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

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310 8 Hybrid Quantum Information Processing

8.3

Hybrid Entanglement

In the preceding section, we encountered the example of an entangled state be-

tween a qubit and a qumode. This state, though deﬁned in a combined qubit–

qumode, hence an inﬁnite-dimensional Hilbert space, can be formally written in

a two-qubit Hilbert space as we shall explain in this section. One consequence of

this is that the entanglement of this state can be conveniently quantiﬁed.

Consider the foll o wing bipartite state,

(

jψ

ij0iCjψ

ij1i

)

2, (8.9)

with an orthogonal qubit basis fj0i, j1ig and a pair of linearly independent qumode

states jψ

i and jψ

i. A speciﬁc example of such a state and a possible way to build

it was presented in Eq. (8.7) and the preceding discussion.

Clearly, the state in Eq. (8.9) becomes a maximally entangled, effective two-qubit

Bell state when hψ

jψ

i!0. For 0 < jhψ

jψ

ij < 1, the state is nonmaximally

entangled, but still can be expressed effectively as a two-qubit state. This can be

seen by writing the two pure, non-orthogonal qumode states in an orthogonal, two-

dimensional basis, fjui, jvig,

jψ

iDµjuiCνjvi ,

jψ

iD(µjuiνjvi)e

iφ

, (8.10)

where ν D

1  µ

with µ D [1 Ce

iφ

hψ

jψ

1/2

2. Then, using this orthogo-

nal basis,

juiD



jψ

iCe

iφ

jψ



/(2µ),

jviD



jψ

ie

iφ

jψ



/(2ν) , (8.11)

the hybrid entangled state of Eq. (8.9) becomes

µjuij0iC

1  µ

jvij1i , (8.12)

a nonmaximally entangled two-qubit state with Schmidt coefﬁcients µ and

1  µ

, where 1 ebit is obtained only for µ ! 1/

2andhψ

jψ

i!0. Quan-

tifying the entanglement is straightforward, as the entropy of the reduced density

matrix is a function of the Schmidt coefﬁcients (see Section 1.5).

It is interesting to compare the state of Eq. (8.9) with a bipartite qumode–qumode

entangled state of the form

(

jψ

ijψ

i˙jψ

ijψ

)

, (8.13)

assuming the overlap hψ

jψ

i is real [325]. First of all, in this case, a normalization

constant N

is needed, depending on hψ

jψ

i. Secondly, and quite remarkably,

such a state may always represent a maximally entangled two-qubit state (in the

8.3 Hybrid Entanglement 311

subspaces spanned by jψ

i and jψ

i), independent of hψ

jψ

i,butdependingon

the relative phase [325–327], that is, the sign in Eq. (8.13).

The prime example for such qumode–qumode entangled states are two of the

so-called quasi-Bell states,

jΨ

i

(

jαijαi˙jαij  αi

)

, (8.14)

with N

 2 ˙ 2e

4jαj

.ThestatejΨ



i is identical to the two-qubit Bell state

(juijviCjvijui)/

2when2µ D

2 C 2e

2jαj

,2ν D

2  2e

2jαj

,andN



8µ

, which i s maximally entangled with exactly 1 ebit of entanglement for any

α ¤ 0. In contrast, the state jΨ

i only equals the one-ebit Bell state (juijuiC

jvijvi)/

2 in the limit of orthogonal coherent states, hαjαi!0forα !1.

The amount of entanglement in the qubit–qumode and qumode–qumode states

of Eqs. (8.9) and (8.13), respectively, is bounded above by one ebit, corresponding

to a maximally entangled two-qubit Bell state. This is different from a “genuine”

CV qumode–qumode entangled state such as a Gaussian two-mode squeezed state

which contains an arbitrary amount of entanglement for sufﬁciently high levels of

squeezing.

Let us ﬁnally consider entanglement witnesses (see Section 1.5) in the hybrid

context. Qumode-qumode entangled states like those in Eq. (8.13) may be identi-

ﬁed through the partial transposition criteria (Section 1.5) adapted to the case of

arbitrary CV states [328, 329]. As a result, all known CV inseparability criteria, in-

cluding those especially intended for Gaussian states and expressed in terms of

second moments (see Section 3.2.1), can be derived from a hierarchy of condi-

tions for all moments of the mode operators Oa and Oa

†

. Moreover, for non-Gaussian

entangled states for which the second-moment criteria may fail to detect entangle-

ment, the higher-moment conditions would work. The concept for these criteria is

as follows.

