Complete hyperentangled Bell state analysis for polarization and time-bin hyperentanglement

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by  Xihan Li

Complete hyperentangled Bell state analysis for polarization and time-bin
hyperentanglement
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 a complete hyperentangled Bell state analysis protocol for two-photon four-qubit
stateswhicharesimultaneouslyentangledinthepolarization andtime-bindegreesoffreedom. The
16 hyperentangled states can be unambiguously distinguished via two steps. In the first step, the
polarization entangled state is distinguished deterministically and nondestructively with the help
6 of the cross-Kerr nonlinearity. Then, in the second step, the time-bin state is analyzed with the
1 aid of the polarization entanglement. Wealso discuss theapplications of ourprotocol for quantum
0 informationprocessing. Comparedwithhyperentanglementinpolarizationandspatial-modedegrees
2
offreedom,thepolarizationandtime-binhyperentangledstatesprovidesavinginquantumresources

n since there is no requirement for two spatial modes for each photon. This is the first complete
a hyperentangledBell stateanalysisschemeforpolarization andtime-binhyperentangledstates, and
J it can providenew avenuesfor high-capacity,long-distance quantumcommunication.

8

] PACS numbers: 03.67.Hk, 03.67.Dd, 03.65. Ud The hyperentangled photons carry information encoded
h
in more than one DOF at the same time and different
p
DOFscanbemanipulatedindependently. Thesedistinct
-
t I. INTRODUCTION features make hyperentanglement useful for many appli-
n
cations in quantum information processing. It can in-
a
u crease the channel capacity and also enhance the secu-
Entanglement is a unique quantum phenomenon and
q rity of quantum communication schemes [16]. For in-
a crucial resource widely used in quantum computation
[ stance,hyperentanglementcanassistinthe conventional
and quantum communication in the past decades. It
Bell state analysis (BSA) by enlarging the Hilbert space
1 plays a key role as the information carrier in quantum
v [17–20]. In 2008, it was exploited to beat the channel
communication schemes such as quantum key distribu-
2 capacity limit in a protocol in which complete BSA of
tion[1,2],quantumdensecoding[3,4],quantumtelepor-
3 polarization states is aided by the orbital angular mo-
tation [5], quantum secure direct communication [9–11]
0 mentum[21]. Hyperentangledstateshavealsobeenused
2 and so on. Among many physical systems proposed for
to accomplish deterministic entanglement purification of
0 quantum communication, the photon is the most com-
polarizationentanglement[22–25],constructhyperparal-
. petitive candidate due to its manipulability and high-
1 lel photonic quantum computing [26, 27] and quantum
0 speed transmission features. Photons have many differ- repeaters [28]. Entanglement concentration and entan-
6 ent degrees of freedom (DOFs) to carry quantum infor-
glement purification protocols for hyperentangled state
1 mation, such as, for instance, the polarization, time-bin,
have also been proposed with the aim of establishing
: spatial-mode, frequency, and orbital angular monmen-
v maximallyhyperentangledchannelsbetweendistantpar-
Xi tum. Entangled states have been prepared in each of ties [29–36].
these DOFs in experiments. Moreover,simultaneous en-
r tanglement in more than one of these DOFs, referred to Inquantumcommunicationschemesthatarebasedon
a
as the hyperentanglement, has also been generated. In entanglement, state analysis is an indispensable step re-
1997, Kwiat et al. proposed the first scheme to generate quired to read out encrypted information. State anal-
an energy-momentum-polarization hyperentangled state ysis is of both theoretical significance and practical im-
[12]. In 2005, Yang et al. generated a two-photon state portance, and thus it has been the focus of much re-
entangled both in polarization and spatial mode DOFs search. Although a set of mutually orthogonal states
to realizethe all-versus-nothingtestoflocalrealism[13]. should in principal be deterministically distinguishable,
In the same year, an experimental demonstration of a this becomesa challengeinphotonic systemssince inter-
photonic hyperentangled system simultaneously entan- actionbetweenphotons isnotaneasytaskusingcurrent
gled in polarization, spatial mode and time-energy was techniques. Itwasprovedthatcomplete BSAis impossi-
first reported [14]. Later, Vallone et al. also realized ble via linear optics alone [37–39]. Hyperentangled Bell
a six-qubit hyperentangled state which was entangled in state analysis (HBSA) in which states of two or more
polarizationandtwolongitudinalmomentumDOFs[15]. DOFs have to be distinguished simultaneously is even
moredifficult, andis thusalsonotpossiblevialinearop-
ticsalone. Ithasbeenshownthat16hyperentangledBell
statescanbeclassedintoonly7groupswithlinearoptics
∗Emailaddress: [email protected] [40, 41]. Therefore, auxiliary states and assistant tools
2
haveto be utilizedto accomplishcompletestate analysis HereAandB denotethetwophotonsandthesubscripts
[42–47]. In2010,Shengetal. proposedthefirstcomplete P and T represent the polarization and time-bin DOFs,
HBSA scheme for polarization and spatial-mode hyper- respectively. Θ is one of four Bell states in the
P AB
| i
entangled states [44]. The two DOFs are distinguished polarization DOF,
withthe helpofthe cross-Kerrnonlinearityintwosteps.
