††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Revealing microcanonical phases and phase transitions of strongly correlated
electrons via time-averaged classical shadows

Gaurav Gyawali  Department of Physics, Cornell University, Ithaca, NY 14853
Department of Physics, Harvard University, Cambridge, MA 02138, USA  Mabrur
Ahmed  Department of Physics, Applied Physics, and Astronomy, Binghamton
University, Binghamton, NY 13902  Eric W. Aspling  Department of Physics,
Applied Physics, and Astronomy, Binghamton University, Binghamton, NY 13902
Luke Ellert-Beck  School of Physics and Applied Physics, Southern Illinois
University, Carbondale, IL 62901  Department of Physics, University of Rhode
Island, Kingston, RI 02881  Michael J. Lawler  Department of Physics, Applied
Physics, and Astronomy, Binghamton University, Binghamton, NY 13902
Department of Physics, Cornell University, Ithaca, NY 14853  Department of
Physics, Harvard University, Cambridge, MA 02138, USA

###### Abstract

Quantum computers and simulators promise to enable the study of strongly
correlated quantum systems. Yet, surprisingly, it is hard for them to compute
ground states. They can, however, efficiently compute the dynamics of closed
quantum systems. We propose a method to study the quantum thermodynamics of
strongly correlated electrons from quantum dynamics. We define time-averaged
classical shadows (TACS) and prove it is a classical shadow(CS) of the von
Neumann ensemble, the time-averaged density matrix. We then show that the
diffusion maps, an unsupervised machine learning algorithm, can efficiently
learn the phase diagram and phase transition of the one-dimensional transverse
field Ising model both for ground states using CS _and state trajectories_
using TACS. It does so from state trajectories by learning features that
appear to be susceptibility and entropy from a total of 90,000 shots taken
along a path in the microcanonical phase diagram. Our results suggest a low
number of shots from quantum simulators can produce quantum thermodynamic data
with a quantum advantage.

††preprint: APS/123-QED

##  I Introduction

Simulation of strongly correlated electrons in the context of quantum
chemistry and condensed matter physics is one of the potential areas in which
quantum computers will have a significant advantage over their classical
counterparts [1, 2, 3]. Strong interaction between the ostensibly simple
electrons can give rise to novel phases, including high-temperature
superconductivity [4, 5], strange metallic behavior[6], fractional excitations
[7], and quantum spin liquids[8]. Condensed matter physics aims to understand
these novel behaviors by studying their phase diagrams and phase-defining
features. However, failure of perturbation theory and exponential scaling of
the Hilbert space for strongly correlated electrons presents a formidable
challenge to classical simulation methods such as exact diagonalization,
density matrix renormalization group(DMRG) [9, 10, 11], quantum Monte-Carlo
[12] and dynamical mean-field theory[13]. Whereas, this same challenge
provides an exciting opportunity for near-term quantum computers.

Harnessing the power of a quantum computer to simulate quantum systems [14]
requires (i) algorithms that can be executed in a reasonable time and (ii) the
ability to learn from quantum experiments without exponentially many
measurements. Studying the phases via ground state preparation is a QMA-
complete problem [15, 16, 17, 18, 19], which cannot be carried out in a
reasonable time, even with quantum resources. However, performing dynamics on
a quantum state is known to be a BQP-hard problem [20, 21], possible within
polynomial time. Likewise, it has been shown that shadow tomography[22]
methods such as classical shadows(CS)[23, 24, 25] are effective at predicting
properties using very few measurements. Thus, if we could combine dynamics
simulations and classical shadows, we would have an efficient algorithm to
simulate condensed matter.

We need to prepare low-energy initial states to use dynamics to simulate
condensed matter and exploit quantum ergodicity [26]. Although preparing
ground states of local Hamiltonians on a physical lattice is a challenging
problem on a quantum device, it is always possible to prepare some low-energy
state with a constant-depth circuit [27, 28]. Ergodicity then provides a link
between statistical averages and time averages obtained from the dynamics of
the low-energy state. It is important that the observables of interest, such
as the order parameter, equilibrate before the qubits decohere. Nevertheless,
rapid equilibration for most local observables is a feature shared by many
interacting quantum systems[29, 30, 31]. Thus, equilibrium dynamics of low-
energy states appears to be a promising route to studying equilibrium quantum
phases and phase transitions.

In this manuscript, we present an algorithm to identifying phase diagrams and
phase transitions of strongly correlated systems _motivated by how physical
quantum systems operate_. It consists of i) identifying an initial state, ii)
generating state trajectories by evolving this state in time, iii) using
shadow tomography to convert the quantum state to classical data, and iv)
applying unsupervised machine learning methods to discover phases of matter
and their phase transitions. A schematic overview of our approach is shown in
Fig. 1.

Figure 1: Schematic overview of our study. (a) Classical shadows(CS) of ground
states and time-averaged classical shadows(TACS) from dynamics of a time-
reversal invariant GHZ state are generated using quantum simulation. (b) The
classical data from quantum simulation is then fed into diffusion map, an
unsupervised machine learning algorithm to learn the phases.

We obtain numerical results using diffusion maps [32, 33], an unsupervised
machine learning(UL) algorithm to learn phase features from unlabeled data.
First, we benchmark diffusion maps on CS data from ground states of a
100-qubit one-dimensional Transverse Field Ising Model(1DTFIM) simulation. It
identifies the magnetic phase transition, its continuous nature, and the
magnetization behavior as a function of the magnetic field (see section
II)—another machine-learning-from-CS success story[23, 24, 34, 25].
Generalizing CS to time-averaged CS (TACS), a shadow tomographic[22]
representation of the time-averaged density matrix[26], we then show, in
section III, on a 20 qubit 1DTFIM, diffusion maps also identify the quantum
critical region and cross-overs along a path in the microcanonical phase
diagram from a total of 90,000 shots on state trajectories. Diffusion maps do
so efficiently by learning features from TACS that appear to be susceptibility
and entropy. Hence, we can efficiently study the phases and phase transitions
of strongly correlated electrons by quantum-simulating state trajectories.

##  II Von Neumann’s Microcanonical Ensemble

A central goal of quantum computing is to build qubits that are completely
isolated from their environment. While this is not the case today, the current
development of quantum error correction techniques[35, 36] suggests it is in
our future. Simulating quantum systems on a quantum computer will therefore
take place _within the microcanonical ensemble_. But quantum microcanonical
dynamics, the evolution of a closed quantum system under Schrödinger’s
equation, does not directly produce the microcanonical ensemble.