It is known that for any positive operator

P  0, we can write

P D

†

f such

that Tr(

†

f ) is non-negative for any operator

f and any physical state .Then,

we may choose the bipartite decomposition

f D

,forwhich

0  Tr





†



ij,kl







†



ij,kl



[M()]

ij,kl

, (8.15)

for any coefﬁcients c

.Hence,thematrixM () is positive-semideﬁnite for any

physical state .Now,anyseparablestate remains a physical state after partial

transposition of either subsystem such that M(

)andM(

) remain positive-

semideﬁnite matrices, where T

and T

denote partial transposition for subsys-

3) Similarly, for the other two (quasi-)Bell states, we then have jαijαijαijαi/juijvijvijui

for any value of α,butjαij  αiCjαijαi/juijuijvijvi only when α !1.

312 8 Hybrid Quantum Information Processing

tem A and B, respectively. Then, negativity of M(

)orM(

), and hence nega-

tivity of any subdeterminant of M(

)orM(

) is a sufﬁcient criterion for entan-

glement from which sets of inequalities can be derived with convenient choices for

the local operators

and

[328].

One such choice for a qumode–qumode state would be each qumode’s position

and momentum operators, eventually reproducing Simon’s criteria in terms of

second-moment correlation matrices. Another choice, adapted to a qubit–qumode

state of the form in Eq. (8.9), is given by f

gDfjφih0j, jφih1jg and f

f1, Ox, Opg, with some generic qubit state jφi [330]. The resulting expectation val-

ue matrix M() then serves again, using partial transposition, as a tool to detect

entanglement, this time for hybrid qubit–qumode states. This can be particularly

useful for verifying the presence of effective entanglement in a binary coherent-

state-based quantum key distribution protocol [331, 332] as a necessary security

requirement (see Section 1.7).

The choices for the local operators

and

discussed so far all only lead to

second-moment conditions. These are experimentally most accessible, but may fail

to detect some form of non-Gaussian entanglement. For qumode–qumode states, a

more general choice is the normally ordered form for

f ,

f D

nmkl

†n

†k

, with mode operators Oa and

b for the two qumodes A and B, respectively.

Inserting this into Tr(

†

f )  0orTr(

†

f )  0 yields a hierarchy of sepa-

rability conditions in terms of the moments of the mode operators. For example,

the quasi-Bell state jΨ



i of Eq. (8.14) leads to a subdeterminant of the matrix

of moments with a sufﬁcient order of the moments such as hOa

†

bi,whichbe-

comes α

(coth(2α

))/(sinh

(2α

))  s for real α [328, 329]. This subdeterminant

is negative for any nonzero α, proving the entanglement of the state jΨ



i for any

α ¤ 0.

To conclude this section, let us summarize that it is straightforward to quanti-

fy the entanglement of hybrid qubit–qumode and non-Gaussian qumode–qumode

states, provided these states can be represented in two-dimensional subspaces. Oth-

erwise, in order to detect the inseparability of such states, the partial transposition

criteria expressed in terms of matrices of moments can be used. In a recent experi-

ment, employing CV quantum information encoded into the spatial wavefunction

of single photons, fourth-order-moment entanglement was detected [333].

8.4

Hybrid Quantum Teleportation

Since photon counting events are deﬁned in the time domain, conventional CV

teleportation schemes with a sideband of an optical carrier [203] cannot be used for

the teleportation of states conditionally prepared through photon counting. There-

4) Note that s ! 0forα !˙1and that s is maximally negative, s 0.125, for jαj < 0.4, even

though we know that jΨ



i has constant entanglement of one ebit for any nonzero α.

8.4 Hybrid Quantum Teleportation 313

∆≈

Ordinary teleportation experiment: side band

fre q .

∆Ω

+Ω−Ω

ΩΩ

∆Ω

+Ω−Ω

ΩΩ

Broad band

fre q .

∆Ω

−

fre q .

∆Ω

−

Time resolution

1/bandwidth

cavity bandwidth

(a)

(b)

Figure 8.7 (a) Frequency mode of the Schrödinger kitten. (b) Frequency mode for conventional

teleportation experiments.

fore, in order to realize CV teleportation of such non-Gaussian states, one needs to

generate EPR correlations in the time domain (see Section 3.2).

