# Correlations and energy in mediated dynamics

Tanjung Krisnanda  School of Physical and Mathematical Sciences, Nanyang
Technological University, 637371 Singapore, Singapore  Su-Yong Lee  Current
address: Agency for Defense Development, Daejeon 34186, Korea School of
Computational Sciences, Korea Institute for Advanced Study, Hoegi-ro 85,
Dongdaemun-gu, Seoul 02455, Korea  Changsuk Noh  Kyungpook National
University, Daegu 41566, Korea  Jaewan Kim  School of Computational Sciences,
Korea Institute for Advanced Study, Hoegi-ro 85, Dongdaemun-gu, Seoul 02455,
Korea  Alexander Streltsov  Centre for Quantum Optical Technologies, Centre of
New Technologies, University of Warsaw, 02-097 Warsaw, Poland  Timothy C. H.
Liew  School of Physical and Mathematical Sciences, Nanyang Technological
University, 637371 Singapore, Singapore  MajuLab, International Joint Research
Unit UMI 3654, CNRS, Université Côte d’Azur, Sorbonne Université, National
University of Singapore, Nanyang Technological University, Singapore  Tomasz
Paterek  Institute of Theoretical Physics and Astrophysics, Faculty of
Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk,
Poland

###### Abstract

The minimum time required for a quantum system to evolve to a distinguishable
state is set by the quantum speed limit, and consequently influences the
change of quantum correlations and other physical properties. Here we study
the time required to maximally entangle two principal systems interacting
either directly or via a mediating ancillary system, under the same energy
constraints. The direct interactions are proved to provide the fastest way to
entangle the principal systems, but it turns out that there exist mediated
dynamics that are just as fast. We show that this can only happen if the
mediator is initially correlated with the principal systems. These
correlations can be fully classical and can remain classical during the
entangling process. The final message is that correlations save energy: one
has to supply extra energy if maximal entanglement across the principal
systems is to be obtained as fast as with an initially correlated mediator.

An evolution of a quantum state into a distinguishable one requires finite
time. The shortest time to achieve this task is governed by the quantum speed
limit (QSL). The first lower bound on the shortest time was derived in a
pioneering work by Mandelstam and Tamm [1]. Thereafter, important advancements
and extensions of the QSL were reported, for example, for pure states [2, 3,
4] as well as mixed states [5, 6, 7, 8]. The applications of these fundamental
findings have been valuable in many areas, e.g., in the analysis for the rate
of change of entropy [9], the limitations in quantum metrology [10] and
quantum computation [11, 12], and the limit on charging capability of quantum
batteries [13, 14, 15]. See also Refs. [16, 17] for studies showing the
application of QSL in the classical regime.

The widely accepted time bound for an evolution of a quantum state $\rho$ (in
general, mixed) to another state $\sigma$ is known as the _unified_ QSL [18,
19], which reads

$\tau(\rho,\sigma)\geq\hbar\frac{\Theta(\rho,\sigma)}{\min\\{\langle
H\rangle,\Delta H\\}},$ (1)

where $\Theta(\rho,\sigma)=\arccos(\mathcal{F}(\rho,\sigma))$ denotes a
distance measure known as the Bures angle,
$\mathcal{F}(\rho,\sigma)=\mbox{tr}\left(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)$
the Uhlmann root fidelity [20, 21], $\langle H\rangle=\mbox{tr}(H\rho)-E_{g}$
the mean energy taken relative to the ground level of the Hamiltonian,
$E_{g}$, and $\Delta H=\sqrt{\mbox{tr}[H^{2}\rho]-\mbox{tr}[H\rho]^{2}}$ the
standard deviation of energy (SDE). Note also that other distances have been
employed [19]. In essence, Eq. (1) is often described as a version of time-
energy uncertainty relation as the evolution time is lower bounded by the
amount of energy (mean or variance) initially accessible to the system.

Here we investigate the evolution speed of two principal objects $A$ and $B$,
which interact either directly or via an ancillary system $C$. While direct
interactions place no restrictions on the joint Hamiltonian $H_{AB}$, the
mediated dynamics is mathematically encoded in the assumption that the
tripartite Hamiltonian is a sum $H_{AC}+H_{BC}$ that excludes the terms
coupling $A$ and $B$ directly. Note that local Hamiltonians, i.e., $H_{A}$,
$H_{B}$, and $H_{C}$, are already included in these general forms. These
scenarios are quite generic and applicable to ample situations. We are
interested in contrasting them and in identifying resources different than
energy that play a role in speeding up the evolution. We therefore impose the
same energy constraint (the denominator in Eq. (1)) in both bipartite and
tripartite settings. Under this condition we show achievable minimal time
required to maximally entangle principal systems starting from disentangled
states. It turns out that the mediated dynamics cannot be faster than the
direct dynamics, but it can be just as fast provided that the mediator is
initially correlated with the principal systems. We show additionally, with an
explicit example, that although entanglement gain between $A$ and $B$ is the
desired quantity, the correlations to the mediator can remain classical at all
times, see also Refs. [22, 23]. These results can be interpreted in terms of
trading correlations for energy. If one starts with an uncorrelated mediator
and aims at entangling the principal systems as fast as with a correlated
mediator, additional energy has to be supplied initially. On the other hand,
due to energy conservation, the same energy must be invested in order to
prepare the correlated mediator, see Refs. [24, 25, 26] for a discussion from
a thermodynamic perspective.

##  I Preliminaries

Figure 1 summarises different considered generic scenarios. We shall refer to
the case of direct interactions as $\mathcal{DI}$ and split the mediated
interactions into two cases where mediator $C$ either interacts with the
principal systems at all times ($\mathcal{CMI}$ for continuously mediated
interactions) or where it first interacts with $A$ and then with $B$
($\mathcal{SMI}$ for sequentially mediated interactions). Note that
$\mathcal{SMI}$ in particular covers the case of commuting Hamiltonians
$H_{AC}$ and $H_{BC}$. We begin by explaining the energy constraints imposed
on these scenarios.