Later, an efficient hyperentangled Greenberger-Horne-
1
Zeilinger(GHZ)stateanalysisschemewaspresented[45].
Moreover, complete HBSA can also be realized with the |Φ±PiAB = √2(|HHi±|VVi)AB, (2)
helpofgiantnonlinearopticsinopticalmicrocavitiesand 1
nitrogen-vacancycentersinresonators[46–48]. Recently, |Ψ±PiAB = √2(|HVi±|VHi)AB. (3)
Liu et al. proposed a complete nondestructive analy-
sis assistedby cross-Kerrnonlinearity oftwo-photonsix-
qubit hyperentangled Bell states in which the photons Here H and V indicate the horizontal and vertical
| i | i
areentangledsimultaneouslyinthe polarizationandtwo polarizations,respectively. The time-bin state Ξ is
S AB
| i
longitudinal momentum DOFs [49]. one of the four Bell states in the time-bin DOF,
Sofar,allhyperentangledstateanalysisprotocolshave
dealtwiththepolarizationandspatial-modehyperentan-
gled state. This kind of hyperentanglement is a promis- 1
ing candidate for quantum communication since both |Φ±TiAB = √2(|SSi±|LLi), (4)
thesetwoDOFscanbemanipulatedwithhighfidelityat 1
present. However, if the spatial-mode DOF is exploited, |Ψ±TiAB = √2(|SLi±|LSi). (5)
each photon requires two paths during the transmission,
which leads to a lot of extra requirements on resources
in long-distance quantum communication. Instead, the Here S and L denote the two different time-bins, the
| i | i
time-binDOFofphotonswithtwodifferentarrivaltimes early (S) and the late (L). Taking the two DOFs to-
asthebasiscansavetheextraresources. Itisalsoasim- gether,thereare16hyperentangledBellstates,whichcan
ple and conventional classical DOF of photons, and can be completely distinguished in the following two steps.
be simply discriminated by the time of arrival. Despite
thedifficultiesinmanipulationoftime-binDOF,wehave
previously proposed a hyperentanglement concentration
scheme for polarizationandtime-bin hyperentanglement
[36]. Here, we propose the first complete HBSA scheme
forpolarizationandtime-binhyperentangledstates. The A. Complete Bell state analysis for the polarization
scheme consists of two steps. In the first step the polar- degree of freedom via cross-Kerr nonlinearity
ization states are distinguished by two quantum nonde-
molition detectors (QNDs) constructed with the cross-
TheprincipleofourproposedpolarizationBSAproto-
Kerr nonlinearity. The parity and phase information of
col is shown in Fig. 1. Two QNDs are used to read the
the polarization state are read without destroying the
parity and phase information of the polarization state.
