#  Perfect quantum-state synchronization

Jakub Czartowski1, Ronny Müller2, Karol Życzkowski1,3, Daniel Braun2
<EMAIL_ADDRESS>1Institute of Theoretical Physics, Jagiellonian
University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
2Institute of Theoretical Physics, University of Tübingen, Auf der
Morgenstelle 14, 72076 Tübingen, Germany
3Center for Theoretical Physics, Polish Academy of Sciences, 02-668 Warszawa,
Poland

(June 21, 2021)

###### Abstract

We investigate the most general mechanisms that lead to perfect
synchronization of the quantum states of all subsystems of an open quantum
system starting from an arbitrary initial state. We provide a necessary and
sufficient condition for such “quantum-state synchronization”, prove tight
lower bounds on the dimension of the ancilla’s Hilbert space in two main
classes of quantum-state synchronizers, and give an analytical solution for
their construction. The functioning of the found quantum-state synchronizer of
two qubits is demonstrated experimentally on an IBM quantum computer and we
show that the remaining asynchronicity is a sensitive measure of the quantum
computer’s imperfection.

††preprint: APS/123-QED

##  I Introduction

Some of the most spectacular and technologically important quantum effects
appear when a large number of quantum systems are in the same quantum state:
Bose-Einstein condensation (BEC), superconductivity, quantum magnetism,
macroscopic quantum tunneling, or arguably even lasing, where a large number
of atoms emmit photons phase-coherently into a mode of an electro-magnetic
resonator. In a superconductor or BEC a macroscopically occupied mode with a
well defined phase arises that manifests itself in macroscopic quantum
interference relevant for instance in superconducting quantum interference
devices (SQUID) for precision measurements of the magnetic field. Particularly
relevant for these macroscopic manifestations of quantum coherence is the
synchronization of a quantum phase, which does not have a classical
counterpart. Such “quantum-phase synchronization” is therefore beyond the much
studied synchronization of quantum dynamics of systems that synchronize
classically [1, 2, 3, 4], such as coupled harmonic or non-linear oscillators
[5, 6, 7, 8, 8, 9, 10]. Synchronization of spins or spins and an oscillator
was studied in [11, 12, 13, 14, 15, 16]. Quantum-phase synchronization for two
qubits was examined in [17], and for three-level systems in [18].

The goal of this work is to investigate the most general quantum channels that
synchronize not only relevant quantum phases but the full quantum states of
all subsystems, understood as the reduced states of the many-body system. We
dub this kind of synchronization “Quantum-State Synchronization” (QSS). We
hence turn around the so far prevailing approach of studying quantum
synchronization for given systems and ask what are the most general quantum
channels that lead to synchronized behavior of quantum systems? The hope is
that this approach will ultimately lead to engineering new macroscopic quantum
effects.

##  II Setting the scene

Consider a composite system with Hilbert space
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{S}$ with a “system” Hilbert
space that is itself composed of $n$ subsystems,
$\mathcal{H}_{S}=\mathcal{H}_{1}\otimes\ldots\otimes\mathcal{H}_{n}$. The
dimension $N=\text{dim}\mathcal{H}_{i}$, $i=1,\ldots,n$ is the same for all
$n$ subsystems, whereas the ancillary Hilbert space $\mathcal{H}_{A}$ might
have different dimension $M$. Let $\mathcal{B}(\mathcal{H}_{i})$ be the set of
bounded linear hermitian operators on $\mathcal{H}_{i}$. A quantum channel
$\Phi:\mathcal{B}(\mathcal{H}_{S})\to B(\mathcal{H}_{S})$,
$\rho\mapsto\rho^{\prime}=\Phi(\rho)$ is a linear, trace-preserving completely
positive map. A partial trace that returns the reduced state of the $i$-th
subsystem will be denoted as $\Tr_{\bar{i}}\rho$, where $\bar{i}$ stands for
the complement of the $i$-th subsystem.

######  Definition 1.

We call a state $\rho_{\text{sync}}\in\mathcal{B}(\mathcal{H}_{S})$ a
“synchronized quantum state” (SQS) if all its reductions to single-party
systems are identical,

$\Tr_{\bar{i}}\rho_{sync}=\Tr_{\bar{j}}\rho_{sync}$ (1)

for all $i,j\in\quantity{1,\ldots,n}$. A quantum channel $\Phi$ is called a
“quantum-state synchronizer” (QSSR) if and only if for arbitrary initial
$\rho\in\mathcal{B}(\mathcal{H}_{S})$, $\rho^{\prime}=\Phi(\rho)$ is a
quantum-synchronized state.

The definition leads to three immediate consequences: 1.) The set $\cal S$ of
all QSSRs for given $n,N$ is closed under concatenation, i.e. with
$\Phi_{1},\Phi_{2}\in\cal S$, also $\Phi_{2}\circ\Phi_{1}\in\cal S$. 2.) $\cal
S$ is a convex set, i.e. $\forall\,\Phi_{1},\Phi_{2}\in\cal S$, also
$p\Phi_{1}+(1-p)\Phi_{2}\in{\cal S}\,\forall\,p\in[0,1]$. 3.) We can restrict
the study of such channels to pure input states. This follows from linearity
of a channel and the partial trace, and the $P$-function representation of any
state – see Appendix.
One trivial example of a SQS is the maximally mixed state, $\mathbb{I}/N^{n}$,
which has all reductions identical to the local maximally mixed state
$\mathbb{I}/N$. Another class of such states are maximally entangled states
for which reduction to any subsystem results in a maximally mixed state. Even
more general SQSs are permutationally invariant states. These are states for
which $P_{ij}\rho P_{ij}=\rho$, where $P_{ij}$ permutes subsystems $i,j$.
Correspondingly, a quantum channel that describes relaxation to thermal
equilibrium at infinite temperature is a special (albeit trivial) case of a
QSSR, and so is a quantum channel that resets all input states to a maximally
entangled state. While illustrating opposite extremes of synchronizing
channels’ spectrum, the mentioned examples are not so interesting, as all
quantum coherences are destroyed. In general, one would like to get not only
QSS, but also keep the synchronized states as pure as possible, and achieve
the synchronization with as small an ancilla as possible.