Figure 1: Different considered scenarios. The principal objects are denoted by
$A$ and $B$. Our goal is to maximally entangle them as fast as possible,
starting with a disentangled initial state. (a) Direct interactions, with
Hamiltonian $H_{AB}$. (b) Continuous mediated interactions with general
Hamiltonians of the form $H_{AC}+H_{BC}$. (c) Sequential mediated interactions
where $C$ first interacts with $A$, and then with $B$.

Consider, for the moment, a unitary evolution of a quantum state $\rho(0)$ to
$\rho_{\text{tar}}$ generated by a Hamiltonian $H$. One can see from the
unified QSL in Eq. (1) that there are two relevant quantities: one being the
fidelity $\mathcal{F}(\rho(0),\rho_{\text{tar}})$ between the initial and
target state and the other $\min\\{\langle H\rangle,\Delta H\\}$, which is the
minimum of the non-negative mean energy or SDE. It is straightforward to check
that scaling of the Hamiltonian, $H\rightarrow kH$, where $k$ is a constant,
leads to the rescaled energy factors $\langle H\rangle\rightarrow k\langle
H\rangle$ and $\Delta H\rightarrow k\Delta H$. A trivial option to speed up
the evolution of the quantum state is therefore to supply more energy, e.g.,
by having stronger coupling. We wish to focus on other quantities playing a
role in the speed of evolution and therefore, in what follows, we put the
strength of all interactions on equal ground by setting $\min\\{\langle
H\rangle,\Delta H\\}=\hbar\Omega$, where $\Omega$ is a frequency unit. This
allows us to write the unified QSL in Eq. (1) as

$\Gamma(\rho(0),\rho_{\text{tar}})\geq\frac{\Theta(\rho(0),\rho_{\text{tar}})}{\min\\{\langle
M\rangle,\Delta M\\}},$ (2)

where $\Gamma=\Omega\tau$ stands for the dimensionless minimal time, whereas
$\langle M\rangle=\langle H\rangle/\hbar\Omega$ and $\Delta M=\Delta
H/\hbar\Omega$ respectively denote the non-negative mean energy and SDE,
normalised with respect to $\hbar\Omega$. Hereafter, we assume the condition

$\min\\{\langle M\rangle,\Delta M\\}=1,$ (3)

which can always be ensured with appropriate scaling $k$. We refer to this
condition as _resource equality_.

To quantify the amount of entanglement, we use negativity, which is a well
known computable entanglement monotone [27, 28, 29, 30, 31]. Negativity is
defined as the sum of negative eigenvalues after the state of a bipartite
system is partially transposed. The bipartite entanglement between objects $X$
and $Y$ is denoted by $N_{X:Y}$ and admits maximum value $(d-1)/2$, where
$d=\min\\{d_{X},d_{Y}\\}$ and $d_{X}$ ($d_{Y}$) is the dimension of object $X$
($Y$). For simplicity, we shall assume that the principal objects have the
same dimension. Maximally entangled states, for any entanglement monotone
[32], are given by pure states of the form

$|\Psi_{XY}\rangle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|x_{j}\rangle|y_{j}\rangle,$
(4)

where $\\{|x_{j}\rangle\\}$ and $\\{|y_{j}\rangle\\}$ are orthonormal bases
for object $X$ and $Y$, respectively.

##  II Direct interactions

Let us begin with optimal entangling dynamics for any dimension $d$, with
direct interactions. Since the initial state we take is disentangled, it has
to be a pure product state as the dynamics is purity preserving and the final
maximally entangled state is pure, see Eq. (4). One easily verifies with the
help of Cauchy-Schwarz inequality that the fidelity between a product state
and maximally entangled state is bounded as
$\mathcal{F}=\langle\alpha\beta|\Psi_{AB}\rangle\leq 1/\sqrt{d}$. From the
resource equality, the time to maximally entangle two systems via direct
interactions follows

$\Gamma_{\mathcal{DI}}\geq\arccos(\mathcal{F})\geq\arccos\left(1/\sqrt{d}\right).$
(5)

This bound is tight and can be achieved with the following exemplary dynamics.
Under an initial state of $|00\rangle$, we take an optimal (to be shown below)
Hamiltonian

$H_{AB}=\frac{\hbar\Omega}{2\sqrt{d-1}}\>\sum_{j=1}^{d-1}(X_{A}^{j}+Y_{A}^{j})\otimes(X_{B}^{j}+Y_{B}^{j}),$
(6)

where the subscripts indicate the corresponding system and we have defined
$X^{j}\equiv|0\rangle\langle j|+|j\rangle\langle 0|$ and
$Y^{j}\equiv-i|0\rangle\langle j|+i|j\rangle\langle 0|$. Note that the
constant factor ensures the resource equality. One can show that the state at
time $t$ takes the form $\left|\psi_{AB}(t)\right\rangle=\cos(\Omega
t)\left|00\right\rangle+\sin(\Omega
t)(\sum_{j=1}^{d-1}\left|jj\right\rangle)/\sqrt{d-1}$, and therefore it
oscillates between the disentangled state $\left|00\right\rangle$ and a
maximally entangled state $|\Psi_{AB}\rangle$. The latter is achieved earliest
at time $T\equiv\Omega t=\arccos(1/\sqrt{d})$, see Fig. 2.

Figure 2: Optimal direct dynamics showing maximum entangling speed between two
objects, each with dimension $d$. Maximum entanglement, $(d-1)/2$, is achieved
at $T=\arccos(1/\sqrt{d})$, indicated by the dots.

