Optimal input states for quantifying the performance of continuous-variable unidirectional and bidirectional teleportation Hemant K. Mishra1 2Samad Khabbazi Oskouei3and Mark M. Wilde1 2

2025-04-29 0 0 662.59KB 26 页 10玖币

侵权投诉

Optimal input states for quantifying the performance of continuous-variable

unidirectional and bidirectional teleportation

Hemant K. Mishra,1, 2 Samad Khabbazi Oskouei,3and Mark M. Wilde1, 2

1Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,

and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA

2School of Electrical and Computer Engineering,

Cornell University, Ithaca, New York 14850, USA

3Department of Mathematics, Varamin-Pishva Branch,

Islamic Azad University, Varamin, 33817-7489, Iran

(Dated: June 6, 2023)

Continuous-variable (CV) teleportation is a fundamental protocol in quantum information sci-

ence. A number of experiments have been designed to simulate ideal teleportation under realistic

conditions. In this paper, we detail an analytical approach for determining optimal input states for

quantifying the performance of CV unidirectional and bidirectional teleportation. The metric that

we consider for quantifying performance is the energy-constrained channel ﬁdelity between ideal

teleportation and its experimental implementation, and along with this, our focus is on determining

optimal input states for distinguishing the ideal process from the experimental one. We prove that,

under certain energy constraints, the optimal input state in unidirectional, as well as bidirectional,

teleportation is a ﬁnite entangled superposition of twin-Fock states saturating the energy constraint.

Moreover, we also prove that, under the same constraints, the optimal states are unique; that is,

there is no other optimal ﬁnite entangled superposition of twin-Fock states.

I. INTRODUCTION

Quantum teleportation is a foundational protocol in

quantum information science that has no classical ana-

logue [1] (see also [2]). It consists of transmitting an un-

known quantum state from one place to another by using

shared entanglement and local operations and classical

communication (LOCC). Quantum teleportation plays

an important role in quantum technologies such as quan-

tum information processing protocols [3], quantum com-

puting [4,5], and quantum networks [6]. Since the in-

vention of this protocol, various modiﬁcations have been

proposed, such as probabilistic teleportation [7–9], con-

trolled teleportation [10–13], and bidirectional telepor-

tation [14–22]. There has also been signiﬁcant progress

in implementing quantum teleportation in laboratories

around the world in the last three decades [23]. Several

experiments have implemented the teleportation proto-

col for simple quantum systems [24–29], and attempts

are being made to extend them to more complex quan-

tum systems [30–33].

The ﬁrst theoretical proposal for quantum telepor-

tation was for two-level quantum systems, also com-

monly called qubits [1]. Later, continuous-variable (CV)

teleportation was devised as an extension of the orig-

inal protocol to quantum systems described by inﬁnite-

dimensional Hilbert spaces [14,34]. This was followed by

many experimental implementations of CV teleportation,

which include teleportation of collective spins of atomic

ensembles [35,36], polarisation states of photon beams

[33], coherent states [37], etc. In standard CV telepor-

tation, the entangled resource state shared between the

sender and receiver, respectively Alice and Bob, is a two-

mode squeezed vacuum (TMSV) state. The protocol be-

gins with Alice mixing an unknown input state with her

share of the entanglement on a balanced beamsplitter and

then performing homodyne detection of complementary

quadratures. Based on the classical measurement out-

comes received by Alice and subsequently transmitted to

Bob, he then performs displacement operations on his

share of the TMSV state and recovers an approximation

of the original state [34].

An ideal implementation of CV teleportation in princi-

ple allows for perfect transmission of quantum states and

hence simulates an ideal quantum channel. However, an

ideal implementation also demands the unphysical condi-

tions of noiseless homodyne detection and inﬁnite squeez-

ing in the TMSV state, which is not possible in practice

because both noiseless homodyne detection and inﬁnite

squeezing require inﬁnite energy. Any experimental im-

plementation of CV teleportation accounts for an unideal

detection and ﬁnite squeezing, which results in an imper-

fect transmission of quantum states, and hence simulates

a noisy quantum channel [34]. It is therefore important

for experimentalists to employ performance metrics, as

well as quantify the performance, for any experimental

simulation of ideal teleportation.