QSS is distinct from quantum cloning, where an arbitrary unknown state
$\ket{\psi}$ is “copied” onto a fixed blank state $\ket{0}$,
$\ket{\psi}\otimes\ket{0}\mapsto\ket{\psi}\otimes\ket{\psi}$, which is
possible only approximately [19, 20]. Note that QSS is not excluded by the
quantum no-broadcasting theorem [21]. In analogy to selfcomplementary channels
[22, 23] QSS maps arbitrary initial states to identical reduced states that
are not necessarily the same as any of the initial ones nor is the total final
state required to be a product state.

##  III Necessary and sufficient conditions for SQSs

The pure state $\ket{\psi}\in\mathcal{H}_{S}$ can be decomposed into a linear
superposition of states that are either even or odd under permutation of two
subsystems
$\ket{\psi}=(\ket{\psi}+P_{ij}\ket{\psi})/2+(\ket{\psi}-P_{ij}\ket{\psi})/2\equiv\ket{s_{ij}}+\ket{a_{ij}}$,
where we use $P_{ij}^{2}=\mathbb{I}$, and $i,j=1,\ldots,n$ (see Appendix for
possible generalizations for “anyonic” states). We are only concerned with
permutations of the system’s subsystems, not the ancilla. The states
$\ket{a_{ij}},\ket{s_{ij}}$ are not normalized in general. In particular, if
$\ket{\psi}$ is (anti-)symmetric, $\ket{a_{ij}}$ ($\ket{s_{ij}}$) vanishes,
respectively. Under partial trace we find $\forall j$

$\displaystyle\rho_{i}$ $\displaystyle=$
$\displaystyle\Tr_{\bar{i}}\quantity(\outerproduct{a_{ij}}{a_{ij}}+\outerproduct{s_{ij}}{s_{ij}})$
(2)
$\displaystyle+\Tr_{\bar{i}}\quantity(\outerproduct{a_{ij}}{s_{ij}}+\outerproduct{s_{ij}}{a_{ij}})\,.$

The reduced state $\rho_{j}$ can be found by first permuting systems $i,j$,
$\rho_{j}=\Tr_{\bar{i}}P_{ij}(|\psi\rangle\langle\psi|P_{ij})\,\,\,\forall i$.
Using the transformation properties of states $\ket{a_{ij}},\ket{s_{ij}}$, we
see that $\rho_{j}$ differs from $\rho_{i}$ only by the sign of the second
partial trace in eq.(2). This implies the following condition for SQSs:

######  Proposition 1.

A state $\ket{\psi}\in\mathcal{H}_{S}$ is a SQS iff
$\Tr_{\bar{i}}(\outerproduct{a_{ij}}{s_{ij}}+\outerproduct{s_{ij}}{a_{ij}})=0$
for all $i,j\in\\{1,\ldots,n\\}$ .

This condition is easily extended to a state in the full Hilbert space,
$\ket{\psi}\in\mathcal{H}$ that serves as purification of a mixed state
$\rho\in\mathcal{B}(\mathcal{H}_{S})$ by simply including the ancilla in
$\bar{i}$. The condition in Proposition 1 is equivalent to requiring that
$\Tr_{\bar{i}}|a_{ij}\rangle\langle s_{ij}|$ is purely anti-hermitian for all
$i,j$ (this expression is still an operator on system $i$). Several sufficient
conditions follow: A state is a SQS, $1.)$ if
$\Tr_{\bar{i}}|a_{ij}\rangle\langle s_{ij}|=0$
$\forall\,i,j\in\\{1,\ldots,n\\}$; or $2.)$ if $\ket{\psi}$ has definite
symmetry or anti-symmetry under $P_{ij}$, i.e. $\ket{a_{ij}}=0$ or
$\ket{s_{ij}}=0$ for all $i,j$. In this case $\outerproduct{\psi}{\psi}$ is a
permutationally symmetric state, but we see that the set of SQSs is larger
than the set of permutationally symmetric states. An example of such a state
that is not permutationally symmetric is the two-qubit state
$\ket{\psi}=(0,1,i,0)/\sqrt{2}$ written in computational basis. It leads to
$\rho_{1}=\rho_{2}=\mathbb{I}/2$, i.e. $\ket{\psi}$ is a (trivial) SQS, but
the full density matrix $\outerproduct{\psi}{\psi}$ is not permutationally
symmetric. More involved examples include absolutely maximally entangled (AME)
states for the systems of five and six qubits, AME(5,2) and AME(6,2)
respectively, for which $\forall_{i}\rho_{i}=\mathbb{I}/2$. Hence these states
are synchronized, despite lack of symmetry or antisymmetry. We prove in
Appendix that one cannot mix different symmetries for different pairs $i,j$ in
condition 2.).

##  IV Minimal ancilla

First focus on condition 2.) with full permutational symmetry of the output
state. We then need to find transformations that map any pure input state to
fully symmetrized states. A synchronizing channel that achieves this will be
called a “symmetrizing QSSR”. To this end we represent the quantum channel
$\Phi$ without restriction of generality as

$\Phi(\rho)=\Tr_{E}\quantity[U\quantity(\outerproduct{0}{0}_{E}\otimes\rho)U^{\dagger}],$
(3)

where $\outerproduct{0}{0}_{E}$ is a fixed state of an ancilla $E$, and $U$ a
joint unitary evolution of the system of the $n$ qudits and the ancilla.
Clearly, to achieve a fully symmetric output state for all pure input states,
it is sufficient and necessary to do so for all computational basis states,
i.e. when including the initial state of the ancilla, all states of the form
$\ket{0;j_{1},\ldots,j_{n}}$, $j_{k}=1,\ldots,N$. This state is mapped to a
column of $U$ with components
$\bra{k;i_{1},\ldots,i_{n}}U\ket{0;j_{1},\ldots,j_{n}}=U_{k;i_{1}\ldots
i_{n},0;j_{1}\ldots j_{n}}$. Such a column then needs to be a vector of the
form

$\ket{\psi_{\text{sym}}}=\sum_{k=1}^{M}\sum_{i_{1},\ldots,i_{n}}a_{k;\quantity{i_{1},\ldots,i_{n}}}\ket{k;i_{1},\ldots,i_{n}},$
(4)

where the expansion coefficients $a_{k;\quantity{i_{1},\ldots,i_{n}}}$ depend
on an unordered set of indices of $S$. We denote the space spanned by such
states as
$\mathcal{H}_{\text{sym}}^{N,n}=\text{span}\quantity(\ket{\psi_{\text{sym}}})$.
These states span a subspace of dimension

