofphotons → √2(|a1b1i±|a2b2i)|α1i=|Φ±iS|α1i(,8)
±
with respect to each other as well as the photons them- 1
selves. The cross-Kerrnonlinearityhasbeenwidelyused Ψ± S α1 = (a1b2 a2b1 )α1
| i | i √2 | i±| i | i
to construct quantum nondemolition detectors (QNDs)
1
for quantum information processing in the past decade (a1b2 α1eiθ a2b1 α1e−iθ )
[23, 30, 41, 46, 47, 52, 53]. In our scheme, it is used in → √2 | i| i±| i| i
two QNDs which read the bit and phase information of = Ψ± S α1e±iθ . (9)
| i | i
the spatial-mode state.
Here we omit the polarization DOF since it is invariant
during the evolution. Note that the final line in (9) fol-
lows from the fact that the X-quadrature measurement
$ !" $ on α1 issetuptoonlydistinguishthephase0from θ.
"#$ ! " Hen|ce,iwith this measurement, Φ can be discri±mi-
! homodyne | ±iS
$#%! ! " anbatoeudt ftrhoemsp|aΨt±iailS-m. oIdneoitshoebrtwaionredds., the bit information
a
) The two spatial modes of each photon are then mixed
a
a ! at the beam splitters (BSs), which act as a Hadamard
operation on the spatial-mode DOF,
a a !

!" # a 1

b |x1i → √2(|x1i+|x2i), (10)

b b ! 1
x2 (x1 x2 ). (11)
| i → √2 | i−| i
b b !
b Here x denotes a or b. The effect of the two BSs is
to transform the input spatial-mode Bell states to dif-
#$ "#$%& "#$%’( ferent spatial-mode Bell states. In detail, Φ+ and
S
Ψ+ are invariant, while Φ ⇀↽ Ψ+ .| Thie form
S − S S
| i | i | i
FIG. 1: Schematic diagram of our complete HBSA protocol. of the nonlinear interactions between the photons and
The four cross-Kerr nonlinear interactions produce a phase the second coherent state α2 is similar to the first
shift of ±θ on the coherent states |α1i and |α2i if photons one, which distinguishes the|origiinal Φ+ S(Ψ+ S) from
areincorrespondingspatialmodes. Thebeamsplitters(BSs) Φ (Ψ ). Thisprovidesthepha|seinifor|matiion,and
− S − S
guide the photons from each input port to these two output | i | i
hence the four spatial-mode Bell states are completely
ports with equal probabilities. The polarizing beam splitters
discriminated. Meanwhile, the spatial-mode entangle-
(PBSs) at 0◦ transmit horizontal polarized states while re- ment is also changed. The relationsbetween the original
flecting vertical polarized states. The PBSs at 45◦ transmit
|+i= 1 (|Hi+|Vi), and reflect |−i= 1 (|Hi−|Vi). The spatial-mode Bell state, the new spatial-mode Bell state
√2 √2 andthephaseshiftsofthetwocoherentbeamsareshown
dashed rectangle represents a single-photon Bell state mea-
in Table. I.
surement (SPBSM), which discriminates four single-photon
Bellstatescompletely. Withthehomodynemeasurementson
thetwocoherentstatesandtwoSPBSMs,16hyperentangled
TABLEI:Correspondingrelationsbetweentheoriginalstate,
Bell states can be completely distinguished.
thenewstateaftertheBS’sinFig1,andthetwophaseshifts
of coherent states.