8.4.1

Experiment: Broadband Qumode Teleportation of a Non-Gaussian Wavepacket

The conditionally prepared, non-Gaussian state of Section 8.2 can be teleported

through a time-domain teleporter using a truly broadband entanglement resource

as introduced in Section 3.2.3.2. Since conventional teleportation experiments were

performed with a frequency sideband of a laser carrier as shown in Section 4.2.1

and Figure 8.7b, this broadband and time-domain teleportation is the next big chal-

lenge.

The “Schrödinger kitten” created by photon subtraction only appears when a

photon is detected at the APD in Figure 8.3. In this sense, the Schrödinger kit-

ten is deﬁned as a wavepacket in the time domain. Moreover, as is shown in Sec-

tion 8.2, the frequency mode should be deﬁned by the OPO resonant band which

is illustrated in Figure 8.7a. To teleport this type of broadband mode, we can use

broadband and time-domain EPR entanglement explained in Section 3.2.3.2 (see

Figure 3.20).

Figure 8.8 shows the experimental setup for broadband and time-domain telepor-

tation. The broadband entangled EPR beams are created with the method explained

in Section 3.2.3.1 and they are shared between Alice and Bob. Alice makes a Bell

measurement for the input kitten and Alice’s EPR beam and sends the result to

Bob. Since Alice’s homodyne detectors for the Bell measurement and Bob’s phase

modulators for the displacement operations have limited bandwidth, it makes a

time delay from the corresponding fraction of Bob’s EPR beam. By using an op-

tical delay line in Bob’s EPR beam path, Bob can compensate the time delay and

can keep the EPR correlation. This is conceptually very different from the neces-

sary phase adjustments in conventional narrow-bandwidth CV teleportation. Bob

makes a displacement operation on his EPR beam with high-reﬂectivity beam split-

ters (T D 1%) and phase modulators (PM), and restores the input kitten at his

314 8 Hybrid Quantum Information Processing

Bob

Creation of a cat

Alice

Output

Alice

Bob

APD

Creation of a kitten

Creation of

EPR beams

Bell measurement

Photon subtraction

Optical

delay line

input

OPO

T=1%

Figure 8.8 Setup for teleportation of a Schrödinger kitten. OPO: optical parametric oscillator for

creation of squeezed vacuum, APD: avalanche photodiode, PD: photodiode, PM: phase modula-

tor.

place. Here, the input kitten i s created with photon subtraction as explained in Sec-

tion 8.2, which is conditioned by a click at APD. Therefore, the output is examined

only at the click with quantum tomography.

Concluding this section, we should emphasize that the setup described above

can be used for teleporting arbitrary wavepackets which can be deﬁned in a broad-

band fashion. Even though the input state to the teleporter may be conditionally

prepared, the whole process of quantum teleportation itself remains determinis-

tic, requiring sufﬁciently high time resolution and synchronization between the

various components (such as Alice’s ﬁnite-bandwidth Bell measurement and the

ﬁnite-bandwidth classical communication between Alice and Bob). Such a syn-

chronization was not n eeded in standard frequency-domain teleportation where

just the phases between Alice’s and Bob’s large wavepackets (each corresponding

to a single frequency mode) had to be adjusted.

Another important example for hybrid, broadband quantum teleportation would

be the teleportation of a single photon, which then can be extended to the telepor-

tation of a polarization qubit. This shall be discussed in the next section.

8.4.2

Experiment: Broadband Qumode Teleportation of a Polarization Qubit

A polarization qubit c

j$i C c

jli can be teleported with two broadband CV tele-

porters which are explained in Section 8.4.1. This protocol was proposed by Ide

et al. [334]. The key is the conversion from a polarization qubit to a dual-rail spatial

qubit with the following relation:

j$i C c

jli ! c

j1i

˝j0i

C c

j0i

˝j1i

, (8.16)

8.5 Hybrid Quantum C omputing 315

CV teleporter

Polarization qubit

PBS

Vertical pol.

Horizontal pol.

output

CV teleporter

input

Figure 8.9 Schematic for teleportation of a polarization qubit with two broadband CV tele-

porters. PBS: polarization beam splitter.

where jni

$,l

(n D 0, 1) denotes a vacuum or single photon state in channel $

or l. This conversion can be realized with a polarization beam splitter (PBS) as

showninFigure8.9.