Several works on characterising the performance of ex-

perimental implementations of the teleportation proto-

col have been conducted for ﬁnite-dimensional quantum

systems in the past few years [38–44], including a more

recent work on bidirectional teleportation, which bench-

marks the performance in terms of normalised diamond

distance and channel inﬁdelity for transmission of arbi-

trary quantum states [45]. There have also been many

theoretical and experimental works on quantifying the

performance of experimental implementations of the CV

teleportation protocol. However, most of them study

the performance by evaluating speciﬁc classes of quan-

tum states, such as coherent states [46–49], pure single-

arXiv:2210.05007v2 [quant-ph] 3 Jun 2023

mode Gaussian states [50,51], squeezed states [52], cat

states [53], etc. All such evaluations are incomplete, in

the sense that they test the performance by transmitting

speciﬁc states rather than arbitrary unknown states. A

true quantiﬁer for CV unidirectional teleportation was

given in [54], which benchmarks the performance of an

experimental implementation in terms of the energy-

constrained channel ﬁdelity between ideal teleportation

and its experimental implementation. We also note here

that [54] is foundational for the present paper.

In this paper, we quantify the performance of any ex-

perimental implementation of CV unidirectional, as well

as bidirectional, teleportation, under certain energy con-

straints. The performance metric that we consider is the

energy-constrained channel ﬁdelity between an ideal tele-

portation and its experimental implementation. We ex-

plicitly ﬁnd optimal input states, i.e., quantum states

whose output ﬁdelity corresponding to the ideal chan-

nel and its experimental approximation is the same as

the energy-constrained channel ﬁdelity between the two

channels. Our method is purely analytical, employing

optimization techniques from multivariable calculus. The

optimal states for unidirectional, as well as bidirectional,

teleportation are ﬁnite entangled superpositions of twin-

Fock states saturating the energy constraint. Further-

more, we prove that the optimal input states are unique;

i.e., there is no other optimal ﬁnite entangled superposi-

tion of twin-Fock states.

Our results on bidirectional teleportation are also re-

lated to one of the most interesting mathematical prob-

lems in quantum information theory: the study of addi-

tive and multiplicative properties of measures associated

with quantum channels [55–58]. Much progress has been

made in addressing these additivity issues [59–63], set-

tling some of the questions posed in [64,65]. The ﬁdelity

of quantum states is well known to be multiplicative for

tensor-product quantum states [63]. This induces an in-

equality for energy-constrained channel ﬁdelity between

two tensor-product channels. As a consequence of our

work, we give examples where the induced inequality is

strict; that is, our results also imply that the energy-

constrained ﬁdelity between the identity channel and an

additive-noise channel is strictly sub-multiplicative.

The rest of our paper is organized as follows. In Sec-

tion II, we review some deﬁnitions. We present a deriva-

tion of the optimal input state for CV unidirectional tele-

portation in Section III, and for CV bidirectional tele-

portation in Section IV. We show in Section Vthat the

energy-constrained ﬁdelity between the ideal swap chan-

nel and the tensor product of two additive-noise chan-

nels is strictly sub-multiplicative. We then discuss pos-

sible extensions and generalizations of the present work

in Section VI. Finally, in Section VII we summarize our

results and outline questions for future work.

The appendices contain necessary calculations for de-

riving the results. In Appendix A, we provide proofs of

some preliminary results required to derive the optimal

input state for CV unidirectional teleportation. Simi-

larly, we prove some preliminary results in Appendix B

that are used to derive the optimal input state for CV

bidirectional teleportation.

II. PRELIMINARIES

Let Hbe a separable Hilbert space, and let Tbe an

operator acting on H. The adjoint of Tis the unique

operator T†acting on Hdeﬁned by ⟨ϕ|T|ψ⟩=⟨ψ|T†|ϕ⟩

for all |ϕ⟩,|ψ⟩ ∈ H;Tis said to be self-adjoint if T=

T†. If Tr(√T†T)<∞then Tis said to be a trace-

class operator, and its trace norm is deﬁned as ∥T∥1:=

Tr(√T†T). A quantum state is a positive semi-deﬁnite,

trace-class operator with trace norm equal to one. We

denote by D(H) the set of all quantum states or density

operators acting on H. Let ρ, σ ∈ D(H). The ﬁdelity

between ρand σis deﬁned by [66]

F(ρ, σ):=

√ρ√σ



1.(1)