Original state New state |α1i |α2i
|Φ+iS |Φ+iS 0 0
The setup of our proposedHBSA protocolis shownin
Fig.1. The process consists of two steps: discrimination |Φ−iS |Ψ+iS 0 ±θ
of the spatial-mode Bell states via the cross-Kerr non- |Ψ+iS |Φ−iS ±θ 0
linearity, followed by discrimination of polarization Bell |Ψ−iS |Ψ−iS ±θ ±θ
states assisted by the spatial-mode entanglement. We
now introduce the process step by step. In the second step, two single-photon Bell state mea-
surements(SPBSMs)areperformedonthe two photons,
Afterthephotonsina1 andb1 interactwiththecoher- whosemeasurementoutcomeswillresultinthecomplete
ent state α1 , the state of the collective system evolves discrimination of four polarization Bell states. The four
| i
4
single-photon Bell states composed of the polarization From the preceding analysis, the 16 hyperentangled
and spatial-mode DOFs are Bell states are completely discriminated with our two-
stepscheme. Thedistinguishingofthepolarizationstate
1 is aided by spatial-mode entanglement. We note that
φ± X = (Hx2 Vx1 )X, (12)
| i √2 | i±| i although our scheme consists of two steps, there is no
1 need to pause the state analysis procedure midway. The
ψ± X = (Hx1 Vx2 )X. (13) informationabout the spatial-mode state can be used to
| i √2 | i±| i
deduce the polarizationstate after all the measurements
Here X(x) can be either A(a) or B(b). After the polar- have been performed.
izing beam splitters (PBSs) at 0 (which transmit hori-
◦
zontal states while reflecting vertical ones) and PBSs at
45 (which transmit + = 1 (H + V ) state and re- III. COMPLETE HYPERENTANGLED
◦ | i √2 | i | i GREENBERGER-HORNE-ZEILINGER STATE
flect = 1 (H V )),foursingle-photonBellstates ANALYSIS
|−i √2 | i−| i
willtriggerthe fourdifferentdetectorsplacedinthe four
outputports,accordingly. Specifically, φ± X goestox±1 In this section, we introduce the complete HGSA
| i
while |ψ±iX goes to x±2. scheme by describing the three-photon hyperentangled
In this step, the spatial-mode state is known and pro- GHZ state as an example first. Then we generalize the
vides importantassistanceinthe analysisofthe four po- scheme to analyze the N-photon hyperentangled GHZ
larization Bell states. For example, if the new spatial- state.
mode stateafterthe firststepis Ψ− S, the fourpossible Generally,theN-photonhyperentangledGHZstatein
| i
correspondinghyperentangledstates will result in differ- bothpolarizationandspatial-modeDOFscanbewritten
ent combinations of SPBSMs as as
1
Φ± P Ψ− S = (ψ± A φ− B + ψ∓ A φ+ B |ΥiAB...Z =|ΘPiAB...Z ⊗|ΞSiAB...Z. (16)
| i ⊗| i 2 | i | i | i | i
φ ψ φ ψ+ )(,14) Here A, B,...,Z denote the N photons. There are 2N
± A − B ∓ A B
−| i | i −| i | i maximally entangled GHZ states in each DOF, which
1
Ψ± P Ψ− S = (ψ− A ψ± B ψ+ A ψ∓ B can be written in a unified form as
| i ⊗| i 2 | i | i −| i | i
−|φ+iA|φ∓iB −|φ−iA|φ±iB).(15) |Ω±ab...ziAB...Z = √12(|ab...zi±|a¯b¯...z¯i)AB...Z.(17)
There are 16 possible measurement combinations, which
Hereab...z 0,1 refertothebitinformationandx¯=
can be collected into four groups. Each group corre- ∈{ }
1 x,(x=ab...z). ForthepolarizationDOF, 0 H ,
spondstoaspecificpolarizationBellstate. Therefore,all − | i≡| i
fourpossiblepolarizationBellstatescanbedeterministi- |1i ≡ |Vi. For the spatial-mode DOF, |0i ≡ |x1i and
cally discriminated by identifying which group the mea- |1i≡ |x2i x = a,b,...z. Since |Ω±ab...zi and |Ω¯±a¯b...¯zi only
differbyanonessentialglobalphase,welimitthenumber
surement outcomes belong to. If the spatial-mode state
of ”1”s in the subscript string “ab...z” of Ω to be no
isoneoftheotherthreeBellstates,thefourpolarization
larger than N/2 to ensure that there are 2N mutually
states can also be distinguished in the same way. The
orthogonal entangled GHZ states in total. (When N is
detailed relations are shown in Table. II. For example,
if the two SPBSM results are ψ and φ+ , the new even, we choose the subscript string with a lower binary
| −iA | iB value.)
state after the first step belongs to the last group. If the
firststepdeterminesthatthespatial-modestateis Φ+ , We discuss the N = 3 situation first. One of these 64
S
| i hyperentangled GHZ states for example is
one can deduce that the polarization state is Ψ .
