# Experimental quantification of spatial correlations in quantum dynamics

Lukas Postler  Institut für Experimentalphysik, Universität Innsbruck,
Technikerstr. 25, A-6020 Innsbruck, Austria  Ángel Rivas  Departamento de
Física Teórica I, Universidad Complutense, 28040 Madrid, Spain  CCS-Center for
Computational Simulation, Campus de Montegancedo UPM, 28660 Boadilla del
Monte, Madrid, Spain  Philipp Schindler  Institut für Experimentalphysik,
Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria  Alexander
Erhard  Institut für Experimentalphysik, Universität Innsbruck, Technikerstr.
25, A-6020 Innsbruck, Austria  Roman Stricker  Institut für
Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck,
Austria  Daniel Nigg  Institut für Experimentalphysik, Universität Innsbruck,
Technikerstr. 25, A-6020 Innsbruck, Austria  Thomas Monz  Institut für
Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck,
Austria  Rainer Blatt  Institut für Experimentalphysik, Universität Innsbruck,
Technikerstr. 25, A-6020 Innsbruck, Austria  Institut für Quantenoptik und
Quanteninformation, Österreichische Akademie der Wissenschaften, Otto-
Hittmair-Platz 1, A-6020 Innsbruck, Austria  Markus Müller  Department of
Physics, College of Science, Swansea University, Singleton Park, Swansea - SA2
8PP, United Kingdom

###### Abstract

Correlations between different partitions of quantum systems play a central
role in a variety of many-body quantum systems, and they have been studied
exhaustively in experimental and theoretical research. Here, we investigate
dynamical correlations in the time evolution of multiple parts of a composite
quantum system. A rigorous measure to quantify correlations in quantum
dynamics based on a full tomographic reconstruction of the quantum process has
been introduced recently [Á. Rivas et al., New Journal of Physics, 17(6)
062001 (2015).]. In this work, we derive a lower bound for this correlation
measure, which does not require full knowledge of the quantum dynamics.
Furthermore we also extend the correlation measure to multipartite systems. We
directly apply the developed methods to a trapped ion quantum information
processor to experimentally characterize the correlations in quantum dynamics
for two- and four-qubit systems. The method proposed and demonstrated in this
work is scalable, platform-independent and applicable to other composite
quantum systems and quantum information processing architectures. We apply the
method to estimate spatial correlations in environmental noise processes,
which are crucial for the performance of quantum error correction procedures.

11footnotetext: These authors contributed equally to this work.

##  1 Introduction

Correlations play a central role in quantum physics. A wide range of quantum
effects including apparently disconnected topics, such as Bell inequalities or
quantum phase transitions, can be analyzed by considering correlations. In
most of the cases these correlations refer to those shared between different
parties of a multipartite quantum state, i.e. describing the statistics of a
system’s observables at a given time. These correlations, both classical and
quantum, have been extensively studied, quantified, and classified (see for
instance [1, 2, 3]). A different kind of correlations, subject of less
attention, are dynamical correlations. These account for the fact that the
dynamics of one part of a system may not be statistically independent from the
dynamics of the other parts. Dynamical correlations in quantum systems are the
basis of many phenomena ranging from super-radiance [4] over super-decoherence
[5] to sub-radiance [6] in atomic gases. Furthermore, the study of dynamical
correlations is of central importance in various research areas, such as e.g.
photosynthesis and excitation transfer dynamics [7, 8, 9, 10, 11, 12], driven-
dissipative phase transitions [13, 14, 15, 16, 17] and quantum metrology [18].

In the context of quantum information, the treatment of spatial correlations
is highly important, but usually limited to the extreme cases of either
completely uncorrelated noise with an independent noise source for each qubit,
or completely correlated noise modeled by a single noise source with equal
strength on all qubits [19]. One consequence of the latter type of
correlations are decoherence-free subspaces [20, 21, 22, 23, 24], which can be
exploited in quantum information processing to extend the storage time of
quantum states in noisy systems. Understanding and quantifying these dynamical
correlations is highly relevant for the performance of quantum error
correction protocols [19], as correlated errors can undermine the fault-
tolerant operation of quantum error correcting procedures [25]. Here, theory
studies focusing on spatially correlated noise [26, 27, 28, 29, 30, 31, 32]
have shown that in particular the distance-dependence of correlated noise can
be crucial as to whether or not modified versions of the threshold theorem
[33, 34] do hold. It is crucial that not only the noise strength is below a
critical value, but also that (unwanted) interactions between qubits decay
sufficiently quickly with increasing distance. In order to assess whether or
not these conditions are met in experimental quantum processors, theoretically
well-founded and practically applicable methods to characterize the strength
as well as the distance-dependence of spatial correlations are required. Such
tools become particularly important in scalable quantum information processing
architectures: There, it is forseeable that the noise environment will not be
fully correlated in processors consisting of multiple smaller units that are
interconnected by quantum channels. Noisy connection channels may introduce
spatial correlations in the system’s dynamics. Similar considerations apply in
distributed quantum systems that are interconnected by flying qubits.

Recently, we proposed and explored a measure to rigorously quantify dynamical
correlations [35]. Inspired by the metrics that can be employed to quantify
the amount of correlations  within quantum states, a method to quantify
dynamical correlations from tomographic data was proposed with the following
features:

  1. 1.

It is a normalized quantity that is zero if and only if the dynamics are
uncorrelated. In particular, it is constructed in terms of an information
measure and does not rely on any assumptions or a priori knowledge of the
underlying dynamics.

  2. 2.

The quantifier introduces a hierarchy of quantum dynamics by enforcing a
partial order relation between dynamics, i.e. a way to quantitatively compare
whether one bipartite dynamic is more or less correlated (with respect to the
evolution of its parts) than another. This partial ordering is known as a
fundamental law of a resource theory [1, 36, 37, 38, 39, 40, 41, 42]. This law
states that the amount of correlations of some given dynamics cannot increase
by adding uncorrelated dynamics to it, i.e. a process for the quantifier
equals zero. This ordering property for the amount of dynamical correlations
is analogous to the fundamental law in the resource theory of entanglement [1,
3]. There, it refers to the fact that entanglement cannot increase under
application of local operations and classical communication (LOCC).

  3. 3.

The measure establishes a rigorous theoretical framework that allows for a
definition and the study of maximally correlated dynamics and the properties
that such dynamics need to fulfill [35]. Again, the concept of maximally
correlated dynamics is analogous to the one of maximally entangled states in
the resource theory of entanglement.

Whereas the proposed quantifier for dynamical correlations [35] allows one to
study correlated dynamics in a variety of contexts, its general applicability
comes at the price of requiring full knowledge of the quantum dynamics.
Experimentally, this requires quantum process tomography of the full system
which is only feasible in small-scale systems, and has been experimentally
demonstrated for up to three qubits [43], and quickly becomes impractical for
quantum systems of larger size. Within this work, we therefore derive a
efficiently measurable lower bound of the quantifier applicable also to larger
systems.

By applying the quantifier to a two-qubit trapped-ion quantum information
processor the amount of correlations is extracted from a reconstructed process
matrix. Furthermore the quantifier’s lower bound is determined for dynamics in
systems consisting of four trapped-ion qubits. We investigate in detail the
noise dynamics and its correlations for different physical encodings of the
qubits that lead to different correlation characteristics. Our findings
underline the importance of experimentally informed choices of qubit encodings
in the presence of spatially correlated noise in the context of quantum
computing and quantum error correction.

The presented work is structured as follows: In Sec. 2 we introduce and review
the correlation measure proposed in [35]. We present the derivation of a lower
bound of the measure in Sec. 3, followed by a generalization to multipartite
systems in Sec. 4. Finally, we present an investigation of noise dynamics in a
trapped-ion system in Sec. 5.

##  2 Measure for Spatial Correlations in Quantum Dynamics

In the following we review the correlation measure suggested in [35], which is
based on the Choi-Jamiołkowski isomorphism [44, 45], providing a one-to-one
mapping of the dynamics in a quantum system to a quantum state of a larger
system. The mathematical construction underlying the isomorphism can be
summarized as follows: consider a bipartite system ${\rm S=S_{1}\otimes
S_{2}}$, as shown in Fig. 1.

Figure 1: Schematic illustration of the measurement of $\bar{I}$ in a
composite system ${\rm S}$. At the beginning of the protocol subsystems
$\rm{S}_{1}$ and $\rm{S}_{2}$ are maximally entangled with
$\rm{S}^{\prime}_{1}$ and $\rm{S}^{\prime}_{2}$, respectively. The dynamics of
interest is subsequently acting on $\rm{S}$ and leaves the whole system in a
product state or a correlated state with respect to ${\rm S}_{1}{\rm
S}^{\prime}_{1}|{\rm S}_{2}{\rm S}^{\prime}_{2}$ depending on the amount of
correlations in the dynamics.

