myctr

# The Role of Localizable Concurrence in Quantum Teleportation Protocols

Mirko Consiglio  Department of Physics, University of Malta
University Ring Road, Msida MSD2080, Malta
<EMAIL_ADDRESS>Louis Zammit Mangion  Department of Physics,
University of Malta
University Ring Road, Msida MSD2080, Malta
<EMAIL_ADDRESS>Tony John George Apollaro  Department of
Physics, University of Malta
University Ring Road, Msida MSD2080, Malta
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

Teleporting an unknown qubit state is a paradigmatic quantum information
processing task revealing the advantage of quantum communication protocols
over their classical counterpart. For a teleportation protocol using a Bell
state as quantum channel, the resource has been identified to be the
concurrence. However, for mixed multi-partite states the lack of computable
entanglement measures has made the identification of the quantum resource
responsible for this advantage more challenging. Here, by building on previous
results showing that localizable concurrence is the necessary resource for
controlled quantum teleportation, we show that any teleportation protocol
using an arbitrary multi-partite state, that includes a Bell measurement,
requires a non-vanishing localizable concurrence between two of its parties in
order to perform better than the classical protocol. By first analyzing the
GHZ channel and GHZ measurement teleportation protocol, in the presence of
GHZ-symmetric-preserving noise, we compare different multi-partite
entanglement measures with the fidelity of teleportation, and we find that the
protocol performs better than the classical protocol when all multi-partite
entanglement measures vanish, except for the localizable concurrence. Finally,
we extend our proof to an arbitrary teleportation protocol with an arbitrary
multi-partite entangled channel.

###### keywords:

localizable concurrence; quantum teleportation; GHZ states; quantum circuits.

##  1 Introduction

Entanglement is the main defining feature of quantum mechanics, allowing
quantum systems to exhibit correlations having no classical counterpart
[Plenio_2007]. These correlations constitute a resource that can be exploited
to perform several Quantum Information Processing (QIP) tasks, such as quantum
computation [Steane_1998], quantum communication [Gisin_2007] and quantum
cryptography [Bennett_2014]. In particular, entanglement is a key resource for
quantum teleportation [Bennett_1993, Pirandola2015], where quantum information
is transmitted from one place to another using a classical communication
channel and the entanglement shared between the sender and receiver.

Since the inception of quantum teleportation via a Bell state [Bennett_1993],
there has been continuous ongoing research in developing protocols allowing
for more complex and elaborate teleportation schemes, such as controlled
quantum teleportation, where one or more agents’ collaboration is needed in
order to complete the teleportation protocol, and quantum error-detection,
where the aim is to ascertain whether the initial shared entanglement has been
subject to noise during the execution of the protocol, undermining the
fidelity of teleportation. Most of these advanced teleportation protocols
involve multi-partite entanglement. The perception that some complex
teleportation protocols are not feasible, or are resource-inefficient, when
relying only on bipartite entanglement, has contributed to the investigation
of the vast, and still largely unknown, field of multi-partite entanglement.

In order to gauge the efficiency of a teleportation protocol, a widely used
figure of merit is the fidelity [Jozsa_1994]. In the ideal case, the fidelity
of teleportation $F$ attains the maximum of $F=1$. However, as the
faithfulness of teleportation using a classical channel can never exceed
$F=\frac{2}{3}$ [Popescu_1994, Massar_1995], every teleportation protocol
achieving a fidelity larger than that classical limit exploits the
entanglement between the sender and the receiver as a resource. This is
evident in the teleportation protocol with a two-qubit mixed channel, where
the fidelity of teleportation is bounded by[Verstraete_2002];

$\max\left\\{\dfrac{3+\mathcal{C}}{6},\dfrac{1+2\mathcal{C}}{3}\right\\}\leq
F\leq\dfrac{2+\mathcal{C}}{3}\,,$ (1)

where $\mathcal{C}$ represents the concurrence, a bipartite entanglement
measure for arbitrary two-qubit states [Wootters_1998]. In this scenario, the
upper bound is attained by the maximally entangled Bell states, whereas, in
the absence of entanglement one obtains the classical upper bound of
$\frac{2}{3}$.

With the theoretical and experimental advances in QIP, great attention has
been dedicated to the investigation of entanglement in multi-partite quantum
systems [Horodecki2009, Guhne2009] and to their use as a resource for
teleportation protocols. The first steps in this direction have involved
creating teleportation schemes utilizing tripartite channels, i.e. using the
entanglement in a three-qubit state as quantum resource [Karlsson_1998,
Lee2005, Agrawal_2006, Yang2009]. Tripartite systems can be represented in
three separate ways; completely separable, biseparable, or tripartite
entangled states [Dur2000]. In the case of bipartite systems, only the four
Bell states, which are equivalent under Local Operations and Classical
Communication (LOCC) [Plenio_2007], are maximally entangled. However, when one
ventures into the realm of three or more qubits, one is able to witness
different classes of maximally entangled states [Ac_n_2001]. For tripartite
systems, we have the family of Greenberg-Horne-Zeilinger (GHZ) states, and the
family of W states,

$\ket{\text{GHZ}}=\dfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})\,,\
\ket{\text{W}}=\dfrac{1}{\sqrt{3}}(\ket{001}+\ket{010}+\ket{100})$ (2)

which are inequivalent under Stochastic LOCC (SLOCC) [Dur2000], that is under
LOCC, but without demanding that the state has to be obtained with certainty.
GHZ states have been utilized in tests of local hidden-variable theories
[greenberger2007], and have also been used in superdense coding applications
and cryptographic conferencing protocols [Bose1998]. In fact, the family of
GHZ states was initially proposed by Greenberger, Horne and Zeilinger to
examine non-locality without making use of an inequality, providing a stronger
statement of non-locality than Bell’s Theorem [QuantumInformation]. One known
benefit of using $n$-qubit GHZ states over a Bell state, to teleport qubits,
is that they can be used for partial error-detection. Implementing the post-
selection after detection of a bit-flip (or phase-flip) error leads to an
overall improved fidelity over the standard teleportation scheme involving a
Bell channel [Moreno_2018].