− P
| i
Ω+ Ω+
| 000iP ⊗| 000iS
1 1
TABLEII:Relationsbetweenthenewstatebeforethesecond = (HHH + VVV ) (a1b1c1 + a2b2c2 ).
step in Fig. 1 and possible detections. √2 | i | i ⊗ √2 | i | i
(18)
New states Possible detections
|Φ+iP ⊗|Φ+iS, |Φ−iP ⊗|Φ−iS, |φ+iA|φ+iB, |φ−iA|φ−iB, The set-up of our complete HGSA scheme for three-
|Ψ+iP ⊗|Ψ+iS,|Ψ−iP ⊗|Ψ−iS. |ψ+iA|ψ+iB, |ψ−iA|ψ−iB. photon states is shown in Fig.2. In this scheme, three
|Φ+iP ⊗|Ψ+iS, |Φ−iP ⊗|Ψ−iS, |φ+iA|ψ+iB, |φ−iA|ψ−iB, coherent states are employed to distinguish the eight
|Ψ+iP ⊗|Φ+iS, |Ψ−iP ⊗|Φ−iS. |ψ+iA|φ+iB, |ψ−iA|φ−iB. spatial-modeGHZstates. Thefirsttwoareusedtocheck
|Φ+iP ⊗|Φ−iS, |Φ−iP ⊗|Φ+iS, |φ+iA|φ−iB, |φ−iA|φ+iB, the parity between AB and AC, respectively, i.e., read
out the bit information. The third one is used to con-
|Ψ+iP ⊗|Ψ−iS, |Ψ−iP ⊗|Ψ+iS. |ψ+iA|ψ−iB, |ψ−iA|ψ+iB.
firm the relative phase information “ ”. If the number
|Φ+iP ⊗|Ψ−iS, |Φ−iP ⊗|Ψ+iS, |φ+iA|ψ−iB, |φ−iA|ψ+iB, of π phase shifts is odd, which result±s in a single over-
|Ψ+iP ⊗|Φ−iS, |Ψ−iP ⊗|Φ+iS. |ψ+iA|φ−iB, |ψ−iA|φ+iB. all π phase shift, the relative phase information of the
5
spatial-mode state is “ ”. Otherwise, an even number
− TABLE III: Corresponding relations between the original
of π shifts leads to zero phase shift, which indicates the
stateand thethreephaseshifts of thecoherent statesin Fig.
“+”relativephaseinformationforthespatial-mode. For
2.
example,
Original state |α1i |α2i |α3i
|Ω+000iS|α3i |Φ+000iS 0 0 0
→ 21(|a1b1c1i+|a2b2c1i+|a2b1c2i+|a1b2c2i)|α3i(,19) ||ΦΦ−0+00001iiSS 00 ±0θ π0
|Ω−000iS|α3i |Φ−001iS 0 ±θ π
→ 12(|a2b2c2i+|a2b1c1i+|a1b2c1i+|a1b1c2i)|α3eiπi. ||ΦΦ−0+01100iiSS ±±θθ 00 π0
(20) |Φ+100iS ±θ ±θ 0
|Φ−100iS ±θ ±θ π
The relations between the original state and these
threemeasuredphaseshiftsareshowninTable. III.Then
three BSs are used to manipulate the spatial-mode state
back to its initial status. In the second step, three SPB- The corresponding relations between the original hyper-
SMs are performed. Based on the SPBSM results and entangled GHZ state and the SPBSM results are
theinformationaboutthespatial-modestateinstepone,
the eightpolarizationGHZ statescanbe completely dis-
ϕ ϕ =ϕ ϕ ϕ , (23)
criminatedandconsequentlythe64hyperentangledGHZ P ⊗ S A⊗ B ⊗ C
states can be distinguished. xP xS =δX.(x=a,b,c,X =A,B,C.) (24)
⊕
With these relations, the polarization state can be de-
duced. For example, if the three single-photon Bell
states are ψ+ ψ φ+ and the spatial-mode state
A − B C
is Ω+ ,|weiha|ve i | i
010 S
| i
ϕ =ϕ ϕ ϕ ϕ = , (25)
P A B C S
⊗ ⊗ ⊗ −
δ δ δ =110, (26)
A B C
a =δ a =1, (27)
P A S
⊕
b =δ b =0, (28)
P B S
⊕
c =δ c =0. (29)
P C S
⊕
The answer is Ω−100 P and the original hyperentangled
GHZ state is |Ω| −100iiP ⊗ |Ω+010iS. Our scheme can dis-
tinguishthe 64 hyperentangledGHZ statesdeterministi-
cally.