$\text{dim}(\mathcal{H}_{\text{sym}}^{N,n})=M\binom{n+N-1}{n}\,.$ (5)

Since the $N^{n}$ computational basis states (extended by the fixed initial
state of the ancilla) map to the first $N^{n}$ columns of $U$, and these need
to be orthonormal, the symmetric subspace must accomodate at least $N^{n}$
linearly independent images of the computational basis states. This sets a
tight lower bound on the dimension $M$ of the ancillary Hilbert space. We have
thus proved the following necessary and sufficient condition for the existence
of a symmetrizing QSSR:

######  Proposition 2.

The tight lower bound on the dimension $M$ of the ancilla necessary to
perfectly quantum-state synchronize with a symmetrizing QSSR $n$ qudits of
dimension $N$ prepared in an arbitrary initial state, is

$M\geq
N^{n}\binom{n+N-1}{n}^{-1}\overset{N\rightarrow\infty}{\longrightarrow}n!\,.$
(6)

For $n=2$, $M=2$ is the tight lower bound for any $N$, i.e. two qudits of
arbitrary dimension can be synchronized with a single qubit as ancilla. And a
single harmonic oscillator as ancilla can lead to perfect QSS for any number
of qudits. A short table giving minimal dimensions for low $n$ and $N$ is
given in Appendix. Our result does not contradict earlier findings [24]
according to which the smallest quantum system that can synchronize is a
spin-1, as that work uses a different definition of synchronization, requiring
the existence of a limit cycle.

Alternatively, the channel $\Phi$ can be represented in terms of $M$ Kraus
operators $K_{a}$ satisfying the identity resolution condition,
$\sum_{a=1}^{M}K_{a}^{\dagger}K_{a}=\mathbb{I}$,

$\Phi(\rho)=\sum_{a=1}^{M}K_{a}\rho K_{a}^{\dagger}.$ (7)

The first $N^{n}$ columns of $U$ define the $M$ Kraus operators, which can be
read off as $N^{n}\times N^{n}$ blocks stacked in those first $N^{n}$ columns
of $U$ by the reshuffling $(K_{a})_{ij}=U_{(a-1)N^{n}+i,j},\
i,j=0,...,N^{n}-1$. Since an entire column vector of $U$ has definite
permutational symmetry, the columns of all Kraus operators must have the same
symmetry.

##  V Construction of quantum-state synchronizers

Next, we provide the construction for a symmetrising QSSR. We illustrate each
step by the simplest possible example of two qubits, $n=N=2$, with minimal
ancilla $M=2$. First, choose an arbitrary unitary matrix $U$ of dimension
$M\binom{n+N-1}{n}=6$ and select the first $N^{n}=4$ columns from it. For
instance

$U=\matrixquantity(1&0&0&0&0&0\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}&0&0&0\\\
0&0&0&0&0&1)\rightarrow\matrixquantity(1&0&0&0\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\ 0&0&0&1\\\ \hline\cr 0&0&0&0\\\
0&\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}&0\\\ 0&0&0&0),$ (8)

written in the basis states $\ket{j;i_{1},\ldots,i_{n}}_{S}$ of the
$M\binom{n+N-1}{n}$-dimensional space, where $0\leq i_{1}\leq\ldots\leq
i_{n}\leq N-1$ and $j\in\quantity{1,\ldots,M}$. Now embed the
$\ket{j;i_{1},\ldots,i_{n}}_{S}$ in the $MN^{n}$-dimensional space as

$\ket{j;i_{1},\ldots,i_{n}}_{S}\rightarrow\frac{1}{\mathcal{N}}\sum_{\sigma\in
S_{n}}\ket{j;\sigma(i_{1},\ldots,i_{n})}\,,$ (9)

with $S_{n}$ the permutation group on $n$ elements and $\mathcal{N}$ the
normalisation factor introduced in order to keep unit norm for all basis
states. For $n=2$, the matrix elements that define the map (9) are SU(N)
Clebsch-Gordan coefficients. The symmetric and antisymmetric cases correspond
to the highest and lowest “total angular momentum”. E.g. for $n=2$, an $N$
state system is a pseudo-spin $(N-1)/2$. In the fully symmetric case,

$\ket{j;i_{1},i_{2}}\mapsto\ket{j}\sum_{M=-(N-1)}^{N-1}\ket{J,M}\innerproduct{J,M}{j_{1},j_{2};m_{1},m_{2}}$

with $J={N-1}$, $m_{k}=i_{k}-j_{k}$, $j_{k}=\frac{N-1}{2}$, $k=1,2$, in
standard $j,m$-notation of angular momenta. For $n=3$, the matrix elements are
$3j$-symbols, and so on. For arbitrary $n$ they can always be calculated by
applying the descending collective angular momentum ladder operator to the
state with $i_{k}=N-1$ $\forall k$ (see Appendix for details). Classically,
this procedure is inefficient in $n$ due to the exponential increase of the
number of computational basis states for which matrix elements are needed, but
an efficient quantum algorithm for the calculation of Clebsch Gordan
coefficients exists [25, 26]. In the 2-qubit case
$\ket{j;00}\rightarrow\ket{j;00}$,
$\ket{j;01}\rightarrow\frac{1}{\sqrt{2}}\quantity(\ket{j;01}+\ket{j;10})$,
$\ket{j;11}\rightarrow\ket{j;11}$. After the embedding, revert from the
description with $MN^{n}$-dimensional column vectors of $U$ to $M$ Kraus
operators $K_{i}$,

Fig. 1 summarizes the action of this channel. Remarkable is the rotational
symmetry about any axis passing through the center of Alice’s Bloch sphere,
implying that already synchronized states are mapped to themselves. The
channel is non-unital and leaves large quantum coherences.

