1. Introduction
Quantum teleportation [
1] shows the power of entanglement as a resource: by jointly measuring the quantum states of two particles, we can transfer, without any actual exchange of matter, a quantum state to a remote station. This process is commonly referenced in the context of quantum communication over long distances, but its applications to quantum computation are also paramount, as demonstrated by the success of the measurement-based model for quantum computing [
2]. The measurement-free teleportation protocol, put forward in Ref. [
3], helped illustrate the centrality of entanglement by entirely removing measurements that, in contrast, are very important for the success of the original scheme [
1].
Ref. [
4] challenged the nearly dogmatic view on the essential role of entanglement to explore the relation between the efficiency of measurement-free teleportation and non-Markovianity [
5,
6,
7]. Specifically, the analysis by Tserkis et al. drew links between the information back-flow from the
instrumental part of the computational register (considered to be an environment) to its
relevant part (the system), and the entanglement present in the environment. It is worth noticing that the original teleportation protocol has previously been studied in the context of non-Markovianity [
8,
9,
10,
11], but such an assessment has normally been done by introducing an external environment. A different take to the
role played by non-Markovianity in teleportation was addressed in Ref. [
8], where non-Markovianity was seen as an additive to performance rather than the mechanism underpinning it. Ref. [
11], instead, studied the use of non-Markovianity to mitigate against the effects of noise on the resource state.
This paper arises from the work of Tserkis [
4] and critically assesses the link between non-Markovianity and efficiency in the measurement-free teleportation protocol. Methodologically, we model the teleportation circuit
as a quantum channel for a system of interest [
12,
13,
14,
15,
16,
17] and analyze the dynamics inherent within it.
While we do not introduce non-Markovianity through any external means, by focusing on the measurement-free teleportation approach, we are able to gain insight into the underlying dynamics of teleportation and analyze the non-Markovianity which could be inherently present in the protocol.
We show that such connections are—at best—very weak and delicately dependent on the way the dynamics underpinning the protocol are implemented and interpreted. Only a very fine-grained assessment of the various stages of the teleportation channel allows us to unveil how non-Markovianity enters the dynamics of the register and, potentially, could play a role in the establishment of the right fluxes of information from the instrumental part of the teleportation register to its relevant part. On the one hand, our results point towards the careful assessment of the way a dynamical map is implemented in all its sub-parts before any conclusions on what embodies a
resource of it can be made. On the other hand, it indirectly points to the need for a deeper understanding of the role played by non-Markovianity in quantum information problems and, in turn, the benefits of developing a comprehensive quantum resource theory of non-Markovianity [
18,
19].
The remainder of this paper is organized as follows: in
Section 2, we illustrate the measurement-free teleportation protocol and address it from the perspective of open-system dynamics, including the effects of its
channel description on distinguishability of input states.
Section 3 reviews the key instruments for our quantitative assessment of non-Markovianity and applies them to the evaluation of the information back-flow entailed by the measurement-free teleportation protocol.
Section 4 assesses in detail the link with quantum correlations shared by the relevant and ancillary part of the register, highlighting the controversial nature of claims linking such physical quantities and the performance of the scheme itself. Finally,
Section 5 offers our conclusions and perspectives.
2. Measurement-Free Teleportation
We follow the three-qubit measurement-free teleportation protocol put forward in Ref. [
3], whose quantum circuit we present in
Figure 1. As our aim is to study the information back-flow and non-Markovianity in the protocol itself, it is appropriate to use the language of open quantum systems when describing it. We thus use the label
S for the
system whose state is being teleported, and
for the
environmental particles that are ancillary for the protocol.
Two agents, conventionally identified as Alice and Bob, hold control of the full register consisting of system and environment as per
Figure 1. Alice aims to teleport to Bob the state
, which she encodes in qubit
S. On the other hand, qubits
make up a resource state.
In the original formulation of the measurement-free protocol in Ref. [
3], the teleported state was encoded in the degrees of freedom of one of the environmental qubits (specifically,
). The authors of Ref. [
4] proposed an altered version of the scheme where
is recovered from qubit
S. This, as mentioned before, put them in a position to argue for a direct relationship between the quality of retrieval of
from the degrees of freedom of
S and the non-Markovian character of the system-environment dynamics. As our scope is to critically assess the actual implications of non-Markovianity for the effectiveness of the protocol, we will adhere to the formulation in Ref. [
4]. We emphasize that, though this modification is beneficial for studying and understanding the underlying dynamics, we would use the original formulation in Ref. [
3] to teleport states in a quantum computation context.
In an ideal teleportation scheme, the resource encoded in the
-
compound would be a maximally entangled Bell state. Here, to study the necessary correlations for the protocol at hand, we weaken this strong requirement and take the resource to be the Werner state
where
and
is a Bell state. Unless
, there are correlations present in
: it is entangled for
and carries quantum discord and classical correlations for
.