Then, the idea is to initially prepare a pair of maximally entangled states
$\left|\psi^{+}\right>=\tfrac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\left|kk\right>$
between every part $\rm{S}_{1}$ and $\rm{S}_{2}$ with two respective ancilla
systems ${\rm S}^{\prime}_{1}$ and ${\rm S}^{\prime}_{2}$ of the same
dimension,

$\left|\Phi_{\rm SS^{\prime}}\right>\equiv\left|\psi^{+}\right>_{\rm
S_{1}S_{1}^{\prime}}\left|\psi^{+}\right>_{\rm
S_{2}S_{2}^{\prime}}=\frac{1}{d}\sum_{k=0}^{d-1}\left|kk\right>\sum_{l=0}^{d-1}\left|ll\right>,$
(1)

where $d$ is the dimension of the subsystems $\rm{S}_{1}$ and $\rm{S}_{2}$,
assumed to be the same for both of them for simplicity. Then the part ${\rm
S=S_{1}\otimes S_{2}}$ of the state $\left|\Phi_{\rm SS^{\prime}}\right>$
evolves according to a dynamical process that is given by the map
$\mathcal{E}_{\rm S}$, and which leaves the part ${\rm
S^{\prime}=S^{\prime}_{1}\otimes S^{\prime}_{2}}$ unperturbed. After the
evolution, the final Choi-Jamiołkowski state $\rho^{\rm CJ}_{\rm
S}=\mathcal{E}_{\rm S}\otimes\mathds{1}_{\rm
S^{\prime}_{1}S^{\prime}_{2}}(|\Phi_{\rm SS^{\prime}}\rangle\langle\Phi_{\rm
SS^{\prime}}|)$ is a product state with respect to the bipartition ${\rm
S}_{1}{\rm S}^{\prime}_{1}|{\rm S}_{2}{\rm S}^{\prime}_{2}$, ${\rho^{\rm
CJ}_{\rm S}=\rho_{\rm S_{1}S^{\prime}_{1}}\otimes\rho_{\rm
S_{2}S^{\prime}_{2}}}$, if and only if the dynamics are uncorrelated,
$\mathcal{E}_{\rm S}=\mathcal{E}_{\rm S_{1}}\otimes\mathcal{E}_{\rm S_{2}}$.
This result follows from the one-to-one correspondence between the map
$\mathcal{E}_{\rm S}$ and $\rho^{\rm CJ}_{\rm S}$. In contrast, if the process
on the composite system is correlated, then the resulting state $\rho^{\rm
CJ}_{\rm S}$ is correlated as well. A quantifier $\bar{I}$ which suitably
reflects the amount of correlations in $\rho^{\rm CJ}_{\rm S}$ and thus in the
process $\mathcal{E}_{\rm S}$ is the quantum mutual information [46], given by
the following definition:

$\bar{I}(\mathcal{E}_{\rm S}):=\frac{1}{4\log d}\left[S\left(\rho^{\rm
CJ}_{\rm S}|_{\rm S_{1}S^{\prime}_{1}}\right)+S\left(\rho^{\rm CJ}_{\rm
S}|_{\rm S_{2}S^{\prime}_{2}}\right)-S(\rho^{\rm CJ}_{\rm S})\right],$ (2)

with $S(\cdot)=-{\rm{Tr}}[(\cdot)\log(\cdot)]$ the von Neumann entropy
evaluated for $\rho^{\rm CJ}_{\rm S}$ and the reduced density operators
$\rho^{\rm CJ}_{\rm S}|_{\rm S_{1}S^{\prime}_{1}}:={\rm{Tr}}_{\rm
S_{2}S^{\prime}_{2}}\left(\rho^{\rm CJ}_{\rm S}\right)$ and $\rho^{\rm
CJ}_{\rm S}|_{\rm S_{2}S^{\prime}_{2}}:={\rm{Tr}}_{\rm
S_{1}S^{\prime}_{1}}\left(\rho^{\rm CJ}_{\rm S}\right)$.

Leaving aside the normalization factor $\tfrac{1}{4\log d}$, this quantifier
can be intuitively understood as the amount of information that is needed to
distinguish the actual dynamics $\mathcal{E}_{\rm S}$ from the individual
dynamics of its parts $\mathcal{E}_{\rm S_{1}}\otimes\mathcal{E}_{\rm S_{2}}$
[47]. Namely, the information that is lost when $\mathcal{E}_{\rm
S_{1}}\otimes\mathcal{E}_{\rm S_{2}}$ is taken as an approximation of
$\mathcal{E}_{\rm S}$. The normalized quantity ${\bar{I}\in[0,1]}$ quantifies
this information relative to the maximum value it can take on all possible
processes.

Furthermore, $\bar{I}$ fulfills the equation

$\bar{I}\left[(\mathcal{L}_{\rm S_{1}}\otimes\mathcal{L}_{\rm
S_{2}})(\mathcal{E}_{\rm S})(\mathcal{R}_{\rm S_{1}}\otimes\mathcal{R}_{\rm
S_{2}})\right]\leq\bar{I}(\mathcal{E}_{\rm S})$ (3)

for all local dynamical maps $\mathcal{L}_{\rm S_{1}}$, $\mathcal{L}_{\rm
S_{2}}$, $\mathcal{R}_{\rm S_{1}}$, and $\mathcal{R}_{\rm S_{2}}$ that might
act before and after the actual, possibly correlated dynamics
$\mathcal{E}_{\rm S}$ [35]. Equation (3) states that the amount of
correlations of the dynamics $\mathcal{E}_{\rm S}$ cannot increase by
composition with uncorrelated maps. In other words, if a process is a
composition of a correlated and an uncorrelated part, the amount of
correlations in the composition has to be equal or smaller than the amount of
correlation that is inherent to the correlated part. In fact, this is the
fundamental law of the resource theory of correlated dynamics [35], where the
correlations are considered as a resource, and the operations which do not
increase the amount of this resource are uncorrelated maps.

For clarity, we remark that the use of an ancilla system ${\rm S^{\prime}}$ is
underlying the mathematical construction of the isomorphism, but is not
required in an experimental determination of $\bar{I}$. Rather than
reconstructing the Choi-Jamiołkowski state $\rho^{\rm CJ}_{\rm S}$ from
quantum state tomography on the enlarged system ${\rm SS^{\prime}}$, one can
equivalently determine $\rho^{\rm CJ}_{\rm S}$ by reconstructing the dynamics
$\mathcal{E}_{\rm S}$ by means of quantum process tomography on the physical
system S [48]. Due to the Choi-Jamiołkowski isomorphism in both cases the
number of real parameters to determine ($4^{N}(4^{N}-1)$) is the same and
grows exponentially with the number of qubits. In the following sections we
therefore provide alternative strategies to estimate $\bar{I}$ avoiding full
tomography.

##  3 Lower Bound to $\bar{I}$

A lower estimate for $\bar{I}$ can be obtained by performing correlation
measurements on the subsystems ${\rm S_{1}}$ and ${\rm S_{2}}$. Our central
result is that the normalized quantity $\bar{I}(\mathcal{E}_{\rm S})$ is lower
bounded by

$\bar{I}(\mathcal{E}_{\rm S})\geq\frac{1}{8\ln
d}\frac{C_{\rho^{\prime}}^{2}(X_{1},X_{2})}{\|X_{1}\|^{2}\|X_{2}\|^{2}},$ (4)

with two local quantum observables $X_{1}$ and $X_{2}$ and
$C_{\rho^{\prime}}(X_{1},X_{2})=\langle X_{1}\otimes
X_{2}\rangle_{\rho^{\prime}}-\langle X_{1}\rangle_{\rho^{\prime}}\langle
X_{2}\rangle_{\rho^{\prime}}$. $\rho^{\prime}=\mathcal{E}_{\rm S}(\rho)$ is
the evolution of an initial product state $\rho$ according to the dynamical
map $\mathcal{E}_{\rm S}$. Here $\|\cdot\|$ denotes the operator norm (the
absolute value of the maximum eigenvalue) and we have taken the logarithms
inside $\bar{I}(\mathcal{E}_{\rm S})$ in Eq. (2) to be binary logarithms
$\log_{2}$ (otherwise the natural logarithm $\ln d$ on the right hand side
becomes multiplied by a different factor).