Figure 1: Action of the channel with Kraus operators (V) on product states,
$\ket{\psi}=\ket{\psi_{A}}\otimes\ket{\phi_{B}}$. The positions of gray
spheres on the central red sphere represent $\ket{\psi_{A}}$. The gray spheres
are Bob’s Bloch ball, i.e. the position of a point on the gray spheres the
initial state of Bob $\ket{\phi_{B}}$. The small black spheres depict the end-
state of the QSSR,
$\rho_{A}=\rho_{B}=\Tr_{B}(\Phi(\outerproduct{\psi}{\psi}))$. The grids on the
small black spheres are exactly aligned with the gray spheres which allows one
to identify the resulting state based on the positions on the red and gray
spheres, i.e. $\rho_{B}$ corresponds to the state on the small black sphere
(given through its Bloch vector with respect to the gray sphere) with
coordinates of $\ket{\phi_{B}}$ on the gray sphere. The entire figure is
rotationally symmetric about any axis through the origin.

For the second possibility in condition 2.), i.e. that the first $N^{n}$
columns of $U$ are fully antisymmetric, the dimension is $M\binom{N}{n}$; the
rest of the argument is the same as in the fully symmetric case. Hence, we
have $M\geq N^{n}/\binom{N}{n}$, which is always larger than in the symmetric
case. One way to obtain more general QSSRs is to exploit the condition in
Proposition 1, without imposing permutational symmetry in the columns of $U$.
A simple example is given by a channel with the same Kraus operators as in
eq.(V) up to an overall minus sign in the third row in $K_{2}$. The channel
maps any state to a mixture of symmetric and anti-symmetric states, hence a
permutationally symmetric state and hence a SQS. Allowing $\pm$ symmetry
independently in $M$ Kraus operators, $M+1$ different types of QSSRs arise
(only the number of (anti-)symmetric Kraus operators matters), where the
largest $M$ to be considered is $N^{2n}$. Even more generally, the
representation of $P_{ij}$ can be different between the Kraus operators. An
example of such behaviour, which leads to a QSSR manifold with additional
$M\quantity(\frac{N(N-1)}{2}+1)\quantity(n-1)$ dimensions, is given in
Appendix.

Necessary conditions for the existence of more general QSSRs can be obtained
from parameter counting. A single normalized pure states
$\ket{\psi}\in\mathcal{H}$ contains $2(MN^{n}-1)$ real parameters, not
counting the global phase. Eq.(1) are $(n-1)(N^{2}-1)$ conditions. This leaves
$2(MN^{n}-1)-(n-1)(N^{2}-1)$ free real parameters in $\ket{\psi}$ in order to
have a SQS after tracing out the ancilla. Even though the eqs.(1) are non-
linear in the components of $\ket{\psi}$, this parameter count sets a mild
(for $n>2$) lower bound on $M$.

##  VI Experimental realization on a quantum computer

We verified the functioning of the 2-qubit QSSR with a single-qubit ancilla
experimentally by constructing a quantum circuit that realizes (V) – see
Appendix – and executing the quantum circuit on the five qubit “Santiago”
quantum computer of IBM, a noisy intermediate-scale quantum (NISQ) device.

We evaluate the quality of synchronization after the action of the QSSR with
an asynchronicity measure $\mathcal{A}$ based on the spread of the Bloch
vector components $\hat{x}^{(i)}_{j}$ of the individual qubits,
$i=1,\ldots,n$. They are estimated as empirical means from $N_{r}$
measurements for each component,
$\hat{x}^{(i)}_{j}=(1/N_{r})\sum_{\nu=1}^{N_{r}}s_{j}^{(i,\nu)}$, where
$s_{j}^{(i,\nu)}\in\\{\pm 1\\}$ is the measurement outcome of measuring the
Pauli matrix $\sigma_{j}^{(i,\nu)}$ in the $\nu$-th run with a given initial
state, and $j=1,2,3$ for $x,y,z$. To prevent strongly mixed states from
contributing little inspite of large angular spread of the Bloch vectors, the
Bloch vectors are rescaled so that the longest Bloch vector has unit length,

$\displaystyle\tilde{x}^{(i)}_{j}$
$\displaystyle=\hat{x}^{(i)}_{j}/r_{\text{max}}\quad\quad
r_{\text{max}}=\max\\{r^{(1)},...,r^{(n)}\\}$ $\displaystyle r^{(i)}$
$\displaystyle=\sqrt{(\hat{x}_{1}^{(i)})^{2}+(\hat{x}_{2}^{(i)})^{2}+(\hat{x}_{3}^{(i)})^{2}}\quad\tilde{\mu}_{j}=\frac{1}{n}\sum_{i=1}^{n}\tilde{x}^{(i)}_{j}\,.$

The measure $\mathcal{A}$ for a given initial state is then defined as

$\mathcal{A}=\sum_{j=1,2,3}\frac{1}{n}\sum_{i=1}^{n}(\tilde{x}^{(i)}_{j}-\tilde{\mu_{j}})^{2}\,.$
(10)

On perfect devices the estimates of the Bloch vectors are still subject to
statistical noise, which results in a finite value of ${\mathcal{A}}$ for
finite $N_{r}$. The noise decays as $1/N_{r}$ for large $N_{r}$, such that
asynchronicity $\mathcal{A}=0$ signals perfect QSS in that case. For imperfect
QSS, to be expected on a NISQ device, $\mathcal{A}$ will remain finite even
for $N_{r}\to\infty$.

Figure 2:  Experimental implementation of the quantum channel (V). Upper part:
Each bar contains the asynchronicity of the initial state
($\mathcal{A}_{init}$, lower end of dark blue lines ), of the final
experimental state ($\mathcal{A}$, black dots), simulated state (light blue),
plotted against ${\mathcal{A}_{init}}$. The simulation data is free of
hardware noise and shows the statistical limit set by the finite number of
shots. The dashed blue line is given by $\mathcal{A}=\mathcal{A}_{init}$.
Lower part: Experimental and simulated mean Bloch vector lengths
$\mu_{r}=\frac{1}{n}\sum_{i=1}^{n}r^{(i)}$ as function of the initial
asynchronicity.