Following von Neumann’s 1929 paper[26, 37] on the quantum ergodic theorem, it
is straightforward to derive a link between time averages and statistical
averages using density matrices. Assuming we start from an initial state
$|\psi(0)\rangle$ and evolve under a Hamiltonian $H$ via a quantum circuit
algorithm to $|\psi(t)\rangle$, the equilibrium distribution is captured by
the von Neumann ensemble, the time average of the density matrix

$\rho_{vn}\\!=\\!\frac{1}{T}\int_{0}^{T}\\!\\!dt|\psi(t)\rangle\langle\psi(t)|\\!\underset{T\to\infty}{\xrightarrow{\hskip
22.76228pt}}\\!\omega\\!=\\!\sum_{n}P_{n}|\psi(0)\rangle\langle\psi(0)|P_{n},$
(1)

where $P_{n}$ is a projector onto the $n^{\text{th}}$ degenerate subspace of
the energy eigenvalues i.e. $P_{n}=\sum_{k\in n}|E_{k}\rangle\langle E_{k}|$.
The $T\to\infty$ limit, obtained in exponential time[26, 38, 39, 31], results
in equilibration of all observables. Existence of $\rho_{vn}$ results in the
ergodic principle that time averages of observables should be captured by the
statistical averaging with respect to $\rho_{vn}$. Specifically, in the
Schrödinger picture,

$\displaystyle\langle O\rangle_{T}$
$\displaystyle=\frac{1}{T}\int_{0}^{T}dt\langle\psi(t)|O|\psi(t)\rangle$ (2)
$\displaystyle=\frac{1}{T}\int_{0}^{T}dt\text{Tr}\left(|\psi(t)\rangle\langle\psi(t)|O\right)$
(3)
$\displaystyle=\text{Tr}\left(\rho_{vn}O\right)\underset{T\to\infty}{\longrightarrow}\Tr(\omega
O)$ (4)

where again the $T\to\infty$ limit produces equilibration, though here a
presumably easier state to reach for it is just necessary for $\rho_{vn}$ and
$\omega$ to be indistinguishable to $O$[31]. Thus, the time-averaged density
matrix is a link between time averages and statistical averages governed by
the von Neumann ensemble $\rho_{vn}$, a link that holds regardless of whether
the system equilibrates.

The connection to Boltzmann’s microcanonical ensemble, obtained by quantizing
the classical microcanonical ensemble, is achieved by taking the thermodynamic
limit, measuring only coarse-grained observables, requiring non-degenerate
energy level spacings/gaps, and considering “typical” initial states. The
coarse-grained observables used by von Neumann were a commuting set of
generators of global symmetries, a restriction more recently generalized[40]
as part of the development of quantum thermodynamic resource theories[41]. The
typical initial states were the first recorded use of typicality
arguments[37]. Under these circumstances, von Neumann obtained

$\omega\sim\rho_{mc}=\frac{1}{\Omega}\sum_{E<E_{n}<E+\Delta}|E_{n}\rangle\langle
E_{n}|.$ (5)

Namely, for the purposes of computing $\text{Tr}(O\omega)$, there is no
difference between using $\omega$ and $\rho_{mc}$, a maximally mixed state
within an energy window $[E,E+\Delta]$ containing $\Omega$ states. Von Neumann
extended this claim to a second ensemble,
$\rho^{\prime}_{vn}=|\psi(T)\rangle\langle\psi(T)|$, that also satisfies
$\rho^{\prime}_{vn}\sim\rho_{mc}$. The equivalence to $\rho_{mc}$ is also
readily proven with the seemingly stronger requirement that each eigenstate
satisfies the eigenstate thermalization hypothesis[42, 43, 44]. So, in this
way, $\rho_{vn}$ reproduces $\rho_{mc}$.

A central new ingredient in von Neumann’s approach to describing the
microcanonical ensemble is the initial state, which is never fully forgotten
in quantum mechanics. For each initial condition, it’s necessary to check
whether the time series was run long enough for relevant observables to reach
equilibrium . The same observable will equilibrate at different times
depending on the initial conditions. It turns out, this time depends on the
effective dimension given by $d_{\text{eff}}=1/\sum p_{k}^{2}$, where
$p_{k}=|\bra{\psi}\ket{E_{k}}|^{2}$[38, 39]. Here, the overlaps $p_{k}$
measure how many energy eigenstates have a significant weight in
$|\psi\rangle$. Remarkably, a large effective dimension results in rapid
equilibration. Furthermore, there is a bound on the equilibration time given
by the second largest $p_{k}$ if it is significantly smaller than
$1/d_{\text{eff}}$[38]. The distribution of the $p_{k}$’s likely also affects
equilibration times[45]. These arguments suggest we choose initial conditions
that exhibit a small overlap with most energy levels and or a macroscopic
occupation of a single energy level.

Because we will use a machine learning method as part of our study of von
Neumann’s microcanonical ensemble, we need to compare it to what we already
know to validate the approach. In the next section, section III, we will turn
to a CS data-driven ground state before continuing to our TACS data-driven
thermodynamic study in section V.

##  III Ground state data

To verify our approach to phase classification and phase-defining feature
identification, we first apply it to ground states of the ferromagnetic 1DTFIM
defined by the Hamiltvon neuonian:

$H_{\text{1DTFIM}}=-\sum_{\langle i,j\rangle}Z_{i}Z_{j}+h_{x}\sum_{i}X_{i},$
(6)

where $\langle.\rangle$ denotes nearest neighbors, $Z_{i}$ is the Pauli$-Z$
operator, and $h_{x}$ is a parameter proportional to the transverse magnetic
field. At zero temperature, this model has a ferromagnetic phase for
$\absolutevalue{h_{x}}<1$, and a paramagnetic phase for
$\absolutevalue{h_{x}}>1$. The ground state study uses diffusion maps to
detect the second-order phase transition at $h_{x}=1$ for a $100$ site 1DTFIM.
The ground states were generated using density matrix renormalization group
(DMRG) [9, 10, 11] with ITensor package[46]. Training datasets were generated
using two kinds of measurements on these ground states- (i) computational
basis measurements to obtain the Z dataset and (ii) measurements on a random
Pauli basis to obtain the CS dataset.

###  III.1 Classical Shadows

Figure 2: Diagrammatic description of classical shadows showing a linear
relationship between $S_{N}[\rho]$ and estimator
$\sigma_{N}[\rho]\approx\rho$. (a) The full-density matrix $\rho$ can be
approximated by summing over reduced classical shadows with a coefficient that
grows exponentially in the number of remaining qubits. (b) A classical shadow
$S_{N}[\rho]$ is obtained by summing over $N$ measurement outcomes on random
Pauli bases. For a given $N$, a reduced density matrix, that involves smaller
coefficients in the expansion, can be approximated more accurately compared to
the full density matrix.

Obtaining any useful information from a quantum computer requires performing
measurements on a quantum state, which is destructive to the quantum
information by nature. Since the dimension of the Hilbert space increases
exponentially in the number of qubits, a naive strategy to learn the state
requires an exponentially large number of copies. Aaronson[22] introduced an
alternative method using the notion of shadow tomography, an approximate
classical description of the quantum state, in which $M$ properties of a
quantum state can be estimated with error $\epsilon$ by only
$O(\frac{M}{\epsilon^{2}})$ copies of the state. We can think of a shadow as
an approximation of a quantum state $\rho$ by summing over measurement
outcomes $x$, obtained by performing measurements on bases $b$ for a quantum
state $x$, i.e.