In order to prove Eq. (4), we use the relation between the mutual information
and the quantum relative entropy ${I(\rho_{\rm AB})=S(\rho_{\rm AB}\|\rho_{\rm
A}\otimes\rho_{\rm B})}$ [49], so that by taking the bipartition ${\rm
A=S_{1}S^{\prime}_{1}}$ and ${\rm B=S_{2}S^{\prime}_{2}}$, we rewrite Eq. (2)
as

$\bar{I}(\mathcal{E}_{\mathrm{S}})=\frac{1}{4\log_{2}d}S\Big{(}\rho^{\rm
CJ}_{\mathrm{S}}\Big{\|}\rho^{\rm CJ}_{\mathrm{S}}|_{{\rm
S_{1}S^{\prime}_{1}}}\otimes\rho^{\rm CJ}_{\mathrm{S}}|_{\rm
S_{2}S^{\prime}_{2}}\Big{)}.$ (5)

Then the fundamental law that composition with uncorrelated dynamics does not
increase the correlatedness of dynamics, as expressed in Eq. (3), yields the
inequality

$\bar{I}(\mathcal{E}_{\mathrm{S}})\geq\frac{1}{4\log_{2}d}S\Big{(}\tilde{\rho}^{\rm
CJ}_{\mathrm{S}}\Big{\|}\tilde{\rho}^{\rm CJ}_{\mathrm{S}}|_{{\rm
S_{1}S^{\prime}_{1}}}\otimes\tilde{\rho}^{\rm CJ}_{\mathrm{S}}|_{{\rm
S_{2}S^{\prime}_{2}}}\Big{)}.$ (6)

Here, the state $\tilde{\rho}^{\rm CJ}_{\rm S}=\left[(\mathcal{E}_{\rm
S})(\mathcal{R}_{\rm S_{1}}\otimes\mathcal{R}_{\rm
S_{2}})\right]\otimes\mathds{1}(|\Phi_{\rm SS^{\prime}}\rangle\langle\Phi_{\rm
SS^{\prime}}|)$ is obtained by composition of the dynamics
$\mathcal{E}_{\mathrm{S}}$ with arbitrary local maps $\mathcal{R}_{\rm S_{1}}$
and $\mathcal{R}_{\rm S_{2}}$, i.e. maps that act only locally on ${\rm
S_{1}}$ and ${\rm S_{2}}$, respectively. Now, monotonicity of the relative
entropy with respect to the partial trace [46] leads to the bound for the
correlation measure

$\bar{I}(\mathcal{E}_{\mathrm{S}})\geq\frac{1}{4\log_{2}d}S\Big{(}\tilde{\rho}^{\rm
CJ}_{\mathrm{S}}|_{\rm S}\Big{\|}\tilde{\rho}^{\rm
CJ}_{\mathrm{S}}|_{S_{1}}\otimes\tilde{\rho}^{\rm
CJ}_{\mathrm{S}}|_{S_{2}}\Big{)}.$ (7)

The trace over the subsystem ${\rm S^{\prime}}$ on the Choi-Jamiołkowski state
yields

$\displaystyle\tilde{\rho}^{\rm CJ}_{\mathrm{S}}|_{\rm S}:$
$\displaystyle={\rm{Tr}}_{\rm S^{\prime}}\left\\{\left[(\mathcal{E}_{\rm
S})(\mathcal{R}_{\rm S_{1}}\otimes\mathcal{R}_{\rm
S_{2}})\right]\otimes\mathds{1}(|\Phi_{\rm SS^{\prime}}\rangle\langle\Phi_{\rm
SS^{\prime}}|)\right\\}$ $\displaystyle=(\mathcal{E}_{\rm S})(\mathcal{R}_{\rm
S_{1}}\otimes\mathcal{R}_{\rm
S_{2}})\left(\frac{\mathbb{1}}{d^{2}}\right)=\mathcal{E}_{\rm S}(\rho_{\rm
S_{1}}\otimes\rho_{\rm S_{2}})=\rho^{\prime}.$ (8)

where the local maps $\mathcal{R}_{\rm S_{1}}$ and $\mathcal{R}_{\rm S_{2}}$
are chosen such that $\rho_{\rm S_{1},S_{2}}:=\mathcal{R}_{\rm
S_{1},S_{2}}(\tfrac{\mathbb{1}}{d})$. Moreover, since $\tilde{\rho}^{\rm
CJ}_{\mathrm{S}}|_{\rm S_{1},S_{2}}={\rm{Tr}}_{\rm
S_{2},S_{1}}[\mathcal{E}_{\rm S}(\rho_{\rm S_{1}}\otimes\rho_{\rm
S_{2}})]:=\rho^{\prime}_{\rm S_{1},S_{2}}$, we write Eq. (7) as

$\bar{I}(\mathcal{E}_{\mathrm{S}})\geq\frac{1}{4\log_{2}d}S(\rho^{\prime}\|\rho^{\prime}_{\rm
S_{1}}\otimes\rho^{\prime}_{\rm S_{2}}).$ (9)

Now we use the quantum Pinsker inequality [50]

$S(\rho\|\sigma)\geq\frac{1}{2\ln 2}\|\rho-\sigma\|_{1}^{2},$ (10)

where the relative entropy is measured in bits, $\rho$ and $\sigma$ are
density matrices and the trace norm is
$\|A\|_{1}={\rm{Tr}}\sqrt{A^{\dagger}A}$. Thus, we obtain

$\displaystyle\bar{I}(\mathcal{E}_{\mathrm{S}})$
$\displaystyle\geq\frac{1}{8(\log_{2}d)(\ln{2})}\|\rho^{\prime}-\rho^{\prime}_{\rm
S_{1}}\otimes\rho^{\prime}_{\rm S_{2}}\|_{1}^{2}$ (11)
$\displaystyle=\frac{1}{8\ln{d}}\|\rho^{\prime}-\rho^{\prime}_{\rm
S_{1}}\otimes\rho^{\prime}_{\rm S_{2}}\|_{1}^{2}.$

Finally, by considering two arbitrary observables on ${\rm S_{1}}$ and ${\rm
S_{2}}$, $X_{1}$ and $X_{2}$ respectively, the inequality
$\|A\|_{1}\geq\frac{{\rm{Tr}}(AB)}{\|B\|}$ implies:

$\displaystyle\bar{I}(\mathcal{E}_{\mathrm{S}})$
$\displaystyle\geq\frac{1}{8\ln{d}}\left\\{\frac{{\rm{Tr}}[X_{1}\otimes
X_{2}(\rho^{\prime}-\rho^{\prime}_{\rm S_{1}}\otimes\rho^{\prime}_{\rm
S_{2}})]}{\|X_{1}\otimes X_{2}\|}\right\\}^{2}$ $\displaystyle=\frac{1}{8\ln
d}\frac{C_{\rho^{\prime}}^{2}(X_{1},X_{2})}{\|X_{1}\|^{2}\|X_{2}\|^{2}}.$ (12)

Therefore the inequality from Eq. (4) is recovered. This result is useful
because it allows us to estimate a lower bound of the amount of dynamical
correlation only by preparing product states and measuring correlation
functions. Specifically, the measurement protocol, also shown in Fig. 2, is as
follows:

  1. 1.

State preparation: The bipartite system is initially prepared in a product
state $\rho=\rho_{\rm S_{1}}\otimes\rho_{\rm S_{2}}$.

  2. 2.

Evolution: The state $\rho$ evolves accordingly to the dynamical map
$\mathcal{E}_{\rm S}$ to some state $\rho^{\prime}=\mathcal{E}_{\rm S}(\rho)$.

  3. 3.

Measurement: Correlation measurements of two local quantum observables $X_{1}$
and $X_{2}$ are carried out, $C_{\rho^{\prime}}(X_{1},X_{2})=\langle
X_{1}\otimes X_{2}\rangle_{\rho^{\prime}}-\langle
X_{1}\rangle_{\rho^{\prime}}\langle X_{2}\rangle_{\rho^{\prime}}$.

Figure 2: Schematic illustration of the measurement of the lower bound of
$\bar{I}$. The system is prepared in a separable state $\rho=\rho_{\rm
S_{1}}\otimes\rho_{\rm S_{2}}$ and correlations of the observables $X_{1}$ and
$X_{2}$ are measured after the evolution.

Of course, the tightness of the inequality depends on the choice of $X_{1}$
and $X_{2}$, so in practice one has to make a well informed choice of
observables, taking into account prior information about the system.
Typically, one would choose observables that are orthogonal to the dominant
noise operator in the system. However there is no simple and universal recipe
to determine the observables giving the tightest lower bound. In particular,
note that finding non-vanishing values of the lower bound as given by Eq. (4)
allows one to reveal unexpected dynamical correlations if suitable observables
have been estimated in any experiment.

##  4 Extension to multipartite systems

Figure 3: Schematic illustration of the multipartite correlation measure. (A)
Choi-Jamiołkowski representation of the dynamics. The system is prepared in a
product of maximally entangled states of $2M$ parties $\\{{\rm S}_{j}|{\rm
S}_{j}^{\prime}\\}$ and the dynamics affect only the subsystems ${\rm S}_{j}$.
If and only if the dynamics are correlated, the bipartitions $\\{{\rm
S}_{i}{\rm S}^{\prime}_{i}|{\rm S}_{j}{\rm S}^{\prime}_{j}\\}$ will be
entangled, yielding a nonzero $\bar{I}$. (B) Schematic depiction of the
procedure to estimate a lower bound of $\bar{I}$. There, the system is
prepared in a separable state $\rho_{{\rm S}_{1}}\otimes\rho_{{\rm
S}_{2}}\cdots\otimes\rho_{{\rm S}_{M}}$ and correlations in the dynamics
appear as correlations C (see page 3) in a measurement of the observables
$X_{j}$.

The approach to measure and estimate bipartite correlations can be extended to
the multipartite case. In this situation, one has to specify what kind of
correlations are the matter of interest. For instance, one may be interested
in the amount of correlations shared between two parties of the system or
between all parties. Figure 3 illustrates a generic situation where
correlations among all systems are investigated.