If one of the quantum states is pure, i.e., say ρ=|ψ⟩⟨ψ|,

then F(ρ, σ) = Tr(ρσ). The sine distance between ρand

σis given by [67–70]

C(ρ, σ):=p1−F(ρ, σ).(2)

The following inequalities relate the ﬁdelity, sine dis-

tance, and trace distance [71, Theorem 1]

1−pF(ρ, σ)≤1

2∥ρ−σ∥1≤C(ρ, σ).(3)

The set of bounded operators on Hforms a C∗-algebra

under the operator norm, and we denote it by L(H). Let

HAdenote the Hilbert space corresponding to a quan-

tum system A. A quantum channel from a quantum

system Ato a quantum system Bis a completely posi-

tive, trace preserving linear map from L(HA) to L(HB).

Let MA→Band NA→Bbe quantum channels. Let HA

be a Hamiltonian corresponding to the quantum system

A, and let Rdenote a reference system. The energy-

constrained channel ﬁdelity between MA→Band NA→B

for E∈[0,∞) is deﬁned by [72,73]

FE(MA→B,NA→B):=

inf

ρRA:Tr(HAρA)≤EF(MA→B(ρRA),NA→B(ρRA)),(4)

where ρRA ∈ D(HR⊗ HA), ρA= TrR(ρRA),and it is

implicit that the identity channel IRacts on the refer-

ence system R. Furthermore, the optimization in (4) is

taken over every possible reference system R. Similarly,

the energy-constrained sine distance between MA→Band

NA→Bfor E∈[0,∞) is deﬁned by [72,73]

CE(MA→B,NA→B):=

sup

ρRA:Tr(HAρA)≤E

C(MA→B(ρRA),NA→B(ρRA)).(5)

Although the optimizations in (4) and (5) are over ar-

bitrary mixed states and arbitrary reference systems, it

suﬃces to restrict the optimization over pure states such

that the reference system Ris isomorphic to the channel

input system A. This is a consequence of puriﬁcation, the

Schmidt decomposition, and data processing [74, Section

3.5.4]. We thus have

FE(MA→B,NA→B) =

inf

ϕRA:Tr(HAϕA)≤EF(MA→B(ϕRA),NA→B(ϕRA)),(6)

CE(MA→B,NA→B) =

sup

ϕRA:Tr(HAϕA)≤E

C(MA→B(ϕRA),NA→B(ϕRA)),(7)

where the optimizations (6) and (7) are taken over pure

states ϕRA with reference system Risomorphic to sys-

tem A.

III. OPTIMAL INPUT STATE FOR CV

UNIDIRECTIONAL TELEPORTATION

The CV quantum teleportation protocol describes how

to transmit an unknown quantum state from Alice to

Bob when their systems are in CV modes and they share

a prior entangled state known as a resource state [34]. In

this protocol, Alice mixes the unknown quantum state

with her share of the resource state (TMSV state) and

performs homodyne detection. The homodyne detection

destroys the input state on Alice’s end. Alice then com-

municates the classical outcomes of the detection to Bob,

based on which he performs unitary operations on his

share of the resource state to generate an approximation

of the input state. Let Adenote the input mode, and let

Bdenote the output mode. An ideal teleportation pro-

tocol requires noiseless homodyne detection and inﬁnite

squeezing in the TMSV state, and it induces the identity

channel IA→Bon the input states [1,34] (see [75] for

further clariﬁcation of the convergence of the protocol to

the identity channel). However, an experimental imple-

mentation of CV teleportation has a noisy detector and

ﬁnite squeezing in the resource state which makes the ex-

perimental implementations of teleportation perform less

than ideal. It realizes an additive-noise channel Tξ

A→B,

where the noise parameter ξ > 0 encodes unideal de-

tection and ﬁnite squeezing [34,76]. The additive-noise

channel Tξis a composition of the quantum-limited am-

pliﬁer A1/η with gain parameter 1/η and the pure-loss

channel Lηwith transmissivity η, where η= 1/(1 + ξ)

[77,78]. See [79, Section II.B] for more details.