The inequivalence of the two entanglement classes is also witnessed from a
resource point of view, as a GHZ state can be utilized for the deterministic
teleportation of one qubit, whereas a W state accomplishes the task only with
probability $\frac{2}{3}$ [Joo2003]. Moreover, the presence of a third qubit
in the GHZ state allows for an increase in the complexity of the teleportation
protocol when compared with the standard one using a Bell state. Karlsson and
Bourennane [Karlsson_1998] proposed the Controlled Quantum Teleportation (CQT)
protocol utilizing a GHZ state. It involves a sender, a receiver and a
controller, with quantum teleportation being realized only by the
participation of the controller. In the absence of the controller’s
collaboration in the protocol, the fidelity of CQT is bounded above by the
classical limit. Interestingly, in Ref. Barasinski_2018, it has been shown
that the localizable concurrence [Verstraete_2004, Popp_2005] between two
qubits of the GHZ-state is a necessary resource in CQT.

In this paper we build on their result by investigating the necessary
resources for the quantum teleportation protocol utilizing a GHZ channel
[Eltschka_2014] and a GHZ measurement [Cunha_2019], while exposing the channel
to GHZ-symmetry-preserving decoherence. The typical protocol setup will
consist of generating a GHZ state capable of achieving teleportation, followed
by applying noise that retains the GHZ symmetry in the quantum channel. We
find that the localizable concurrence is a necessary resource for every
teleportation protocol which can be decomposed into a quantum circuit
including a Bell measurement. Our result encompasses the original CQT protocol
[Karlsson_1998, Barasinski_2018], and in addition applies to teleportation
protocols involving an arbitrary number of qubits, and is able to extend CQT
to involve an arbitrary number of controllers while still utilizing the same
quantum resource.

The paper is organized as follows; In Section 2, we will review the GHZ
channel and measurement teleportation protocol, followed by an overview of the
GHZ-symmetric states in Section 2.1, and the relevant entanglement monotones
and teleported states in the presence of dephasing and depolarizing noise in
Section 2.2. In Section 3 we present our main results and discuss the concept
of localizable concurrence in the GHZ channel and measurement teleportation
protocol, and a comparison is made with CQT and other teleportation protocols.
Here we demonstrate that any general multi-partite teleportation scheme
involving a Bell measurement as a component of the protocol, necessitates the
use of localizable concurrence as a resource in quantum teleportation.
Finally, in Section 4 conclusions are drawn.

##  2 GHZ Channel and Measurement Teleportation Protocol

A teleportation protocol is a QIP task able to transfer an unknown quantum
state from one place to another using classical communication and pre-
established entangled states between the sender and receiver. The first
teleportation protocol that has been proposed involves a Bell state shared
between two parties, generally named Alice and Bob [Bennett_1993], aimed at
teleporting the unknown state of a qubit. A Bell measurement is performed on
the qubit containing the message, and Alice’s qubit being part of the Bell
state. Then, two bits of classical information containing the measurements’
outcome are transmitted to Bob, enabling him to perform a unitary
transformation on his qubit to complete the protocol. The GHZ channel and
measurement teleportation protocol achieves the same results, however it
operates by using both a GHZ channel and a GHZ measurement [Karlsson_1998] as
outlined in the following derivation.

The GHZ state is defined as

$\ket{\text{GHZ}}=\dfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})=\dfrac{1}{\sqrt{2}}\sum\limits_{i=0}^{1}\ket{iii}\,.$
(3)

Suppose Alice wants to teleport a single qubit, denoted by the subscript $1$,
to Bob, through a GHZ state. In this teleportation protocol Alice possesses
two of the GHZ qubits, denoted by the subscripts $2$ and $3$, while Bob has
the other GHZ qubit, denoted by the subscript $4$. The general state of qubit
1 can be written as

$\ket{\psi}_{1}=\alpha_{0}\ket{0}_{1}+\alpha_{1}\ket{1}_{1}=\sum\limits_{i=0}^{1}\alpha_{i}\ket{i}_{1}\,.$
(4)

Combining Eqs. (3) and (4), the state of the entire system reads

$\ket{\Psi}_{1234}=\dfrac{1}{\sqrt{2}}\sum\limits_{i,j=0}^{1}\alpha_{i}\ket{ijjj}_{1234}\,.$
(5)

Rewriting the above equation by separating the qubits that will be part of the
GHZ measurement, we end up with

$\ket{\Psi}_{1234}=\dfrac{1}{\sqrt{2}}\sum\limits_{i,j=0}^{1}\alpha_{i}\ket{ijj}_{123}\otimes\ket{j}_{4}\,.$
(6)

A GHZ measurement can be performed by taking the inner product of

$\ket{\phi_{lmn}}=\dfrac{1}{\sqrt{2}}\sum\limits_{k=0}^{1}(-1)^{lk}\ket{k,k\oplus
m,k\oplus n}\,,$ (7)

with the first three qubits of the state $\ket{\Psi}_{1234}$ [Cunha_2019].
This results in Bob acquiring the state

$\ket{\eta_{lmn}}_{4}=\dfrac{1}{2}\sum\limits_{i,j,k=0}^{1}\alpha_{i}(-1)^{kl}\braket{k,k\oplus
m,k\oplus
n}{ijj}_{123}\otimes\ket{j}_{4}=\sum\limits_{k=0}^{1}\alpha_{k}(-1)^{kl}\ket{k\oplus
n}_{4}.$ (8)

Thus, Bob needs to perform unitary operations depending on the measured values
of $l,m$ and $n$.