We used $N_{r}=204800$ shots to estimate each Bloch vector component of
$N_{s}=77$ random initial states. The experimental results of ${\mathcal{A}}$
and the mean Bloch vector lengths are shown in Fig. 2. Despite the imperfect
hardware, the QSSR reduces the asynchronicity of initial states by up to
several orders of magnitude. The drop in ${\mathcal{A}}$ is largest for
initially not well synchronized states, as the final value of ${\mathcal{A}}$
shows little variation. Simulated results for perfect hardware are included to
act as a reference to the statistical limit, set by the finite number of
shots. The difference in simulation and experiment is due to the noisy
hardware, which is not accounted for in the simulation. We see that the
asynchronicity measure is a much more sensitive measure of the hardware errors
than the reduction in purity of the qubits given by the mean Bloch vector
lengths. To assess the overall performance of the QSSR, we average
$\mathcal{A}$ over $N_{s}$ randomly chosen Haar distributed initial states and
define the mean asynchronicity
$\overline{\mathcal{A}}=\frac{1}{N_{s}}\sum_{\gamma=1}^{N_{s}}\mathcal{A}_{\gamma}$,
where $\gamma$ labels the initial states. In our experiments it drops by more
than an order of magnitude, $\text{initial:
}\overline{\mathcal{A}}=0.44\rightarrow\text{final:
}\overline{\mathcal{A}}=0.027\,.$

##  VII Summary

We have found necessary and sufficient conditions and an analytical
construction of non-trivial QSSRs that perfectly synchronize the quantum
states of an arbitrarily large number of quantum systems in arbitrary initial
states. They need an ancilla of minimal lowest dimension that we determined.
They may be considered as templates for end-states of continuous dissipative
processes that lead to quantum state synchronization, opening the path to
identifying hitherto unknown interactions and damping mechanisms that generate
macroscopic quantum phenomena. We demonstrated QSS experimentally on an IBM
quantum computer for the simplest set-up of two working qubits with a single
ancilliary qubit, and introduced an asynchronicity measure capable of
quantifying the hardware errors.

After completion of this work we became aware of [27], in which quantum
synchronization of Markovian open quantum systems was classified based on the
dynamics.

###### Acknowledgements.

We thank Roberta Zambrini, Gianluca Giorgi, and Christoph Bruder for useful
communication. Financial support by Narodowe Centrum Nauki under the grant
numbers DEC-2015/18/A/ST2/00274 and 2019/35/O/ST2/01049 and by Foundation for
Polish Science under the Team-Net NTQC project is gratefully acknowledged. We
also acknowledge use of the IBM Q for this work. The views expressed here are
those of the authors and do not reflect the official policy or position of IBM
or the IBM Q team.

##  Appendix A Restriction to pure states

To see that it is enough to consider pure states as input for finding the most
general quantum-state synchronizer (QSSR), first represent the state of a
single subsystem in the SU(2)-coherent state representation,

$\rho=\int d^{2}\alpha P(\alpha)\ket{\alpha}\bra{\alpha}\,.$ (11)

Here, the $\ket{\alpha}$ are SU(2) coherent states labelled with the complex
parameter $\alpha$. By stereographic projection, we can consider $\alpha$
represented alternatively in terms of polar and azimuthal angle on the unit
sphere. The function $P$ can then be chosen as a real smooth function on the
sphere that exists for all states in arbitrary finite dimensions, see e.g.
[28]. In particular, we can expand all hermitian $\rho_{i}^{(j)}$ operators
that form a basis of $\mathcal{B}(\mathcal{H}_{j})$ in the form (11) with a
$P$-function $P_{i_{j}}^{(j)}$. An arbitrary state in
$\mathcal{B}(\mathcal{H}_{S})$ can then be written as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{i_{1},\ldots,i_{n}=1}^{N^{2}}c_{i_{1},\ldots,i_{n}}\rho_{i_{1}}^{(1)}\otimes\ldots\otimes\rho_{i_{n}}^{(n)}$
(12) $\displaystyle=$ $\displaystyle\int d^{2n}\alpha
P(\bm{\alpha})\ket{\bm{\alpha}}\bra{\bm{\alpha}}\,,$ (13) $\displaystyle
P(\bm{\alpha})$ $\displaystyle\equiv$
$\displaystyle\sum_{i_{1},\ldots,i_{n}=1}^{N^{2}}c_{i_{1},\ldots,i_{n}}P_{i_{1}}^{(1)}(\alpha_{1})\cdot\ldots\cdot
P_{i_{1}}^{(n)}(\alpha_{n})\,,$ (14)

with $c_{i_{1},\ldots,i_{n}}\in\mathbb{R}$ and hence
$P(\bm{\alpha})\in\mathbb{R}$. From linearity of the partial trace it then
follows immediately that if a QSSR synchronizes all pure product states, hence
$\Tr_{\bar{i}}\Phi(\outerproduct{\bm{\alpha}}{\bm{\alpha}})=\Tr_{\bar{j}}\Phi(\outerproduct{\bm{\alpha}}{\bm{\alpha}})$
for all tensor products of SU(2) coherent states $\ket{\bm{\alpha}}$ and all
$i,j=1,\ldots,n$, then also $\Tr_{\bar{i}}\Phi(\rho)=\Tr_{\bar{j}}\Phi(\rho)$
for arbitrary input states $\rho$ and all $i,j$. If a quantum channels
synchronizes all pure product states it synchronizes all input states. The
converse holds trivially.

##  Appendix B Proof of the sharpening of condition 2

Suppose that in a set of three subsystems we have a state $\ket{\psi}$ with an
even symmetry under $P_{12}$, and odd symmetry under $P_{13}$,

$\displaystyle P_{12}\ket{\psi}$ $\displaystyle=\ket{\psi}$ $\displaystyle
P_{13}\ket{\psi}=-\ket{\psi}$ (15)

noted as (12+) and (13-). Then, starting with the three subsystems ordered
initially as 123, we get no sign change under $P_{12}$, noted as (213+).
Continuing a series of alternating (13-) and (12+) swaps, we have (231-), with
the antisymmetry indicated by the - sign, (321-), (312+), (132+), (123-), i.e.
the state has changed the sign even though the ordering of the systems is the
same as at the beginning. In other words,

$\displaystyle\ket{\psi}$ $\displaystyle\rightarrow
P_{12}\ket{\psi}=\ket{\psi}$
$\displaystyle\rightarrow\underbrace{P_{13}P_{12}}_{P_{123}}\ket{\psi}=-\ket{\psi}$
$\displaystyle\rightarrow\ldots$
$\displaystyle\rightarrow\underbrace{\quantity(P_{13}P_{12})^{3}}_{P_{123}^{3}=\mathbb{I}}=-\ket{\psi}$

where it can be proven that the eigenvalues of the 3-cycle operator $P_{123}$
are $\quantity{1,e^{i\frac{2\pi}{3}},e^{i\frac{-2\pi}{3}}}$ and cubing them
gives identity, forcing the cube of the operator to be the identity operator.
Hence, the symmetry must be the same for all pairs $(i,j)$.