$S[\rho]=\sum_{b,x}P(b)\mathcal{P}_{b,x}\rho\mathcal{P}_{b,x},$ (7)

where $\mathcal{P}_{b,x}$ is a projector onto the measurement outcome $x$ on
basis $b$, and $P(b)$ is the probability of choosing $b$.

Based on this notion, Huang et al.[47, 24, 25] developed an algorithm called
classical shadows and showed that it is highly successful at learning the
properties of a many-body system. Two kinds of measurement protocols were
proposed to construct classical shadows- (i) random Clifford measurements on
the entire Hilbert space; (ii) random single-qubit Pauli measurements.
Protocol (ii) results in very shallow measurement circuits and thus is more
suitable for the NISQ-era[48] hardware. After measuring each of the qubits in
some random Pauli basis $X$, $Y$ or $Z$ with outcomes $\pm 1$, the post-
measurement wavefunction is given by the product state
$\left|s^{(n)}\right\rangle=\bigotimes_{l=1}^{L}\left|s_{l}^{(n)}\right\rangle$.
Here,
$\left|s_{l}^{(n)}\right\rangle\in\\{|0\rangle,|1\rangle,|+\rangle,|-\rangle,|\mathrm{i}+\rangle,|\mathrm{i}-\rangle\\}$
is a Pauli basis state to which the $l^{th}$ qubit has collapsed. The
classical shadow $S_{N}[\rho]$ is obtained by summing over $N$ such randomized
measurement outcomes as follows (also see Fig. 2(b))

$\displaystyle S_{N}[\rho]$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\ket{s^{(n)}}\bra{s^{(n)}}$ (8)
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\ket{s_{1}^{(n)}}\bra{s_{1}^{(n)}}\otimes\cdots\otimes\ket{s_{L}^{(n)}}\bra{s_{L}^{(n)}}.$
(9)

The underlying quantum state $\rho$ can be approximated by adding the reduced
classical shadows (see Fig. 2(a)). This sum simplifies to the following
expression from Ref. [47, 24, 25]

$\rho\approx\sigma_{N}(\rho)=\frac{1}{N}\sum_{n=1}^{N}\sigma_{1}^{(n)}\otimes\cdots\otimes\sigma_{L}^{(n)},$
(10)

where

$\sigma_{l}^{(n)}=3\left|s_{l}^{(n)}\right\rangle\left\langle
s_{l}^{(n)}\right|-\mathbb{I}.$ (11)

The definition of $S_{N}(\rho)$ presented above is different from Refs. 47, 24
which defines it to be the dataset of shots itself and not the density matrix
obtained from these shots. But both definitions are complete for Fig. 2 (whose
derivation from tensor network diagrams is presented in Appendix A) shows the
density matrix $S_{N}(\rho)$ defined above is linearly related to the
estimator $\sigma_{N}(\rho)$ of the quantum state $\rho$ obtained by Refs. 47,
24, 25. Hence, the two definitions are informationally equivalent.

Although estimating the exact density matrix requires $N\rightarrow\infty$, we
still desire to predict various linear as well as nonlinear functions of
$\rho$ (e.g., $\Tr(O\rho)$ and $\Tr(\rho\text{log}(\rho))$ respectively). This
can be achieved with $N\propto\log(L)4^{k}/\epsilon^{2}$ copies of the state,
where $k$ is the locality of operator $O$ [34]. It was shown in Ref. [24] that
classical machine learning algorithms can efficiently predict the ground state
properties of gapped Hamiltonians in finite spatial dimensions after learning
the classical shadows from a training set. An example of interest is
classifying the quantum phases of matter. Classifying the symmetry-breaking
phases is conceptually simple because it involves calculating $\text{tr}(\rho
O)$ for some k-local observable $O$, such that $\text{tr}(\rho O)\geq
1\>\forall\>\rho\in\text{ phase A}$ and $\text{tr}(\rho
O)\leq-1\>\forall\>\rho\in\text{ phase B}$.

In contrast to classifying symmetry-breaking phases, capturing continuous
phase transitions and classifying topological phases involves nonlinear-
in-$\rho$ observables like critical exponents and entropy, which are harder to
estimate than linear observables. Learning such nonlinear functions requires
an expressive ML model. A central object in kernel-based ML is the kernel
function, a local similarity measure in the feature space where the samples
live. Ref. [24] proposed a kernel based on mapping from classical shadows to a
high-dimensional feature space that includes the polynomial expansion of many-
body reduced density matrices. Learning nonlinear functions requires access to
k-body reduced density matrices, where k may be large, but with enough shots,
classical shadows can accomplish this.Using such a kernel, Ref. 24 found a
rigorous guarantee that a classical ML algorithm can efficiently classify
phases of matter, including the topological phases. We will employ this kernel
to study the continuous phases transition in the 1DTFIM.

###  III.2 Machine Learning Method: Diffusion Maps

Let’s now turn to the final step in our approach: applying an unsupervised
machine learning method called diffusion maps [32, 33] to extract features
from the shadow tomography data. A diffusion map is a nonlinear dimensionality
reduction technique that relies on learning the underlying manifold from which
the data points have been generated. Recently, this method was used to
identify phases in systems with complex order parameters, which are difficult
to learn using linear methods (such as principal component analysis
(PCA)[49]). Examples of such phase identification studies include: topological
phases and phase transitions [50], incommensurate phases, and many-body
localized phases in quantum systems [51].

In the application of diffusion maps, we imagine a random walk on a dataset
$X\left(x_{1},x_{2},\ldots,x_{N}\right)$, where the $x_{n}$ are estimators
$\sigma_{N}(\rho)$ of density matrices $\rho$ obtained from different points
in the phase diagram. The transition probability $P(j\backslash i)$ of jumping
from $x_{i}$ to $x_{j}$ in a single “timestep” is proportional to the kernel
function $k(x_{i},x_{j})$, a non-negative similarity measure between the two
data points. Here we use the classical shadow kernel function prescribed in
[24], defined to be for two points $x$ and $\tilde{x}$ in the dataset

$\displaystyle
k^{(shadow)}(x,\tilde{x})=k^{(shadow)}(\sigma_{N}(\rho),\sigma_{N}(\tilde{\rho})){}$
(12)
$\displaystyle=\exp\\!\left(\\!\sum_{n,n^{\prime}=1}^{N}\\!\frac{\tau}{N^{2}}\\!\exp\\!\left(\\!\frac{\gamma}{L}\\!\sum_{l=1}^{L}\\!\Tr[\sigma_{l}^{(n)}\tilde{\sigma}_{l}^{(n^{\prime})}]\\!\right)\\!\right).$

This kernel measures the local similarity between $x$ and $\tilde{x}$ by
comparing the trace distance between the CS estimates of all k-reduced density
matrices. For the diagonal components ($x=\tilde{x}$) of the kernel matrix,
the trace distance between the k-reduced density matrices is the $2^{nd}$
Renyi-entropy. We then construct a transition probability matrix $P$ such that

${P(j\backslash i)}=\frac{k^{(shadow)}(i,j)}{\sum_{l}k^{(shadow)}(i,l)}.$ (13)

After $t$ timesteps of the random walk, the transition probabilities are given
by the matrix $P^{t}$, where $P_{ij}^{t}$ gives the probability of going from
$x_{i}$ to $x_{j}$ in $t$ timesteps, it’s a sum of the probabilities
associated with all of the possible paths to go from $x_{i}$ to $x_{j}$ in $t$
timesteps. As $t$ increases, the diffusion process unfolds, where data points
situated along the overall geometric structure of the dataset become more
strongly connected because of the abundance of strongly connected intermediate
points along the way.