state. Then the time-bin state is analyzed with the help
Each QND is composed of two polarizing beam splitters
of the polarization entanglement, without resorting to
(PBSs), two nonlinearities and a coherent probe beam
any nonlinearity. The 16 hyperentangled Bell states can
α . The PBS transmits the horizontal state H and
be completely and deterministically discriminated. We | i | i
reflects the vertical one V . Two photons are guided
also give two examples of the application of our HBSA | i
into two input ports labeled A and B, and then interact
scheme for quantum information processing. Our pro-
with the nonlinear medium after the first PBS. The in-
tocol is single-shot and requires less nonlinearities com-
teraction between the photons in the two paths and the
pared with previous HBSA schemes for polarization and
coherent probe beam causes the coherent state to pick
spatial-mode hyperentanglement, and is thus useful for
up a phase shift α αeiNθ when there are N pho-
practical long-distance quantum communication. | i → | i
tonsinthe correspondingspatialmode [50]. Hereθ =χt
where χ is the coupling strength of the nonlinearity and
t is the interaction time, both of which can be set in ad-
II. COMPLETE HYPERENTANGLED BELL vance. Forexample,ifthepolarizationstateis V H
A B
STATE ANALYSIS FOR POLARIZATION AND (H V ),boththe twophotonsgothrough|thie u|ppier
A B
TIME-BIN HYPERENTANGLEMENT | i | i
(lower) path and the coherent state α1 picks up a phase
shift2θ(-2θ). WecanchoosetheX-quadraturemeasure-
The two-photon four-qubit hyperentangled Bell state mentsuchthatitcannotdistinguishphaseshiftsdiffering
can be written as onlyinsign“ ”. Thisfeaturepreservesthecoherenceof
±
photonswithrespecttoeachotheraswellasthephotons
Υ = Θ Ξ . (1) themselves. TheevolutionsofthesefourpolarizationBell
AB P AB T AB
| i | i ⊗| i
3
states in the first QND are |Ψ+Pi states will lead to no phase shift on |α2i while the
other two states put 2θ on the coherent state. Hence,
1 ±
|Φ±Pi|α1i = √2(|HHi±|VVi)|α1i tchriemfionuartepdo.laTrihzeatrieolnatBioenllbsettawteesenarteheunoarmigibniagluostuastley,dthise-
1 twophase shifts ofthe two coherentbeams, andthe new
→ √2(|HHi±|VVi)|α1i=|Φ±Pi|α1i, (6) state are shown in Table. I. The preserved polarization
1 entanglementwill playanimportantroleinthe time-bin
|Ψ±Pi|α1i = √2(|HVi±|VHi)|α1i state analysis.
1
→ √2(|HVi|α1e−2iθi±|VHi|α1e2iθi) TABLEI:Relationsbetweentheoriginalstate,thenewstate
of the polarization DOF, and the two phase shifts of the co-
= |Ψ±Pi|α1e±2iθi. (7) herent states.
Original state |α1i |α2i New state
|aæ +q -q |Φ+Pi 0 0 |Φ+Pi
2 |cæÆc| |Φ−i 0 ±2θ |Ψ+i
P P
homodyne |Ψ+i ±2θ 0 |Φ−i
|a1æ +q -q |ΨP−i ±2θ ±2θ |ΨP−i
P P
A
B B. Complete Bell state analysis for the time-bin
state assisted by the polarization entanglement
HWP PBS Inthesecondstep,thetwophotonsareeachprojected
onto single-photonBellbasis states and the time-bin en-
tangled states are distinguished with the help of the in-
FIG. 1: Schematic diagram of thecomplete polarization Bell
formation about the polarization entanglement.
stateanalyzer. Thepolarizingbeamsplitters(PBSs)transmit
Each photon has two DOFs and can be measured in
horizontalpolarized stateswhile reflectingverticalones. The
half-waveplates(HWPs)implementtheHadamardoperation, an entangled basis composed of the two DOFs,
which transform the phase information of the state into the 1
parityinformation. Thefourcross-Kerrnonlinearinteractions φ± X = (HL VS )X, (10)
| i √2 | i±| i
put phase shift ±θ on the coherent states |α1i and |α2i if
a photon appears in the corresponding spatial modes. The 1
ψ = (HS VL ) . (11)
first QND is used to distinguish |Φ±Pi from |Ψ±Pi while the | ±iX √2 | i±| i X
secondQNDreadstherelativephaseinformation “±”. After
Here X can be either A or B. The single-photon Bell
thehomodyne measurements on thetwo coherent states, the
statescanbecompletelydistinguishedbyasingle-photon
four polarization Bell states can be completely distinguished
without destroying theentanglement or losing thephotons. Bell state analyser (SPBSA), shown in Fig. 2.