##  Appendix C Generalized representation of $P_{12}$ swap on qubits

In the main text we have assumed that $P_{12}^{2}\ket{\psi}=\ket{\psi}$. In
projective Hilbert space, where all states with the same global phase are
identified, one might think about generalizing this to
$P_{12}^{2}\ket{\psi}=e^{2i\Delta}\ket{\psi}$. A special case is the anyonic
behavior given by $P_{12}\ket{\psi}=e^{i\Delta}\ket{\psi}$ with any phase
$\Delta\in\mathbb{R}$ [29, 30]. Different computational basis states might
have different phases under the action of $P_{12}$,

$\displaystyle P_{12}\ket{00}=e^{i\Delta}\ket{00}$ $\displaystyle
P_{12}\ket{01}=e^{i(\Delta+\phi_{01})}\ket{10}$ $\displaystyle
P_{12}\ket{10}=e^{i(\Delta+\phi_{10})}\ket{01}$ $\displaystyle
P_{12}\ket{11}=e^{i(\Delta+\phi_{11})}\ket{11}\,,$ (16)

where the phase related to $\ket{00}$ defines a global phase w.r.t. which the
others are measured. We will make two natural assumptions, however. First, for
an arbitrary state $\ket{\psi}=\sum_{i,j=0}^{1}a_{ij}\ket{ij}$, $P_{12}^{2}$
should return the same state in projective Hilbert space, i.e.

$\displaystyle P_{12}^{2}\ket{\psi}=$ $\displaystyle
e^{2i\Delta}\left(a_{00}\ket{00}+e^{i(\phi_{01}+\phi_{10})}\quantity(a_{01}\ket{01}+a_{10}\ket{10})+e^{2i\phi_{11}}a_{11}\ket{11}\right)$
$\displaystyle=$ $\displaystyle e^{2i\Delta}\ket{\psi}\,.$

This is a physical request: after permuting back the state must differ at most
by a global phase. This implies

$\displaystyle\phi_{11}=n\pi\,,\phi_{01}+\phi_{10}=2m\pi\,,\,\,n,m\in\mathbb{Z}\,.$
(17)

The second one is that for “double states”,
$\ket{\psi_{2}}=a_{00}\ket{00}+a_{11}\ket{11}$ should stay identical up to a
global phase, $P_{12}^{2}\ket{\psi_{2}}=e^{2i\Delta}\ket{\psi_{2}}$, as
physically one cannot keep track of the number of permutations of such states
even modulo 1, such that any differing phases for different double states
would be ill defined. From this we get $\phi_{11}=0$, i.e. in the subspace
$\text{span}(\ket{00},\ket{11})$, $P_{12}$ is represented by
$e^{i\Delta}\mathbb{I}$, whereas in the subspace
$\text{span}(\ket{01},\ket{10})$, $P_{12}$ is represented by

$P_{12}=e^{i\Delta}\matrixquantity(0&e^{i\phi_{01}}\\\
e^{-i\phi_{01}}&0)=e^{i\Delta}(\cos(\phi_{01})\sigma_{x}+\sin(\phi_{01})\sigma_{y})\,.$
(18)

Hence, the eigenvalues of $P_{12}$ are $\pm e^{i\Delta}$ (with $e^{i\Delta}$
three-fold degenerate). The corresponding subspaces correspond to the usual
triplet and singlet states of two spins-1/2, which now aquire the global
phases $\pm e^{i\Delta}$. If we define the $\ket{s_{ij}}$ and $\ket{a_{ij}}$
as basis vectors in these two subspaces, we recover up to the global phases
the situtation studied in the main text, and Proposition 1 applies
accordingly. Additional freedom arises, however, from the fact that the
permutations act on the state in the full Hilbert space $\mathcal{H}$ when we
act on columns of $U$. I.e. there is an additional label for the ancilla, and
so $P_{12}$ can act differently in different subspaces labelled by different
ancillary states, see the first example below. In the second example we use
two different representations of $P_{12}$ that differ by the phases
$\phi_{01},\phi_{10}$ and hence lead to different subspaces with eigenvalue 1
that are then mixed. The considerations extend to higher dimension, yielding a
single $\Delta$ global phase and $\frac{N(N-1)}{2}$, which corresponds to
$\frac{1}{2}\text{dim}\quantity(\text{span}\quantity{\ket{ij}:i\neq j})$.
Moreover, for $n$ primary subsystems one can select $n-1$ exchange operators
which generate the whole permutation group $S_{n}$ and for each of them
representation is independent. Thus, the total number of degrees of freedom
provided by the SWAP operators is given by
$\quantity(n-1)\quantity(\frac{N(N-1)}{2}+1)$.

##  Appendix D  Connection to representations of SU(N)

Here we give more details on the construction of the QSSRs, eq.(9) in the main
text. As mentioned there, the fully symmetric or fully anti-symmetric QSSRs
can be constructed as maximal or minimal angular momentum representations of
SU(N). The fully anti-symmetric QSSRs exist only for sufficiently large local
dimensions, as can be seen by using Young tableaus. Consider the following
SU(2) examples:

$\displaystyle\ydiagram{1}\otimes\ydiagram{1}$
$\displaystyle=\underbrace{{\color[rgb]{1,0,0}\ydiagram{2}}}_{\text{symmetric
space}}\oplus\underbrace{{\color[rgb]{1,0,0}\ydiagram{1,1}}}_{\text{antisymmetric
space}}$ (19)
$\displaystyle\ydiagram{1}\otimes\ydiagram{1}\otimes\ydiagram{1}$
$\displaystyle=\quantity(\ydiagram{2}\oplus\ydiagram{1,1})\otimes\ydiagram{1}$
$\displaystyle=\underbrace{{\color[rgb]{1,0,0}\ydiagram{3}}}_{\text{symmetric
space}}\oplus\ydiagram{2,1}\oplus\ydiagram{2,1}$ (20)

and there is no antisymmetric space beyond $n=2$.