For definiteness, suppose we consider the total amount of correlations, i.e.
the amount of correlations shared by all parties (other cases can be analyzed
in a similar manner). In that case, if the system $\mathrm{S}$ has $M$ parties
$\mathrm{S}_{1},\mathrm{S}_{2},\ldots,\mathrm{S}_{M}$, we introduce respective
ancillary systems
$\mathrm{S}^{\prime}_{1},\mathrm{S}^{\prime}_{2},\ldots,\mathrm{S}^{\prime}_{M}$
and prepare a collection of $M$ maximally entangled states between
$\mathrm{S}_{1}$ and $\mathrm{S}^{\prime}_{1}$, $\mathrm{S}_{2}$ and
$\mathrm{S}^{\prime}_{2}$, etc. [see Fig. 3(A)]. The dynamics are then applied
on the system $\mathrm{S}$ we want to study. The amount of total normalized
correlations in the dynamics can be assessed by

$\displaystyle\bar{I}(\mathcal{E}_{\mathrm{S}})$
$\displaystyle:=\frac{1}{2M\log d}S\Big{(}\rho^{\rm
CJ}_{\mathrm{S}}\Big{\|}\rho^{\rm
CJ}_{\mathrm{S}}|_{\mathrm{S}_{1}\mathrm{S}^{\prime}_{1}}\otimes\ldots\otimes\rho^{\rm
CJ}_{\mathrm{S}}|_{\mathrm{S}_{M}\mathrm{S}_{M}^{\prime}}\Big{)}$
$\displaystyle:=\frac{1}{2M\log d}\left\\{\left[\sum_{i=1}^{M}S\left(\rho^{\rm
CJ}_{\mathrm{S}}|_{\mathrm{S}_{i}\mathrm{S}^{\prime}_{i}}\right)\right]-S\left(\rho^{\rm
CJ}_{\mathrm{S}}\right)\right\\},$ (13)

where $\rho^{\rm
CJ}_{\mathrm{S}}|_{\mathrm{S}_{i}\mathrm{S}^{\prime}_{i}}={\rm{Tr}}_{\\{\forall\mathrm{S}_{j\neq
i}\mathrm{S}^{\prime}_{j\neq i}\\}}(\rho^{\rm CJ}_{\mathrm{S}})$.

The lower bound for the multipartite setting can be applied as shown in Fig.
3(B), by measuring correlations. Mathematically the same steps as in the
bipartite case [see Eq. (4)] can be applied, resulting in

$\bar{I}(\mathcal{E})\geq\frac{1}{4M\ln
d}\frac{C_{\rho^{\prime}}^{2}(X_{1},\ldots,X_{M})}{\|X_{1}\|^{2}\ldots\|X_{M}\|^{2}}.$
(14)

Here, $\rho^{\prime}$ is the joint state after the evolution of an initial
product state, $X_{1},\ldots,X_{M}$ are local observables for the parties
$\mathrm{S}_{1},\ldots,\mathrm{S}_{M}$, respectively, and

$C_{\rho^{\prime}}(X_{1},\ldots,X_{M})=\langle X_{1}\ldots
X_{M}\rangle_{\rho^{\prime}}-\langle X_{1}\rangle_{\rho^{\prime}}\ldots\langle
X_{M}\rangle_{\rho^{\prime}}.$ (15)

This multipartite bound makes investigating correlation dynamics accessible in
systems that are too large for full quantum process tomography, as the number
of measurements increases only linearly compared to an exponential scaling for
full quantum process tomography.

##  5 Application to experiments

In the following, we study the dynamics of spatial correlations of noise
processes in a trapped ion quantum information processor [51]. In Sec. 5.1 we
analyze the temporal development of the spatial correlation estimator
$\bar{I}$ to determine the degree of spatial correlations in a two-qubit
register. For this, we perform full quantum process tomography on qubit
registers with varying degree of correlations which yields following
behaviour:

  1. a)

Fully correlated noise enables decoherence free subspaces (DFS) [52] and thus
the correlation quantifier increases with decoherence, eventually reaching a
steady state for times larger than the single qubit coherence time when all
single qubit coherences have vanished but coherences in the DFS survive. For
purely dephasing noise the saturation value is $\bar{I}=0.125$ (see Appendix).

  2. b)

Partially correlated dynamics do not feature a full DFS and thus also, after
an initial stage of increasing spatial correlations, the correlations will
vanish in the limit of infinite waiting times. More precisely, the
correlations will start to decrease when the individual constituents of the
imperfect DFS suffer substantial dephasing.

  3. c)

For uncorrelated dynamics the quantifier should not detect any statistically
significant correlations.

In 5.2 we utilize the lower bound to $\bar{I}$ to characterize dynamics in a
system consisting of four qubits. Initially, we investigate the two-body
correlators as a function of qubit distance in the register. Following that,
we investigate the four-body spatial correlations for different qubit
encodings.

In the experimental platform used to implement the two protocols each qubit is
encoded in the $4S_{1/2}$ and $3D_{5/2}$ states of a single 40Ca+ ion of a
string of ions trapped in a macroscopic linear Paul trap [51]. Doppler cooling
of the ion crystal is performed on a short-lived cycling transition between
the $4S_{1/2}$ and the $4P_{1/2}$ levels, as illustrated in Fig. 4. The same
transition is used to detect the qubit state via the electron shelving scheme
[51]. Two additional repumping lasers ensure that the ion does not get trapped
in a dark state and enable resetting from the long-lived $3D_{5/2}$ state. A
more detailed description of the toolset and the experimental setup used can
be found in [51].

To manipulate the state of the qubit two different laser beams are used: A
global beam effectively illuminates all ions in the chain with equal power and
allows rotations on the Bloch sphere of all qubits simultaneously. Therefore
interactions of the following form are possible:

$R_{\phi}(\theta)=e^{-i\frac{\theta}{2}S_{\phi}},$ (16)

where
$S_{\phi}=\sum_{i=0}^{N}(\sigma_{x}^{(i)}\cos\phi+\sigma_{y}^{(i)}\sin\phi)$
with $\sigma_{x,y}^{(i)}$ being single-qubit Pauli matrices acting on qubit
$i$.

To perform local operations on single qubits an addressed beam is available.
This tightly focused beam is steered along the ion chain via an electro-
optical deflector. By driving the qubit transition on resonance or in a
detuned way, two types of rotations can be realised:

$\begin{split}R_{\phi}^{(j)}(\theta)=&e^{-i\frac{\theta}{2}(\sigma_{x}^{(j)}\cos\phi+\sigma_{y}^{(j)}\sin\phi)}\\\
&\mathrm{and}\\\
S_{z}^{(j)}(\theta)&=e^{-i\frac{\theta}{2}\sigma_{z}^{(j)}}.\end{split}$ (17)

With this control toolset at hand we are able to prepare the qubits in the
required initial state, encode them in different Zeeman sublevels and perform
quantum process tomography.

The degree of noise correlations between individual qubits can be tuned by
encoding them in Zeeman states with differing magnetic field susceptibility.
In 40Ca+, there exist multiple possibilities to encode a qubit in the Zeeman
levels of the $4S_{1/2}$ and $3D_{5/2}$ states as shown in Fig. 4. The
susceptibility of the qubits to the magnetic field ranges from -2.80 MHz/G to
+3.36 MHz/G, which allows us to tune not only the coherence time of the
individual qubits but also the correlations between qubits, when magnetic
field fluctuations are the dominant noise source.  Understanding the dephasing
dynamics, and in particular noise correlations, in registers containing qubits
in different encodings is essential in the context of error mitigation and
quantum error correction: this understanding will be needed to determine the
viability of an approach to build, e.g. functional logical qubits, in
complementary approaches either based on the use of spectroscopic decoupling
of ion qubits, as compared to, e.g., shuttling-based protocols [53].

###  5.1 Determining the degree of spatial correlation

In the following we consider dephasing dynamics that is caused by a magnetic
field acting on a string of two ions. The various qubit-states have different
susceptibility to magnetic field fluctuations, given by the Landé g factors
$g_{i}$ of the involved Zeeman substates. The phase that qubit $i$ accumulates
during the time evolution is

$\phi_{i}(t)=\int_{0}^{t}d\tau B(\tau)\mu_{b}g_{i}$

with the magnitude of the magnetic field $B(\tau)$ and the Bohr magneton
$\mu_{b}$. The magnetic field fluctuations are modeled by multiple random
implementations of $B(t)$. The time evolution for a single implementation can
then be expressed as:

$U=e^{-i\phi_{1}(\sigma_{1}^{z}+g\sigma_{2}^{z})}.$ (18)

with the ratio of the Landé factors $g=g_{2}/g_{1}$. In order to estimate the
dynamics under a dephasing decay, one needs to average the evolution over many
noise realizations with random phases. A detailed analysis of the expected
decay for qubits with different susceptibility to magnetic fields is given in
the Appendix, Sec. A.

In the experiment we are investigating the following qubit configurations that
implement dephasing and spontaneous decay dynamics:

  1. a)

Configuration 1: For the realization of maximally correlated dephasing
dynamics, both qubits are encoded in the
$\left|4S_{1/2},m_{S}=-1/2\right\rangle$ and
$\left|3D_{5/2},m_{S}=-5/2\right\rangle$ states. This encoding is referred to
as encoding A hereinafter, and corresponds to the green markers in Fig. 4.
Both qubits have a susceptibility to the magnetic field of -2.80 MHz/G,
leading to identical susceptibility coefficients ($g=1$) (see Eq. (18)).

