Furusawa A., van Loock P. Quantum Teleportation and Entanglement: A Hybrid Approach to Optical Quantum Information Processing

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250 6 Quantum Teleportation of Gates

Figure 6.5 A schematic of conditional CNOT

gate proposed by Ralph et al. [274]. B

is a

beam splitter whose reﬂectivity is η

.Thecon-

trol qubit is represented by modes C

and

with the dual-rail fashion, and the target

qubit is represented by modes t

and t

.The

modes v

and v

are ancillary vacuum inputs.

follows [274],

out



2 Oa

COa



out



Oa

COa



out



COa



out



COa

Oa



out



Oa

2 Oa



out



Oa



, (6.2)

where the control qubit is represented by modes C

and C

with the dual-rail

fashion, and the target qubit is represented by modes t

and t

.Themodesv

and

are ancillary vacuum inputs.

The general input state to the gate (the control and target qubits) can be described

in the Schrödinger picture as

jψi

(

αjHHiCβjHViCγjVHiCδjVVi

)

j00i



α Oa

†

C β Oa

†

C γ Oa

†

Cδ Oa

†



j0000ij00i , (6.3)

wheretheorderingintheketsisjn

ijn

i with On

†

,andsoon,andweusej1010iDjHHi,andsoon,whereappropriate.

6.1 Teleporting Qubit Gates 251

Thus, we can derive the output state of the beam splitter network by using

Eqs. (6.2) and (6.3),

jψi

out



α Oa

†

out

†

out

C β Oa

†

out

†

out

C γ Oa

†

out

†

out

Cδ Oa

†

out

†

out



j0000ij00i

[

αjHHiCβjHViCγ jVViCδjVHi

]

j00i

2(α C β)j0100ij10iC

2(α  β)j0000ij11i

C (α C β)j1100ij00iC(α  β)j1000ij01i

2αj0010ij10iC

2βj0001ij10i



2(γ C δ)j0200ij00i(γ  δ)j0100ij01i

2γ j0020ij00iC(γ  δ)j0010ij01i

C (γ C δ)j0011ij00iC(γ  δ)j0001ij01i

2δj0002ij00i

. (6.4)

From this result, we obtain (after renormalization)

jψi

post-select

D αjHHiCβjHViCγjVViCδjVHi , (6.5)

by measuring jn

out

iDj00i, exactly one photon in total for fn

out

, n

out

and the same for fn

out

, n

out

g. These patterns can be postselected in the experi-

ment and they occur with a success probability of 1/9. The state of Eq. (6.5) is the

CNOT output for the input of Eq. (6.3); the target qubit only ﬂips when the control

qubit is V.

O’Brien et al. realized this postselected gate as shown in Figure 6.4c. The two

input polarization qubits are converted to spatial dual-rail qubits using the trick

shown in Figure 6.4b. Then, the dual-rail qubits enter a beam splitter network, as

depicted in Figure 6.4a (Figure 6.5), which is implemented with half wave plates

(HWPs) and polarization beam splitters (PBSs). Especially, a 1/2-beam splitter is

realized through a polarization rotation by 45

(HWP rotation by 22.5

)anda1/3-

beam splitter is realized by means of a polarization rotation by 125

(HWP rotation

by 62.5

), where 

1/3 ' cos 125

Figure 6.6 shows the experimental results for the CNOT gate operation as it was

demonstrated by O’Brien et al. [273]. The results agree well with the ideal opera-

tion of the gate. Note that postselection was a crucial part of this experiment. In

particular, the protocol for this gate implementation (achieving a gate efﬁciency of

1/9) relies upon a ﬁnal step that inclu des destroying the two-qubit output state of

the gate. More precisely, it is absolutely necessary to detect (and hence destroy) the

photon number states of all six modes in Eq. (6.4) in order to accomplish the gate

operation; including postselection in the gate output ports themselves.

In this sense, the current CNOT gate is a destructive gate, conceptually different

from, for example, those gates obtainable in the KLM scheme. There, the output

252 6 Quantum Teleportation of Gates

(a) (b)

Figure 6.6 Experimental results for the CNOT gate operation demonstrated by O’Brien

et al. [273]. (a) Ideal logical basis operation of a CNOT gate. (b) Measured operation of the gate.

ports need not be detected and the two-qubit output state of the gate can be further

exploited in subsequent quantum circuits, potentially leading to a fully scalable

quantum computation.