Alice sends
S and
through the circuit to Bob. The operations
undergone by the
S-
compound can be grouped into three
blocks to highlight their roles in the process. The first is
Here
stands for the Hadamard gate and
is a controlled-NOT gate (with
X and
Z the Pauli
x and
z matrix, respectively). Owing to the interaction entailed by such a gate, quantum correlations might be established between
S and
through the action of
. At this point, at least some of the information about the system state is encoded in the form of system-environment correlations. The degree to which this happens, though, depends on the initial state of the system: for
,
S and the environment remain uncorrelated, while the total correlations are maximized for
. At this stage, due to the initial correlations within the environment, all elements of the register would be quantum correlated, in general.
The second block takes the form of the unitary gate
with
the swap gate which, for any state
, acts as
In the ideal case where the environment is initially maximally entangled (i.e., for ), the operation acts to decouple S and , therefore localizing the information encoded through between S and only. Some correlations do remain between all three systems for .
The final block of unitaries of the protocol is
2.1. Effective Depolarizing-Channel Description
In the original protocol with
[
3], all system-environment correlations vanish after the application of
, and all the information about the input state is localized in the desired system. In the version discussed here, with an imperfect resource, the success of the protocol grows with
p.
To see this quantitatively, we resort to an effective description of the dynamics undergone by system
S as a result of the action of the quantum circuit and its coupling to the environmental qubits. We call
the initial state of the system qubit, label as
the total unitary of the circuit and decompose the identity
in the Hilbert space of the environmental compound over the Bell basis
, where
and
and we introduce the remaining Bell states
The final state of
S thus reads
Upon inserting Equation (
1) into this expression, we have
where the symbol
stands for the summation over all the elements of the Bell basis except
and we have introduced the operators
of the open-system dynamics undergone by
S. An explicit calculation leads to the results summarized in
Table 1.
Using such expressions, we are finally able to recast the final state of the system in the form of the operator-sum decomposition
with
which immediately gives
and allows us to conclude that the action of the measurement-free teleportation protocol on the state of the system is that of a depolarizing channel acting with a resource-dependent rate
. The corresponding state fidelity with
reads
thus increasing linearly from
when
, to 1 when
. The role of the
gate in
is to transfer the information on
otherwise encoded in the state of
to the system qubit
S. As already anticipated, the inclusion of this gate in the protocol allows us to characterize the quality of the teleportation performance in terms of state-revival in the system qubit.
2.2. Distinguishability and Non-Markovianity Resulting from the Dynamics
While this analysis shows the non-trivial nature of the overall action of the quantum circuit on the state of
S, it is instructive to dissect the effects of the individual
, particularly in terms of the degree of distinguishability of different input states of
S. To do this quantitatively, we make use of the instrument embodied by the trace distance between two quantum states. This is defined as
where
are two arbitrary density matrices and
is the trace norm of an arbitrary matrix
A.
First, let us consider the action of
on the initial state of
S. Following an approach fully in line with the one formalized in Equation (
8) but for
and by labeling the state of
S resulting from the application of this block of unitaries alone as
, so as to emphasize the dependence on the initial-state parameter
, we have
where we have introduced the Kraus operators
and
, which are written in terms of the eigenstates
of
such that
. We thus consider
where
(without loss of generality) identifies two different initial states of the system. Having in mind the analysis of the degree of non-Markovianity that will be presented later in this work, we take
and
(so as to prepare
S in eigenstates of
) and thus consider fully distinguishable input states. For such a choice, we have
, achieving again full distinguishability regardless of the properties of the environmental system (as
does not depend on
p).
As for
, it is clear from
Figure 1 that this block of unitaries is
local with respect to the bipartition
S-
, i.e.,
does not contain degrees of freedom of
S, which implies that the corresponding operator-sum decomposition of the effective channel acting on the system involves only the identity operator
. The evolved state
after
is thus identical to Equation (
12). Notice, though, that the state of the environment will be changed by this part of the circuit.
Finally, block
will need to be applied to the—in general quantum correlated—joint state of
S-
. This immediately gives evidence of the fundamental difference between the action entailed by
and the other blocks of operations: while, as for
, this operation couples
S to the environment, the input state to
is a state that features, as mentioned above, system-environment correlations that
may play a key role in determining the nature of the dynamics of
S. Technically, such correlations prevent us from using the same approach as above to identify the effective channel acting on
S. Instead, we will have to calculate
with
the output state of the system-environment compound after application of
, and
the
p-dependent dynamical map resulting from taking the trace over the environmental degrees of freedom. The trace distance between two input states of
S reads
This shows that the last block of the quantum circuit at hand is the only one that could change the degree of distinguishability between the input state and the evolved one, which, in general, shrinks linearly with the depolarization rate.