Note that Alice needs only to send two bits of information since we require
that $k\oplus m=j=k\oplus n$, meaning that the measurement results of qubits 2
and 3 must be the same, i.e. $n=m$. Should this not be the case, a bit-flip
has occurred on one of those qubits, meaning that the teleported state can be
rejected. This means that partial bit-flip errors can be detected in the
teleportation protocol, 111It is useful to note that the teleportation
protocol can be adjusted to detect phase-flip errors instead of bit-flip
errors, which can prove to be advantageous if the frequency of phase-flip
errors is higher than the corresponding bit-flip ones resulting in an improved
overall fidelity [Moreno_2018], which is fundamental for teleportation
protocols in ascertaining that the transmitted information is correct.
Finally, Bob can retrieve the state in Eq. (4) by applying the unitary
${\sigma_{x}}^{n}{\sigma_{z}}^{l}$ to the state in Eq. (8), resulting in

$\ket{\psi}_{4}={\sigma_{x}}^{n}{\sigma_{z}}^{l}\ket{\eta_{lmn}}_{4}=\sum\limits_{k=0}^{1}\alpha_{k}\ket{k}_{4}\,,$
(9)

which is exactly the state as required.

###  2.1 GHZ-Symmetric States

Let us move to a specific family of tripartite high-ranked mixed states,
namely the GHZ-symmetric states [Eltschka_2012, Eltschka_2014]. These states
are ideal for analyzing the effects of noise on QIP tasks as they have well-
defined analytical entanglement monotones, allowing us to derive quantitative
relations between the fidelity of the teleportation protocol and different
entanglement measures.

The GHZ-symmetric family contains all tripartite mixed states which are
invariant under the operations listed below;

  1. 1.

Qubit permutations.

  2. 2.

Simultaneous three-qubit-flips.

  3. 3.

Qubit rotations around the $z$ axis of the form

$U(\phi_{1},\phi_{2})=e^{i\phi_{1}\sigma_{z}}\otimes
e^{i\phi_{2}\sigma_{z}}\otimes e^{-i(\phi_{1}+\phi_{2})\sigma_{z}}\,.$ (10)

The two GHZ-symmetric pure states are
$\ket{\text{GHZ}^{+}}=\frac{(\ket{000}+\ket{111})}{\sqrt{2}}$, and the sign-
flipped state
$\ket{\text{GHZ}^{-}}\equiv\frac{(\ket{000}-\ket{111})}{\sqrt{2}}$. The
complete set of GHZ-symmetric states consists of mixtures of the above two
states and the mixed state $\rho=\sum_{ijk=001}^{110}\ket{ijk}\bra{ijk}$. A
GHZ-symmetric state $\rho^{GS}$ can be fully described by two real parameters;

$\displaystyle x\left(\rho^{GS}\right)$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\bra{\text{GHZ}^{+}}\rho^{GS}\ket{\text{GHZ}^{+}}-\bra{\text{GHZ}^{-}}\rho^{GS}\ket{\text{GHZ}^{-}}\right)\,,$
(11) $\displaystyle y\left(\rho^{GS}\right)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{3}}\left(\bra{\text{GHZ}^{+}}\rho^{GS}\ket{\text{GHZ}^{+}}+\bra{\text{GHZ}^{-}}\rho^{GS}\ket{\text{GHZ}^{-}}-\dfrac{1}{4}\right)\,.$
(12)

A general GHZ-symmetric state can then be written as

$\begin{split}\rho^{GS}(x,y)=&\left(\dfrac{2}{\sqrt{3}}y+x\right)\ket{\text{GHZ}^{+}}\bra{\text{GHZ}^{+}}+\\\
&\left(\dfrac{2}{\sqrt{3}}y-x\right)\ket{\text{GHZ}^{-}}\bra{\text{GHZ}^{-}}+\dfrac{\sqrt{3}-4y}{8\sqrt{3}}\mathbb{1}_{8}\,,\end{split}$
(13)

where $\mathbb{1}_{8}$ is the $8\times 8$ identity matrix. This
parameterization constructs Fig. 1, where the completely mixed state is
located at the origin.

Figure 1: Entanglement classes of the GHZ-symmetric states, where on the top
two corners lie the $\ket{\text{GHZ}^{\pm}}$ states. The red kite in the
middle represents the separable states. The two blue triangles are the
biseparable states. The two yellow triangle-like shapes are the W-class
entangled states, bounded by the orange, curved, GHZ-W line. The last two gray
triangle-like shapes represent the GHZ-class entangled states. The dashed
magenta line represents the path taken by a GHZ state exposed to depolarizing
noise as described in Section 2.2.2. The solid green line separates GHZ-
symmetric states which have zero localizable concurrence from those with non-
zero value. Figure adapted from Ref. Eltschka_2014.

Any point in this triangle describes a GHZ-symmetric state, with each region
describing a different entanglement class with respect to SLOCC. Particularly,
the GHZ-class states are bounded below by the parameterized line;

$x^{W}=\dfrac{v^{5}+8v^{3}}{8(4-v^{2})},\
y^{W}=\dfrac{\sqrt{3}}{4}\dfrac{4-v^{2}-v^{4}}{4-v^{2}}\,,$ (14)

where $v\in[-1,1]$. Given the coordinates specified in Eqs. (11) and (12), one
is able to determine the three-tangle, an entanglement monotone that measures
genuine tripartite entanglement of the GHZ-class states [Coffman_2000] as
follows; one first determines the straight line that connects the GHZ+ state
at $(\frac{1}{2},\frac{\sqrt{3}}{4})$ with the point $(x,y)$. This line
intersects the GHZ-W line at the point $(x^{W},y^{W})$, from which the three-
tangle $\tau_{3}\left(\rho^{GS}(x,y)\right)$ is determined by

$\tau_{3}(x,y)=\begin{cases}0\,,&\text{for }x<x^{W}\,,y<y^{W}\,,\\\
\dfrac{x-x^{W}}{\frac{1}{2}-x^{W}}\,,&\text{otherwise.}\end{cases}$ (15)

Below this line the GHZ-symmetric state turns into a W-class state. The curve
described by Eq. (14) is in fact typically called the GHZ-W line. The lower
bound of the W-class states is characterized by the vanishing of the Genuine
Multi-partite Entanglement (GME) concurrence [D_r_1999, de_Vicente_2011,
Huber_2014], which is given as

$\mathcal{C}_{\text{GME}}\left(\rho^{GS}\right)=\max\left\\{0,2|x|+\sqrt{3}y-\dfrac{3}{4}\right\\}.$
(16)

This line separates the W-class states from the Biseparable-class states,
which is denoted as the W-B line. Finally, the lower bound for Biseparable-
class states is characterized by the vanishing of the negativity
[Horodecki_1998], given by

$\mathcal{N}\left(\rho^{GS}\right)=\max\left\\{0,|x|+\dfrac{1}{2\sqrt{3}}y-\dfrac{1}{8}\right\\}.$
(17)

Below this line lie the separable-class states.