Given this random walk, we can define a ‘diffusion distance’ to quantify this
idea of connectivity between two data points:

$D_{t}(x_{i},x_{j})^{2}=\sum_{m=1}^{N}|P_{im}^{t}-P_{mj}^{t}|^{2},$ (14)

where the bigger the diffusion distance, the weaker the connection between
them. This allows us to map the data points onto a new ‘diffusion space’ so
that the diffusion distance in data space is equal to the Euclidean distance
in this new space. Following Ref. 32, we will do so with the map:

$x_{i}\rightarrow
y_{i}=\begin{bmatrix}\lambda_{1}^{t}\psi_{1}(i),&\lambda_{2}^{t}\psi_{2}(i),&\cdots,&\lambda_{N-1}^{t}\psi_{N-1}(i)\end{bmatrix}$
(15)

where $\lambda_{k}$ and $\psi_{k}$ are eigenvalues and right eigenvectors of
the matrix $P^{t}$, $\psi_{k}(i)$ is the $i$-the element of the $k$-th
eigenvector. Then the diffusion distance is,

$D_{t}^{2}(x_{i},x_{j})=|y_{i}-y_{j}|^{2}\\\
=\sum_{k=1}^{N-1}(\lambda_{k}^{t})^{2}\left[\psi_{k}(i)-\psi_{k}(j)\right]^{2}.$
(16)

Plotting data in this new space provides an intuitive geometric picture of the
data manifold.

The map provides several features we can exploit when interpreting the data.
In equations (15) and (16) the $k=0$ component is ignored because the leading
eigenvector, $\psi_{0}(i)=\frac{1}{\sqrt{N}},\;\lambda_{0}=1$, is constant for
all $i$ by the Perron-Frobenius theorem. But this constant eigenvector impacts
the other eigenvectors: We note add a constant term
$\psi_{k}(i)\to\psi_{k}(i)+C$ to all other eigenvectors and still preserve the
diffusion distance. Additionally, equation (16) suggests a dimensionality
reduction, as the terms with bigger $\lambda_{k}$ will dominate the sum
increasingly as $t\rightarrow\infty$. So plotting the data $x_{i}$ in the
truncated space
$[\lambda_{1}^{t}\psi_{1}(i),\lambda_{2}^{t}\psi_{2}(i),\lambda_{M}^{t}\ldots,\psi_{M}(i)$
with $M$ determined by keeping only the significant eigenvalues
$\lambda_{m}\gg\lambda_{M+1}$, implies they are accurately separated by
distance $D_{t}(x_{i},x_{j})$. Lastly, we see that $t$ is arbitrary. Choosing
different $t$ rescales the lengths of each component of the vector $y$. Hence,
the data exists on a hyperplane in Euclidean space up to a certain shift in
the origin and a one-parameter rescaling of the axes.

So, using the properties of the diffusion space, we can define _diffusion
coordinates_
$\text{dc1}(i)=A(\psi_{1}(i)+C),\text{dc2}(i)=B(\psi_{2}(i)+C),\ldots$ that
map the data $x_{i}$ onto a Euclidean space that through the choice of
constants $A,B,C,\ldots$ _allow us to interpret the coordinates of each point
and visualize the geometry of the data_.

##  IV Phase Classification of Ground States

For the ground state study, we used two datasets, one the computational basis
measurements and the other generated via CS tomography. The first (Z-dataset)
contains qubit measurements only along the Z-axis [in Eq. 7, $P(b)=1$ for
$b=Z$, $P(b)=0$ for all other b]. While the other (CS-dataset) has randomized
Pauli measurements using the CS method [$P(b)=\frac{1}{3}$ for
$b\in\\{X,Y,Z\\}$]. Both of them contain 200 100-spin 1DTFIM ground state
shots for each state obtained from different $h_{x}$ values ($h_{x}$ ranging
from 0.1 to 100). Since the $Z$-magnetization is the order parameter for TFIM,
the UL algorithm should be able to learn the phases of the model from the
Z-dataset. Using this knowledge, we compare the Z-dataset and the CS-data to
see if the UL algorithm can successfully identify phases in each case and if
so how it does so.

Figure 3: Learning phases from ground state (CS and Z) data. (a) and (c) show
the 10 largest eigenvalues of the $P$-matrix (excluding the trivial $k=0$) for
(a) Z-data and (c) CS-data datasets. (b) Z-data points in 5D diffusion space
visualized in 2D, using metric MDS. Clustering clearly emerges based on the
two phases of the model. (d) CS-data points in 2D diffusion space, the figure
reveals the symmetry-breaking phase transition. In this case, there is a
direct correlation of relevant parameters, the z-magnetization and the
$h_{x}$-values, with machine-learned diffusion coordinates dc1 and dc2
respectively. Notice the CS-data shows one cluster in (d) with non-trivial
geometry associated with the critical point, while the Z-data shows two
clusters in (b) with trivial geometry and no understanding of the critical
point.

By deploying diffusion maps armed with the shadows kernel function (Eq. 12)
utilized for both data sets as our UL model, we are able to identify the
phases from both the Z data and the CS data. In both cases, we set $\tau=1$,
$\gamma=1$ in Eq. 12. For the Z data, we chose the first five non-trivial
eigenvectors as the diffusion space basis vectors because the $P$-matrix (Eq.
13) eigenvalue spectrum shows the first five eigenvalues to be larger than
others (Fig. 3(a)). Mapping the states from Z data onto this five-dimensional
diffusion space, we found that clear clustering emerges based on the phases of
the states. We used multidimensional scaling (MDS), a dimensionality reduction
method [52] that seeks to preserve point-to-point distances, to project these
states onto a 2D plane. We see a clear separation of the two phases even on
this 2D reduced space (Fig. 3(b)), indicating the machine’s success in
identifying the two phases.