Two Pockel cells (PCs) [51] are used to flip the polar-
By measuring the phase shift via the X-quadrature izations of the photons at specific times, i.e., the PCL
measurement, the parity information of the polarization (PCS) is activated only when the L (S) component is
statecanbedistinguished,i.e.,|Φ±Piisdistinguishedfrom pofretswenotP. BTShsenartweousuendbatolanadcejudsitnttehrefetriommee-bteinrssctoamtepsousecdh
|inΨa±Plis.paAtniaolthsteartPusB,Si.ec.h,aonngeesphthoetosntapteerbpaactkht.oTith’es oprhiog-- that the path length difference between the long path L
and the short one S cancels the time difference between
tons and polarization states are preserved after the first
QND.Thentwohalf-waveplates(HWPs)implementthe the two time-bins. The HWPs effect the Hadamard op-
eration, and the PBSs then allow a measurement in the
Hadamard operation
diagonal basis = 1 (H V ). As shown in Fig.
1 |±i √2 | i±| i
H (H + V ), (8) 2,differentsingle-photonBellstateswilltriggerdifferent
| i → √2 | i | i detectorsplacedonthe fouroutput ports. Thusthe four
1 single-photon Bell states can be deterministically distin-
V (H V ). (9)
| i → √2 | i−| i guished.
Using two SPBSAs for each of the two photons, the
Thepolarizationstatechangesasfollows: both Φ+ and time-bin state can be discriminated with the help of the
|Ψ−Pi are invariant while |Φ−Pi ⇀↽ |Ψ+Pi. In othe|rPwiords, undestroyed polarization entanglement. Each of the po-
thetwoHWPstransferthephaseinformationofthestate tential hyperentangledBell states will resultin four pos-
to the parity one. After the second QND, the phase in- sibledetections. Thereare16possiblemeasurementcom-
formation is obtained. That is, the original Φ+ and binations, which can be classed into four groups. Each
| Pi
4
L |f-æ III. APPLICATIONS OF OUR COMPLETE
HBSA SCHEME
PC |f+æ
L
S Hyperentangled states have a lot of applications in
PC |y+æ quantumcommunicationinwhichaHBSAmayrequired
S to read out the information. Here we demonstrate the
L applications of our HBSA for two protocols: quantum
|y-æ teleportation and entanglement swapping. Conventional
quantum teleportation can transmit an unknown quan-
tum state through a pre-established entangled channel
HWP PBS
without transmitting the photon itself. Entanglement
swapping is an important constituent for quantum re-
FIG.2: Schematicdiagramofthesingle-photonBellstatean- peaters in long-distance communication. The channel
alyzer (SPBSA). PCL (PCS) is a Pockel cell which effects a capacity of both protocols can be increased with hyper-
bitflipoperationwhentheL(S)componentispresent. Then
entangledstates. Here wediscuss the applicationsofour
two unbalanced interferometers composed of two PBSs are
HBSA scheme in quantum teleportation and entangle-
used to adjust the time-bin states: the length difference be-
ment swapping using hyperentanglement in polarization
tween the long (L) path and the short S one is set to cancel
and time-bin DOFs.
thetimeintervalbetweentwotime-bins. Thenthetwopaths
intersectataPBSandthephotonismeasuredinthediagonal
polarization basis.
A. Teleportation of a single-photon two-qubit state
Suppose the sender Alice andthe receiverBobshare a
group corresponds to four specific hyperentangled Bell hyperentangled channel in advance
states. The detailed relations are shown in Table. II.