By taking the performance metric to be the energy-

constrained channel ﬁdelity between ideal teleportation

and the additive-noise channel, the performance of ex-

perimental implementations has been studied in [54]. By

choosing the Hamiltonian HAto be the photon number

operator ˆnA=P∞

n=0 n|n⟩⟨n|A, the energy-constrained

channel ﬁdelity in (6) for the identity channel IA→Band

the additive-noise channel Tξ

A→Bcan be further simpli-

ﬁed, as a consequence of phase averaging and joint phase

covariance of these channels [54,73], as

FE(IA→B,Tξ

A→B) =

inf

ψRA

F(IA→B(ψRA),Tξ

A→B(ψRA)),(8)

where the inﬁmum is taken over pure and entangled su-

perpositions of twin-Fock states ψRA =|ψ⟩⟨ψ|RA such

that

|ψ⟩RA =

∞

n=0

λn|n⟩R|n⟩A,(9)

λn∈R+for all n, P∞

n=0 λ2

n= 1,and P∞

n=0 nλ2

n≤E. An

analytical solution to the energy-constrained channel ﬁ-

delity in (8), using Karush–Kuhn–Tucker conditions, was

given in [54] for small values of ξand arbitrary values of

E. The optimal input state so obtained was

|ψ⟩RA =p1− {E}|⌊E⌋⟩R|⌊E⌋⟩A+p{E}|⌈E⌉⟩R⌈E⌉⟩A,

(10)

where {E}:=E− ⌊E⌋. Another contribution of [54]

was to provide a method, using a combination of numer-

ical and analytical techniques, for ﬁnding optimal input

states to test the performance of unidirectional CV tele-

portation under the energy-constrained channel ﬁdelity

measure.

In this section, we show that an optimal input state for

the energy-constrained channel ﬁdelity (8) is a ﬁnite en-

tangled superposition of twin-Fock states saturating the

energy constraint for arbitrary values of ξand Esatisfy-

ing E≤(1 + ξ)/(1 + 3ξ),and it is given by

|ψ⟩RA =√1−E|0⟩R|0⟩A+√E|1⟩R|1⟩A.(11)

Observe that the optimal state in (11) is the same as that

in (10) under the common conditions of E≤(1+ξ)/(1+

3ξ) and small ξ. Our method also shows that the optimal

state in (11) is unique; i.e., there is no other optimal ﬁnite

entangled superposition of twin-Fock states for (8). We

emphasize that our method is purely analytical. For, we

use optimization techniques from multivariable calculus,

and the constraint E≤(1 + ξ)/(1 + 3ξ) is needed in our

analysis in the proof of Proposition 2that plays a major

role in establishing the result. We also note that it is

still an open problem to ﬁnd the optimal state for larger

values of Eanalytically.

In order to compute the energy-constrained channel

ﬁdelity between the ideal channel IA→Band its experi-

mental implementation Tξ

A→B,we deﬁne the M-truncated

energy-constrained channel ﬁdelity between IA→Band

Tξ

A→Bas

FE,M (IA→B,Tξ

A→B):= inf

ψRA

F(IA→B(ψRA),Tξ

A→B(ψRA)),

(12)

where the inﬁmum is taken over pure states ψRA =

|ψ⟩⟨ψ|RA of the form

|ψ⟩RA =

n=0

√pn|n⟩R|n⟩A,(13)

such that pn≥0 for all n, PM

n=0 pn= 1,and PM

n=0 npn≤

E. In Proposition 1in Appendix A, we show that

F(IA→B(ψRA),Tξ

A→B(ψRA))

≥1

(1 + ξ)

 M

n=0

(1 + ξ)n!2

+ M

n=1

pnξ

(1 + ξ)n!2

,

(14)

for every state of the form in (13). Deﬁne the real-valued

function

fM,ξ(p):=

(1 + ξ)

 M

n=0

(1 + ξ)n!2

+ M

n=1

pnξ

(1 + ξ)n!2

,(15)

for all p∈RM+1. By (14) and (15), we thus have

F(ψRA,Tξ

A→B(ψRA)) ≥fM,ξ(p).(16)

The minimizer of the function fM,ξ subject to pn≥0 for

all n∈ {0, . . . , M},

n=0

pn= 1,

n=0

npn≤E, (17)

is the unique point given by p0= 1 −E, p1=E, and

pn= 0 for all n≥2,whenever E≤(1 + ξ)/(1 + 3ξ). See

Proposition 2in Appendix A. It thus follows from (14)

that the optimal input state to the M-truncated energy-

constrained channel ﬁdelity is unique, and it is given by

(11), whenever E≤(1+ξ)/(1+3ξ). See Lemma 3in Ap-

pendix A. From the solution to the M-truncated energy-

constrained channel ﬁdelity, and the inequality (A35) in

Appendix A, it follows that the optimal input state in

(8) is given by (11), whenever E≤(1 + ξ)/(1 + 3ξ). The

uniqueness follows from the uniqueness of the optimal

state for the M-truncated energy-constrained channel ﬁ-

delity. See Theorem 4in Appendix A.