However, the KLM two-qubit entangling gate is much harder to implement ex-

perimentally; and, again from a conceptual point of view, the KLM gates represent-

ing non-destructive gates are fundamentally limited in their efﬁciencies depending

on the resources employed. Here, the bounds on the nondestructive gates are nat-

urally lower than those on the destructive gates. For instance, the nondeterministic

implementation of a non-destructive CNOT gate according to KLM achieves an ef-

ﬁciency of 1/16, and this value is near 2/27, which is conjectured to be the theoreti-

cal optimum when no feedforward operations are permitted (like in the destructive

gate protocol described above).

6.2

Teleporting Qumo d e Gates

6.2.1

Experiment: Gaussian Qumode Gates

6.2.1.1 Universal Squeezer

In this section, we will describe a universal squeezer as an example for off-line

Gaussian quantum information processing on qumodes, focusing on the experi-

ment demonstrated by Yoshikawa et al. [275] based upon the theoretical proposal

by Filip et al. [276].

The implementation of a direct nonlinear quantum operation is often hampered

by decoherence due to inevitable practical imperfections in physical systems. Be-

cause of the necessity of invoking such unitary transformations in a fault-tolerant

quantum information processor, the future of developing such units was not too

bright. However, new optimism arose from the introduction of the so-called off-line

schemes where a n onlinear transformation is executed on a quantum state through

simple linear interference with some off-line prepared ancillas followed by detec-

6.2 Teleporting Qumode Gates 253

tion and feedforward [84, 170, 242, 249, 268]. The signiﬁcance of that approach is

that the nonlinear transformation need not be performed directly onto the fragile

quantum state, but is accomplished by tailoring the off-line resource states that can

be prepared at anytime.

The ﬁrst simple example of such an off-line scheme is teleportation which

demonstrates the implementation of the most trivial unitary quantum opera-

tion [17, 163, 277], namely, the identity operation: The off-line resource is a bipar-

tite entangled state which is detected jointly with the fragile quantum information

in a Bell measurement and the classical outcomes are fed forward to achieve the

identity (or teleportation) operation. Remarkably, it was found that by manipul at-

ing the off-line entangled state in the teleporter, it is possible to implement any

unitary transformation through teleportation. This was ﬁrst realized for qubits [84]

and subsequently used in the linear optical quantum computer [242], and later

extended to continuous variables (CVs) which beneﬁt from the practical CV Bell

measurement [268].

Such a teleportation-based off-line scheme can, for example, be used for the im-

plementation of a squeezing operation. It was however proposed in [276] that a

EOM

99:1

50:50

PBS

HWP

displacement

classical

channel

pump OPO

squeezed vacuum

input

50:50

output

squeezing

operation

input

squeezed

vacuum

ancilla

squeezed

output

Disp

beam

splitter

entangled

measurement

feedforward

(a)

(b)

Figure 6.7 (a) Schematic of high-ﬁdelity

squeezing [275]. (b) Experimental setup for

high-ﬁdelity squeezing [275]. A variable beam

splitter is realized by a half-wave plate (HWP)

and two polarizing beam splitters (PBS).

EOM: electro-optic modulator, LO: local os-

cillator, OPO: optical parametric oscillator.

254 6 Quantum Teleportation of Gates

much simpler off-line scheme relying only on a single vacuum squeezed ancilla

sufﬁces to implement the squeezing operation (see Figure 6.7a). In essence, this

simple setup allows for the experimentally feasible and noise-resistant squeezing

transformation of optical quantum states, and it can be seen as the CV analog to

the one-qubit teleportation approach in [278].

The single-mode squeezer is ideally described by a single-mode LUBO trans-

formation (see Chapter 2) which maps the input Wigner function W(x, p )on-

to W

(x, p) D W(xe

, p e

r

)wherex and p represent the amplitude and phase

quadrature of the ﬁeld and r is the squeezing factor. Although this simple trans-

formation is standard in any text book on quantum optics, its experimental re-

alization for arbitrary inputs (that is quantum information) remained extremely

challenging. Previously demonstrated squeezing transformations either were suf-

fering from large decoherence (as is the case for ﬁber or cavity implementations),

thus corrupting the fragile quantum information of a quantum state, or were using

an input dependent nondeterministic approach [279]. In contrast to previous im-

plementations, the squeezing transformation explained here is deterministic and

it processes quantum information with very high ﬁdelity.