The dual-rail spatial qubit can be teleported with two broadband CV teleporters

also shown in Figure 8.9. In this process, superpositions of a vacuum and a single

photon are teleported in each CV teleporter. As the ﬁnal step, the teleported dual-

rail spatial qubit is converted back to a polarization qubit with another PBS, to

obtain the output of teleportation.

Ide et al. calculated the teleportation ﬁdelity as a function of the resource squeez-

ing level [334]. From the calculation, it is found that one needs around 10dB of

squeezing for the resource of CV teleportation to exceed the classical limit of qubit

teleportation of 2/3 [335], which is available with current technology [218, 219].

8.5

Hybrid Quantum Computing

In this section, we shall now discuss a few explicit examples for hybrid quantum

processing, the most prominent of which is the GKP scheme [249]. This scheme, in

principle, achieves fault-tolerant quantum computing on logical qubits embedded

into physical qumodes. We will discuss it in detail in Section 8.5.3 and, in particular,

we shall expose its relation with CV cluster computation. Before discussing the

GKP proposal, we will introduce the notion of hybrid Hamiltonians in Section 8.5.1

and some natural qubit-into-qumode encodings in Section 8.5.2.

8.5.1

Hybrid Hamiltonians

One particularly interesting hybrid approach is based upon unitary evolutions, that

is, Hamiltonians which are hybrid. These Hamiltonians contain DV qubit and CV

qumode operator combinations such as, for instance, the controlled rotation in

Eq. (8.5) corresponding to a dispersive light-matter interaction Hamiltonian. More

generally, we may consider a unitary gate of the form,

U D exp



iλ f (σ

, σ

) ˝ g( Ox, Op)



, (8.17)

316 8 Hybrid Quantum Information Processing

acting on the composite system of a qubit and a qumode. In fact, in the context of

combining the DV and CV approaches, it has been pointed out [336] that a suitable

set of elementary Hamiltonians, including the controll ed rotations and additional

uncontrolled displacements,

†

Oa, σ

†

Oa, Ox



, (8.18)

is, i n principle, sufﬁcient for universal quantum computation on qubits.

Even

earlier, Lloyd considered a universal set containing only controlled displace-

ments [337],

f˙σ

Ox, ˙σ

Opg . (8.19)

Typically, in quantum optics, controlled rotations are easier to achieve than con-

trolled displacements. Recall, for instance, the cat engineering scheme of Sec-

tion 8.1.

There are two ways to understand the universality of the above Hamiltonian sets.

One approach is based upon the decomposition of Eq. (1.119). Recall from that de-

composition in Section 1.8 that by applying the Hamiltonians H

and H

for some

short time, we can also approximately implement the Hamiltonian i[H

, H

], pro-

vided the interaction times are sufﬁciently short.

Now , ﬁrst of all, it can be shown

that by using Eq. (1.119) and the elementary Hamiltonians of Eq. (8.19), one can

generate any single-qubit, any single-qumode as well as any qubit–qumode uni-

tary [337] through commutation . The extra two-qubit and two-qumode entangling

gates in order to complete the universal sets for DV and CV universality, respec-

tively, are then achieved through the following commutators,

i

(1)

Ox, σ

(2)

D σ

(1)

(2)

/2 , (8.20)

i

[

, σ

]

D 2σ

, (8.21)

respectively. Here, the superscripts and subscripts denote operators acting upon

one of the two qubits or qumodes. In other words, the C

-gates of the sets in

Eqs. (1.121) and (1.127) can be enacted approximately by applying some of the el-

ementary Hamiltonians in Eq. (8.19). However, there is a crucial difference be-

tween the above two commutators. The commutator in Eq. (8.20) commutes with

the elementary Hamiltonians from which it is built, whereas the commutator in

Eq. (8.21) does not. As a consequence, in the l atter case, the decomposition for-

mula in Eq. (1.119) is indeed only an approximation that requires inﬁnitesimally

small interaction periods. However, since i[σ

(1)

Ox, σ

(2)

Op]commuteswithσ

(1)

Ox and

(2)

Op, and al l higher-order commutators vanish as well, the two-qubit C

-gate ac-

cording to Eq. (1.119) with Eq. (8.20) no longer depends on small interaction times.

5) Note that compared to the discussion on universal sets for DV and CV quantum computation in

Section 1.8, we are now writing a universal set in terms of elementary Hamiltonians instead of

elementary gates.