From the CS data, the unsupervised learning algorithm was also able to learn
about the phases and the underlying parameters of the model. We can see in
Fig. 3(c) that the $P$ matrix has two non-trivial eigenvalues larger than the
rest. The eigenvectors corresponding to these two eigenvalues are the basis
vectors of the reduced diffusion space. Figure (3(d)) shows all of the ground
state classical shadows projected onto this two-dimensional plane. It shows
three groups- the top left and the bottom left are the all-up and all-down
states, whereas the group on the center right are states in the disordered
phase. This closely resembles the spontaneous symmetry-breaking phase
transition of 1DTFIM[53]. The learned diffusion coordinates (dc1 and dc2) have
direct correlations with the magnetization $<M_{z}>$ and the field values
$h_{x}$ respectively, as shown in appendix C (Fig. 8).

The above-mentioned clustering is dependent on the number of snapshots $N$ for
a given state. However, the dimensionality reduction and the subsequent
clustering will settle down after a minimum value of $N$ has been reached
($N_{c}$). Increasing the value of $N$ beyond that point does not change the
results in any significant way. We find $N_{c}$ (Z data) $<N_{c}$ (CS data),
so it is easier for the algorithm to learn the phase space structure of the
Z-data (hence fewer snapshots are required) than the CS-data.

A striking feature of the diffusion map results presented in Fig. 3 is the
geometry it reveals about the data. In the case of Z-data, it finds the data
is separated into two distinct clusters(Fig. 3(b)) while in the CS-data case
it finds only one cluster but that this cluster has a non-trivial geometry
with three curves meeting at the critical point (Fig. 3(d)). This geometry is
directly a consequence of visualizing of the data space through the lens of
the kernel function that defines distances between data points via Eqs. 13 and
14. Another kernel function might see the same CS-data as separate clusters.
Hence, it is a striking feature of the kernel of Eq. 12 that it can capture
the full geometry of phase defining features in the TFIM model.

##  V Microcanonical Dynamics of the 1DTFIM

Despite the successful identification of the ground state phases with the
CS+ML model, the fact remains that the problem of calculating ground-states is
a QMA-hard problem [18]. We now turn to an algorithm built using Schrödinger
dynamics, a known BQP class algorithm, that aims to reveal the microcanonical
phase diagram as a proxy for a ground state study.

Consider now Fig. 4, a sketch of the thermodynamic phase diagram of the 1DTFIM
relating internal energy $E=\langle\psi(0)|\hat{H}|\psi(0)\rangle$ to the
transverse magnetic field $h_{x}$ inspired by Ref. [54]. This phase diagram is
relevant for a microcanonical dynamics study governed by the entropy
$S(E,h_{x})$. It exists even for a simulation over a finite time $T$ and with
a finite number of spins $N$ and a specific choice of initial conditions but
with finite $T$, finite $N$, and initial choice-dependent errors that round
phase transitions. We present in this figure our expectations for the phase
diagram in this context, pointing out phase transitions where the phase
diagram will sharpen in the thermodynamic limit. We further highlight the path
through the phase diagram carried out by our simulations below, showing that
we expect it to cross the quantum critical region and so be sensitive to the
phase diagram at a rounded level even in the long-$T$, large-$N$ limit.

Figure 4:  A sketch of the expected 1DTFIM phase diagram at finite $T$ and
finite $N$ as a function of internal energy $E$ and transverse magnetic field
$h_{x}$. This diagram is a modification of the canonical ensemble
representation of the phase diagram in Ref. 54, adapted to the microcanonical
ensemble. The spectrum is mirror symmetric about $E=0$ due to the chiral
symmetry $\mathcal{C}=ZYZYZY...$

To reveal the phase diagram expected from microcanonical dynamics presented in
Fig. 4, we need an experimentally producible classical representation of the
quantum data obtained from a microcanonical dynamics simulation. Noticing that
the time-averaged integral amounts to an expectation value of the pure state
density matrix $|\psi(t)\rangle\langle\psi(t)|$ over the probability
distribution $P_{T}(t)=(1/T)\left(\Theta(T-t\rangle)-\Theta(-t)\right)$, where
$\Theta(x)$ is the Heaviside step function, we see we can construct _time-
averaged classical shadows_(TACS) by the quantum channel

$TACS\left[\rho\right]=\lim_{T\to\infty}\int
dt\sum_{b,\sigma}P_{T}(t)P(b)\mathcal{P}_{b,\sigma}|\psi(t)\rangle\langle\psi(t)|\mathcal{P}_{b,\sigma}.$
(17)

Hence by sampling the joint probability distribution $P_{T}(t)P(b)$ to obtain
$(t_{i},b_{i})$, $i=1\ldots N$, and then measuring one shot $\sigma_{i}$ from
$|\psi(t_{i})\rangle$ in basis $b_{i}$ we obtain a finite-shot TACS via

$TACS_{N}\left[\rho\right]=\sum_{i=1}^{N}|t_{i}b_{i}\sigma_{i}\rangle\langle
t_{i}b_{i}\sigma_{i}|$ (18)

This approach captures the power of CS tomography and enables an experimental
study of microcanonical thermodynamics.

With this shadow tomography method in mind, we ran quantum dynamics
simulations of 1DTFIM using the TDVP algorithm[55, 56] starting from the GHZ
state $\ket{\psi(0)}=\frac{\ket{000\cdots}+\ket{111\cdots}}{\sqrt{2}}$ to
generate TACS data from 500 randomly sampled dimensionless time values between
$t=10.0$ to $t=20.0$ and 187 randomly sampled $h_{x}$ field values between
$h_{x}=0.1$ to $h_{x}=10.0$. An example code to generate TACS dataset for
1DTFIM is available in our github repository [57]. These 187 TACS were the
data points with which we performed unsupervised learning by constructing the
$187\times 187$ kernel matrix using the shadow kernel in equation (12) and
then using diffusion maps for dimensionality reduction.

A key element needed to obtain reasonable results from the above calculation
is an initial state that equilibrates within the chosen time window for
observables of interest that are accurately captured by the chosen method of
shadow tomography. In the above case, we started from a GHZ state because it
equilibrated efficiently for local observables, as shown in Appendix C.
Presumably, this equilibration would occur even faster if we broke the
integrability of the 1D TFIM model by adding certain additional terms to the
Hamiltonian. Hence, up to possibly finite size effects, thermodynamic
observables in our results should behave as expected.

Fig. 5 shows our results. The first two eigenvectors were chosen as our
diffusion space basis vectors because as Fig. 5(inset) shows, those are the
two dominant non-trivial eigenvalues in the spectrum. The rest of the
eigenvectors go to zero as the number of data points increases. Projecting the
states onto this two-dimensional diffusion space, we see that the states all
fall on a curve in this hyperplane along which the value of $h_{x}$ increases
monotonically, and the inflection point neighborhood of the curve coincides
with the quantum critical region [see D.2]. Therefore, as in the ground state
study, it is apparent that the unsupervised learning algorithm was able to
infer two phases from the data via a single cluster with non-trivial geometry.
Presumably, taking paths through the phase diagram closer to the critical
point, we would see the inflection point sharpen, leading to a singular point
in the data manifold at the critical point. However, unlike the ground state
study, it is not obvious what the two diffusion coordinates dc1 and dc2
correspond to. To identify these, we need to study observables capable of
capturing the phase-defining features and see which correlate with these
learned coordinates.