The scheme is illustrated in Figure 6.7 and goes as follows. The input state under

interrogation is combined with a squeezed vacuum at a beam splitter. A quadrature

to be anti-squeezed is measured using homodyne detection, and after appropriate

rescaling of the outcomes the remaining ﬁeld is displaced accordingly. Mathemat-

ically, the transformation can be easily derived in the Heisenberg picture. First, we

consider the input–output relations for the beam splitter:

T Ox

1  T Ox

, (6.6)

T Op

1  T Op

, (6.7)

T Ox



1  T Ox

, (6.8)

T Op



1  T Op

, (6.9)

where Ox and Op represent the quadratures to be squeezed and anti-squeezed, the

indices “i” and “a” refer to the input and ancillary mode, respectively, and T is the

transmittance of the beam splitter. The quadratures of the ancilla are written as

( Ox

, Op

) D ( Ox

(0)

r

, Op

(0)

)wherer

is the squeezing parameter and Ox

(0)

and Op

(0)

represent vacuum ﬂuctuations. In the reﬂected part, the quadrature Op

is measured

using homodyne detection. The measurement outcomes are subsequently rescaled

by a factor denoted by g and ﬁnally used to displace the remaining part of the

system which is equivalent to the transformation Ox

!Ox

DOx

and Op

!Op

C g Op

. By choosing g D

(1  T)/T, we arrive at the following input–output

relations

T Ox

1  T Ox

(0)

r

, (6.10)

. (6.11)

6.2 Teleporting Qumode Gates 255

In the limit of the inﬁnitely squeezed ancilla corresponding to r

!1,the

transformation coincides with perfect unitary squeezing operation with the actual

squeezing parameter r Dln

T which is directly controlled by the transmit-

tance of the beam splitter. Furthermore, the quadrature being squeezed can also

be easily controlled through adjustment of the relative phase between the signal

and the squeezed ancilla and correspondingly the measured quadrature in the

feedforward loop [276]. Therefore, full control of the squeezing process is accessed

through simple operations on linear passive devices. Let us note that by changing

some of the settings of the setup (such as the local oscillator phase, the feedfor-

ward gain and the ancilla state), the setup can function as a non-unitary noiseless

ampliﬁer [280], a non-unitary quantum nondemolition measurement device [281]

or as a squeezed state puriﬁer [282].

In a realistic situation, the ancilla state is not inﬁnitely squeezed and some extra

quantum noise will inevitably be added to the squeezed quadrature as indicated

by the second term in Eq. (6.10). Note that the noise suppression performance

never goes further than that of the ancilla. In contrast, the imperfections of the

ancilla state do not degrade the quality of the transformation of th e anti-squeezed

quadrature as well as the mean values: The excess noise of the ancilla is not coupled

intothemodenordoesitdisturbthemeanvaluetransformation.

The operation described above is universal and thus squeezes all input states.

In the following experimental investigation, however, the squeezing of particular

states is considered, namely, coherent states. To ensure that the coherent states

are truly pure, they are deﬁned as a sideband at a radio-frequency relative to the

carrier of a laser beam. This beam as well as other auxiliary beams are delivered by

a Ti:sapphire laser operating at 860 nm. The experiment is divided in three parts;

preparation, processing and veriﬁcation which will now be discussed.

Preparation: In the preparation stage, the input coherent state and the squeezed

ancilla state are generated. The coherent state is prepared by traversing a part of

the laser beam through an electro-optic modulator operating at 1 MHz and set to

modulate the amplitude and phase simultaneously. As a result, a true coherent

state is generated at a 1 MHz sideband and the bandwidth is assumed to be 30 kHz.

The power of the optical carrier is about 3 µW, whereas the power of the sideband is

about 15 dB above the corresponding shot noise level. The ancillary squeezed state

is produced in an optical parametric oscillator (OPO). It is a 500 mm long bow-

tie shaped cavity consisting of two plane mirrors and two mirrors with a 50 mm

radius of curvature. The nonlinear crystal is a 10 mm periodically-poled KTiOPO

(PPKTP) crystal (see [283] for details). The OPO is pumped with light at 430 nm,

stemming from a second harmonic generator with the same conﬁguration as the

OPO cavity but with a KNbO

crystal. The second harmonic generator is pumped

with the light from the Ti:sapphire laser. To monitor and lock the squeezing phase,

we inject a weak coherent beam to the OPO. The output from the OPO and the

coherent state are then directed to the processing part. They have 97 and 143 kHz

modulation sidebands for phase locking.