To gauge the efficiency of the teleportation protocol in the presence of
noise, we employ the fidelity [Jozsa_1994] between the initial state Alice
wants to teleport,
$\ket{\psi}=\cos\left(\frac{\vartheta}{2}\right)\ket{0}+e^{i\varphi}\sin\left(\frac{\vartheta}{2}\right)\ket{1}$,
and Bob’s state after the teleportation protocol is completed, $\rho_{T}$.

$F(\ket{\psi},\rho_{T})=\bra{\psi}\rho_{T}\ket{\psi}.$ (18)

When using a GHZ-symmetric state channel for the teleportation protocol, the
fidelity reads

$F(\vartheta)=\dfrac{1}{6}\left(3+3|x|+2\sqrt{3}y+(2\sqrt{3}y-3|x|)\cos\left(2\vartheta\right)\right)\,.$
(19)

Notice that the state dependence of the fidelity can be removed by considering
the average fidelity of teleportation instead:

$\bar{F}=\dfrac{1}{18}\left(9+12|x|+4\sqrt{3}y\right)\,,$ (20)

resulting in a fidelity dependent only on the GHZ-symmetric quantum channel
used in the protocol.

###  2.2 Fidelity of teleportation with noisy channels

The fidelity of teleportation with noisy tripartite channels has been
extensively investigated. [Jung2008, Li2010a, Chun_2010, Hu2011, Liang2015]
Here we will restrict our attention to GHZ-symmetry-preserving noise acting on
each qubit of the GHZ state before the teleportation protocol is applied. The
effect of both dephasing, and depolarizing noise, on the entanglement measures
and the fidelity of the teleportation protocol will be analyzed.

####  2.2.1 Dephasing noise

The first type of noise to be discussed is dephasing noise. The Kraus
operators of a phase damping channel are the following;

$K_{0}=\begin{pmatrix}1&0\\\ 0&\sqrt{1-p}\end{pmatrix}\,,\
K_{1}=\begin{pmatrix}0&0\\\ 0&\sqrt{p}\end{pmatrix}$ (21)

where $p\in[0,1]$ [NC2011]. Thus, homogeneous phase damping acting on each
qubit of a GHZ state results in the following mixed state:

$\rho=\dfrac{1}{2}\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&(1-p)^{\frac{3}{2}}\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
(1-p)^{\frac{3}{2}}&0&0&0&0&0&0&1\end{array}\right)\,.$ (22)

Using equations (11) and (12), for the above state we get the parameters

$x(\rho)=\dfrac{1}{2}(1-p)^{\frac{3}{2}}\,,\ y(\rho)=\dfrac{\sqrt{3}}{4}\,.$
(23)

Now, using the procedure described in Section 2.1, we find that every point
described with the above equation lies on a horizontal line between the GHZ+
and GHZ- points, intersecting with $(x^{W},y^{W})=(0,\frac{\sqrt{3}}{4})$.
Applying Eq. (15), we find that the three-tangle results

$\tau_{3}(\rho)=(1-p)^{\frac{3}{2}}\,.$ (24)

Similarly using Eqs. (16) and (17) to find the GME concurrence and negativity,
respectively, we get

$\displaystyle\mathcal{C}_{\text{GME}}(\rho)$ $\displaystyle=$
$\displaystyle(1-p)^{\frac{3}{2}}\,,$ (25) $\displaystyle\mathcal{N}(\rho)$
$\displaystyle=$ $\displaystyle\frac{1}{2}(1-p)^{\frac{3}{2}}\,.$ (26)

The teleported state $\rho_{T}$, utilizing a dephased GHZ channel, becomes

$\rho_{T}=\left(\begin{array}[]{cc}\cos^{2}\left(\frac{\vartheta}{2}\right)&\frac{1}{2}(1-p)^{\frac{3}{2}}e^{i\varphi}\sin(\vartheta)\\\\[4.30554pt]
\frac{1}{2}(1-p)^{\frac{3}{2}}e^{-i\varphi}\sin(\vartheta)&\sin^{2}\left(\frac{\vartheta}{2}\right)\end{array}\right)\,,$
(27)

with an average fidelity of

$\bar{F}=\dfrac{1}{3}\left(2+(1-p)^{\frac{3}{2}}\right)\,.$ (28)

Thus, when $p$ goes to unity, $\bar{F}=\frac{3}{2}$, which is exactly the
average fidelity achieved by using only a classical communication channel to
transmit information.

Clearly the dephasing channel does not allow for discerning the quantum
resource in use for attaining quantum teleportation, since all three
entanglement measures, given in Eqs. (24), (25) and (26), vanish
simultaneously at $p=1$. As such we require investigating another type of
noise capable of discriminating among the quantum resources in the
teleportation protocols via a GHZ channel.