Alternatively, the optimality of this dynamics can be understood from the
triangle inequality of the Bures angle [33]:
$\Theta(0,T)+\Theta(T,\arccos(1/\sqrt{d}))\geq\Theta(0,\arccos(1/\sqrt{d}))$,
where we have used a short notation
$\Theta(T_{1},T_{2})\equiv\Theta(\rho(T_{1}),\rho(T_{2}))$. Under the resource
equality, the optimal time should be equal to the Bures angle. Indeed this is
the case for the above dynamics as $\Theta(T_{1},T_{2})=T_{2}-T_{1}$,
_saturating_ the triangle inequality. Therefore, not only the maximally
entangled state is reached in the shortest time, but also all intermediate
states as well.

The described fastest entangling dynamics has the following special features.
(i) The Bures angle between any two states in the dynamics is proportional to
entanglement gain, so that QSL directly translates to the limits on
entanglement generation. (ii) This generation has its origin in components
$(\sum_{j=1}^{d-1}\left|jj\right\rangle)/\sqrt{d-1}$ and the high entangling
speed comes from the fact that already the linear term in the expansion of the
evolution operator $\exp{(-i\Delta tH_{AB}/\hbar)}$ introduces these
components. That is, the rate of change of entanglement is strictly positive
$\dot{N}_{A:B}(t)>0$, for all times up to maximally entangling time.

##  III Can mediator speed up entangling process?

At first sight, one might wonder whether the use of quantum mechanical
mediator could speed up the evolution by utilising non-commuting Hamiltonians,
as revealed through the Baker-Campbell-Hausdorff (BCH) formula. Namely, the
dynamics generated by direct coupling $H_{AB}=A\otimes B$ could be
reconstructed through the mediator system $C$ interacting via
$H_{AC}+H_{BC}=A\otimes p_{C}+x_{C}\otimes B$, where $x_{C}$ and $p_{C}$ are
the position and momentum operators acting on the mediator. Due to the
canonical commutation relation the BCH equation reduces to:

$\displaystyle e^{-it(A\otimes p_{C}+x_{C}\otimes B)/\hbar}$ $\displaystyle=$
$\displaystyle e^{-itA\otimes p_{C}/\hbar}\,e^{-itx_{C}\otimes B/\hbar}$ (7)
$\displaystyle e^{-it^{2}A\otimes B/2\hbar}.$

Effective direct coupling is now identified in the last term on the right-hand
side. Since the corresponding exponent is proportional to squared time, it is
legitimate and interesting to enquire about the speeding up possibility.

On the other hand, the special features described at the end of the previous
section make it unlikely that any other dynamics is faster than the fastest
direct one. Indeed, this is shown in Theorem 1 presented in the Appendix. Any
dynamics (direct or mediated) that starts with disentangled principal systems
can maximally entangle them in time lower bounded as

$\Gamma_{\text{any}}\geq\arccos\left(1/\sqrt{d}\right),$ (8)

where the resource equality is assumed. One then wonders whether mediated
dynamics can achieve the same speed as the direct one. At this stage initial
correlations with the mediator enter the picture.

We shall show that if the mediator is initially completely uncorrelated from
the principal systems, the time required to reach the maximally entangled
state is _strictly_ larger than $\arccos(1/\sqrt{d})$. Then we provide
explicit examples of mediated dynamics, with initially correlated mediators,
that achieve the shortest possible entangling time.

Consider the initial tripartite state of the form
$\rho(0)=\rho_{AB}\otimes\rho_{C}$ (with separable $\rho_{AB}$) and, to give a
vivid illustration first, take a Hamiltonian
$H_{AC}+H_{BC}=(H_{A}+H_{B})\otimes H_{C}$, or any commuting Hamiltonians for
which one can identify common eigenbasis $\\{|c\rangle\\}$. Let us take a
specific product state $\left|\alpha\beta\gamma\right\rangle$ in the
decomposition of the initial state $\rho(0)$, and write
$\left|\gamma\right\rangle=\sum_{c}\lambda_{c}\left|c\right\rangle$. Since
$[H_{AC},H_{BC}]=0$ the evolution is mathematically equivalent to
$U_{BC}U_{AC}=\exp(-itH_{BC}/\hbar)\exp(-itH_{AC}/\hbar)$ and the initial
product state evolves to
$\left|\psi(t)\right\rangle=\sum_{c}\lambda_{c}|\alpha_{c}(t)\rangle|\beta_{c}(t)\rangle|c\rangle$,
where
$|\alpha_{c}(t)\rangle=\exp(-itE_{c}H_{A}/\hbar)\left|\alpha\right\rangle$ and
$|\beta_{c}(t)\rangle=\exp(-itE_{c}H_{B}/\hbar)\left|\beta\right\rangle$ with
the corresponding eigenvalue $E_{c}$ of the Hamiltonian $H_{C}$. By tracing
out system $C$ we note that the state of $AB$ is a mixture of product states
and hence not entangled. Application of this argument to all the product
states in the decomposition of $\rho(0)$ shows that this evolution cannot
generate any entanglement between the principal systems whatsoever, i.e.,
$\Gamma_{\mathcal{CMI}}=\infty$ in this case. This stark contrast with the QSL
comes from the fact that the Bures angle is no longer related to the amount of
entanglement in the subsystem $AB$.