1 1
With the knowledge of the polarization entanglement, Υ = (HH + VV ) (SS + LL )(.12)
AB
| i √2 | i | i ⊗ √2 | i | i
thetime-binstatecanbedeterministicallyidentified. For
example, if the two measurement results are ψ and
− A The unknown state of photon X which Alice wants to
φ+ , the new state after the first step belo|ngsito the
B send to Bob is
| i
last group. If the first step determines that the original
polarization state is |Φ−Pi (the new state is |Ψ+Pi ), one |ϕiX =(α|Hi+β|Vi)⊗(δ|Si+η|Li). (13)
can deduce that the time-bin state is |Φ−Ti. The initial
hyperentangled state is |Φ−Pi⊗|Φ−Ti. noAisylicechcaannneelitohrertaskeendadtvhaenptahgoetoonf dthireecsthlyartehdrohuygphera-
entanglement. If she chooses the latter, she performs
the HBSA on photons X and A. The state of the whole
system can be rewritten in terms of the hyperentangled
TABLEII:Relationsbetweenthenewstatebeforethesecond
step and possible detections. state as
New states Possible detections ϕ Υ
X AB
||ΦΨ+P+Pii⊗⊗||ΦΨ+T+Tii,,||ΨΦ−P−Pii⊗⊗||ΨΦ−T−Tii., ||ψφ++iiAA||ψφ++iiBB,, ||φψ−−iiAA||φψ−−iiBB,. = |14[i|Φ+P⊗iX|Ai(α|Hi+β|Vi)B +|Φ−PiXA(α|Hi−β|Vi)B
||||ΨΦΨΦ+P+P+P+Piiii⊗⊗⊗⊗||||ΨΨΦΦ−T+T−T+Tiiii,,,,||||ΦΨΨΦ−P−P−P−Piiii⊗⊗⊗⊗||||ΨΦΦΨ+T−T−T+Tiiii,,.. ||||ψφψφ++++iiiiAAAA||||ψψφφ−−++iiiiBBBB,,,,||||φψφψ−−−−iiiiAAAA||||ψφφψ+−−+iiiiBBBB,.,. +⊗+|[|Ψ|ΨΦ+P+T+TiiiXXXAAA(((αδδ|||LVSiii+++ηηβ||S|LHii))iBB)B+++||ΨΦ|Ψ−T−T−PiiXXiXAAA((δδ(|α|LS|iVi−−i−ηη||SβLi|iH))BBi]).B]
|Φ+Pi⊗|Ψ−Ti, |Φ−Pi⊗|Ψ+Ti, |φ+iA|ψ−iB, |φ−iA|ψ+iB, (14)
|Ψ+Pi⊗|Φ−Ti, |Ψ−Pi⊗|Φ+Ti. |ψ+iA|φ−iB, |ψ−iA|φ+iB. Alicehas16possiblemeasurementresults,correspond-
ing to which there are 16 potential single-photon two-
qubit states for Bob’s photon. With our HBSA scheme,
From the preceding analysis, the 16 hyperentangled the 16 hyperenatngled states can be completely distin-
Bell states are unambiguously discriminated with our guished, according to which Bob knows the state of
two-stepscheme. Thedistinguishingofpolarizationstate his own photon with certainty. For example, if Alice’s
resorts to the cross-Kerr nonlinearity while the discrim- measurement result is Ψ+ Φ+ , Bob’s state is
| PiXA⊗| TiXA
ination of time-bin state can be accomplished with the (αV +β H ) (δ S +η L ) . Thentheoriginalstate
B B
| i | i ⊗ | i | i
helpofthe preservedpolarizationentanglement,without can be obtained by Bob with proper single-photon uni-
any nonlinear optics. tary operations on both DOFs.
5
Compared with conventional quantum teleportation, IV. DISCUSSION AND SUMMARY
the use of polarization and time-bin hyperentanglement
provides a way for teleporting a single-photon two-qubit In this paper, we have proposed a HBSA protocol for
state which carries more information. In conventional polarization and time-bin hyperentanglement, in which
quantumteleportation,twooffourBellstatescanbedis- the hyperentangledstates aredistinguishedintwosteps.
tinguished by linear optics alone. Thus Bob can get the The polarization state is analyzed with the help of the
desiredstatewith50%probability. Inoursceme,without cross-Kerr nonlinearity. Both the photons and the po-
acompleteHBSA,the16hyperentangledstatescanonly larization entanglement are preserved after the discrim-
be classedinto 7 groups and eachof them contains more ination. Then the time-bin state is distinguished with
thanone state. Thus the uncertaintyofAlice’s measure- thehelpoftheinformationaboutthepolarizationentan-
ment outcomes results in a mixed state in Bob’s hand. glement. No nonlinear interaction is required to analyze
This shows that a complete HBSA is indispensable for the time-bin entanglement. The two photons are each
teleporting a single-photon two-qubit state. measured in the single-photon Bell basis. Based on this,
the16hyperentangledstatescanbediscriminatedunam-
biguously.