Figure 5: Phase identification from dynamics data. The eigenvalue spectrum
(inset) obtained from diffusion maps shows the two largest eigenvalues
corresponding to the two dominant diffusion coordinates dc1 and dc2 (the
trivial point $k=0$, is not shown). The TACS data points largely fall on a
two-stranded curve parameterized by $h_{x}$ in this 2D reduced diffusion
space. The quantum critical region (in green) coincides with the inflection
point neighborhood of the curve, with points on the left strand belonging
mostly to the ordered phase while points on the right strand belong to the
disordered phase.

As a preliminary exploration of phase-defining observables, a straightforward
first approach is to check whether the diffusion coordinates obey power laws
consistent with the known quantum critical point. In Fig. 6(a), we plot dc2,
which diverges as it approaches the critical point with critical exponents
$p_{-}=0.58\pm 0.05$ and $p_{+}=0.7\pm 0.1$. We have shifted the diffusion
coordinates by $C=-0.0027$ since the diffusion distance is invariant under an
overall shift of the origin as mentioned in Sec. III.2 and this renders it
positive. Remarkably, this shift _simultaneously renders both dc1 and dc2
positive_. However, our errors in the exponents are hard to estimate. Suppose
we view the unknown variable $C$ as a Gaussian distribution. In that case, the
corresponding distribution of $p_{\pm}$ from our predictions is highly non-
Gaussian (see Appendix D.4). The closest known critical exponent is $\nu=1$
[58] (see also wikipedia[59]). However, we found that the observable that
qualitatively matches the diverging behavior at $h_{x}=1$ is xx component of
the susceptibility, which we define as:

$\chi_{ab}=\frac{1}{L^{2}}\sum_{ij}\expectationvalue{\sigma^{a}_{i}\sigma^{b}_{j}},$
(19)

where $\sigma^{a}\in\\{X,Y,Z\\}$ is a $2\times 2$ Pauli matrix. This is
different than the usual definition obtained by summing over the connected
correlations. The resemblance between dc2 and susceptibility here is only
qualitative, so the critical exponents do not match. Since there are a number
of observables, such as $\chi_{xx},\chi_{yy}$ and $\chi_{zz}$, that are
equally likely candidates to define the phases, we conjecture that dc2 could
be some combination of these.

This leaves the puzzle of determining dc1, which neither diverges nor shows a
power law behavior. When inverted, it appears qualitatively similar to the ZZ
component of the susceptibility (see Sec. D.3) but we turn to calculate the
$2^{nd}$ Renyi entropy given the diagonal components of the kernel function
capture this quantity. Entropy is the key thermodynamic potential of the
microcanonical ensemble that certainly captures phase transitions and from
which all important phase-defining features could be extracted. It is also
known that the quantum critical region features the interplay of equilibrium
and quantum fluctuations leading to the entropy being maximized [60, 54].
Renyi entropy is a lower bound for the von Neumann entropy and has the same
limits—it vanishes for pure states and reaches $N$ for the maximally mixed
state. We do so by first computing the purity using eqn 10 as follows

$\displaystyle\gamma\left[\rho\right]$
$\displaystyle=\text{Tr}\left[\rho^{2}\right]$ (20)
$\displaystyle\approx\frac{1}{N^{2}}\sum_{n\neq
n^{\prime}=1}^{N}\text{Tr}\left[\sigma_{1}^{(n)}\sigma_{1}^{(n^{\prime})}\right]\times\cdots\times\text{Tr}\left[\sigma_{L}^{(n)}\sigma_{L}^{(n^{\prime})}\right],$
(21)

then the Renyi entropy is $S_{2}=-\log_{2}\gamma$.

Figure 6: Interpretation of dc1 and dc2 for microcanonical dynamics of 1DTFIM.
The divergent behavior of dc2 qualitatively matches the xx component of the
susceptibility, computed using 100k shot TACS data for a 10-site 1DTFIM,
denoted by orange circles (b) dc1 matches the Bayesian inference estimate for
the second Renyi entropy per site ($S_{2}/n$) in the thermodynamic limit
($n=\infty$). Bayesian inference is performed on $n$-body entropies for
$n=1-5$, also computed using the 10-site dataset.

Figure 6(b) presents the Renyi entropy as calculated from TACS shots on a
$L=10$ qubit system using exact diagonalization. We use Bayesian inference, as
detailed in appendix E (see also Ref. 61) to extract the large-$n$, large-$N$
predictions with $N=100,000$ shots per $h_{x}$ value and 35 $h_{x}$ values
between $h_{x}=0.1$ and $h_{x}=3.3$. The range of time sampled for this
particular calculation was between $t=5$ and $t=25$. The results show clear
evidence that entropy is maximized in the quantum critical region around
$h_{x}=1$ as expected from Fig. 2 of [54]. Error estimates for these values
were obtained in Ref. 62, and given by:

$N\geq\frac{4^{n+1}\gamma}{\epsilon^{2}\delta}$ (22)

Where $1-\delta$ is the probability of obtaining a good TACS dataset and
$\epsilon$ is the additive error. For a $\delta=0.33$ (67 percentile), and
$N=100,000$ shots, we find an additive error for the $n=5$ entropy curve
plotted in Fig. 6 at $h_{x}=1.0$ of $S_{2}/n\approx 0.5\pm 0.24$. This is
larger than the observed error shown via error bars in Fig. 6 but only within
a factor of order 1. Hence, by using $3,300,000$ shots, we have estimated the
thermodynamic entropy as a function of $h_{x}$ that reproduces the expected
maxima at the critical point.

Given an estimate of the entropy, we lastly turn to plotting it alongside dc1
to discover that it is highly correlated with this observable. Although dc1
was calculated using only 500 shots per $h_{x}$ value, orders of magnitude
less than the number of shots needed for accurate Bayesian inference
estimation, the diffusion process is able to combine information across
different $h_{x}$ values without any supervision.

In summary, the diffusion map was able to learn phase-defining features from
TACS and used these features to map the data points as a function of the model
parameter $h_{x}$ onto a curve in the two-dimensional plane with geometry that
reveals the quantum phase transition.

##  VI Outlook

In this paper, we have identified an approach to studying quantum
thermodynamics on a quantum computer in a way that is suitable for studying
quantum materials, their phases, and their phase transitions. This approach
consists of

  1. (i)

preparing a low-depth initial state for which relevant observables are
observed to equilibrate within the coherence time of the quantum computer,

  2. (ii)

time evolving this state using a quantum algorithm to a randomly chosen time
point $t$ within some time interval,

  3. (iii)

extending shadow tomography methods to obtain a physically useful
representation of the von Neumann ensemble such as TACS used in this paper,
and

  4. (iv)

employing an unsupervised machine learning method to discover the phase
diagram, with kernel methods such as diffusion maps employing well designed
kernels showing promise.