Consider now a general Hamiltonian $H_{AC}+H_{BC}$. In Theorem 2 presented in
the Appendix we show that starting with $\rho(0)=\rho_{AB}\otimes\rho_{C}$ the
mediated dynamics has non-positive entanglement rate at time $t=0$, i.e.,
$\dot{N}_{A:B}(0)\leq 0$ if the three systems are open to their local
environments and $\dot{N}_{A:B}(0)=0$ for any closed mediated tripartite
system. This delay is causing a departure from the optimal entangling path and
cannot be compensated in the future. We show rigorously in Theorem 3 presented
in the Appendix that starting with an uncorrelated mediator, i.e.,
$\rho(0)=\rho_{AB}\otimes\rho_{C}$ the time required to maximally entangle $A$
and $B$ via $\mathcal{CMI}$ satisfies a strict bound

$\Gamma_{\mathcal{CMI}}>\arccos{(1/\sqrt{d})}.$ (9)

Furthermore, we have performed numerical checks with random initial states and
Hamiltonians (see the Appendix for details) and conjecture that the actual
time to maximally entangle the principal systems with initially uncorrelated
mediator is $\Gamma_{\mathrm{conj}}\geq 2\arccos(1/\sqrt{d})$. The following
two examples with three quantum bits shed light on the origin of this
hypothetical lower bound. As initial state, consider $\left|000\right\rangle$,
in the order $ABC$, and first take a Hamiltonian
$H=\hbar\Omega(X_{A}Y_{C}+Y_{B}X_{C})/\sqrt{2}$, where $X$ and $Y$ denote
Pauli operators for the respective qubits. One verifies that the resource
equality holds and the state at time $t$ reads
$\left|\psi(t)\right\rangle=\cos(\Omega t)\left|000\right\rangle+\sin(\Omega
t)|\psi^{+}\rangle\left|1\right\rangle$, where
$|\psi^{+}\rangle=(\left|01\right\rangle+\left|10\right\rangle)/\sqrt{2}$ is
the Bell state. The maximally entangled state is obtained at time $\Omega
t=\pi/2$ because one has to wait until the dynamics completely erases the
$\left|000\right\rangle$ component. In contradistinction, the direct dynamics
introduces $\left|11\right\rangle$ component (already in linear time $\Delta
t$) and hence the evolution can stop at $\Omega t=\pi/4$. Another natural way
to entangle two systems via mediator is to entangle the mediator with one of
the systems first and then swap this entanglement. Each of these processes
takes time at least $\arccos(1/\sqrt{d})$ and hence again we arrive at the
bound anticipated above (the swapping step actually takes a bit longer, see
Appendix E). A rigorous proof of this bound is left as an open problem.

We finally give examples of mediated dynamics, starting with a correlated
mediator, that entangles as fast as the fastest direct dynamics. One may think
of utilising an extreme option where the dynamics is initialised with a
maximally entangled mediator. This is indeed possible but it is also possible
to utilise purely classical correlations with the mediator. Let us start with
the entangled mediator first. Consider three qubits with an initial state and
the Hamiltonian written as

$\displaystyle\left|\psi(0)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\left|000\right\rangle+\left|111\right\rangle),$
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{\hbar\Omega}{2\sqrt{2}}(Z_{A}\otimes
H_{C_{1}}+Z_{B}\otimes H_{C_{2}}),$ (10)

where $H_{C_{1}}=-(\openone+X_{C}+Y_{C}+Z_{C})$ and $H_{C_{2}}=\openone-
X_{C}-Y_{C}+Z_{C}$. The principal system is initially disentangled but the
mediator is maximally entangled with the rest of the systems,
$N_{AB:C}(0)=1/2$. One verifies that $N_{A:B}$ follows the curve for $d=2$ in
Fig. 2.

As mentioned, quantum correlations to the mediator are not necessary. Consider
the following example:

$\displaystyle\rho(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left|\psi_{m}\right\rangle\left\langle\psi_{m}\right|\otimes\left|0\right\rangle\left\langle
0\right|+\frac{1}{2}|\tilde{\psi}_{m}\rangle\langle\tilde{\psi}_{m}|\otimes\left|1\right\rangle\left\langle
1\right|,$ $\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{\hbar\Omega}{2}(Z_{A}\otimes Z_{C}+Z_{B}\otimes Z_{C}),$
(11)

where
$\left|\psi_{m}\right\rangle=(\left|+-\right\rangle+\left|-+\right\rangle)/\sqrt{2}$
and
$|\tilde{\psi}_{m}\rangle=(\left|--\right\rangle+\left|++\right\rangle)/\sqrt{2}$
are two Bell-like states of $AB$ with
$|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$. This example is similar to
those in Refs. [22, 23] used to demonstrate entanglement localisation via
classical mediators and to indicate that controlled quantum teleportation can
be realised without genuine multipartite entanglement [34]. Note that
initially the principal system is disentangled (an even mixture of Bell
states) and this time the mediator is only classically correlated — its states
flag in which maximally entangled state is the principal system [35].
Furthermore, Hamiltonians $H_{AC}$ and $H_{BC}$ in Eq. (III) commute, with the
common $Z$ eigenbasis, and hence in the absence of initial correlations with
the mediator entanglement in the principal system would be impossible. One can
now verify via standard computations that the dynamics of $N_{A:B}$ resulting
from Eq. (III) is the same as in Fig. 2 for $d=2$. Note that the states of the
mediator are the eigenstates of $H$ and hence they are stationary.
Accordingly, only the Bell-like states evolve in time. It has been shown
recently in a general case of $\mathcal{CMI}$ where the state contains only
classical correlations in the partition $AB:C$ at all times, that the
entanglement gain, quantified by the relative entropy of entanglement [36], is
bounded by the initial mutual information, i.e., $E_{A:B}(t)-E_{A:B}(0)\leq
I_{AB:C}(0)$ [23]. In the particular example of Eq. (III) this bound is
achieved as we initially have $I_{AB:C}(0)=1$ and $E_{A:B}(0)=0$ which get
converted to maximal entanglement $E_{A:B}(T)=1$. More generally, for the task
discussed here an immediate strategy is to start with at least $I_{AB:C}(0)$
equal to the entanglement $E_{A:B}$ of the target state $|\Psi_{AB}\rangle$.