  2. b)

Configuration 2: To introduce an asymmetric dephasing dynamics, one qubit is
encoded in A and the second is encoded in the states
$\left|3D_{1/2},m_{S}=-1/2\right\rangle$ and
$\left|3D_{5/2},m_{S}=-5/2\right\rangle$ respectively. This encoding is
referred to as encoding B hereinafter, and corresponds to the blue markers in
Fig. 4. Their different susceptibilities to magnetic field noise of -2.80
MHz/G and +3.36 MHz/G introduce unequal dephasing and therefore affect
correlations between the qubits, corresponding to the susceptibility
coefficients ($g=-0.83$).

  3. c)

Configuration 3: Uncorrelated dynamics can be engineered by introducing
spontaneous decay. In this scenario, both qubits are encoded in Encoding A. A
laser pulse resonant with the $3D_{5/2}\leftrightarrow 4P_{3/2}$ transition at
854 nm shortens the effective lifetime of the exited state by inducing a
spontaneous decay to the $4S_{1/2},m_{S}=-1/2$ level via the
$3P_{3/2},m_{S}=-3/2$ level. This pump process implements an uncorrelated
noise process that can be modeled as spontaneous decay. The effective lifetime
depends on the laser power and is in our case set to be
$T_{spont}=7(1)\,\mu$s.

Figure 4: Level scheme of 40Ca+. The green and blue squares and circles
indicate different qubit encodings, denoted A and B, respectively. Squares are
marking the qubit state $\left|1\right>$ whereas the state $\left|0\right>$ is
highlighted with circles. The corresponding frequency shifts of the
transitions caused by the magnetic field are -2.80 MHz/G and +3.36 MHz/G for
the qubits marked with green and blue symbols respectively. For configuration
1 described in the enumeration in the main text for both qubits the encoding
marked in green is used. The asymmetry in scenario 2 is introduced by encoding
one of the qubits in in the states illustrated in blue. For the third
configuration both qubits again use the encoding marked in green and the
spontaneous decay from $\left|0\right>$ to $\left|1\right>$ is enhanced.

The system size of two qubits allows us to perform full process tomography to
estimate the correlation measure $\bar{I}$ (see Eq. (2)). In our platform for
full process tomography in an $N$ qubit system, $12^{N}$ measurement settings,
each providing $2^{N}-1$ measurements, are required. The amplitude of the
magnetic field fluctuations is non-stationary as it depends on the entire
laboratory environment which cannot be controlled accurately. We engineer a
stationary magnetic field noise as the dominating noise source (a situation
where laser and magnetic field noise have to be taken into account is
described in the Appendix). Thus we can control the single qubit coherence
time as shown in Fig. 5. The stationary magnetic field noise is engineered by
applying a white-noise current to the coils that generate the magnetic field
at the ions’ positions. We set the noise amplitude such that the coherence
time of the qubit encoded in $\left|4S_{1/2},m_{S}=-1/2\right\rangle$ and
$\left|3D_{5/2},m_{S}=-5/2\right\rangle$ is reduced from $59(3)\,$ms to
$1.98(7)\,$ms. The increase of magnetic field noise by a factor of
$\approx\,30$ ensures that laser phase noise is negligible. From the measured
data, a process matrix is reconstructed using an iterative Maximum Likelihood
method (see Ref. [48]) that ensures trace preservation and positivity of the
process matrix.

Figure 5: Coherence decay of the qubit in encoding A without (blue triangles)
and with (green circles) additional magnetic field noise.

The estimated quantifier for spatial correlations as defined in Eq. (2)
$\bar{I}$ is shown in Fig. 6 for the decoherence processes of the different
configurations described above. These processes are described by an
exponential decay and show different timescales. To compare the data from the
different configurations we express the free evolution time in units of the
respective decay times $\tau$. The temporal development of $\bar{I}$ is
studied for evolution times of up to 5 times the decoherence time for
configurations 1 and 2 and up to 1.6 times the lifetime for configuration 3,
as the differences in the dynamics of different correlation strength are most
pronounced on those timescales.

It can be seen in Fig. 6 that the symmetric configuration (Configuration 1),
depicted with blue triangles and labeled with ”sym.”, shows the highest degree
of correlations that reaches a steady state for long evolution times. The
correlations converge to a saturation value of 11.2(8) %, which is in
agreement with the theoretical value of 12.5 % (as expected in the limit of
perfectly correlated dephasing) within 2 standard deviations (see Appendix).

Measurements using the asymmetric configuration (Configuration 2), depicted
with green circles and labeled with ”asym.”, show similar dynamics to the
symmetric setting for times up to twice the coherence time. For longer
evolution times, however, a significant decrease in correlations is observed
as no DFS is available in the system.

The third investigated scenario (Configuration 3) implementing engineered
uncorrelated dynamics by adding spontaneous decay, is depicted with red
diamonds. The correlations do not exceed a value of 3.1(6) % in this case.
This is significantly lower than the maximum of $\bar{I}$ for fully and
partially correlated dephasing dynamics.

The blue shaded area in the figure shows simulated results where random phase
fluctuations are acting on a two-qubit system. From the resulting output state
of the simulation we generate data including projection noise and the same
analysis as for the experimental data is performed. To simulate the asymmetric
configuration the applied random phase fluctuations are acting on the qubit
weighted according to the different susceptibilities to the magnetic field.
For the simulation of uncorrelated spontaneous emission, instead of phase
fluctuations, probabilistic, uncorrelated quantum jump trajectories of the
individual qubits are simulated. A more detailed description of the simulation
can be found in the Appendix. There is qualitative agreement between
simulations and measurements, but still there are significant deviations,
especially in the case of uncorrelated dynamics, of up to approximately
$4\,\sigma$. We assume that this overestimation of the spatial correlations in
the system dynamics by the quantifier is due to miscalibration and drifts of
experimental parameters. For instance a mismatch between the actual and the
calibrated Rabi frequency would lead to additional correlated errors during
the process tomography. This effect is best visible for Configuration 3, where
the dynamics are expected to show no correlations at all.

Figure 6: Dynamics of the spatial correlation quantifier $\bar{I}$ for
different qubit encodings. Three cases are depicted: Both qubits encoded in
$\left|4S_{1/2},m_{S}=-1/2\right\rangle$ $\leftrightarrow$
$\left|3D_{5/2},m_{S}=-5/2\right\rangle$ (green triangles), one qubit encoded
in $\left|4S_{1/2},m_{S}=-1/2\right\rangle$ $\leftrightarrow$
$\left|3D_{5/2},m_{S}=-5/2\right\rangle$ and
$\left|3D_{1/2},m_{S}=-1/2\right\rangle$ $\leftrightarrow$
$\left|3D_{5/2},m_{S}=-5/2\right\rangle$ (blue circles) and both qubits
subject to uncorrelated dynamics via spontaneous decay (red diamonds). The
horizontal axis is normalized to the coherence time for the first two cases
and to the decay time for the third case. Results from a Monte Carlo based
simulation with 500 samples are depicted with shaded areas in the
corresponding color.

###  5.2 Spatial correlations in multi-qubit systems

Due to the exponential scaling of the number of measurement settings for a
single full process tomography an analysis for a four qubit system would take
about 24 hours per waiting time setting and configuration in our system.
Therefore, the feasible method to investigate correlations is based on the
provided lower bound for the correlation measure as described in Sec. 4. For
these experiments, the free-evolution time is fixed to 10 ms, approximately
corresponding to 5 times the coherence time. Inferring from the two-qubit
measurements, this evolution time renders a reliable discrimination of
encoding configurations, leading to differing degrees of correlations,
possible.

First, we investigate the distance dependence of pairwise spatial correlations
along the register. For this we evaluate the lower bound for $\bar{I}$ (Eq.
(14)) between the outermost qubit and subsequently all other qubits. In a
four-qubit system this corresponds to evaluating the observables $X_{1}X_{i}$,
with $X_{i}=\bigl{(}\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\bigr{)}_{i}$ acting on ion $i\in\\{2,3,4\\}$, yielding
the lower bound $\bar{I}_{LB}=\frac{1}{4\cdot 2\ln 2}\left[\langle
X_{1}X_{i}\rangle-\langle X_{1}\rangle\langle X_{i}\rangle\right]^{2}$. By
applying a global $\pi/2$ pulse around the x-axis of the Bloch sphere
$R_{0}(\pi/2)$ the system is prepared in the state
$\left|+\right>~{}:=~{}\frac{1}{\sqrt{d}}\sum_{k=1}^{d}\left|k\right>$, where
$d=2^{4}$ for four ions†. After the preparation the system undergoes a free
evolution. With this preparation the qubits are in an eigenstate of $X_{i}$,
the expectation values of these observables are decaying quickly due to the
dominant dephasing caused by the excess magnetic field fluctuations. Therefore
the lower bound of the dynamical correlations is the tightest for this choice
of observables. The data as shown in Fig. 7 indicate that the spatial
correlations do not show any distance dependence. We compare the measured
values by numerical simulations which are shown as shaded bars in Fig. 7 (see
also Appendix). The measured values are close to the expected value for a
fully dephased state under perfectly correlated noise of 4.5% (see Appendix).
22footnotetext: In that notation the computational basis is numbered
consecutively, so $\left|1\right>=\left|0000\right>$,
$\left|2\right>=\left|0001\right>$, …, $\left|16\right>=\left|1111\right>$.