####  2.2.2 Depolarizing noise

We turn our focus to a GHZ state under depolarizing noise. The Kraus operators
of a depolarizing channel are

$\begin{split}K_{0}&=\dfrac{1}{\sqrt{2}}\left(\begin{array}[]{lc}1&0\\\
0&\sqrt{1-p}\end{array}\right)\,,\
K_{1}=\dfrac{1}{\sqrt{2}}\left(\begin{array}[]{lc}0&\sqrt{p}\\\
0&0\end{array}\right)\,,\\\\[4.30554pt]
K_{2}&=\dfrac{1}{\sqrt{2}}\left(\begin{array}[]{cl}\sqrt{1-p}&0\\\
0&1\end{array}\right)\,,\
K_{3}=\dfrac{1}{\sqrt{2}}\left(\begin{array}[]{cl}0&0\\\
\sqrt{p}&0\end{array}\right)\,,\end{split}$ (29)

where $p\in[0,1]$ [NC2011]. The noisy GHZ-symmetric state reads

$\rho=\left(\begin{array}[]{cccccccc}a&0&0&0&0&0&0&c\\\ 0&b&0&0&0&0&0&0\\\
0&0&b&0&0&0&0&0\\\ 0&0&0&b&0&0&0&0\\\ 0&0&0&0&b&0&0&0\\\ 0&0&0&0&0&b&0&0\\\
0&0&0&0&0&0&b&0\\\ c&0&0&0&0&0&0&a\end{array}\right)\,,$ (30)

where $a=\frac{(4-3(2-p)p)}{8}$, $b=\frac{(2-p)p}{8}$,
$c=\frac{(1-p)^{3}}{2}$. The mixed state traces the dashed magenta path in
Fig. 1 as $p$ goes from zero to unity. Once again, using equations (11) and
(12), for the above state we get

$x(\rho)=\dfrac{1}{2}(1-p)^{3}\,,\ y(\rho)=\dfrac{\sqrt{3}}{4}(1-p)^{2}\,.$
(31)

Here we require the calculation of the three-tangle numerically, since the
resulting equations require the solution of a quintic polynomial. In addition,
for an analytical comparison, an accurate formula for the three-tangle derived
from the analytical lower bound in Ref. Eltschka_2014, is given here;

$\tilde{\tau}_{3}(\rho)=\max\left\\{0,\dfrac{1}{7}(7-54p+39p^{2}-8p^{3})\right\\}\,.$
(32)

Note that the difference between the exact numerical value and the lower bound
is less than $2\times 10^{-3}$, and when the lower bound is zero, the exact
three-tangle is also zero. Now, using Eqs. (16) and (17) to find the GME
concurrence and negativity, respectively, we get

$\displaystyle\mathcal{C}_{\text{GME}}(\rho)$ $\displaystyle=$
$\displaystyle\max\left\\{0,\dfrac{1}{4}(4-18p+15p^{2}-4p^{3})\right\\}\,,$
(33) $\displaystyle\mathcal{N}(\rho)$ $\displaystyle=$
$\displaystyle\max\left\\{0,\dfrac{1}{8}(4-14p+13p^{2}-4p^{3})\right\\}\,.$
(34)

Fig. 2 (left panel) shows the plots of these three entanglement measures along
with the localizable concurrence, which shall be derived in the following
section. Given a depolarized GHZ channel, the teleported state $\rho_{T}$
becomes

$\rho_{T}=\dfrac{1}{2}\left(\begin{array}[]{cc}1+(1-p)^{2}\cos(\vartheta)&(1-p)^{3}e^{i\varphi}\sin(\vartheta)\\\\[4.30554pt]
(1-p)^{3}e^{-i\varphi}\sin(\vartheta)&1-(1-p)^{2}\cos(\vartheta)\\\
\end{array}\right)\,,$ (35)

with an average fidelity of

$\bar{F}=\dfrac{1}{6}\left(6-8p+7p^{2}-2p^{3}\right)\,.$ (36)

Here we find that $\bar{F}=\frac{1}{2}$ as $p$ goes to unity.

Figure 2: (left) Entanglement measures as a function of the rate of
depolarization, $p$. (right) Teleportation fidelity (36) as a function of
different entanglement measures. One can see that the entanglement measures
decay to zero in the order determined by the SLOCC classes the depolarizing
GHZ state passes through. Looking at the right panel, a tripartite channel can
have GME concurrence and three-tangle zero, yet the teleported state has a
fidelity greater than $\frac{2}{3}$. The classical bound is only reached when
the localizable concurrence is equal to zero.

Letting $G,W,B$ and $S$ denote the sets of GHZ-, W-, biseparable- and
separable-class states, we find that the three-tangle (32) equals zero when
$p_{G\backslash W}\approx 0.144204$, which achieves an average fidelity of
$\bar{F}_{G\backslash W}\approx 0.830989$, which is larger than $\frac{2}{3}$.
The same procedure can be applied for the GME concurrence (33) and the
negativity (34). The GME concurrence equals zero when $p_{W\backslash
B}\approx 0.284595$, achieving an average fidelity of $\bar{F}_{W\backslash
B}\approx 0.707350$, while the negativity equals zero when $p_{B\backslash
S}\approx 0.443307$, with an average fidelity of $\bar{F}_{B\backslash
S}\approx 0.609159$. Only when $p_{Q}\approx 0.342702$ does
$\bar{F}=\frac{2}{3}$.

Fig. 2 (right panel) shows the teleportation fidelity as a function of
different entanglement measures. As one can see, the maximum classical
fidelity is reached when the depolarized GHZ state is in the biseparable-
class, yet not all biseparable mixed states are able to achieve a fidelity
higher than $\frac{2}{3}$. This implies that GME concurrence, and consequently
three-tangle, are not necessary for the GHZ channel and measurement
teleportation protocol. The quantum resource necessary to outperform classical
communication protocols turns out to be the localizable concurrence, which
shall be discussed in the following section.

##  3 Localizable Concurrence

It has been already established that localizable concurrence is a necessary
resource for the CQT protocol [Barasinski_2018]. Localizable concurrence is
concretely defined in Section IIA of Ref. Popp_2005, by considering the
concurrence as the entanglement measure applied to the definition. The
localizable concurrence of a quantum channel can then be roughly defined as
the maximum possible concurrence generated between two qubits, specifically
the sender and the receiver, given that optimal local measurements are
performed on all other qubits present in the quantum channel. Here we prove
that the localizable concurrence is also the necessary resource for the
quantum teleportation protocol using a GHZ state and GHZ measurement.