We compare two classes of experimentally relevant

quantum states, namely coherent states and TMSV

states, with the optimal state under the given energy

constraint. See [80] for further background on CV quan-

tum information. Let |α⟩denote a coherent state, which

is given by

|α⟩:=e−|α|2

∞

n=0

αn

√n!|n⟩.(18)

The energy of the coherent state |α⟩is E=|α|2,and

its covariance matrix is I2,the 2 ×2 identity matrix.

The covariance matrix of Tξ(|α⟩⟨α|) is (1 + 2ξ)I2. Let

ψ(n)RA =|ψ(n)⟩⟨ψ(n)|RA be the TMSV state given by

|ψ(n)⟩RA :=1

√n+ 1

∞

n=0 sn

n+ 1n

|n⟩R|n⟩A.(19)

The energy of its reduced state is E=n, and its covari-

ance matrix is

Vψ(n)RA =(2n+ 1)I22pn(n+ 1)σz

2pn(n+ 1)σz(2n+ 1)I2,(20)

where σzis the Pauli-zmatrix. The covariance matrix

of Tξ(ψ(n)RA) is given by

VTξ(ψ(n)RA)=(2n+ 1)I22pn(n+ 1)σz

2pn(n+ 1)σz(2n+ 1 + 2ξ)I2.(21)

We then have

F|α⟩⟨α|,Tξ(|α⟩⟨α|)=2

pDet (2(1 + ξ)I2)(22)

1 + ξ,(23)

and also,

Fψ(n)RA,Tξ(ψ(n)RA)

=22

qDet Vψ(n)RA +VTξ(ψ(n)RA)

(24)

1 + (2n+ 1)ξ(25)

1 + (2E+ 1)ξ.(26)

These ﬁdelity expressions are evaluated using Eq. (4.51)

of [80].

In Figure 1, we plot the output ﬁdelity FEbetween

the ideal channel and an additive-noise channel versus

the input energy E, corresponding to a coherent state,

a TMSV state and the optimal state, with input energy

E∈[0,0.5]. The noise parameter is taken as ξ= 0.5. In

order, the dotted (orange), dashed (magenta), and solid

(blue) lines indicate the output ﬁdelity for the coherent

state (23), the TMSV state (26), and the optimal state

(A33). The graph indicates that coherent and TMSV

states are not optimal states in general. Interestingly,

however, the TMSV state is very close to being an opti-

mal input state for CV unidirectional teleportation. This

was observed in a diﬀerent regime for the energy con-

straint, in Figure 2 of [54].

IV. OPTIMAL INPUT STATE FOR CV

BIDIRECTIONAL TELEPORTATION

The CV bidirectional teleportation protocol consists

of a two-way transmission of unknown quantum states

0.1 0.2 0.3 0.4 0.5

Energy E

0.50

0.55

0.60

0.65

Fidelity FE

Coherent TMSV Optimal

Figure 1: The graph plots the output ﬁdelity FEbetween

the ideal channel and an additive-noise channel versus the

input energy E, corresponding to a coherent state, a TMSV

state, and the optimal state (each having input energy E).

The noise parameter for the additive-noise channel is taken

as ξ= 0.5 and the states have energy E∈[0,0.5]. The

dotted (orange), dashed (magenta), and solid (blue) lines

represent the output ﬁdelity for the coherent state, the

TMSV state, and the optimal state, respectively.

between Alice and Bob. One implementation of the pro-

tocol, which we consider here, allows for a simulation of

two ideal CV unidirectional quantum channels with the

help of shared entanglement and LOCC. See [45] for a dis-

cussion of more general implementations. The protocol

that we consider here can be thought of as a combination

of two CV unidirectional teleportations, one from Alice

to Bob and the other from Bob to Alice. Let Aand B

denote the input modes for Alice and Bob, and let A′and

B′denote the output modes for Alice and Bob, respec-

tively. An ideal CV bidirectional teleportation between

Alice and Bob is represented by the following unitary

swap channel

SAB→A′B′(·):= SWAP(·) SWAP†,(27)

where the unitary swap operator SWAP is deﬁned as

SWAP :=

∞

m,n=0 |m⟩A′⟨n|A⊗ |n⟩B′⟨m|B.(28)