Our approach parallels statistical mechanics calculations on classical
Hamiltonians, where a random initial state is prepared, a Metropolis Monte-
Carlo algorithm is run beginning from this state, and data is collected and
analyzed using traditional observables and more recently machine learning
methods. Our results, demonstrating the existence of a quantum phase
transition and the ability to map out regions of the phase diagram by a
careful choice of initial conditions, show promise.

There are several resources needed to carry out the microcanonical dynamics
simulations. A key resource is a low-energy state that equilibrates within the
accessible time scale and can also be prepared easily. For local Hamiltonians
on physical lattices, we can always find low-energy states which can be
prepared with constant-depth circuits [27]. Another resource to carry out
dynamics simulations to a time $T$ in which the relevant local observables
equilibrate ($T=25$ in our simulations). It allows one to exploit advances in
variational time evolution algorithms[63, 64, 65], which are especially
suitable for the NISQ era due to robustness to noise and ability to go beyond
the coherence time of quantum computers. Finally, we need the ability to
perform time averages by sampling the state $|\psi(t)\rangle\langle\psi(t)|$
at least at the Nyquist rate determined by the bandwidth, which is linear in
the system size. We produced a TACS dataset consisting of 500 shots from the
equilibrium dynamics starting from a GHZ state at each of 187 points in the
phase diagram; these resources were all that were required for diffusion maps
to learn the phases and identify the phase transition for a 20-qubit system.
Somewhat different resources, a system size of $L=10$, and $N=100,000$ shots
at $33$ points in the phase diagram, were required for Bayesian inference to
obtain reasonable estimates for the thermodynamic entropy.

We believe these resources are significantly smaller than those of other
proposed methods for studying thermodynamics on quantum computers. For
example, the overhead from using ancillas as a heat bath in the existing
methods to study thermodynamics on a quantum computer [66, 67, 68] is not an
issue with our approach. The resources required also open up an exciting
possibility of employing new generation of quantum simulators [69, 70] to
study quantum thermodynamics as they too can simulate quantum dynamics and are
capable of performing randomized measurements. Finally, a possible direction
for future research would be to identify and benchmark a strongly interacting
system with initial states that will yield quantum advantage in the near term.

###### Acknowledgements.

We thank Mikhail Lukin, Katherine Van Kirk, Nishad Maskara, Yanting Teng,
Subir Sachdev, Daniel Parker, and Anurag Anshu for useful discussions. This
material is based upon work supported by the National Science Foundation under
Grants No. OAC-1940243 and OAC-1940260.

##  Appendix A Diagramatic understanding of classical shadows

In this section, we will develop a diagrammatic understanding of the classical
shadows. We will work in the superoperator formalism where the indices of the
density matrix $\rho_{ij}$ are grouped together to make a vector
$|\rho\rangle\rangle$ and the product $A\rho B$ translates to an operator
$A\otimes B^{T}$ acting on the vector $|\rho\rangle\rangle$.

In general, we can view the outcomes obtained from many classical shadows
measurements on the same prepared state $\rho$ as defining an ensemble of
states $S\left[\rho\right]$ where

$S[\rho]=\sum_{b,x}P(b)\mathcal{P}_{b,x}\rho\mathcal{P}_{b,x}=\bigg{[}\sum_{b,x}P(b)\mathcal{P}_{b,x}\otimes\mathcal{P}_{b,x}^{T}\bigg{]}|\rho\rangle\rangle,$
(23)

and $\mathcal{P}_{b,x}$ is the projector in basis $b$ onto qubit state
$|x\rangle$. To break this down into manageable parts, let’s start from the
one qubit case and work our way up to $N$ qubits.

_One qubit case._ In the one qubit case, we generate classical shadows samples
for bases $b\in\\{X,Y,Z\\}$ with uniform probabilities i.e. $P(b)=1/3$. We can
thus express the one-qubit version of 23 with the following diagram:

$\displaystyle
S\left[\rho\right]=\frac{1}{3}\sum_{b,x}\begin{array}[]{cc}\leavevmode\hbox
to114.61pt{\vbox to17.87pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-8.93582pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
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Using cap and cup notation, we can redraw this in the superoperator form:

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S[\rho]=\frac{1}{3}\sum_{b,x}\begin{array}[]{cc}\leavevmode\hbox
to128.84pt{\vbox to71.93pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@rect{18.4133pt}{19.91693pt}{20.07892pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.7463pt}{26.00832pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{bx}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{62.59607pt}{-8.53583pt}{17.07166pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{68.54683pt}{-1.18056pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{56.90552pt}\pgfsys@lineto{113.81104pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{18.0133pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{18.0133pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{38.89221pt}{0.0pt}\pgfsys@lineto{62.19608pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {{}{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{38.89221pt}{28.45276pt}\pgfsys@lineto{99.58466pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {{}{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{80.06772pt}{0.0pt}\pgfsys@lineto{99.58466pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{99.58466pt}{0.0pt}\pgfsys@curveto{106.69785pt}{0.0pt}{113.81104pt}{21.33957pt}{99.58466pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@curveto{-7.11319pt}{28.45276pt}{-14.22638pt}{49.79233pt}{0.0pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{-14.22638pt}\pgfsys@moveto{7.11319pt}{-14.22638pt}\pgfsys@lineto{7.11319pt}{42.67914pt}\pgfsys@lineto{49.79233pt}{42.67914pt}\pgfsys@lineto{49.79233pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{49.79233pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}$ (27)

The highlighted box can be viewed as a superoperator acting on the space of
linear operators $\rho$. Remarkably, this particular superoperator consisting
of a product of two projection operators simplifies substantially i.e.

$\displaystyle\sum_{b,x}\begin{array}[]{cc}\leavevmode\hbox to57.71pt{\vbox
to46.32pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-8.93582pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{18.4133pt}{-8.53583pt}{20.07892pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.7463pt}{-2.44444pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{bx}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{18.4133pt}{19.91693pt}{20.07892pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.7463pt}{26.00832pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{bx}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{}{{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{18.0133pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{18.0133pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {{}{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{38.89221pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {{}{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{38.89221pt}{28.45276pt}\pgfsys@lineto{56.90552pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}=\left[\;\begin{array}[]{cc}\leavevmode\hbox
to15.03pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}+\begin{array}[]{cc}\leavevmode\hbox
to29.25pt{\vbox to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@lineto{28.45276pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}\;\right].$
(31)

They amount to the sum of an identity and a cup-cap product. Using this
simplification we recognize the one qubit case as the depolarizing map