B. Entanglement swapping between
The cross-Kerr nonlinearity is used in our scheme in
hyperentangled pairs
thefirststep. ThefeasibilityofourHBSAschememainly
depends on the cross-Kerr nonlinearity. Although it re-
Entanglementswappingenablestwoindependent pho-
mainsachallengewithcurrenttechnology,ithasbeenex-
tons to be entangled with each other without any direct ploited in many quantum information processing proto-
interactions between them. This plays a crucial role in
cols[22,35,44,45,50]. Recentstudiesalsoshowpromis-
quantum repeaters. Conventionalquantumrepeaters es-
ing results for using the effect in the near future [52–58].
tablish entanglement in one DOF between two distant
Moreover,wenotethatinourHBSAscheme,onlyasmall
parties. Sincehyperentanglementsharedbetweentwore-
nonlinearityisrequired,aslongasthe phaseshiftcanbe
motepartiescanimprovethechannelcapacitygreatly,it
distinguished fromthe zero phase shift case. This makes
is useful to establishhyperentanglementbetween remote
itmorepromisingtoimplementwithpresenttechniques.
parties. Suppose two remote parties Alice and Bob each
It is interesting to compare our HBSA with previous
share hyperentanglement with a central node Charlie.
protocolsthatusedifferentDOFs. First,inalltheHBSA
1 1 schemes the two DOFs are distinguished in two steps. It
|ΥiAC1 = √2(|HHi+|VVi)AC1 ⊗ √2(|SSi+|LLi)AC1, is important to point out that our scheme is one-shot
in that no pause is required between two steps. In con-
(15)
trast, the other HBSA schemes need to confirm the spa-
1 1
tial mode of the photons before they move on to the po-
Υ = (HH + VV ) (SS + LL ) .
| iC2B √2 | i | i C2B ⊗ √2 | i | i C2B larization state analysis [44, 46]. Otherwise, setups with
(16) more nonlinearities need to be prepared in advance for
all possible situations. Second, in our scheme the dis-
HereAandB belongtoAliceandBob,respectively,and crimination of the second DOF is realized without any
C1,C2areheldbyCharlie. AfterCharlieperformsHBSA nonlinear optics while in other schemes the analysis of
onC1andC2,AliceandBob’sphotonswillcollapsedinto both DOFs resorts to nonlinearities [44, 46]. These two
a hyperentangled state. advantages make our scheme time-saving and resource-
saving and thus more useful and practical.
Υ Υ
| iAC1 ⊗| iC2B In summary, we have described an efficient scheme
1
= 4[(|Φ+PiC1C2 ⊗|Φ+PiAB +|Φ−PiC1C2 ⊗|Φ−PiAB for complete HBSA of a two-photonsystem hyperentan-
gled in the polarization and time-bin DOFs. The 16 hy-
+|Ψ+PiC1C2 ⊗|Ψ+PiAB +|Ψ−PiC1C2 ⊗|Ψ−PiAB) perentangledstatescanbedistinguishedunambiguously.
+⊗|(Ψ|Φ+T+TiiCC11CC22⊗⊗|Ψ|Φ+T+TiiAABB++||ΨΦ−T−TiiCC11CC22⊗⊗||ΨΦ−T−TiiAABB)(].17) OibziauntriDoHnOBaFSnArdeqtpiumriorete-osbclioenslshisyreptsehoreuenrfictreasntingollenomengetn-odt.idseStaainnlccweeittqhhueapntoitmluamer--
From the expressionwe find the state of AB depends on communicationcomparedwiththe popularspatial-mode
the measurement result of C1C2. For instance, if Char- DOF,webelieveitwillfindmoreapplicationsinquantum
lie’s result is Φ+ Ψ+ , the state shared by communication schemes, thus making our HBSA proto-
AliceandBob|isPΦiC+1C2 ⊗|ΨT+iC1C2. WithCharlie’sinfor- col useful and relevant for future applications.
| PiAB⊗| TiAB
mation,AliceandBobcansharethedesiredhyperentan-
gledstate withorwithout someadditionalsingle-photon
operations. With ourHBSAscheme, hyperentanglement Acknowledgement
canbe establishedbetweendistantparties,whichwillbe
useful for long-distance, high-capacity quantum commu- XL is supported by the National Natural Science
nication. Foundation of China under Grant Nos. 11574038 and
6
11547305. SG acknowledges support from he Natural Sciences and Engineering Research Council of Canada.
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