Here {|m⟩A}∞

m=0 is the photonic number basis corre-

sponding to the system A, and so on. The swap channel

acts on product states by swapping them, i.e.,

SAB→A′B′(ϕA⊗ψB) = ψA′⊗ϕB′.(29)

Thus, the swap channel can be thought of as the tensor

product of the ideal channels IA→B′and IB→A′. An

experimental implementation of CV bidirectional tele-

portation realizes an approximate swap channel given

by the tensor product of two additive-noise channels

Tξ

A→B′⊗T ξ′

B→A′. The Hamiltonian for the composite sys-

tem AB is the total photon number operator:

ˆnAB := ˆnA⊗IB+IA⊗ˆnB.(30)

Given any state ρAB ∈ D(HA⊗ HB),the inequality

Tr(ˆnAB ρAB)≤2Eimplies that the average photon num-

ber in ρAB over each of the modes Aand Bis at most E.

So, the energy-constrained channel ﬁdelity (6) between

the ideal bidirectional teleportation SAB→A′B′and its

experimental implementation Tξ

A→B′⊗ T ξ′

B→A′is given

FE(SAB→A′B′,Tξ

A→B′⊗T ξ′

B→A′) = inf

ϕRAB : Tr(ˆnAB ϕAB )≤2E

FSAB→A′B′(ϕRAB),Tξ

A→B′⊗ T ξ′

B→A′(ϕRAB ),(31)

where ϕRAB is a pure state and ϕAB = TrR(ϕRAB ). As

a consequence of the joint phase covariance of SAB→A′B′

and Tξ

A→B′⊗ T ξ′

B→A′,and the arguments given for (8),

the inﬁmum in (31) can be recast as

FE(SAB→A′B′,Tξ

A→B′⊗ T ξ′

B→A′) =

inf

ψRAB

FSAB→A′B′(ψRAB),Tξ

A→B′⊗ T ξ′

B→A′(ψRAB ),

(32)

where ψRAB =|ψ⟩⟨ψ|RAB is a pure, entangled superpo-

sition of twin-Fock states given by

|ψ⟩RAB =

∞

m,n=0

λm,n|m, n⟩R|m, n⟩AB ,(33)

such that λm,n ≥0 for all mand n,P∞

m,n=0 λ2

m,n = 1,

and P∞

m,n=0(m+n)λ2

m,n ≤2E. We further simplify

the ﬁdelity expression in (32) as follows. Since we are

working with CV modes, the systems A, B, A′, B′are all

isomorphic to each other. So, the energy constrained

channel ﬁdelity between SAB→A′B′and Tξ

A→B′⊗ T ξ′

B→A′

must be the same as that of IA→A′⊗IB→B′and Tξ

A→A′⊗

Tξ′

B→B′. For simplicity of notations, we shall denote any

channel MC→C′by MC,where Cand C′are isomorphic

systems. Recall that an additive-noise channel Tξcan be

written as a composed channel A1/η ◦ Lηfor η= 1/(1 +

ξ). Also, the adjoint of the quantum-limited ampliﬁer is

related to the pure-loss channel by (A1/η)†=ηLη[81].

For the given pure state in (33), we have

F(SAB→A′B′(ψRAB),Tξ

A→B′⊗ T ξ′

B→A′(ψRAB ))

=F((IA⊗ IB)(ψRAB ),(Tξ

A⊗ T ξ′

B)(ψRAB )) (34)

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Optimalinputstatesforquantifyingtheperformanceofcontinuous-variableunidirectionalandbidirectionalteleportationHemantK.Mishra,1,2SamadKhabbaziOskouei,3andMarkM.Wilde1,21HearneInstituteforTheoreticalPhysics,DepartmentofPhysicsandAstronomy,andCenterforComputationandTechnology,LouisianaStateUniversity,B...

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Optimal input states for quantifying the performance of continuous-variable unidirectional and bidirectional teleportation Hemant K. Mishra1 2Samad Khabbazi Oskouei3and Mark M. Wilde1 2

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