The SWAP Imposter Bidirectional Quantum Teleportation and its Performance Aliza U. Siddiqui12and Mark M. Wilde32

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The SWAP Imposter:

Bidirectional Quantum Teleportation and its Performance

Aliza U. Siddiqui1,2 and Mark M. Wilde3,2

1Division of Computer Science and Engineering, Louisiana State University, Baton Rouge, Louisiana 70803, USA

2Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, and Center for Computation and

Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA

3School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA

October 21, 2022

Abstract

Bidirectional quantum teleportation is a fundamental protocol for exchanging quantum information be-

tween two parties. Speciﬁcally, the two individuals make use of a shared resource state as well as local

operations and classical communication (LOCC) to swap quantum states. In this work, we concisely high-

light the contributions of our companion paper [Siddiqui and Wilde, arXiv:2010.07905]. We develop two

diﬀerent ways of quantifying the error of nonideal bidirectional teleportation by means of the normalized di-

amond distance and the channel inﬁdelity. We then establish that the values given by both metrics are equal

for this task. Additionally, by relaxing the set of operations allowed from LOCC to those that completely

preserve the positivity of the partial transpose, we obtain semideﬁnite programming lower bounds on the

error of nonideal bidirectional teleportation. We evaluate these bounds for some key examples—isotropic

states and when there is no resource state at all. In both cases, we ﬁnd an analytical solution. The second

example establishes a benchmark for classical versus quantum bidirectional teleportation. Another example

that we investigate consists of two Bell states that have been sent through a generalized amplitude damping

channel (GADC). For this scenario, we ﬁnd an analytical expression for the error, as well as a numerical

solution that agrees with the former up to numerical precision.

1 INTRODUCTION

Quantum teleportation is one of the most prominent protocols in quantum information due to its ability to

communicate a quantum state between two individuals who share entanglement. In this protocol, there is no

need to transmit the physical system. While direct transmission of a qubit is possible (shown in Figure 1), its

fragile nature is well known. Environmental noise will either corrupt the information encoded in the qubit or

prevent it from arriving at its destination altogether. As a result, the quantum teleportation protocol serves as

an alternative to physical transmission and utilizes shared entanglement as well as local operations and classical

communication (LOCC) to achieve this goal. Teleportation is now commonly used as a fundamental building

block in quantum information science, with applications in quantum communication, quantum error correction,

quantum networking, etc.

As a reminder, the procedure of standard quantum teleportation is as follows:

1. Two parties, Alice and Bob, are spatially separated and share a maximally entangled state Φ ˆ

Aˆ

Bdeﬁned

Φˆ

Aˆ

B:=1

i,j=0 |iihj|ˆ

A⊗ |iihj|ˆ

B.(1)

2. Alice wishes to send her system Ato Bob. So, she performs a projective Bell measurement on her systems

Aand ˆ

arXiv:2210.10882v1 [quant-ph] 19 Oct 2022

Figure 1: Ideally, we would want to send quantum information from one party to another directly via a quantum

channel.

Φ

Μ1

Μ2

ΧΜ1

ΖΜ2

Figure 2: Due to the fragile nature of quantum bits, the unidirectional quantum teleportation protocol (shown

above) was devised as a method for simulating an ideal unidirectional quantum channel, i.e., to transmit quantum

information from one party Alice, to another party Bob.

3. Alice obtains two classical values from her measurement and transmits them to Bob via a classical com-

munication channel.

4. Bob, based on the classical results, performs corrective operations on his system of the shared entangled

state ˆ

Bto recover the original state Alice wished to transfer.

See Figure 2for a quantum circuit depiction of the teleportation protocol.

Teleportation has been extended in various ways, and one way is through bidirectional quantum teleportation.

It should be noted that standard quantum teleportation—–also known as unidirectional teleportation—realizes

a one-way ideal quantum communication channel from one party Alice to another party Bob. The idea of

bidirectional teleportation is to provide a two-way quantum communication channel. Instead of only Alice

having the ability to transmit quantum information to Bob, individuals can now exchange quantum information.

In the ideal version of this protocol, our two parties share two pairs of maximally entangled qubits (ebits) and

teleport qubits to each other, performing standard quantum teleportation twice in opposite directions. We

will refer to any state shared by individuals to perform any variation of teleportation as a resource state. The

ideal protocol therefore utilizes two ebits as its resource state and is equivalent to a perfect swap channel

between two individuals, as shown in Figure 3. This extension of teleportation was observed early on in [1],

and it was subsequently considered in [2,3]. There has recently been a ﬂurry of research on the topic with

various proposals for bidirectional teleportation [4,5]. There has been even more interest in a variation called

bidirectional, controlled teleportation, using ﬁve qubit [6,7,8,9,10], six qubit [11,12,13,14,15], seven qubit

[16,17,18], eight qubit [19,20], and nine qubit [21] entangled resource states (see also [22]). Bidirectional

controlled teleportation is a tripartite protocol in which three individuals, typically called Alice, Bob, and

Charlie, share an entangled resource state and use LOCC to exchange qubits between Alice and Bob. In other

words, Charlie is present to assist Alice and Bob, who wish to swap quantum information. See also [23] for

other variations of bidirectional teleportation.