We shall consider the teleportation protocol from the perspective of a quantum
circuit. The quantum circuit for the tripartite channel teleportation protocol
with GHZ measurement is given in Fig. 3, and we highlight the decomposition of
the GHZ measurement into a Bell measurement and a preceding $CX$ gate. As in
the standard teleportation scheme [Bennett_1993] concurrence is the necessary
quantum resource and a Bell measurement is an essential part of the protocol,
identifying in a teleportation scheme via multi-partite entanglement a multi-
qubit measurement decomposition involving Bell measurements paves the way to
the identification of the necessary quantum resource.

This measurement decomposition constitutes the key argument of our proof as it
reduces the GHZ channel and measurement protocol to an effective standard
teleportation protocol where the localizable concurrence is the necessary
quantum resource utilized for teleportation. Note that the last measurement
also serves to ascertain that both of the measurements on Alice’s tripartite
state qubits give the same measurement value, that is both ones or zeros. If
they differ, then one can be certain that a bit-flip error has occurred in one
of the qubits [Moreno_2018], allowing for the post-selection of noisy
measurements, which cannot be carried out by using a Bell channel. This
partial error detection leads to a better overall fidelity of teleportation.

Figure 3: Quantum circuit for teleporting a qubit containing the state
$\ket{\psi}$, using a tripartite state $\rho$. The top three lines of the
circuit represent Alice’s message and tripartite qubits. The bottom line is
Bob’s tripartite qubit.

Localizable concurrence is not a quantity that is straightforward to evaluate,
however following on the lines of Ref. Barasinski_2018, we can calculate the
average localized concurrence between qubits 2 and 4, being the sender qubit
and receiver qubit, respectively. By applying the $CX$ gate between qubits 1
and 3, and then applying an optimal one-qubit orthogonal measurement on qubit
3 (as if qubit 3 is playing the role of the controller within the GHZ
measurement circuit decomposition given in Fig. 3), the localizable
concurrence between the sender and receiver qubits is calculated as follows:

$\mathcal{C}_{L}(\rho)=\max_{U}\sum\limits_{t=0}^{1}\bra{t}U\rho_{3}U^{\dagger}\ket{t}\mathcal{C}\left(\rho_{24}^{t}\right)\,,$
(37)

where $\rho^{\prime}=\ket{\psi}\bra{\psi}\otimes\rho$ represents the state of
the message qubit and tripartite channel, and $\rho_{24}^{t}$ represents the
state of qubits 2 and 4 after applying a $CX$ gate between qubits 1 and 3, and
obtaining the measurement outcome $t$ on qubit 3;

$\rho_{24}^{t}=\dfrac{\text{Tr}_{3}\left\\{\left(U^{\dagger}\ket{t}\bra{t}U\right)_{3}\text{Tr}_{1}\left\\{CX_{13}\rho^{\prime}CX_{13}\right\\}\left(U^{\dagger}\ket{t}\bra{t}U\right)_{3}\right\\}}{\bra{t}U\rho_{3}U^{\dagger}\ket{t}}\,,$
(38)

where the subscripts on the operators denote the qubit on which it is acting.
$\bra{t}U\rho_{3}U^{\dagger}\ket{t}$ represents the probability of achieving
the outcome $t$ on qubit 3 after applying the $CX$ gate, and $U$ is a $2\times
2$ unitary matrix.

Taking the general form of a $2\times 2$ unitary matrix (ignoring global
phase),

$U=\left(\begin{array}[]{cc}\cos\left(\frac{\theta}{2}\right)&-e^{i\lambda}\sin\left(\frac{\theta}{2}\right)\\\\[4.30554pt]
e^{i\phi}\sin\left(\frac{\theta}{2}\right)&e^{i(\phi+\lambda)}\cos\left(\frac{\theta}{2}\right)\end{array}\right)\,,$
(39)

where $\theta\in[0,4\pi],\phi,\lambda\in[0,2\pi]$, so that we are able to
maximize Eq. (37), we find that for a GHZ-symmetric state $\rho^{GS}$,

$\rho_{24}^{t}=\left(\begin{array}[]{cccc}a+b&0&0&c\\\\[4.30554pt]
0&d&0&0\\\\[4.30554pt] 0&0&d&0\\\\[4.30554pt]
c^{*}&0&0&a-b\end{array}\right)\,,$ (40)

where $a=\frac{1}{4}+\frac{y}{\sqrt{3}}$,
$b=(-1)^{t}\frac{2y\cos(\vartheta)\cos(\theta)}{\sqrt{3}}$,
$c=-(-1)^{t}x\sin(\theta)(\cos(\lambda)-i\cos(\vartheta)\sin(\lambda))$, and
$d=\frac{1}{4}-\frac{y}{\sqrt{3}}$, with
$\bra{t}U\rho_{3}U^{\dagger}\ket{t}=\frac{1}{2}$ since
$\rho_{3}=\frac{1}{2}\mathbb{1}_{2}$ and $t=0,1$. From this we can calculate
the average bipartite concurrence between qubits 2 and 4, which is maximized
when $\theta=\frac{\pi}{2}$ and $\lambda=0,\pi$:

$\mathcal{C}_{L}\left(\rho^{GS}\right)=2\max\left\\{0,|x|+\dfrac{y}{\sqrt{3}}-\dfrac{1}{4}\right\\}\,.$
(41)

This is exactly the result found in Ref. Barasinski_2018. For a depolarized
GHZ state $\rho$,

$\mathcal{C}_{L}\left(\rho\right)=\max\left\\{0,\dfrac{1}{4}\left(2-8p+7p^{2}-2p^{3}\right)\right\\}\,,$
(42)

where $\mathcal{C}_{L}=0$ when $p=p_{Q}$ for the depolarized GHZ channel,
implying that localizable concurrence between qubits 2 and 4 is a necessary
resource for the tripartite channel teleportation protocol utilizing a GHZ
measurement. Clearly, because of the permutation symmetry of the GHZ-symmetric
states, the role of the qubits is interchangeable.