6) So this is the same asymptotic, approximate model for universal quantum computation as it was

used for discrete [85] and continuous [86] variables on their own, see Section 1.8.

8.5 Hybrid Quantum C omputing 317

Qubit 1

Qubit 2

Qumode

Figure 8.10 A quantum optical illustration of the qubus principle. A CV probe pulse (qumode)

subsequently interacts with two matter qubits. After a sequence of interactions, an entangling

gate between the two qubits can be mediated through the qubus.

Instead, we obtain the exact formula

iσ

(2)

Opt

iσ

(1)

Oxt

iσ

(2)

Opt

iσ

(1)

Oxt

D e

iσ

(1)

(2)

. (8.22)

This observation leads us to an alternative way of understanding how the hybrid

gates of Eq. (8.19) can be used to achieve universal quantum computation on

qubits. In particular, a two-qubit entangling gate of sufﬁcient strength (i.e., t



2π) is then possible without direct interaction between the two qubits; a single

qumode subsequently interacting with each qubit would rather mediate the qubit–

qubit coupling – as a kind of quantum bus (so-called qubus [338, 339], see Fig-

ure 8.10). The sudden exactness of the gate sequence can be explained by interpret-

ing it as a controlled geometric phase gate [336, 340].

Therefore, even though the original hybrid scheme of Lloyd [337] achieves uni-

versality using a ﬁnite gate set, it appears unrealistic to switch between the ele-

mentary Hamiltonians over an arbitrarily short time. Accomplishing the univer-

sal gates and hence the Hamiltonian simulation exactly over a ﬁnite number of

steps, as described by Eq. (8.22), is thus an essential extension of these hybrid

approaches.

Let us ﬁnally note that the two-qumode C

gate, as discussed in Section 1.8, of

course, can be implemented directly using beam splitters and squeezers (through

Bloch–Messiah reduction, see Chapter 2), independent of a supposed asymptotic

scheme based on Eq. (1.119) with Eq. (8.21). Such a simple realization, however, is

not available for qubits.

8.5.2

Encoding Qubits into Qumodes

There are various ways to encode a photonic qubit into optical modes such as po-

larization or spatial modes, as we discussed in Chapter 2. In particular, the photon

occupation number in a single-rail, single-mode Fock state may serve as a qubit or

a more general DV basis. In dual-rail or, more generally, multiple-rail encoding, a

single photon encoded into multi-mode states can even be universally processed

through linear optical elements; though, in an unscalable fashion, unless compli-

cated ancilla states and feedforward are employed [242].

Another natural way to encode a logical DV state into a physical, optical multi-

mode state would be based upon at least two qumodes and a constant number of

318 8 Hybrid Quantum Information Processing

photons distributed over the physical qumodes such that, for example, a logical,

2 J C 1-dimensional spin-J state, jJ, m

i, m

DJ, J C 1,..., J  1, J,canbe

represented by two physical qumodes in the two-mode Fock state,

jJ  (n

C n

)/2, m

 (n

 n

)/2i , (8.23)

where n

and n

denote the photon numbers of the two modes.

In fact, for n

D 1, we obtain a dual-rail encoded spin-1/2 qubit: fjJ D 1/2, m

D1/2i, jJ D

1/2, m

DC1/2ig  fjn

D 0, n

D 1i, jn

D 1, n

D 0ig. Similarly, a spin-1

qutrit, fjJ D 1, m

D1i, jJ D 1, m

D 0i, jJ D 1, m

D 1ig corresponds to

fj0, 2i, j1, 1i, j2, 0ig in the Fock basis.

These natural encodings may even automatically provide some resilience against

certain errors such as photon losses. For i nstance, the dual-rail encoding directly

serves as an error detection code [5, 242]. However, for the purpose of fault-tolerant,

universal quantum information processing, other encodings could be preferable,

though they might be much harder to realize. One example is the fault-tolerant

quantum computation proposal based upon cat states such as the qubit-type states

in Eq. (8.1) [308]. Although some universal DV gates such as the Hadamard gate

(possibly using auxiliary hybrid entangled states of the form in Eq. (8.14) for a

teleportation-based realization) are hard to implement for this encoding, the effect

of photon losses on these cat-type states corresponds to random phase ﬂips in the

coherent-state basis such that repetition codes known from DV qubit quantum

error correction can be directly applied [341]. Nonetheless, these codes would still

require Hadamard gates for encoding and decoding.