##  IV Sequential mediated interactions

At last we move to the $\mathcal{SMI}$ scenario, where system $C$ first
interacts only with $A$ and then only with $B$. This setting was studied to
some degree in Ref. [37] where, in the present context, it was found that in
order to prepare a maximally entangled state between the principal systems the
dimension of $C$ has to be at least $d$. We therefore set it to $d$ and take
as initial state $\rho(0)=\rho_{AB}\otimes\rho_{C}$. Under these conditions
Theorem 4 in the Appendix shows the following lower bound on the entangling
time:

$\Gamma_{\mathcal{SMI}}\geq\arccos(1/\sqrt{d})+\arccos(1/d).$ (12)

Our numerical simulations indicate that this bound is tight. Note that this is
even longer than $2\arccos(1/\sqrt{d})$ already demonstrated to be achievable
with $\mathcal{CMI}$.

##  V Discussion

We wish to conclude with a few comments on the obtained results. Since a
maximally entangled state $|\Psi_{AB}\rangle$ is pure and the direct closed
dynamics preserves the purity, the maximal entanglement cannot be achieved via
direct coupling if one starts with a mixed state. After introducing an
ancillary system, the reduced $AB$ dynamics is, in general, not unitary and
hence the purity of $\rho_{AB}$ may change. For a concrete example see below
Eq. (III), where the initial purity of $1/2$ is increased to $1$ while the
disentangled initial state becomes maximally entangled. Therefore, for states
of $AB$ that are initially mixed, the only way to achieve maximum entanglement
and saturate the time bound of $\mathcal{DI}$ is to make use of a correlated
mediator.

Having said this, a possibility emerges to maximally entangle initially mixed
principal systems by opening just one of them to a correlated local
environment. This is reasonable because the incoherent evolution may increase
the purity of $\rho_{AB}$ and previously established entanglement with the
environment can flow to the principal systems. A simple example is as follows.
Suppose $A$ and $B$ are qubits and only qubit $A$ interacts with its single-
qubit environment $C$. As the initial state, we take the one in Eq. (III) and
consider a Hamiltonian $H=\hbar\Omega\>Z_{A}\otimes Z_{C}$ for the local
interaction with environment. One verifies that the resulting dynamics gives
the same entanglement $N_{A:B}$ as in Fig. 2 for $d=2$.

The last example is interesting from the point of view of open quantum
systems. Note that the mutual information in the principal system grows from
the initial value $I_{A:B}(0)=1$ to the final value $I_{A:B}(\pi/4)=2$. Yet,
subsystem $B$ has not been operated on — only system $A$ interacts with its
local environment. One therefore asks what happens to the data processing
inequality stating that information can only decay under local operations
[33]. The answer is that the inequality is derived for local maps which are
completely positive and trace preserving. Accordingly, the example just given
is likely one of the simplest of non-completely-positive dynamics. Violation
of data processing inequality has already been discussed as a witness of such
forms of evolution [38].

Our main result shows that correlations play a similar role to energy in
speeding up dynamics. In tripartite mediated system A-C-B, where principal
systems $A$ and $B$ are coupled via mediator $C$, it takes strictly longer to
maximally entangle $AB$ when the evolution is initialised with uncorrelated
mediator than when it begins with a correlated mediator. We conjecture that
the required minimal time for the case of uncorrelated mediator is twice as
long. In other words, if one would like to start with an uncorrelated mediator
and reach a maximally entangled state at the same time as with a correlated
mediator, one has to supply twice as much energy.

###### Acknowledgements.

We thank Felix Binder and Varun Narasimhachar for stimulating discussions.
T.K. and T.C.H.L. acknowledge the support from the Ministry of Education
(Singapore) project T2EP50121-0006. C.N. was supported by the National
Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
(NRF-2022R1F1A1063053). J.K. was supported by KIAS Individual Grants
(CG014604) at Korea Institute for Advanced Study. A.S. was supported by the
“Quantum Coherence and Entanglement for Quantum Technology” project, carried
out within the First Team programme of the Foundation for Polish Science co-
financed by the European Union under the European Regional Development Fund.
T.P. was supported by the Polish National Agency for Academic Exchange NAWA
Project No. PPN/PPO/2018/1/00007/U/00001.

##  Appendix A No speeding up with mediators

######  Theorem 1.

Consider dynamics described by a Hamiltonian $H$, involving three objects $A$,
$B$, and $C$ (direct or mediated). For initial states $\rho(0)=\rho_{ABC}$,
having disentangled $\rho_{AB}$, the lower bound on the time required to
maximally entangle $AB$ satisfies

$\Gamma_{\mathrm{any}}\geq\arccos\left(1/\sqrt{d}\right),$ (13)

where the resource equality is assumed.

###### Proof.

In the target state the principal systems are maximally entangled, which
implies that their state is pure and uncorrelated with the mediator $C$, i.e.,
$\rho_{\text{tar}}=|\Psi_{AB}\rangle\langle\Psi_{AB}|\otimes\rho_{C}$. We
evaluate the fidelity of the initial and target states:

$\displaystyle\mathcal{F}(\rho(0),\rho_{\text{tar}})$ $\displaystyle=$
$\displaystyle\mathcal{F}(\rho_{ABC},\left|\Psi_{AB}\right\rangle\left\langle\Psi_{AB}\right|\otimes\rho_{C})$
(14) $\displaystyle\leq$
$\displaystyle\mathcal{F}(\rho_{AB},\left|\Psi_{AB}\right\rangle\left\langle\Psi_{AB}\right|)$
$\displaystyle\leq$
$\displaystyle\max_{p_{j},\left|a_{j}b_{j}\right\rangle}\sqrt{\sum_{j}p_{j}|\langle
a_{j}b_{j}|\Psi_{AB}\rangle|^{2}}$ $\displaystyle\leq$
$\displaystyle\max_{\left|a_{j}b_{j}\right\rangle}|\langle
a_{j}b_{j}|\Psi_{AB}\rangle|=\frac{1}{\sqrt{d}},$

where the steps are justified as follows. The first inequality is due to
monotonicity of fidelity under trace-preserving completely positive maps [39]
(here, tracing out $C$). Then we expressed the disentangled state as
$\rho_{AB}=\sum_{j}p_{j}\>\left|a_{j}b_{j}\right\rangle\left\langle
a_{j}b_{j}\right|$ and used its convexity properties. The final equation
follows from the form of maximally entangled state. Finally, by having the
resource equality, one gets
$\Gamma_{\mathrm{any}}\geq\arccos\left({\mathcal{F}(\rho(0),\rho_{\text{tar}})}\right)\geq\arccos{(1/\sqrt{d})}$.
∎