This can be generalized to $n$-partite teleportation protocols of a qubit from
a sender to a receiver. Below we provide a proof that the localizable
concurrence is a necessary resource for any teleportation scheme requiring a
Bell measurement within the set of measurements carried out during the
protocol. A quantum circuit scheme, similar to Fig. 3 for the GHZ measurement
protocol, is given in Fig. 4. We generalize Proposition 1 of Ref.
Barasinski_2018 using the same line of reasoning, in order to go beyond the
applications of localizable concurrence in CQT protocols only. The main
difference is that our proof includes the possibility of applying multi-qubit
gates between the message state and the $n$-partite entangled state, and in
between the entangled state itself, before any measurements are performed.

###### Proposition 1

Suppose we have a teleportation protocol that can be constructed such as in
Fig. 4. The protocol utilizes an entangled $n$-partite state $\rho$ capable of
achieving the teleportation of one qubit. Without loss of generality, the
initial state $\ket{\psi}$ is stored in qubit 1, with the sender being qubit 2
and the intended receiver being qubit $n+1$, acquiring the state $\rho_{T}$
after teleportation. Qubits 3 to $n$ can be under the possession of any
combination of the sender and any number of controllers. $V$ contains all the
(non-)local operators that are necessary for teleportation, while $W$ is the
classically-conditioned operator required to complete the protocol, which
depends on the measurement results from the other qubits. Then the fidelity of
the teleportation protocol using the n-partite state $\rho$ is

$\max\left\\{\dfrac{3+\mathcal{C}_{L}(\rho)}{6},\dfrac{1+2\mathcal{C}_{L}(\rho)}{3}\right\\}\leq
F_{T}(\rho)\leq\dfrac{2+\mathcal{C}_{L}(\rho)}{3}\,,$ (43)

where $\mathcal{C}_{L}(\rho)$ is the localizable concurrence of $\rho$ and
$F_{T}(\rho)\equiv F(\ket{\psi},\rho_{T})$.

Figure 4: Teleportation scheme following the definition for Prop. 1.

######  Proof 3.1.

Define $X$ to be the set of qubits which shall be measured, after applying the
necessary set of gates $V$ required in the protocol to induce localized
entanglement between the sender $S$ and receiver $R$, i.e. in this case qubits
$S=2$, $R=n+1$ and $X=\\{3,\dots,n\\}$. Define
$\rho^{\prime}=\ket{\psi}\bra{\psi}\otimes\rho$ and
$\rho_{X}=\text{Tr}_{[n]\backslash
X}\left\\{V_{[n]}\rho^{\prime}V^{\dagger}_{[n]}\right\\}$, where we define
$[n]\equiv\\{1,\dots,n\\}$. Then

$\rho_{SR}^{t}=\dfrac{\text{Tr}_{X}\left\\{\left(U^{\dagger}\ket{t}\bra{t}U\right)_{X}\text{Tr}_{1}\left\\{V_{[n]}\rho^{\prime}V^{\dagger}_{[n]}\right\\}\left(U^{\dagger}\ket{t}\bra{t}U\right)_{X}\right\\}}{\bra{t}U\rho_{X}U^{\dagger}\ket{t}}\,,$
(44)

where the subscripts on the operators denote the qubit on which it is acting.
$U$ is the unitary matrix to be maximized, such that

$\mathcal{C}_{L}(\rho)=\max_{U}\sum\limits_{t}\bra{t}U\rho_{X}U^{\dagger}\ket{t}\mathcal{C}\left(\rho_{SR}^{t}\right)\,,$
(45)
$F_{T}(\rho)=\max_{U}\sum\limits_{t}\bra{t}U\rho_{X}U^{\dagger}\ket{t}F\left(\rho_{SR}^{t}\right)\,.$
(46)

Let us first prove the upper bound. Suppose there exists $\tilde{U}$ which
maximizes Eq. (45), such that $\tilde{\rho}_{SR}^{t}$ is an optimal two-qubit
state. Now from Eq. (1),

$F_{T}\left(\rho_{SR}^{t}\right)\leq\dfrac{2+\mathcal{C}\left(\rho_{SR}^{t}\right)}{3}\,,$
(47)

where $\mathcal{C}\left(\rho_{SR}^{t}\right)$ is the concurrence of the
localized state $\rho_{SR}^{t}$. Thus

$\displaystyle F_{T}(\rho)$ $\displaystyle=$
$\displaystyle\sum\limits_{t}\bra{t}\tilde{U}\rho_{X}\tilde{U}^{\dagger}\ket{t}F\left(\tilde{\rho}_{SR}^{t}\right)$
(48) $\displaystyle\leq$
$\displaystyle\sum\limits_{t}\bra{t}\tilde{U}\rho_{X}\tilde{U}^{\dagger}\ket{t}\dfrac{2+\mathcal{C}\left(\tilde{\rho}_{SR}^{t}\right)}{3}$
$\displaystyle\leq$
$\displaystyle\max_{U}\sum\limits_{t}\bra{t}U\rho_{X}U^{\dagger}\ket{t}\dfrac{2+\mathcal{C}\left(\rho_{SR}^{t}\right)}{3}$
$\displaystyle=$ $\displaystyle\dfrac{2+\mathcal{C}_{L}(\rho)}{3}\,.$

Now for the lower bound. Suppose that there also exists $\tilde{U}$ which
maximizes Eq. (45), then from Eq. (1)

$\max\left\\{\dfrac{3+\mathcal{C}\left(\tilde{\rho}_{SR}^{t}\right)}{6},\dfrac{1+2\mathcal{C}\left(\tilde{\rho}_{SR}^{t}\right)}{3}\right\\}\leq
F\left(\tilde{\rho}_{SR}^{t}\right)\,.$ (49)