Figure 7: Lower bound for pairwise spatial correlations in a four-qubit system
as a function of the distance (in terms of ion index difference in the ion
string) to qubit 1, depicted with black circles. The corresponding simulations
are shown with shaded bars. As all qubits have the same encoding, the
correlations between qubit 1 and qubit 2, 3 and 4 respectively are the same
within errorbars.

We then analyze the spatial correlations as given by the four-body observable
$X_{1}X_{2}X_{3}X_{4}$. We investigate three different configurations of qubit
encodings which give rise to different decoherence free subspaces:

  * •

Configuration 1: All four qubits encoded in Encoding A.

  * •

Configuration 2: Qubits 1 and 2 encoded in Encoding A, qubits 3 and 4 encoded
in Encoding B.

  * •

Configuration 3: Qubits 1, 2 and 3 encoded in Encoding A, qubit 4 encoded in
Encoding B.

The preparation of the system is the same as in the measurement of the
pairwise correlations, but after the waiting time, a four qubit state
tomography measurement is performed. From the reconstructed density matrix, we
estimate the expectation values of the observables $\langle X_{1}\rangle$,
$\langle X_{2}\rangle$, $\langle X_{3}\rangle$, $\langle X_{4}\rangle$ and
$\langle X_{1}X_{2}X_{3}X_{4}\rangle$, where
$X_{i}=\bigl{(}\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\bigr{)}_{i}$ is
a Pauli matrix acting on qubit $i$. The experimental results for the lower
bound $\bar{I}_{LB}=\frac{1}{4\cdot 4\ln 2}\left[\langle
X_{1}X_{2}X_{3}X_{4}\rangle-\langle X_{1}\rangle\langle X_{2}\rangle\langle
X_{3}\rangle\langle X_{4}\rangle\right]^{2}$ but also for the individual
expectation values are presented in Fig. 8. In the left subplot in the top row
the lower bound is plotted in green, blue and red for the qubit encoding
configurations 1, 2 and 3, respectively. The theoretically expected values
from numerical simulations of the microscopic noise are depicted with shaded
bars. These estimated correlations and single-qubit expectation values lead to
a lower bound of 1.66(18) %, 1.05(18) % and 0.84(16) % for the three qubit
encoding patterns, respectively. These results show agreement with simulations
within 1 standard deviation. Note furthermore that in particular the lower
bound value of 1.66(18) % for the first encoding pattern in which all four
qubits reside in the Encoding A (Configuration 1) is a signature of the almost
perfectly correlated dynamics - it is close to the theory value predicted for
the long time limit of perfectly correlated dephasing dynamics of 1.27 %. In
contrast, for two qubits in Encoding A and two qubits in B (Configuration 2),
the observed bound of 1.05(18) % is lower than for Configuration 1 and to be
compared with the theory value of 0.56 % predicted for the long time limit
(see Appendix). For the asymmetric Configuration 3 (three qubits in Encoding A
and one in B), one observes with 0.84(16) % the lowest value of the lower
bound. As shown in the Appendix, theory predicts the lower bound to fully
vanish in the limit of even longer waiting times.

Instead of performing state tomography the expectation values necessary to
calculate the lower bound of the correlation quantifier could be measured
directly, leading to a linear scaling of the number of measurements with the
number of qubits in the worst case. In exchange for the better scaling the
correlation estimation gets vulnerable to projection noise. By increasing the
number of repetitions of the experiment this error source can be reduced to
arbitrary low levels.

Figure 8: a) Lower bound $\bar{I}_{LB}$ for $\bar{I}$ for 3 different qubit
encoding configurations described in Section 5.2. b) and c) show the
underlying expectation values from which $\bar{I}_{LB}$ is calculated. The
expected results from numerical simulations are depicted with shaded bars in
the corresponding color.

##  6 Conclusions and Outlook

We have presented a lower bound for the measure for correlations of quantum
dynamics proposed in Ref. [35]. We have shown how this lower bound can be
evaluated without full knowledge of the quantum process and how it can be
extended to multipartite systems. We have applied both the full measure
$\bar{I}$, which requires full tomographic information, as well as the lower
bound $\bar{I}_{LB}$ to different electronic qubit encodings in a trapped ion
quantum information processor.

Our experimentally measured values are in agreement with expected values from
theoretical simulations which are based on a modelling of the various noise
sources for different types of qubit encoding patterns. For strings of up to
four qubits our measurements confirm that the natural noise in the quantum
processor characterized in this work is dominated by perfectly correlated
dephasing noise. Notably, both the values of the exact measure $\bar{I}$, as
obtained from quantum process tomography in the two-ion case, as well as the
values of the lower bounds obtained for up to four ions, clearly reveal that
the noise dynamics is no longer perfectly correlated once a subset of ions is
encoded in different electronic states than the others. Furthermore, the
observed values for $\bar{I}$ for an asymmetric encoding also reveal the time
scale on which this breaking of the perfectly correlated dephasing dynamics
takes effect. This quantitative information is valuable, if one is e.g.
interested in using sets of ion-qubits for the exploitation of decoherence-
free subspaces.

In fact, one of the most intriguing applications of the correlation measure
$\bar{I}$ is to characterize noise processes in the context of quantum error
correction. It should be noted that the measure itself cannot be directly used
to assess the influence of the correlations on a QEC protocol, but it is able
to test a specific noise model and to furthermore estimate the model’s
parameters. In particular, the measure can be used to experimentally determine
the behaviour of correlated noise as a function of the distance between the
constituent particles. The noise model can then inform a microscopic model to
estimate the QEC protocol’s performance. This is due to the fact that the
effect of correlations on error correction procedures depends strongly on the
interplay of the actual form of the spatial correlations. The correlation
measure quantifies the amount of correlations and thus cannot be directly
connected to the threshold of an error correcting code without knowledge of
the actual form of the correlations.

In addition, the exact correlation measure (and also the lower bound) can be
used to certify that the amount of spatial correlations is reduced in qubit
pairs at positions $\mathbf{r}_{i}$ and $\mathbf{r}_{j}$ with increasing
distance according to certain scalings: Observing the way it decreases, e.g.
as a powerlaw $\propto|r_{i}-r_{j}|^{-\alpha}$ with a sufficiently large
exponent $\alpha$, allows one then to establish a connection to quantum error
correction protocols. Such decay behaviour enters error correction protocols
as a necessary condition for the provable existence of a regime in which the
error correcting codes allow for fault-tolerant quantum computing [26, 27, 28,
29, 30, 31, 32].

Complementary, the lower bound is useful to detect unexpected correlations in
systems that do not allow full process tomography. It should be noted that the
lower bound can be estimated using data that is taken routinely during system
tune-up, such as Ramsey experiments, in many different physical quantum
information processing architectures. Based on prior knowledge about the
experimental system, one can also design the measured observable to be
sensitive to a certain kind of errors.

The presented methods can also be applied to characterize the noise
environment in precision measurements. For example in [54], two spatially
separated ions are used for dc magnetometry. The authors also investigated the
coherence time of the singlet state as a function of the ion distance. For
short distances ($\approx 4\,\mu$m) the ions see almost perfectly correlated
noise resulting in no measurable decay (an almost perfect DFS). For larger ion
separation ($\approx 6$ mm), the coherence time is reduced by at least one
order of magnitude, indicating only partially correlated noise. The spatial
correlation measure could be readily applied to this experiment to
characterize the correlations in the noise as a function of ion distance. The
distance of the qubits in the present work is in the order of $5\mu$m and thus
we expect and observe completely correlated noise.

## Acknowledgments

We gratefully acknowledge support by the Austrian Science Fund (FWF), through
the SFB FoQuS (FWF Project No. F4002-N16), as well as the Institut für
Quanteninformation GmbH. This research was funded by the Office of the
Director of National Intelligence (ODNI), Intelligence Advanced Reasearch
Projects Activity (IARPA), through the Army Research Office grant
W911NF-16-1-0070. All statements of fact, opinions or conclusions contained
herein are those of the authors and should not be construed as representing
the official views or policies of IARPA, the ODNI, or the U.S. Government. We
also acknowledge support by U.S. A.R.O. through grant W911NF-14-1-0103, the
Spanish MINECO grant FIS2015-67411, and the CAM research consortium QUITEMAD+
S2013/ICE-2801.