Processing: At this stage the actual squeezing transformation is implemented.

First, the two states from the preparation stage merge at a variable beam splitter

256 6 Quantum Teleportation of Gates

composed of a half wave plate (HWP) sandwiched between two polarizing beam

splitters (PBS). The beam splitting ratio is thus easily controlled via a wave plate

rotation. One output of the beam splitter is directed to a homodyne detector which

measures the p quadrature. The visibility between the output and a local oscilla-

tor is 96% and the quantum efﬁciency of the detectors is more than 99%. The

measurement outcomes are ampliﬁed electrically in a low-noise ampliﬁer and sub-

sequently used to drive a phase modulator which displaces an auxiliary beam in

phase space. Finally, the displacement of the signal is achieved by combining it

with the displaced auxiliary ﬁeld using a highly asymmetric beam splitter ( 99/1).

Veriﬁcation: In the ﬁnal stage of the experiment, the protocol is veriﬁed by mea-

suring the input states as well as the squeezed output states. The states are fully

characterized by balanced homodyne detection. The visibility between the squeezed

output beam and a local oscillator is 96% and the total propagation efﬁciency is

96%. The electronic noise is always 19 dB smaller than the optical noise. After de-

tection, the photocurrents are used to reconstruct the quantum states: The 1 MHz

component of the measured output signal is extracted by means of a lock-in detec-

tion scheme. The signal is mixed with a 1 MHz sine-wave signal from a function

generator, low pass ﬁltered (30 kHz) and ﬁnally digitized and fed into a computer

with the sampling rate of 300 kHz.

Figure 6.8 shows the raw data of the time resolved measurements of the input

states and the output states. The time series for the input coherent states (Fig-

ure 6.8a) and the vacuum squeezed states (Figure 6.8e) are measured by adjust-

ing the beam splitter t ransmittance to unity and zero, respectively (and blocking

the displacement beam). Then, the squeezing transformation is activated and the

time series for three different transmittances, namely, 0.75, 0.50, and 0.25, are

measured, the results of which are shown in Figure 6.8b–d respectively. It is ev-

ident from the plots that the input coherent states become more and more de-

formed as the transmittance decreases (and thus the squeezing degree increases).

In Figure 6.8f, the reconstructed Wigner function of the transformed states with

T D 0.25 is presented.

As indicated by the reconstructed Wigner function, the involved states are Gaus-

sian. With this a priori information, the states are completely characterized by the

ﬁrst two moments. Due to the symmetry of the states (squeezed in x and anti-

squeezed in p), it sufﬁces to evaluate the mean values and variances of x and p.

Results of such evaluations are shown in Figures 6.9 and 6.10.

In Figure 6.9, the phase space diagrams of the input coherent states as well as

the output states are shown by ellipses which correspond to the cross sections of

their respective Wigner functions. When a coherent state is unitarily squeezed, the

amplitude is transformed along a hyperbolic curve, as shown by the dotted line. The

four ellipses correspond to (from the right) the input coherent states, the squeezed

outputs with T D 0.75, T D 0.50, T D 0.25, respectively, and their centers, marked

by dots, represent the measured averages. The circles represent the data obtained

without the feedforward. The lengths of the major and minor axes of the ellipses

are the measured standard deviations of x and p.Obviously,themeanvaluesare

transformed almost ideally.

6.2 Teleporting Qumode Gates 257

-6

-5

-4

-3

-2

-1

0 0.5 1 1.5 2

quadrature data

phase ( ×

rad )

input coherent state

-6

-5

-4

-3

-2

-1

0 0.5 1 1.5 2

quadrature data

phase ( ×

rad )

squeezed with T=0.75

-6

-5

-4

-3

-2

-1

0 0.5 1 1.5 2

quadrature data

phase ( ×

rad )

squeezed with T=0.50

-6

-5

-4

-3

-2

-1

0 0.5 1 1.5 2

quadrature data

phase ( ×

rad )

squeezed with T=0.25

-6

-5

-4

-3

-2

-1

0 0.5 1 1.5 2

quadrature data

phase ( ×

rad )

ancilla squeezed vacuum

Wigner function for T=0.25

-4

-2

-4

-2

-0.1

0.1

0.2

0.3

0.4

0.5

(a)

(d)(c)

(f)

(e)

(b)

Figure 6.8 Results of the homodyne measurements [275]. (a)–(e) are the raw quadrature data

as a function of the phase of the local oscillator and (f ) is the reconstructed Wigner function

(using inverse radon transformation [113]) for one realization of the experiment.