The applications of bidirectional teleportation align with those of standard teleportation. Speciﬁcally, it

applies in a basic quantum network setting in which two parties would like to exchange quantum information.

Although the ideal version of bidirectional teleportation is a trivial extension of the original protocol in which the

latter is simply conducted twice (but in opposite directions), the situation becomes less trivial and more relevant

to experimental practice when the quantum resource state deviates from the ideal resource of two maximally

entangled states. Much of the prior work focuses on precisely this kind of case, when the resource state is

diﬀerent from two maximally entangled states, either by being a diﬀerent pure state (such as cluster states),

a mixed state, or a state with insuﬃcient entanglement to accomplish the task. These kinds of investigations

are essential for understanding ways to simulate or mimic the ideal protocol approximately in an experimental

setting.

Despite the many works listed above on the topic of bidirectional teleportation, which try to perform the

protocol using various resource states, a systematic method for quantifying its performance has been missing.

In other words, there is no procedure to concretely determine how well a certain protocol of bidirectional

teleportation performs in comparison to the ideal protocol. How shall we know we have found our perfect

imposter? Any experimental implementation of bidirectional teleportation will necessarily be imperfect and

therefore, there is a need for such a metric. Indeed, entangled states generated in experimental settings using

methods such as spontaneous parametric down-conversion are only approximations to ideal maximally entangled

states [24]. Our aim in [25] and in the present paper is to ﬁll this void.

While this paper will give basic insight to the work, all proofs as well as additional material are present in

our main paper. The present paper instead aims to highlight the essential contributions of our companion paper

[25] and give some additional clariﬁcation.

2 PRELIMINARIES

Before proceeding further, we establish some notation and concepts that will be used throughout this paper.

In our work, instead of considering only qubits, we generalize all of our scenarios to qudits with dimension d.

Speciﬁcally, we consider two diﬀerent dimensions—the dimension of the resource state and the dimension of the

unitary swap channel we are trying to mimic. Given two parties Alice and Bob, the dimension of Alice’s qudit

that she wishes to send is denoted as dA(similarly dBfor Bob). The dimensions of Alice and Bob’s qudits

are also equivalent to the dimension of the swap channel d. The dimension of the shared resource state when

Bob’s system is discarded is denoted as dˆ

A(similarly dˆ

Bif Alice’s system was discarded). While we consider

the case where dA=dB=d, we make no assumptions about dˆ

Aand dˆ

Bother than the fact that they are

ﬁnite-dimensional. In other words, it need not be the case that dˆ

A=dˆ

B. Additionally, instead of maximally

entangled qubits, individuals will now share maximally entangled qudits (e-dits).

We also make use of the following bilateral unitary twirl channel in our paper

TCD(XCD):=ZdU (UC⊗UD)(XCD),(2)

where U(·):=U(·)U†,U(·):=U(·)UT(the overline indicates the complex conjugate), XCD is the bipartite

quantum state, and dU denotes the Haar measure (uniform distribution on unitary operators). The bilateral

twirl channel is an LOCC channel, in the sense that Alice can pick a unitary at random according to the Haar

measure, apply it to her system, report to Bob via a classical channel which one she selected, and Bob can then

apply the complex conjugate unitary to his system.

The bilateral twirl is typically utilized to symmetrize quantum states. Speciﬁcally, depending on the type

of twirl performed, the output state will be invariant under any unitary channel of the form U ⊗U or the form

U ⊗ U. For example, given a quantum state eσAB prepared by the isotropic bilateral twirl

ZdU (UA⊗ UB)(σAB ) = eσAB ,(3)

the following holds for every unitary channel U:

(UA⊗ UB)(eσAB ) = eσAB .(4)

Additionally, states prepared by this operation can be described by fewer parameters. Therefore, the twirled

state eσAB now has a sparse density matrix and can be characterized by fewer variables.

3 IDEAL BIDIRECTIONAL TELEPORTATION

Let us ﬁrst examine the case of ideal bidirectional teleportation on two qudits in detail. Doing so is helpful in

establishing a basic metric for when we consider nonideal bidirectional teleportation later. As stated in the intro-

duction, a trivial way to conduct quantum teleportation bidirectionally between two spatially separated parties,

Alice and Bob, is by performing two standard quantum teleportations, once in each direction. This method

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TheSWAPImposter:BidirectionalQuantumTeleportationanditsPerformanceAlizaU.Siddiqui1,2andMarkM.Wilde3,21DivisionofComputerScienceandEngineering,LouisianaStateUniversity,BatonRouge,Louisiana70803,USA2HearneInstituteforTheoreticalPhysics,DepartmentofPhysicsandAstronomy,andCenterforComputationandTechnolo...

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