##  Appendix B No initial entanglement gain with uncorrelated mediator

######  Theorem 2.

Consider the case of $\mathcal{CMI}$, where all objects can be open to their
own local environments (for generality). For initial states where the mediator
is uncorrelated, i.e., $\rho(0)=\rho_{AB}\otimes\rho_{C}$, the rate of any
entanglement monotone follows $\dot{E}_{A:B}(0)\leq 0$.

###### Proof.

We take the evolution of the whole tripartite system following the Lindblad
master equation to include the contribution from interactions with local
environments:

$\displaystyle\frac{\rho(\Delta t)-\rho(0)}{\Delta t}$ $\displaystyle=$
$\displaystyle-i[H,\rho(0)]+\sum_{X=A,B,C}L_{X}\rho(0),$ (15) $\displaystyle
L_{X}\rho(0)$ $\displaystyle\equiv$
$\displaystyle\sum_{k}Q^{X}_{k}\rho(0)Q^{X{\dagger}}_{k}-\frac{1}{2}\\{Q^{X{\dagger}}_{k}Q^{X}_{k},\rho(0)\\}.$

We set $\hbar$ to unity in this proof for simplicity. Note that the first term
in the RHS of Eq. (15) corresponds to the coherent part of the dynamics, while
the second constitutes incoherent processes from interactions with local
environments, that is, the operator $Q^{X}_{k}$ only acts on system $X$. We
take the total Hamiltonian as $H=H_{A}\otimes H_{C}+H_{B}\otimes
H_{C^{\prime}}$ without loss of generality, and note that the proof easily
follows for a general Hamiltonian $H=\sum_{\mu}H_{A}^{\mu}\otimes
H_{C}^{\mu}+\sum_{\nu}H_{B}^{\nu}\otimes H^{\nu}_{C^{\prime}}$.

Following Eq. (15), the state of the principal objects at $\Delta t$ reads

$\displaystyle\rho_{AB}(\Delta t)$ $\displaystyle=$
$\displaystyle\mbox{tr}_{C}(\rho(\Delta t))$ (16) $\displaystyle=$
$\displaystyle\mbox{tr}_{C}(\rho(0)-i\Delta t[H,\rho(0)]+\Delta
t\sum_{X}L_{X}\rho(0))$ $\displaystyle=$ $\displaystyle\rho_{AB}-i\Delta
t[H_{A}E_{C}+H_{B}E_{C^{\prime}},\rho_{AB}]$ $\displaystyle+\Delta
t(L_{A}+L_{B})\rho_{AB},$

where $E_{C}=\mbox{tr}(H_{C}\rho_{C})$ and
$E_{C^{\prime}}=\mbox{tr}(H_{C^{\prime}}\rho_{C})$ denote the initial mean
energies, and we have used $\rho(0)=\rho_{AB}\otimes\rho_{C}$. Also,
$\mbox{tr}_{C}(Q^{C}_{k}\rho_{C}Q^{C{\dagger}}_{k}-\frac{1}{2}\\{Q^{C{\dagger}}_{k}Q^{C}_{k},\rho_{C}\\})=0$
follows from the cyclic property of trace.

Effectively, the evolution of the principal objects leading to
$\rho_{AB}(\Delta t)$, as written in Eq. (16), consists of local Hamiltonians
weighted by the corresponding mean energies $H_{A}E_{C}+H_{B}E_{C^{\prime}}$,
and interactions with respective local environments. Therefore, for any
entanglement monotone, a measure that is non-increasing under local operations
and classical communication, one concludes that $E_{A:B}(\Delta t)\leq
E_{A:B}(0)$, and hence, $\dot{E}_{A:B}(0)\leq 0$. In particular, this holds
for negativity used in the main text. ∎

Unitary dynamics is a special case of Theorem 2 without incoherent
interactions with local environments. Since entanglement monotones are
invariant under local unitary operations $E_{A:B}(\Delta t)=E_{A:B}(0)$ or
$\dot{E}_{A:B}(0)=0$. As a consequence, changes in entanglement between the
principal objects (positive or negative) are only possible if the mediator $C$
is correlated with them.

By applying this argument to the final state
$|\Psi_{AB}\rangle\langle\Psi_{AB}|\otimes\rho_{C}$ and backwards in time, we
conclude that any dynamics (direct or mediated) approaches the final state at
a rate $\dot{E}_{A:B}(T)=0$, clearly seen in Fig. 2.

##  Appendix C Strict bound for uncorrelated mediator

We revisit the condition where $C$ is initially uncorrelated, i.e.,
$\rho(0)=\rho_{AB}\otimes\rho_{C}$, which is a special case of Theorem 1. In
this case, we have

$\mathcal{F}(\rho(0),\rho_{\text{tar}})=\mathcal{F}(\rho_{AB},\left|\Psi_{AB}\right\rangle\left\langle\Psi_{AB}\right|)\>\mathcal{F}(\rho_{C},\rho_{C}^{\prime}),$
(17)

where $\rho_{C}^{\prime}$ is the state of $C$ in the target
$\rho_{\text{tar}}$. The only way to saturate the optimal bound of direct
dynamics is to set $\mathcal{F}(\rho_{C},\rho_{C}^{\prime})=1$, i.e.
$\rho_{C}=\rho_{C}^{\prime}$. Accordingly, the initial state of $AB$ has to be
in a pure product form. Having this in mind, the theorem below shows that the
time bound is still strict.