$\displaystyle
S\left[\rho\right]=\frac{1}{3}\left[\begin{array}[]{cc}\leavevmode\hbox
to57.71pt{\vbox to17.87pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-8.93582pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{19.91693pt}{-8.53583pt}{17.07166pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.86769pt}{-1.18056pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.51694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{37.38858pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}+\begin{array}[]{cc}\leavevmode\hbox
to29.25pt{\vbox to6.49pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{5.69046pt}\pgfsys@lineto{28.45276pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}\right],$
(34)

where we further simplified using

$\displaystyle\begin{array}[]{cc}\leavevmode\hbox to57.71pt{\vbox
to30.68pt{\pgfpicture\makeatletter\hbox{\hskip 28.85275pt\lower-8.93582pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-7.11319pt}{21.33957pt}\pgfsys@lineto{7.11319pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-21.33957pt}{0.0pt}{-28.45276pt}{21.33957pt}{-7.11319pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{21.33957pt}{0.0pt}{28.45276pt}{21.33957pt}{7.11319pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{-8.53583pt}{-8.53583pt}{17.07166pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.58507pt}{-1.18056pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}=1$ (36)

Inverting, we can extract the original density matrix (full tomography) via

$\displaystyle\begin{array}[]{cc}\leavevmode\hbox to57.71pt{\vbox
to17.87pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-8.93582pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{19.91693pt}{-8.53583pt}{17.07166pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.86769pt}{-1.18056pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.51694pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{37.38858pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}=3\begin{array}[]{cc}\leavevmode\hbox
to57.71pt{\vbox to17.87pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-8.93582pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{16.40274pt}{-8.53583pt}{24.10004pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.73575pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S\left[\rho\right]$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.00275pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{40.90277pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}-\begin{array}[]{cc}\leavevmode\hbox
to29.25pt{\vbox to6.49pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{5.69046pt}\pgfsys@lineto{28.45276pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}$ (40)

Going from diagrammatic results to equations gives

$S\left[\rho\right]=\frac{1}{3}(\rho+I)\,\implies\,\rho=3S\left[\rho\right]-I,$
(41)

which is a single-qubit depolarizing channel.

_Two-qubit case._ In the two-qubit case, the CS map of Eq. 23 takes the form

$\displaystyle\frac{1}{9}\sum_{b,x}\sum_{b^{\prime},x^{\prime}}\begin{array}[]{cc}\leavevmode\hbox
to143.06pt{\vbox to57.71pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{17.53331pt}{-8.53583pt}{21.8389pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.86632pt}{-2.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$P_{b^{\prime}x^{\prime}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{18.4133pt}{19.91693pt}{20.07892pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.7463pt}{26.00832pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{bx}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{}{{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{17.13332pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{18.0133pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {{}{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{39.7722pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {{}{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{38.89221pt}{28.45276pt}\pgfsys@lineto{56.90552pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{
{}{}{}{}{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{56.90552pt}{-14.22638pt}\pgfsys@moveto{56.90552pt}{-14.22638pt}\pgfsys@lineto{56.90552pt}{42.67914pt}\pgfsys@lineto{85.35828pt}{42.67914pt}\pgfsys@lineto{85.35828pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{85.35828pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{68.54683pt}{13.04582pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{85.35828pt}{0.0pt}\pgfsys@lineto{113.81104pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{85.35828pt}{28.45276pt}\pgfsys@lineto{113.81104pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{102.89159pt}{-8.53583pt}{21.8389pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{106.2246pt}{-2.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$P_{b^{\prime}x^{\prime}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{103.77158pt}{19.91693pt}{20.07892pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{107.10458pt}{26.00832pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{bx}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}
{{}{}}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{125.13048pt}{0.0pt}\pgfsys@lineto{142.2638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{} {{}{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{124.25049pt}{28.45276pt}\pgfsys@lineto{142.2638pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(43)

Again, we can add cups and caps to express it in superoperator form:

$\displaystyle\begin{array}[]{cc}\leavevmode\hbox to143.06pt{\vbox
to143.06pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-57.30551pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@rect{17.53331pt}{-8.53583pt}{21.8389pt}{17.07166pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.86632pt}{-2.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$P_{b^{\prime}x^{\prime}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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This diagram shows that the two-qubit projection operators produce the same
structure on each qubit as they did in the one-qubit case. Applying the same
simplification as before, we arrive at

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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}{
{}{}{}{}{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.64131pt}{13.04582pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{-19.91684pt}\pgfsys@lineto{42.67914pt}{-19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{-28.45276pt}\pgfsys@lineto{42.67914pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@curveto{-17.78297pt}{0.0pt}{-21.33957pt}{-14.22638pt}{-14.22638pt}{-19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{0.0pt}\pgfsys@curveto{46.23573pt}{0.0pt}{49.79233pt}{-14.22638pt}{42.67914pt}{-19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}+\begin{array}[]{cc}\leavevmode\hbox
to57.71pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
28.85275pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45276pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}\right],$
(51)

where again we used $\text{Tr}\left[\rho\right]=1$. This expression amounts to
a simple sum of all possible reduced density matrices. We can invert our
diagram for the two qubit case by first tracing over one of its qubits to get

$\displaystyle\begin{array}[]{cc}\leavevmode\hbox to86.16pt{\vbox
to71.93pt{\pgfpicture\makeatletter\hbox{\hskip
28.85275pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}{
{}{}{}{}{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.50937pt}{11.72638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$S\left[\rho\right]$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{56.90552pt}\pgfsys@lineto{42.67914pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@curveto{49.79233pt}{28.45276pt}{56.90552pt}{49.79233pt}{42.67914pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{28.45276pt}\pgfsys@curveto{-21.33957pt}{28.45276pt}{-28.45276pt}{49.79233pt}{-14.22638pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}=\frac{1}{3}$
$\displaystyle\left[\begin{array}[]{cc}\leavevmode\hbox to86.16pt{\vbox
to71.93pt{\pgfpicture\makeatletter\hbox{\hskip
28.85275pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}{
{}{}{}{}{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.64131pt}{13.04582pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-14.22638pt}{56.90552pt}\pgfsys@lineto{42.67914pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@curveto{49.79233pt}{28.45276pt}{56.90552pt}{49.79233pt}{42.67914pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{28.45276pt}\pgfsys@curveto{-21.33957pt}{28.45276pt}{-28.45276pt}{49.79233pt}{-14.22638pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(54) $\displaystyle+$ $\displaystyle\left.\begin{array}[]{cc}\leavevmode\hbox
to57.71pt{\vbox to0.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-0.4pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(56)

This allows us to rewrite the reduced density matrices in terms of the
partially summed $S[\rho]$ . Using this relation, we then arrive at the
inverse of the two-qubit classical shadow

$\displaystyle\begin{array}[]{cc}\leavevmode\hbox to57.71pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}{
{}{}{}{}{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.64131pt}{13.04582pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\rho$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}=\>$
$\displaystyle 9\>\begin{array}[]{cc}\leavevmode\hbox to57.71pt{\vbox
to57.71pt{\pgfpicture\makeatletter\hbox{\hskip
14.62637pt\lower-14.62637pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@lineto{28.45276pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@fillstroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.50937pt}{11.72638pt}\pgfsys@invoke{
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{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
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