8.5.3

GKP

The proposal by Gottesman, Kitaev, and Preskill (GKP) [249] may be referred to as

a hybrid scheme for the following reason. It achieves universal quantum comput-

ing with logical qubits which are embedded in physical qumodes. This qubit-into-

qumode encoding, however, is conceptually different from those that we have con-

sidered so far. It is not naturally given in a subspace of the optical Fock space, but

would require a rather complicated, highly non-G aussian encoding step. Nonethe-

less, physical operations on the qumodes eventually correspond to universal, log-

ical gates on the encoded qubits. Moreover, this can be, in principle, done in a

fault-tolerant fashion, as the GKP scheme includes, at the same time, a quantum

error correction encoding of the qubit into the qumode.

7) The choice of constant total number

COn



and varying number

differences On

On



corresponds to

a speciﬁc basis in the so-called Schwinger

representation. For example, in order to

faithfully represent the SU(2) algebra by

the Lie algebras of two inﬁnite-dimensional

oscillators, i.e., two qumodes Oa

and Oa

one may replace the usual Pauli matrices

 1, σ

 σ

, σ

 σ

,andσ

 σ

by the so-called quantum Stokes operators

D (Oa

†

, Oa

†

)σ

( Oa

, Oa

)

, i D 0, 1, 2, 3,

satisfying the SU(2) Lie algebra commutators

[

] D 2i

,while[

] D 0, for

j D 1, 2, 3.

8.5 Hybrid Quantum C omputing 319

…

Figure 8.11 Encoding a qubit into a qumode according to

GKP [249]. The logical, computational qubit basis states with

kiD(1)

ki, k D 0, 1 are inﬁnite superpositions of delta

peaks in position space. This encoding is particularly suited to

protect the qubit against small, diffusive shift errors as arising

from, for instance, weak amplitude damping of the qumode.

νµ+

Encoding

Syndr+Corr

Decoding

Figure 8.12 The GKP encoding of a logical qubit into a physical qumode to protect the logical

qubit against small shift errors on the physical qumode. Such small shift errors would include

Gaussian errors such as amplitude damping.

The GKP encoding (see Figure 8.11) is primarily intended to protect a logical

qubit against small errors such as small shifts of the physical qumode in phase

space (see Figure 8.12). Direct translations of standard qubit quantum error cor-

rection codes against rarely occurring Pauli errors into the CV regime, as dis-

cussed in Section 1.9 and in Chapter 5, fail to provide any protection against such

small, diffusive, typically Gaussian errors [93]. Provided the error shifts are suf-

ﬁciently small, that is, smaller than ∆/4 with ∆ the distance between the delta

peaks in Figure 8.11, the encoded states remain sufﬁciently intact, and suitable

syndrome measurements enable one, in principle, to detect the errors and correct

the states.

Besides having this special error correction capability, the GKP scheme also in-

cludes universal gate operations. The link between logical and physical gate op-

erations can be easily understood from Figure 8.11.

AlogicalPaulioperatorU 2

X ,

Y ,

Zgis transformed under conjugation by a logical qubit Clifford unitary C in-

to a different logical Pauli operator, C

†

UC D U

2f˙

X , ˙

Y , ˙

Zg.Thus,asthe

logical Pauli gates correspond to WH shifts of the physical qumode, any physical

Gaussian operation (since it preserves such WH gates under conjugation) can only

lead to a logical Clifford operation. Therefore, a physical non-Gaussian operation is

needed in order to achieve logical non-Clifford qubit gates and hence DV qubit universali-

ty. GKP demonstrate two such non-Clifford gates of which one uses a controlled ro-

tation through a dispersive atom-light interaction as in Eq. (8.5), whereas the other

one is based upon a cubic phase gate on the physical qumode, D

(

) D exp(i

8) In Figure 8.11, we do not show the logical

eigenstates of the logical

X Pauli operator .

Similar to the logical eigenstates of the logical

Z Pauli operator, which are superpositions

of delta peaks along the position axis as

shown, the

X eigenstates are delta-peak

superpositions along the momentum axis

in the physical qumode ’s phase space. The

small shift errors can then be shifts along x

and p, and as these WH errors form a basis,

any sufﬁciently small error may be corrected.

This reasoning is similar to that for the

standard CV qumode codes (see Section 1.9

and Chapter 5) against sufﬁciently large,

stochastic shift errors in a single channel.