It is possible to generalize our scheme to analyze N-
FIG. 2: Schematic diagram of our complete HGSA protocol photon hyperentangled GHZ states as shown in Fig.3.
forthree-photonhyperentangledGHZstates. Threecoherent Firstly,N 1paritychecksareperformedonphotonpairs
−
statesareutilizedtodistinguishthespatial-modeGHZstates. AB,AC,... AZ withthehelpofN 1auxiliarycoherent
The circles represent nonlinear interactions and the phases states α (n=1,2,...N 1)to rea−dthe bit information
n
inside the circles are the phase shift of the coherent states. ofthes|patiial-modestates−. AfterX-quadraturemeasure-
SPBSM denotes a single-photon Bell state measurement as ments on these N 1 coherent states, 2N spatial-mode
showninFig.1,whichcandiscriminatefoursingle-photonBell GHZstatescanbe−placedinto2N 1 groups,andthepar-
states with certainty. −
ity information “a b ...z ” can be identified. Then af-
S S S
tertheeffectofN BSsandinteractionsbetweenphotons
We denote ϕ to be the phase information of a state
andacoherentstate α ,therelativephaseinformation
N
| i
“ ” of the states in each group can be further detected.
ϕΩ± ,ϕφ± ,ϕψ± . (21) ±
| i ≡± | i ≡± | i ≡± The polarization GHZ state can be deduced based on
We also define the bit information of single-photon Bell the spatial-mode state information and N SPBSMs fol-
states as lowing a similar procedure as described in the preceding
three-photon HGSA scheme. Thus the 4N hyperentan-
δψ± 1,δφ± 0. (22) gled GHZ states can be completely distinguished.
| i ≡ | i ≡
6
! ! " " " also promote the feasibility of the nonlinearity. In 2010,
! NN! " ! " $ !" $ Wittmannet al showedthedisplacement-controlledpho-
homodyne
tonnumberresolvingdetectorsurpassedthestandardho-
modyne detector [59]. It is fortunate that our complete
$#! ! "
% HBSA scheme only requires a small phase shift, and the
a scheme will succeed as long as the small phase shift can

!" # be distinguished from zero. For a weak cross-Kerr non-
a
linearity, a sufficiently large amplitude of the coherent
b state that satisfies αθ2 >>1 canmake it possible to dis-
!" #
b tinguish a small phase shift in the coherent state from

! " 0 phase shift. In Ref.[23], the estimated value of α2 is
1.3 104 provided that a cryogenic NV-diamond system
z can×generate a phase shift of more than 0.1 rad per sig-
!" #
z nalphoton. In2009,thefirstexperimentalobservationof
optical-fiber Kerr nonlinearity at the single-photon level
FIG. 3: Schematic diagram of our complete HGSA protocol was demonstrated [60]. Most recently, an implementa-
for N-photonhyperentangled GHZstates. N coherentstates tion of strong optical nonlinearity using electromagneti-
are used to discriminate the spatial-mode states without de- callyinducedtransparencywaspresentedandanonlinear
stroyingtheentanglement. ThenN SPBSMsareusedtomea-
phase shift was measured [61]. All these studies indicate
sure the polarization state with the help of the information
that our HBSA scheme is feasible with current technol-
about thespatial-mode state.
ogy. On the other hand, our complete HGSA protocol
requires nonlinearities that can generate a π phase shift.