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##  Appendix A Simulations

It is common to find more than one source of dephasing in ion trap
architectures. Here we consider the case where one type of dephasing is caused
by the external and global magnetic field fluctuations and the other due to
the laser frequency fluctuations of the addressing laser source. In
particular, with respect to the contribution from magnetic field fluctuations,
it is important to note that trapped ion qubits encoded in the electronic
qubit levels $S_{1/2}(m_{j}=-1/2)=|1\rangle$ $\rightarrow$
$D_{5/2}(m_{j}=-1/2)=|0\rangle$ will undergo a different dephasing dynamics
than temporarily spectroscopically decoupled (“hidden”) qubits.
Spectroscopical decoupling has been commonly employed e. g. in the context of
repetitive quantum error correction [55], entangling subsets of qubits [56]
and quantum teleportation [57].

Here, we consider a string of two ions, the first of which is
spectroscopically decoupled, and the second one residing in the standard qubit
subspace. Thus, we effectively consider two ions with a laser field addressing
the second one. The joint state of the ions $\rho$ undergoes the evolution
which corresponds to accumulating random phases,

$e^{-i\phi_{B}(a\sigma_{1}^{z}+b\sigma_{2}^{z})}e^{-i\phi_{L}\sigma_{2}^{z}},$
(19)

where $\phi_{B}$ is due to the global magnetic field fluctuations and
$\phi_{L}$ to frequency fluctuations of the laser source addressing the second
ion. The constants $a$ and $b$ depend on the specific energy levels on each
ion taken into consideration. If those are the same, we have $a=b=1$. Denoting
by $p_{B}(\phi_{B})$ and $p_{L}(\phi_{L})$ the probability distributions for
$\phi_{B}$ and $\phi_{L}$, respectively, we have the noisy dynamics

$\mathcal{E}(\rho)=\int d\phi_{B}p_{B}(\phi_{B})\int
d\phi_{L}p_{L}(\phi_{L})[e^{-i\phi_{B}(a\sigma_{1}^{z}+b\sigma_{2}^{z})}e^{-i\phi_{L}\sigma_{2}^{z}}]\rho[e^{i\phi_{B}(a\sigma_{1}^{z}+b\sigma_{2}^{z})}e^{i\phi_{L}\sigma_{2}^{z}}].$
(20)

We can write

$\displaystyle
e^{-i\phi_{B}a\sigma_{1}^{z}}=\cos(a\phi_{B})-i\sin(a\phi_{B})\sigma_{1}^{z},$
(21) $\displaystyle
e^{-i(b\phi_{B}+\phi_{L})\sigma_{2}^{z}}=\cos(b\phi_{B}+\phi_{L})-i\sin(b\phi_{B}+\phi_{L})\sigma_{2}^{z},$
(22)

so that

$\displaystyle
e^{-i\phi_{B}a\sigma_{1}^{z}}e^{-i(b\phi_{B}+\phi_{L})\sigma_{2}^{z}}$
$\displaystyle=\cos(a\phi_{B})\cos(b\phi_{B}+\phi_{L})$
$\displaystyle-i\sin(a\phi_{B})\cos(b\phi_{B}+\phi_{L})\sigma_{1}^{z}$
$\displaystyle-i\cos(a\phi_{B})\sin(b\phi_{B}+\phi_{L})\sigma_{2}^{z}$
$\displaystyle-\sin(a\phi_{B})\sin(b\phi_{B}+\phi_{L})\sigma_{1}^{z}\sigma_{2}^{z},$
(23)

and then

$\mathcal{E}(\rho)=\sum_{\alpha,\beta=0,\dots
3}\chi_{\alpha\beta}G_{\alpha}\rho G_{\beta},$ (24)

with $G_{0}=\mathbb{1}\otimes\mathbb{1}$, $G_{1}:=\sigma_{1}^{z}$,
$G_{2}:=\sigma_{2}^{z}$, and $G_{3}:=\sigma_{1}^{z}\sigma_{2}^{z}$. The
coefficients $\chi_{\alpha\beta}$ form a self-adjoint matrix with the
following components:

$\displaystyle\chi_{00}$ $\displaystyle=\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\cos^{2}(a\phi_{B})\cos^{2}(b\phi_{B}+\phi_{L}),$
$\displaystyle\chi_{01}$ $\displaystyle=\frac{i}{2}\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\sin(2a\phi_{B})\cos^{2}(b\phi_{B}+\phi_{L}),$
$\displaystyle\chi_{02}$ $\displaystyle=\frac{i}{2}\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\cos^{2}(a\phi_{B})\sin[2(b\phi_{B}+\phi_{L})],$
$\displaystyle\chi_{03}$ $\displaystyle=-\frac{1}{4}\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\sin(2a\phi_{B})\sin[2(b\phi_{B}+\phi_{L})],$
$\displaystyle\chi_{11}$ $\displaystyle=\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\sin^{2}(a\phi_{B})\cos^{2}(b\phi_{B}+\phi_{L}),$
$\displaystyle\chi_{12}$ $\displaystyle=\frac{1}{4}\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\sin(2a\phi_{B})\sin[2(b\phi_{B}+\phi_{L})],$
$\displaystyle\chi_{13}$ $\displaystyle=\frac{i}{2}\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\sin^{2}(a\phi_{B})\sin[2(b\phi_{B}+\phi_{L})],$
$\displaystyle\chi_{22}$ $\displaystyle=\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\cos^{2}(a\phi_{B})\sin^{2}(b\phi_{B}+\phi_{L}),$
$\displaystyle\chi_{23}$ $\displaystyle=\frac{i}{2}\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\sin(2a\phi_{B})\sin^{2}(b\phi_{B}+\phi_{L}),$
$\displaystyle\chi_{33}$ $\displaystyle=\int\int
d\phi_{B}d\phi_{L}p_{B}(\phi_{B})p_{L}(\phi_{L})\sin^{2}(a\phi_{B})\sin^{2}(b\phi_{B}+\phi_{L}).$
(25)

If we consider a Gaussian distribution for every random phase

$f(\phi_{B})=\frac{1}{\sqrt{2\pi}\sigma_{B}}e^{-\tfrac{\phi_{B}^{2}}{2\sigma_{B}^{2}}}\quad\text{and}\quad
f(\phi_{L})=\frac{1}{\sqrt{2\pi}\sigma_{L}}e^{-\tfrac{\phi_{L}^{2}}{2\sigma_{L}^{2}}},$
(26)

we obtain $\chi_{01}=\chi_{02}=\chi_{13}=\chi_{23}=0$ because of the odd
parity of the integrating functions.

As we assumed to have pure dephasing dynamics the Choi-Jamiołkowski state can
be written as:

$\displaystyle\rho^{\rm CJ}_{\rm S}$ $\displaystyle=\mathcal{E}_{\rm
Z}\otimes\mathds{1}(|\Phi_{\rm SS^{\prime}}\rangle\langle\Phi_{\rm
SS^{\prime}}|)$
$\displaystyle=\frac{1}{d^{2}}\sum^{d}_{k,l,m,n=1}\alpha_{klmn}\left|kl\right>\left<mn\right|\otimes\left|kl\right>\left<mn\right|$

All diagonal elements are $\alpha_{klkl}=1$, and the remaining integrals in
Eq. (25) can analytically be performed, yielding the following components of
the Choi-Jamiołkowski state $\rho^{\rm CJ}_{\rm S}$:

$\displaystyle\alpha_{1112}$
$\displaystyle=\alpha_{2122}=\chi_{00}+\chi_{11}-\chi_{22}-\chi_{33}=e^{-2(b^{2}\sigma^{2}_{B}+\sigma_{L}^{2})}$
$\displaystyle\alpha_{1121}$
$\displaystyle=\alpha_{1222}=\chi_{00}-\chi_{11}+\chi_{22}-\chi_{33}=e^{-2a^{2}\sigma^{2}_{B}}$
$\displaystyle\alpha_{1122}$
$\displaystyle=\chi_{00}+2\chi_{03}-\chi_{11}-2\chi_{12}-\chi_{22}+\chi_{33}=e^{-2[(a+b)^{2}\sigma^{2}_{B}+\sigma_{L}^{2}]}$
$\displaystyle\alpha_{1221}$
$\displaystyle=\chi_{00}-2\chi_{03}-\chi_{11}+2\chi_{12}-\chi_{22}+\chi_{33}=e^{-2[(a-b)^{2}\sigma^{2}_{B}+\sigma_{L}^{2}]}.$
(27)

This allows us to compute the measure of correlation $\bar{I}$ from the CJ
state. The results are shown in Fig. 9 for $a=\pm b$, and for $a=1$,
$b=-0.83$. For $a\neq\pm b$ the amount of correlations has a maximum for some
value of $\sigma_{B}$ and $\sigma_{L}=0$, and decays for large $\sigma_{B}$ or
$\sigma_{L}$. In the case of $a=\pm b$, where both systems have the same
susceptibility to magnetic field fluctuations, a maximum value for $\bar{I}$
of 0.125 is reached in the limit of $\sigma_{B}\rightarrow\infty$ and
$\sigma_{L}=0$. For the experimental implementations from section 5 enhanced
magnetic field noise, engineered by applying white current noise to coils in
the ion’s surrounding, was added, rendering the laser phase noise described by
$\sigma_{L}$ negligible. Therefore, the presented experimental results
correspond to a cut through these 3D figures at $\sigma_{L}=0$. Configurations
1 in section 5.1 corresponds to $a=b$ (see left part of Fig. 9). The
asymptotic limit of 0.125 is in agreement with the experimental results in
Fig. 6. Configuration 2 is corresponding to $\frac{b}{a}=-0.83$.