4. Information Back-Flow and Correlations
In the previous section, we discovered that the relation between non-Markovianity, the performance of the teleportation scheme, and the entanglement in the initial state of the environment depends on whether the implementation of the circuit allows for the consideration of the individual
gates rather than the blocks of unitaries playing key roles in the evolution of the state of
S. In the latter arrangement, the dynamics of
S is non-Markovian for
; in the former, non-Markovianity is present in the map evolving
S even when the environment is in a separable state, thus breaking the connection established in Ref. [
4]. We now study the relation between non-Markovianity of the dynamics and system-environment correlations as time evolves.
Initially, correlations are only present in the environmental Werner state. As done previously, we begin by describing the dynamics using the Hamiltonian in Equation (
24).
Figure 5a displays the entanglement between the system and environment as time evolves from an input state of
, as measured by the logarithmic negativity [
24,
25]
to quantify the entanglement in the bipartition
S-vs-
, where
is the partial transpose of the evolved state
S-
compound with respect to
S. As might have been expected, the larger the initial environmental entanglement (as related to
p), the more entanglement is shared between
S and
during the protocol, and this corresponds to a larger degree of non-Markovianity.
However, the growth in entanglement when
is quite surprising:
S and the environment are more entangled when
than when such parameter takes a small (
) yet non-zero value, even though the environment has more quantum and classical correlations, initially, in this case. This could also relate to correlations of a nature that are different from entanglement. To address this, we use figures of merit for quantum and classical correlations defined as in Refs. [
26,
27]. First, we quantify classical correlations in a bipartite system composed of
A and
B using the generalized conditional entropy
where
is a POVM on system
B,
and
is the state of
after system
B has been measured with
. This enables us to find the maximum information we can gain about system
A by measuring system
B. As for quantum correlations, we resort to discord [
28], namely the difference between total correlations (as measured by the quantum mutual information) and classical correlations
For simplicity, the maximum entailed by the definition of
will be sought over all projective measurements only, following the examples in Refs. [
29,
30]. While this is accurate and rigorous only for two-qubit systems, for our three-qubit problem, we will only be able to quantify lower (upper) bounds to classical (quantum) correlations.
Starting from the same initial state of
, discord and classical correlations for the bipartition
S-vs-
are shown in
Figure 5b,c, where we can appreciate a behavior that is, qualitatively, the inverse of entanglement: larger degrees of discord and mutual information are found in the state at hand as
p decreases, which is somewhat counterintuitive. Therefore, while entanglement and non-Markovianity may be connected, we can conclude that discord and classical correlations are not linked to non-Markovianity.
It is important to note that at the end of the protocol (i.e., for ), only classical system-environment correlations remain for . The information about is encoded in such correlations, and thus, information back-flow is prevented. This is the reason behind the reduced success of the protocol as p diminishes.
As the initial state of the system directly affects how entanglement is shared during the protocol (as highlighted in
Section 2), without affecting the performance of the protocol, we addressed the case of inputting state
rather than
. However, the results were similar to all the same features visible in the behavior of each figure of merit of correlations.
As in
Section 3.2, we now change the dynamics to that in Equation (
25), and thus assume that each gate can be independently performed one by one. We begin, as before, with the initial system state
. The correlations are shown in
Figure 6, which displays some similarities with the study performed in
Figure 5. As in the previous case, the entanglement between
S and
is larger for larger
p [cf.
Figure 6a]. However, for this implementation of the quantum circuit operations, there is no unexpected growth in entanglement for
. Moreover, entanglement only appears when the final gate of the circuit is performed, which is precisely when the trace distance rises in
Figure 3, signaling non-Markovianity. This all heavily implies that entanglement is necessary for non-Markovian dynamics in the protocol.
The discord and classical correlations in
Figure 6b,c also share features of those in
Figure 5; they are both larger for smaller
p. When
, these correlations grow and vanish only during
, the SWAP gate between
S and
. However, they can also appear during
when
.
At first glance, the two types of dynamics seem to result in similar dynamics. However, we see stark changes when we change the system’s initial state. Although the features of the correlation dynamics remain much the same for the dynamics in Equation (
24), they are remarkably different when we change the initial state from
to
when the Hamiltonian is that in Equation (
25). This can be easily seen by comparing
Figure 6 and
Figure 7. After the initial CNOT operation, the system and environment become entangled; this is reflected in both
Figure 7a,b. This means that we now see more discord between
S and
for larger
p rather than smaller; the opposite trend when the initial state is
. However, entanglement is similar; the more entanglement, the more non-Markovianity. Now, we see a small spike during the final gate of the circuit, similar to the unusual resurgence of entanglement when
in the overlapping gates case.