The SU(3) case is slightly different:

$\displaystyle\ydiagram{1}\otimes\ydiagram{1}$
$\displaystyle={\color[rgb]{1,0,0}\ydiagram{2}}\oplus{\color[rgb]{1,0,0}\ydiagram{1,1}}$
(21) $\displaystyle\ydiagram{1}\otimes\ydiagram{1}\otimes\ydiagram{1}$
$\displaystyle={\color[rgb]{1,0,0}\ydiagram{3}}\oplus\ydiagram{2,1}\oplus\ydiagram{2,1}\oplus\underbrace{{\color[rgb]{1,0,0}\ydiagram{1,1,1}}}_{\text{Antisymmetric
space}}\,.$ (22)

Here, an antisymmetric subspace appears, as we can antisymmetrize three
different symbols.

As a final case, consider four parties from SU(4)

$\ydiagram{1}\otimes\ydiagram{1}\otimes\ydiagram{1}\otimes\ydiagram{1}={\color[rgb]{1,0,0}\ydiagram{4}}\oplus
3\,\ydiagram{3,1}\oplus 2\,\ydiagram{2,2}\oplus
3\,\ydiagram{2,1,1}\oplus{\color[rgb]{1,0,0}\ydiagram{1,1,1,1}}\,.$ (23)

In general, the full columns and full rows of the Young tableaus generate the
fully symmetric and fully antisymmetric representations, respectively, which
constitute SQSs and thus a basis for QSSRs.

##  Appendix E Examples of mixed symmetry synchronizing channels

_Mixing $\pm$ symmetries between Kraus operators_ — The symmetry of subspaces
to which the columns of Kraus operators are related to can be different for
each operator. An example of such a QSSR is given by the following Kraus
representation

_Mixing representation of exchange operator between the Kraus operators_ —
Here we consider two very specific, perfectly admissible representations of
the exchange operator:

_Total space of QSSRs_ — In summary, for a fixed local dimension $N$, number
of primary subsystems $n$ and the ancillary dimension $M$, we find at most
$M+1$ disjoint manifolds, distinguished by the number of symmetric Kraus
operators $M_{sym}$ and the antisymmetric ones $M_{asym}=M-M_{sym}$. The real
dimensionality of each manifold is given by

$\underbrace{M\quantity(n-1)\quantity(\frac{N(N-1)}{2}+1)}_{\text{SWAP
operators'
freedom}}+\underbrace{N^{n}\quantity(2\quantity(M_{sym}\binom{n+N-1}{n}+M_{asym}\binom{N}{n})-N^{n})}_{\text{Choice
of $N^{n}$ orthonormal vectors from proper-dimensional $\mathbb{C}$ vector
space}}$

##  Appendix F Quantum circuit

In order to execute the quantum channel (10) on a quantum device, the
corresponding unitary must be written as a quantum circuit consisting of
quantum gates. This unitary can be constructed by using only CNOT and
controlled Hadamard gates. A CNOT gate acts on two qubits by swapping the
states $\ket{0}$ and $\ket{1}$ of the target qubit if the control qubit is in
the state $\ket{1}$. A Hadamard gate acts on one qubit by transforming
$\ket{0}\rightarrow\frac{1}{\sqrt{2}}\quantity(\ket{0}+\ket{1})$ and
$\ket{1}\rightarrow\frac{1}{\sqrt{2}}\quantity(\ket{0}-\ket{1})$. A controlled
Hadamard gate is simply a Hadamard gate that acts on the target qubit only if
the control qubit is in the state $\ket{1}$. The constructed circuit can be
seen below in Fig. 3.

Figure 3:  The constructed quantum circuit acting on 3 qubits that implements
the quantum channel (10) on qubits 0 and 1, consisting only of CNOTs and
controlled Hadamard gates.

Unfortunately, this circuit cannot be directly implemented on the IBM Santiago
quantum device but must first be transpiled. This is due to the linear qubit
layout of the quantum computer which only allows for the application of two
qubit gates onto direct neighbours. Furthermore, the device has a specific set
of basis gates of which the circuit needs to consist of. The constructed
circuit must therefore be decomposed into those basis gates while respecting
the qubit layout. The implementable, transpiled circuit can be seen in Fig. 4.
The full circuit has two more gates for state preparation and up to six more
gates for the measurement of the Bloch vector components. The initial states
were chosen by creating a product state out of uniformly sampled pure single-
qubit states with respect to the Haar measure.

Figure 4:  Transpiled circuit of the exemplary quantum channel (10) run on the
IBM Santiago. Note the changed assignment of logical qubits to physical ones.
To extract the Bloch vector components, we need to append up to two more gates
for measurements as well as one gate for state preparation on each wire, thus
resulting in a critical path length of 15.

The $U_{3}$ gate is defined as follows:

$U_{3}(\theta,\phi,\lambda)=\begin{pmatrix}\cos(\theta/2)&-e^{i\lambda}\sin(\theta/2)\\\
e^{i\phi}\sin(\theta/2)&e^{i(\phi+\lambda)}\cos(\theta/2)\end{pmatrix},$

while $U_{2}(\phi,\lambda)=U_{3}(\pi/2,\phi,\lambda)$.

$n$ |  $M$ for $N=2$ |  $M$ for $N=3$ |  $M$ for $N=4$
---|---|---|---
2 | 2 | 2 | 2
3 | 2 | 3 | 4
4 | 4 | 6 | 8
5 | 6 | 12 | 19
6 | 10 | 27 | 49
7 | 16 | 61 | 137
8 | 29 | 146 | 398
9 | 52 | 358 | 1192
Table 1:  Overview over the minimal needed dimensionality $M$ of the ancilla
for a different number $n$ of subsystems and dimensionality $N$ of those
subsystems.

If the synchronizing quantum channels in the 2$\times 2\times 2$ case are
additionaly optimized with respect to creating maximum purity, a new property
emerges for a single-qubit ancilla: The ancilla qubit ends up in the initial
state of one of the system qubits in analogy to the process of teleportation.
A similar experiment on a 4-qubit system was conducted but no conclusive
results could be found due to the large circuit depth, incompatible with the
noisy device.