In Figure 6.10, the noise powers of the squeezed and anti-squeezed quadratures

are plotted as a function of the transmittance. The three curves represent theo-

retical predictions for the noise power of the anti-squeezed quadrature (curve i),

258 6 Quantum Teleportation of Gates

0.5

1.5

2.5

3.5

0 0.5 1 1.5 2 2.5 3 3.5

Figure 6.9 Phase space representation of the

experimental results [275]. The phase space

is normalized so that the standard deviation

of the vacuum ﬂuctuation is 1/2 („D1/2).

Note that the measured results are directly

plotted without accounting for detection and

propagation losses.

-6

-4

-2

0 0.25 0.5 0.75 1

noise power (dB)

(i)

(ii)

(iii)

Figure 6.10 The noise powers of the squeez-

er outputs relative to the shot noise lim-

it [275]. We measure 0.7, 1.6 and 2.5 dB for

the squeezed quadrature and 1.3, 3.0 and

5.8 dB for the anti-squeezed quadrature. The

two quadrature variances of the input coher-

ent states are equal to those of vacuum within

˙0.1 dB.

6.2 Teleporting Qumode Gates 259

the squeezed quadrature with the ancilla 5.1 dB squeezed (curve ii) and inﬁnitely

squeezed (curve iii). Note again that the anti-squeezed noise does n ot depend on

the ancilla. Experimental data taken with and without the feedforward in place are

also shown in Figure 6.10: The noise powers of x(p ) with feedforward are indicated

by dots (ﬁlled diamonds), and without feedforward by circles (open diamonds). We

see that the anti-squeezed noise of the ancilla is canceled and the transformation

in p becomes almost ideal after the feedforward. The noise powers of the squeezed

quadrature, however, deviate from the ideal operation due to the ﬁnite squeezing

in the ancilla states. Furthermore we observe a small degradation of the noise sup-

pression due to some imperfections of the feedforward, such as phase ﬂuctuation.

Now, the ﬁdelities [68] of these transformations are calculated. For the case of

Gaussian states, the ﬁdelity between the ideal squeezed state, jψ

i,andtheactual

obtained mixed state, O

out

, is given by (in the unit of „D1/2)

F Dhψ

jO

out

jψ



out

C V



out

C V



 exp



(

out

ihx

)



out

C V





(

out

ihp

)



out

C V



, (6.12)

where the subscripts “id” and “out” denote the ideal squeezing and the experimen-

tal output, respectively, and V denotes the variance. Actually, due to small propaga-

tion and detection losses in the experiment, the ﬁdelity ultimately depends on the

input state. We therefore quantify the individual single shot ﬁdelities for the inputs

considered in the experiment, though the average ﬁdelity will be found by inte-

grating the ﬁdelity in Eq. (6.12) over all possible input states. From the measured

means and variances, the ﬁdelities between the ideally squeezed states of the in-

ferred inputs (accounting for losses) and the directly measured squeezed states are

computed, and 94%˙1% for T D 0.75 (1.2 dB squeezing), 89%˙1% for T D 0.50

(3.0 dB squeezing), and 78% ˙2% for T D 0.25 (6.0 dB squeezing) are found. Note

that the ﬁdelity between the measured input states and the inferred o nes is found

to be 97% ˙ 1%. For comparison, the theoretically calculated ﬁdelities with vacu-

um ancilla states (which correspond to the classical limits) are 93, 82, 63%, for the

transformations corresponding to 1.2, 3.0, 6.0 dB squeezing, respectively.

6.2.1.2 Quantum Non-demolition (QND) Sum Gate

In this section, we explain the demonstration of quantum non-demolition (QND)

sum gate by Yoshikawa et al. [284] by using two universal squeezers explained in

the previous section. Here, the proposal was made by Filip et al. [276] in the same

paper as the universal squeezer.

The analogue of a two-qubit C-NOT gate when continuous quantum variables

are considered is the so-called sum gate. It represents the canonical version of a

two-mode entangling gate for universal quantum computation in the regime of

continuous variables [90] (see Section 1.8). When applied to two optical, bosonic