Althoughitisnoteasytorealizewithcurrenttechnology,
IV. DISCUSSION AND SUMMARY it is necessaryin our scheme for reading the phase infor-
mation of the spatial-mode state without destroying its
entanglement. In this paper, we have demonstrated the
Thispaperpresentstwocompletehyperentangledstate
principle by using the cross-Kerr nonlinearity as an ex-
analysis schemes and gives a general protocol for distin-
ample. There are many other kinds of interaction which
guishing N-photon dual hyperentanglement in polariza-
can also provide feasible ways to realize the function we
tion and spatial-mode states. Using our schemes, en-
need [62–66].
tangled states in both DOFs can be unambiguously dis-
Hyperentanglement analysis plays an important role
criminated. Inourprotocols,the spatial-modestatesare
inquantuminformationprocessingbasedonhyperentan-
analyzed by QNDs constructed using cross-Kerr nonlin-
gled states. Typical applications are hyperentanglement
earities, which is a challenging task with current tech-
swapping, hyperdense coding and teleportation via hy-
nology. However, although the natural cross-Kerr non-
perentangled channels [46, 48, 50, 51]. Moreover it is
linearities are weak and the Kerr phase shift is small at
also useful in establishing quantum repeaters and high-
the single-photon level, recent research shows promising
capacityquantumcommunication. Ourschemessimplify
progress towards practical use of the effect in the near
future. Themagnitudeofθ 10 18 ofthenaturalcross- the analysisprocessand reduce the resources,whichwill
−
Kerrnonlinearitiescanbeim≈provedtomagnitude 10 2 make these applications more feasible and economical.
−
∼
by electromagnetically induced transparencies and other In our protocols, the spatial-mode state is discrimi-
means[23]. In2003,Hofmannet al. showedthataphase nated first followed by the polarization state. In theory,
shiftofπcanbeachievedwithasingletwo-levelatomina the two DOFs are equivalent and the order can be re-
one-sidedcavity[54]. In2011,Feizpouretal showedthat versed. However, distinguishing polarization states first
an observable value amplified from a single-photon-level will consume more resources due to the uncertainty of
cross-Kerrphase shift by using weak-value amplification spatial modes. It is interesting to compare our schemes
ispossible[55]. Later,adevicethatcanamplifythenon- with the previously proposed complete HBSA [46] and
linearityeffectwasalsoproposedtoconstructatwo-qubit completeHGSA[47]schemes. Inboththesetwoschemes,
paritygatewithtinycross-Kerrnonlinearity[56]. In2013, the bit and phase information of the spatial-mode state
agiantcross-Kerreffectinducedbyanartificialatomwas andthebitinformationofthepolarizationstateareread
reported in which average cross-Kerr phase shifts of up out using QNDs. After the analysis of the spatial-mode
to 20 degrees per photon with coherent microwavefields state,thecoherenceofthespatial-modestateisnolonger
atthe single-photonlevelweredemonstrated[57]. More- maintained, i.e., the state collapses to a product state.
over,Ref. [58]hasshownthatgiantcrossKerrnonlinear- Therefore, the process should be paused to confirm the
ities of the probe and the signal pulses can be obtained photons’ spatial modes before discrimination of the po-
withnearlyvanishingopticalabsorption,basedonwhich larization states. Otherwise, more QNDs should be pre-
two-qubit quantum polarization phase gates can be con- pared in advance for all possible spatial-mode states be-
structed. In addition, to enhance the nonlinearities, im- fore reading out the bit information of the polarization
provementofthemeasurementonthe coherentstatecan state. However, in our schemes, the spatial-mode en-
7
tanglement is kept intact after its analysis and is sub- and HGSA protocols. In conclusion, our schemes reduce
sequently used to assist in the discrimination of the po- the number of nonlinearities required and save time and
larization state. Compared with previous schemes, our quantum resources, making them simple and feasible in
protocols thus have some distinctive features: (i) There practical applications.
is no requirementto pausethe processsince ourschemes
can be implemented in one-shot. (ii) The polarization
state is discriminated without resorting to any nonlin-
Acknowledgement
earities, which greatly reduces the requirement on non-
linearinteractions. (iii)AlthoughQNDsareusedtoread
out the phase information of spatial-mode states, which XL is supported by the National Natural Science
lookssimilartothepreviousprotocols,onlytwopotential Foundation of China under Grant Nos. 11574038 and
phase shifts need to be distinguished in our schemes as 11547305. SG acknowledges support from the Natural
opposed to four phase shifts in both the previous HBSA Sciences and Engineering Research Council of Canada.
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