## References

  * Pikovsky  _et al._ [2001] A. Pikovsky, M. Rosenblum, and J. Kurths,  _Synchronization. A Universal Concept in Nonlinear Sciences_ (Cambridge University Press, Cambridge, UK, 2001).
  * Walter  _et al._ [2014] S. Walter, A. Nunnenkamp, and C. Bruder, Quantum Synchronization of a Driven Self-Sustained Oscillator, Phys. Rev. Lett. 112, 094102 (2014).
  * Walter  _et al._ [2015] S. Walter, A. Nunnenkamp, and C. Bruder, Quantum synchronization of two Van der Pol oscillators, Annalen der Physik 527, 131 (2015).
  * DeVille [2018] L. DeVille, Synchronization and stability for quantum Kuramoto, Journal of Statistical Physics 174, 160 (2018).
  * Heinrich  _et al._ [2011] G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Collective dynamics in optomechanical arrays, Phys. Rev. Lett. 107, 043603 (2011).
  * Giorgi  _et al._ [2012] G. L. Giorgi, F. Galve, G. Manzano, P. Colet, and R. Zambrini, Quantum correlations and mutual synchronization, Phys. Rev. A 85, 052101 (2012).
  * Manzano  _et al._ [2013] G. Manzano, F. Galve, G. L. Giorgi, E. Hernández-García, and R. Zambrini, Synchronization, quantum correlations and entanglement in oscillator networks, Scientific Reports 3 (2013).
  * Ludwig and Marquardt [2013] M. Ludwig and F. Marquardt, Quantum many-body dynamics in optomechanical arrays, Phys. Rev. Lett. 111, 073603 (2013).
  * Sonar  _et al._ [2018] S. Sonar, M. Hajdušek, M. Mukherjee, R. Fazio, V. Vedral, S. Vinjanampathy, and L.-C. Kwek, Squeezing enhances quantum synchronization, Phys. Rev. Lett. 120, 163601 (2018).
  * Kato  _et al._ [2019] Y. Kato, N. Yamamoto, and H. Nakao, Semiclassical phase reduction theory for quantum synchronization, Phys. Rev. Research 1, 033012 (2019).
  * Giorgi  _et al._ [2013] G. L. Giorgi, F. Plastina, G. Francica, and R. Zambrini, Spontaneous synchronization and quantum correlation dynamics of open spin systems, Phys. Rev. A 88, 042115 (2013).
  * Zhirov and Shepelyansky [2008] O. V. Zhirov and D. L. Shepelyansky, Synchronization and Bistability of a Qubit Coupled to a Driven Dissipative Oscillator, Phys. Rev. Lett. 100, 014101 (2008).
  * Zhirov and Shepelyansky [2009] O. V. Zhirov and D. L. Shepelyansky, Quantum synchronization and entanglement of two qubits coupled to a driven dissipative resonator, Phys. Rev. B 80, 014519 (2009).
  * Cattaneo  _et al._ [2020] M. Cattaneo, G. L. Giorgi, S. Maniscalco, G. S. Paraoanu, and R. Zambrini, Synchronization and subradiance as signatures of entangling bath between superconducting qubits, arXiv:2005.06229 (2020).
  * Koppenhöfer  _et al._ [2020] M. Koppenhöfer, C. Bruder, and A. Roulet, Quantum synchronization on the IBM Q system, Phys. Rev. Research 2, 023026 (2020).
  * Roulet and Bruder [2018a] A. Roulet and C. Bruder, Quantum synchronization and entanglement generation, Phys. Rev. Lett. 121, 063601 (2018a).
  * Fiderer  _et al._ [2016] L. J. Fiderer, M. Kuś, and D. Braun, Quantum-phase synchronization, Phys. Rev. A 94, 032336 (2016).
  * Jaseem  _et al._ [2020] N. Jaseem, M. Hajdušek, V. Vedral, R. Fazio, L.-C. Kwek, and S. Vinjanampathy, Quantum synchronization in nanoscale heat engines., Phys. Rev. E 101, 020201 (2020).
  * Wootters and Zurek [1982] W. K. Wootters and W. H. Zurek, A single quantum state cannot be cloned, Nature 299, 802 (1982).
  * Gisin and Massar [1997] N. Gisin and S. Massar, Optimal quantum cloning machines, Phys. Rev. Lett. 79, 2153 (1997).
  * Barnum  _et al._ [1996] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Noncommuting mixed states cannot be broadcast, Phys. Rev. Lett. 76, 2818 (1996).
  * Smaczyński  _et al._ [2016] M. Smaczyński, W. Roga, and K. Życzkowski, Selfcomplementary quantum channels, Open Systems Inform. Dynamics 23 (2016).
  * Czartowski  _et al._ [2019] J. Czartowski, D. Braun, and K. Życzkowski, Trade-off relations for operation entropy of complementary quantum channels, Int. J. Quantum Inf. 17, 1950046 (2019).
  * Roulet and Bruder [2018b] A. Roulet and C. Bruder, Synchronizing the smallest possible system, Phys. Rev. Lett. 121, 053601 (2018b).
  * Bacon  _et al._ [2006] D. Bacon, I. L. Chuang, and A. W. Harrow, Efficient quantum circuits for Schur and Clebsch-Gordan transforms, Phys. Rev. Lett. 97, 170502 (2006).
  * Satya Sainadh [2013] U. Satya Sainadh, An Efficient Quantum Algorithm and Circuit to Generate Eigenstates of SU(2) and SU(3) Representations, arXiv:1309.2736 (2013).
  * Buca  _et al._ [2021] B. Buca, C. Booker, and D. Jaksch, Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation, arXiv:2103.01808 (2021).
  * Giraud  _et al._ [2008] O. Giraud, P. Braun, and D. Braun, Classicality of spin states, Phys. Rev. A 78, 042112 (2008).
  * Wilczek [1982a] F. Wilczek, Magnetic flux, angular momentum, and statistics, Phys. Rev. Lett. 48, 1144 (1982a).
  * Wilczek [1982b] F. Wilczek, Quantum mechanics of fractional-spin particles, Phys. Rev. Lett. 49, 957 (1982b).