Figure 9: Amount of correlations for $a=\pm b$ (left) and $a=1$, $b=-0.83$
(right), as a function of $\sigma_{B}$ and $\sigma_{L}$.

To simulate the dynamics of the build-up of space correlations in Fig. 6 a
different simulation method is used: Random phase fluctuations are acting on
both qubits, where the experimental waiting time is corresponding to the width
of the phase distribution from which the samples are drawn. After 1000
realizations of random dephasing dynamics, the resulting density matrix is
used to generate simulated measurement results by calculating expectation
values for all combinations of Pauli operators acting on the two qubits, which
correspond to the probabilities to measure a certain output state. From this
set of probabilities measurement results are generated using a multinomial
distribution. We estimate the error arising from this statistical error
(projection noise) by performing 100 realizations of the simulation. The
methods used to simulate results for the asymmetric configuration are the
same, apart from the fact that the phase fluctuation of random strength is
applied to qubit 1 directly and multiplied by the factor to include the
different susceptibilities to the magnetic field due to the different Landé g
factors of the included states before acting on qubit 2. For the simulation of
the uncorrelated case a slightly different procedure is used: Instead of
random dephasing, an independent decay to the ground state of the two qubits
is applied to reflect the uncorrelated dynamics due to enhanced spontaneous
decay.
Analytical expectations for the long-time limit under dephasing dynamics

4-qubit correlations -  It is instructive to consider also the long-time
dynamics in the case of perfectly correlated dephasing dynamics, and the
situation in which one is interested in obtaining the lower bound of the
correlation measure. Let us first focus on Configuration 1, in which all four
qubits are encoded in Encoding A. The initial product state of the four
qubits, $\left|\psi\right>=\left|+\right>^{\otimes 4}$, can be written as

$\displaystyle\left|\psi\right>=\frac{1}{4}(\left|\psi_{0}\right>+\left|\psi_{4}\right>)+\frac{1}{2}(\left|\psi_{1}\right>+\left|\psi_{3}\right>)+\sqrt{\frac{3}{8}}\left|\psi_{2}\right>$
(28)

with the Dicke states $\left|\psi_{j}\right>$ of $j=0$ up to $j=4$
excitations,

$\displaystyle\left|\psi_{0}\right>$ $\displaystyle=\left|0000\right>,$
$\displaystyle\left|\psi_{1}\right>$
$\displaystyle=\frac{1}{2}(\left|1000\right>+\left|0100\right>+\left|0010\right>+\left|0001\right>),$
$\displaystyle\left|\psi_{2}\right>$
$\displaystyle=\frac{1}{\sqrt{6}}(\left|0011\right>+\left|0101\right>+\left|0110\right>+\left|1001\right>+\left|1010\right>+\left|1100\right>),$
$\displaystyle\left|\psi_{3}\right>$
$\displaystyle=\frac{1}{2}(\left|0111\right>+\left|1011\right>+\left|1101\right>+\left|1110\right>),$
$\displaystyle\left|\psi_{4}\right>$ $\displaystyle=\left|1111\right>.$

Under spatially perfectly correlated dephasing, the initial state
$\left|\psi\right>$ evolves for times much longer than the single-qubit
coherence time, but still shorter than the life-time of the metastable qubit
state, into

$\displaystyle\left|\psi\right>\left<\psi\right|\stackrel{{\scriptstyle
t\rightarrow\infty}}{{\longrightarrow}}\frac{1}{16}(\left|\psi_{0}\right>\left<\psi_{0}\right|+\left|\psi_{4}\right>\left<\psi_{4}\right|)+\frac{1}{4}(\left|\psi_{1}\right>\left<\psi_{1}\right|+\left|\psi_{3}\right>\left<\psi_{3}\right|)+\frac{3}{8}\left|\psi_{2}\right>\left<\psi_{2}\right|,$
(29)

i.e. the subspaces of a fixed excitation number $j$ are decoherence-free, so
that the initial coherences between basis states within one and the same
excitation number $j$ subspace are preserved, whereas coherences between
subspaces of different $j$ and $j^{\prime}$ are eventually fully lost. From
this it is straightforward to see that the single-qubit coherences vanish for
all four qubits, $\langle X_{j}\rangle=0$. In contrast, the four-qubit
operator $X_{1}X_{2}X_{3}X_{4}$ has a non-zero expectation value, $\langle
X_{1}X_{2}X_{3}X_{4}\rangle=3/8$, which results in a lower bound

$\displaystyle\bar{I}_{LB}=\frac{1}{4\cdot 4\ln 2}\left[\langle
X_{1}X_{2}X_{3}X_{4}\rangle-\langle X_{1}\rangle\langle X_{2}\rangle\langle
X_{3}\rangle\langle X_{4}\rangle\right]^{2}=0.0127.$ (30)

For Configuration 2 (qubits 1 and 2 encoded in Encoding A, qubits 3 and 4
encoded in Encoding B) the initial four-qubit state
$\left|\psi\right>=\left|+\right>^{\otimes 4}$ will envolve into the density
matrix $\rho=\rho_{12}\otimes\rho_{34}$ for long times. The state of the first
and second, and third and fourth qubits, respectively, is given by

$\displaystyle\rho_{12}=\rho_{34}=\frac{1}{4}(\left|00\right>\left<00\right|+\left|11\right>\left<11\right|)+\frac{1}{2}(|\Psi^{+}\rangle\langle\Psi^{+}|),$
(31)

Therefore, all four single-qubit coherences vanish, $\langle X_{j}\rangle=0$.
However, due to the presence of the component which corresponds to the pair of
Bell-states in the partially decohered four-qubit density matrix, the four-
qubit operator $X_{1}X_{2}X_{3}X_{4}$ has again a non-zero expectation value,
$\langle X_{1}X_{2}X_{3}X_{4}\rangle=1/4$, which in this case leads to

$\displaystyle\bar{I}_{LB}=\frac{1}{4\cdot 4\ln 2}\left[\langle
X_{1}X_{2}X_{3}X_{4}\rangle-\langle X_{1}\rangle\langle X_{2}\rangle\langle
X_{3}\rangle\langle X_{4}\rangle\right]^{2}=\frac{1}{256\cdot\ln
2}=0.0056=0.56\%.$ (32)

Finally, for Configuration 3, with qubits 1, 2 and 3 encoded in Encoding A,
and qubit 4 in Encoding B, one can show that the initial state
$\left|\psi\right>=\left|+\right>^{\otimes 4}$ evolves for long enough times
into

$\displaystyle\rho=\left[\frac{1}{8}\left(\left|000\right>\left<000\right|+\left|111\right>\left<111\right|\right)+\frac{3}{8}\left(\left|\psi_{1}^{\prime}\right>\left<\psi_{1}^{\prime}\right|+\left|\psi_{2}^{\prime}\right>\left<\psi_{2}^{\prime}\right|\right)\right]_{123}\otimes\frac{1}{2}\mathbb{1}_{4}$
(33)

with the Dicke-type 3-qubit states

$\displaystyle\left|\psi_{1}^{\prime}\right>$
$\displaystyle=\frac{1}{\sqrt{3}}(\left|100\right>+\left|010\right>+\left|001\right>),$
$\displaystyle\left|\psi_{2}^{\prime}\right>$
$\displaystyle=\frac{1}{\sqrt{3}}(\left|110\right>+\left|101\right>+\left|011\right>).$

Since for this state both the single-qubit coherences and $\langle
X_{1}X_{2}X_{3}X_{4}\rangle$ vanish, one expects a vanishing spatial
correlation measure, $\bar{I}_{LB}=0$, in this limit.

2-qubit correlations -  Similarly, it is straightforward to obtain the
expected behaviour for the long-time dynamics of two qubits undergoing
perfectly correlated dephasing. In this case, the initial state
$\left|\psi\right>=\left|+\right>^{\otimes 2}$ of a pair of qubits evolves for
long times into

$\displaystyle\left|\psi\right>\left<\psi\right|\stackrel{{\scriptstyle
t\rightarrow\infty}}{{\longrightarrow}}\frac{1}{4}(\left|00\right>\left<00\right|+\left|11\right>\left<11\right|)+\frac{1}{2}(|\Psi^{+}\rangle\langle\Psi^{+}|),$
(34)

with the Bell state
$\left|\Psi^{+}\right>=\frac{1}{\sqrt{2}}(\left|01\right>+\left|10\right>)$.
Again, the single-qubit coherences vanish, $\langle X_{1}\rangle=\langle
X_{2}\rangle=0$, whereas the two-qubit operator $X_{1}X_{2}$ saturates at
$\langle X_{1}X_{2}\rangle=1/2$, resulting in a lower bound for the two-qubit
correlations of

$\displaystyle\bar{I}_{LB}=\frac{1}{4\cdot 2\ln 2}\left[\langle
X_{1}X_{2}\rangle-\langle X_{1}\rangle\langle X_{2}\rangle\right]^{2}=0.0451.$
(35)