######  Theorem 3.

For the initial state of the form
$\rho(0)=|\alpha\beta\rangle\langle\alpha\beta|\otimes\rho_{C}$, the time
required to maximally entangle the principal systems via $\mathcal{CMI}$
follows a strict bound

$\Gamma_{\mathcal{CMI}}>\arccos(1/\sqrt{d}).$ (18)

###### Proof.

Recall that the dynamics identified in the $\mathcal{DI}$ case saturates the
triangle inequality and is characterised by a straight line in Bures angles.
Any other optimal dynamics (e.g. generated by other Hamiltonians) has to
follow the same straight line. Along the line the states of $AB$ remain pure
at all times. However, Theorem 2 shows that entanglement gain between $A$ and
$B$ is possible only when the mediating system is correlated with the
principal systems at some time $t$ during the dynamics. In the present case,
this means that at $t$, the state of $AB$ is not pure, in particular, the
mediator is not in a decoupled form
$|\psi_{AB}(t)\rangle\langle\psi_{AB}(t)|\otimes\rho_{C}$, where
$|\psi_{AB}(t)\rangle$ is the state from the optimum $\mathcal{DI}$. Since
$\mathcal{F}(\rho_{AB}(t),|\psi_{AB}(t)\rangle\langle\psi_{AB}(t)|)<1$, we use
the triangle inequality of the Bures angle to conclude the strict bound:

$\displaystyle\Gamma_{\mathcal{CMI}}$ $\displaystyle=$
$\displaystyle\Gamma_{1}+\Gamma_{2}$ (19) $\displaystyle\geq$
$\displaystyle\Theta(0,t)+\Theta(t,\arccos(1/\sqrt{d}))$ $\displaystyle>$
$\displaystyle\Theta(0,\arccos(1/\sqrt{d}))=\arccos(1/\sqrt{d}),$

where $\Gamma_{1}$ and $\Gamma_{2}$ respectively denote the minimum time for
evolution $0\rightarrow t$ and $t\rightarrow\arccos(1/\sqrt{d})$. In other
words, the dynamics strictly does not follow the optimum (straight line) path,
where at time $t$ the state is uniquely
$|\psi_{AB}(t)\rangle\langle\psi_{AB}(t)|\otimes\rho_{C}$. ∎

##  Appendix D Numerical simulations for uncorrelated mediator

Here we present results of numerical simulations behind the conjectured
minimal time of $2\arccos(1/\sqrt{d})$ to maximally entangle the principal
systems with initially uncorrelated mediator. Based on the discussion prior to
Theorem 3, we consider initial states of the form
$\rho(0)=|\alpha\beta\rangle\langle\alpha\beta|\otimes\rho_{C}$ and
Hamiltonians $H=H_{AC}+H_{BC}$ scaled to satisfy the resource equality
condition.

Let us first describe the case of three qubits. We random the initial state,
i.e., $|\alpha\rangle$, $|\beta\rangle$, and $\rho_{C}$ as well as the
Hamiltonians $H_{AC}$ and $H_{BC}$ using the quantinf package by Toby Cubitt.
Fig. 3 presents entanglement generated by such evolutions, with $10^{6}$
random instances at each time. As seen, the fastest time to reach maximum
entanglement of $0.5$ is indeed $2\arccos(1/\sqrt{2})$. We also performed
simulations for the same number of random instances for three qutrits ($d=3$).
In this case, entanglement does not even come close to the maximum possible
value at time $2\arccos(1/\sqrt{3})$, indicating that this is a correct lower
bound on the entangling time.

Figure 3:  Numerical simulations of $\mathcal{CMI}$ with three qubits and
initially uncorrelated mediator. We generated $10^{6}$ random initial states
and Hamiltonians for each time. The points highlighted in blue correspond to
evolution time $2\arccos(1/\sqrt{2})$. The dashed line indicates maximum
entanglement between two qubits.

##  Appendix E Sequential mediated dynamics

######  Theorem 4.

Starting with $\rho(0)=\rho_{AB}\otimes\rho_{C}$, maximal entanglement in AB
is achieved via $\mathcal{SMI}$ in time

$\Gamma_{\mathcal{SMI}}\geq\arccos(1/\sqrt{d})+\arccos(1/d).$ (20)

###### Proof.

The final state has the form
$\rho_{f}=|\Psi_{AB}\rangle\langle\Psi_{AB}|\otimes\rho_{C}$. In this scenario
it is to be obtained by the sequence of operations
$\rho_{f}=U_{BC}U_{AC}\rho(0)U_{AC}^{\dagger}U_{BC}^{\dagger}$. We start with
the following argument

$E_{A:B}(\rho_{f})\leq
E_{A:BC}(\rho_{f})=E_{A:BC}(\,U_{AC}\rho(0)U_{AC}^{\dagger}\,)$ (21)

where the inequality is due to the monotonicity of entanglement under local
operations (here, tracing out $C$) and the equality is due to the fact that
the second unitary, $U_{BC}$, is local in the considered bipartition. Thus the
only way to establish maximal final entanglement between the principal systems
is to already prepare it with operation $U_{AC}$. This consumes time
$\arccos(1/\sqrt{d})$ and requires initial state of $A$ and $C$ to be pure,
i.e. $\left|\alpha\gamma\right\rangle$ because $C$ is not correlated with $AB$
initially (note that it does not pay off to start with partial entanglement in
$\rho_{AB}$). Furthermore, since the final state is pure and we are left with
application of $U_{BC}$ only, the state of particle $B$ also has to be pure.
Summing up, after the first step the tripartite state reads
$|\Psi_{AC}\rangle\left|\beta\right\rangle$. In the remaining step we need to
swap this maximal entanglement into the principal systems. To estimate the
time required by the swapping we compute the fidelity:

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle|\langle\Psi_{AC}|\langle\beta|\Psi_{AB}\rangle|\gamma\rangle|$
(22) $\displaystyle=$
$\displaystyle\frac{1}{d}|\sum_{j=1}^{d}\sum_{k=1}^{d}\langle
a_{j}|a^{\prime}_{k}\rangle\langle\beta|b^{\prime}_{k}\rangle\langle
c_{j}|\gamma\rangle|$ $\displaystyle\leq$
$\displaystyle\frac{1}{d}\sqrt{\sum_{j}|\sum_{k}\langle
a_{j}|a^{\prime}_{k}\rangle\langle\beta|b^{\prime}_{k}\rangle|^{2}}\sqrt{\sum_{j}|\langle\gamma|c_{j}\rangle|^{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{d}\sqrt{\sum_{j,k,l}\langle
a^{\prime}_{l}|a_{j}\rangle\langle
a_{j}|a^{\prime}_{k}\rangle\langle\beta|b^{\prime}_{k}\rangle\langle
b^{\prime}_{l}|\beta\rangle}=\frac{1}{d},$

where we have written $|\Psi_{AC}\rangle=\sum_{j}|a_{j}c_{j}\rangle/\sqrt{d}$
and $|\Psi_{AB}\rangle=\sum_{k}|a^{\prime}_{k}b^{\prime}_{k}\rangle/\sqrt{d}$
as the maximally entangled states (note possibly different Schmidt bases).
Then we used the Cauchy-Schwarz inequality to obtain the third line. Since
$\\{|c_{j}\rangle\\}$ form a complete basis the last square root in the third
line equals $1$ (sum of probabilities). Rewriting the remaining mod-squared
and using the completeness of the bases $\\{|a_{j}\rangle\\}$ and
$\\{|b_{k}^{\prime}\rangle\\}$ we arrive at the final result. The total time
required by both steps is therefore at least
$\Gamma_{\mathcal{SMI}}=\Gamma_{1}+\Gamma_{2}\geq\arccos(1/\sqrt{d})+\arccos(1/d)$.
∎

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${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$. The possibilities found by means
of the computer are given by the following five cases:

$Case$ $(1):$ |  ${{\tilde{f}}^{23}}_{\;\;\>3}=\alpha,\;\;\;\;~{}~{}~{}~{}{{\tilde{f}}^{23}}_{\;\;\>4}=\beta,\;\;~{}~{}\;\;{{\tilde{f}}^{24}}_{\;\;\>4}=\gamma,\;\;\;\;~{}~{}{{\tilde{f}}^{12}}_{\;\;\>1}=\gamma-\alpha.$
---|---
$Case$ $(2):$ |  ${{\tilde{f}}^{23}}_{\;\;\>3}=-\alpha,\;\;\;\;~{}~{}{{\tilde{f}}^{23}}_{\;\;\>4}=\beta,\;\;~{}~{}\;\;{{\tilde{f}}^{24}}_{\;\;\>4}=\alpha,\;\;\;\;~{}~{}{{\tilde{f}}^{12}}_{\;\;\>1}=2\alpha,~{}~{}~{}~{}~{}~{}{{\tilde{f}}^{33}}_{\;\;\>1}=\gamma.$
$Case$ $(3):$ |  ${{\tilde{f}}^{23}}_{\;\;\>4}=\alpha,\;\;\;\;~{}~{}~{}~{}{{\tilde{f}}^{33}}_{\;\;\>1}=\beta,\;\;~{}~{}\;\;{{\tilde{f}}^{34}}_{\;\;\>1}=\gamma,\;\;\;\;~{}~{}{{\tilde{f}}^{12}}_{\;\;\>1}=\frac{2\alpha\gamma}{\beta},~{}~{}~{}~{}~{}~{}{{\tilde{f}}^{23}}_{\;\;\>3}=-\frac{2\alpha\gamma}{\beta}.$
$Case$ $(4):$ |  ${{\tilde{f}}^{33}}_{\;\;\>1}=\alpha,\;\;\;\;~{}~{}~{}~{}{{\tilde{f}}^{33}}_{\;\;\>2}=\beta,\;\;~{}~{}\;\;{{\tilde{f}}^{34}}_{\;\;\>1}=\gamma,\;\;\;\;~{}~{}{{\tilde{f}}^{44}}_{\;\;\>1}=\lambda.$
$Case$ $(5):$ | ${{\tilde{f}}^{24}}_{\;\;\>4}=\alpha,\;\;\;\;~{}~{}~{}~{}{{\tilde{f}}^{34}}_{\;\;\>1}=\beta,\;\;~{}~{}\;\;{{\tilde{f}}^{44}}_{\;\;\>1}=\gamma,\;\;\;\;~{}~{}{{\tilde{f}}^{12}}_{\;\;\>1}=-2\alpha,\;\;\;\;~{}~{}{{\tilde{f}}^{23}}_{\;\;\>3}=3\alpha,$
|
${{\tilde{f}}^{23}}_{\;\;\>4}=-\frac{2\alpha\beta}{\gamma},\;\;\;\;~{}~{}{{\tilde{f}}^{33}}_{\;\;\>1}=\frac{\beta^{2}}{\gamma}.$

Using the formula of isomorphism transformation presented in [22] (see also
[17, 29]) one can simply show that the above dual solutions are isomorphic
with some of the Lie superalgebras listed in Tables 2 to 7. Finally, one may
employ the automorphism Lie supergroup of
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$ and use the method demonstrated
in [22] to obtain all inequivalent Lie superbialgebra structures on
${\mathscr{C}}^{3}\oplus{\mathscr{A}}_{1,1}$.

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