Therefore,

$\displaystyle\max\left\\{\dfrac{3+\mathcal{C}_{L}(\rho)}{6},\dfrac{1+2\mathcal{C}_{L}(\rho)}{3}\right\\}=$
(50) $\displaystyle=$
$\displaystyle\max_{U}\sum\limits_{t}\bra{t}U\rho_{X}U^{\dagger}\ket{t}\max\left\\{\dfrac{3+\mathcal{C}\left(\rho_{SR}^{t}\right)}{6},\dfrac{1+2\mathcal{C}\left(\rho_{SR}^{t}\right)}{3}\right\\}$
$\displaystyle=$
$\displaystyle\sum\limits_{t}\bra{t}\tilde{U}\rho_{X}\tilde{U}^{\dagger}\ket{t}\max\left\\{\dfrac{3+\mathcal{C}\left(\tilde{\rho}_{SR}^{t}\right)}{6},\dfrac{1+2\mathcal{C}\left(\tilde{\rho}_{SR}^{t}\right)}{3}\right\\}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{t}\bra{t}\tilde{U}\rho_{X}\tilde{U}^{\dagger}\ket{t}F\left(\tilde{\rho}_{SR}^{t}\right)$
$\displaystyle\leq$
$\displaystyle\max_{U}\sum\limits_{t}\bra{t}U\rho_{X}U^{\dagger}\ket{t}F\left(\rho_{SR}^{t}\right)$
$\displaystyle=$ $\displaystyle F_{T}(\rho)$

This scheme encompasses the GHZ channel and measurement teleportation, CQT,
and W-class teleportation protocols [Agrawal_2006, Li_2007]. Note that for the
standard teleportation scheme utilizing a Bell state and Bell measurement,
which is also contained within the proposition above, the concurrence replaces
the localizable concurrence. An important remark is that while the unitary
matrix in Eq. (39) allows for the determination of the localizable
concurrence, for general $n$-partite states, maximizing Eq. (45) over all
possible measurements (by means of a general $(n-2)$-qubit unitary matrix) is
not a trivial task. At the same time, from Fig. 1 it is evident that neither
W- nor GHZ-entanglement is necessary for obtaining localizable concurrence,
whereas biseparability seems to be a necessary although not sufficient
condition. The same appears to hold for $n>3$ qubit states [Barasinski_2018],
although the boundary for non-zero localizable concurrence within the
biseparable class of states is unknown.

We already mentioned some advantages of using a GHZ channel over a Bell
channel to teleport a single qubit, such as the capability to carry out
partial error detection [Moreno_2018] and CQT [Karlsson_1998]. Additionally
there is the secret sharing protocol [Hillery1999], consisting of splitting
quantum information between two or more party members, quantum state sharing
[Man2007], controlled dense coding [Hao2001], synchronization of clocks
[Komar2014], and many others. This implies that there are certain instances
where physically preparing [Roos2004, DeLimaBernardo2019] and utilizing a GHZ
state is more practical, given their extended applications over Bell states.
Moreover, analysis of noisy GHZ channels [Miao2010, Yang2019] demonstrates
that theoretical and experimental improvements on QIP protocols increasingly
allows the implementation of new technologies that limit the effects of noise
on GHZ states. Most of these considerations can be extended to $n$-partite GHZ
states, as well as other multi-partite states designed to work with other QIP
protocols.

A particularly interesting protocol to consider is the teleportation of a Bell
state utilizing a GHZ channel and measurement [Gorbachev2000]. Alice possesses
one qubit of the GHZ channel and the Bell state to be teleported, while Bob
possesses the other two qubits from the GHZ state. The GHZ measurement is
performed over Alice’s qubits. Alice then sends two bits of classical
information to Bob prompting him to perform local operators on his own two
qubits to complete the teleportation protocol. Now assuming that the channel
is exposed to GHZ-symmetry-preserving noise, one can evaluate that the
fidelity of teleportation is equal to

$F_{T}\left(\rho^{GS}\right)=\dfrac{(3+12|x|+4\sqrt{3}y)}{12}\,,$ (51)

such that the concurrence of the teleported Bell state is equal to unity if
the fidelity is also equal to unity, and the concurrence is equal to zero if
the fidelity is equal to $\frac{1}{2}$. Moreover, one can also find that the
concurrence of the teleported Bell state is equal to the localizable
concurrence of the quantum channel. In the case of GHZ-symmetric channels, by
referring to the solid green line in Fig. 1, the concurrence of the teleported
Bell state goes to zero at the points where the localizable concurrence of the
GHZ-symmetric state vanishes. Note that this teleportation protocol would
require two Bell channels needing four qubits in total, as opposed to
utilizing one GHZ channel requiring the use of only three qubits, thus making
this protocol resource efficient.

##  4 Conclusion

In this paper we have investigated the necessary quantum resource in order to
achieve a better-than-classical teleportation fidelity when using a GHZ
channel and GHZ measurement protocol. In line with the results of Ref.
Barasinski_2018, where it has been shown that the localizable concurrence is
the necessary resource for the controlled quantum teleportation protocol, we
found that for the teleportation protocol we investigated, the localizable
concurrence is also the necessary resource required by the protocol. By
further analyzing the role of localizable concurrence in quantum teleportation
protocols, we have first reviewed and outlined the effects of dephasing and
depolarizing noise in GHZ channels. The GHZ-symmetric mixed states obtained
through the effect of the considered noise have well-defined entanglement
measures, which were utilized in relating quantitatively the fidelity of
teleportation and different entanglement measures. We have found that better-
than-classical fidelity of teleportation can be achieved whenever the
localizable concurrence between two qubits, with one being the sender and the
other the receiver, of the GHZ-symmetric state is non-zero. After specifically
utilizing a GHZ channel and measurement quantum teleportation protocol, a
general proposition extending the work carried out in Ref. Barasinski_2018 was
derived showing that localizable concurrence is necessary, not only for
controlled quantum teleportation protocols, but for any quantum teleportation
protocol of a single qubit relying on a Bell measurement. Our work stresses
the importance of localizable concurrence as a quantum resource for
teleportation protocols via multi-partite entangled states, calling for more
investigation on its quantification and relation to multi-partite
entanglement. Practical scenarios where the localizable concurrence has to be
quantified include $n$-party collaborative teleportation protocols, where up
to $n-2$ agents have to perform optimal local measurements on their qubits in
order to allow, possibly not predetermined, $n$-th agent to successfully
complete the protocol.

## Acknowledgments

We acknowledge fruitful discussions with Quantumalta group members André
Xuereb and Jake Xuereb.

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