# Constraints on magic state protocols from the statistical mechanics of
Wigner negativity

Nikolaos Koukoulekidis<EMAIL_ADDRESS>Department of Physics, Imperial
College London, London SW7 2AZ, UK  David Jennings  Department of Physics,
Imperial College London, London SW7 2AZ, UK  School of Physics and Astronomy,
University of Leeds, Leeds, LS2 9JT, UK

###### Abstract

Magic states are key ingredients in schemes to realize universal fault-
tolerant quantum computation. Theories of magic states attempt to quantify
this computational element via monotones and determine how these states may be
efficiently transformed into useful forms. Here, we develop a statistical
mechanical framework based on majorization to describe Wigner negative magic
states for qudits of odd prime dimension processed under Clifford circuits. We
show that majorization allows us to both quantify disorder in the Wigner
representation and derive upper bounds for magic distillation. These bounds
are shown to be tighter than other bounds, such as from mana and thauma, and
can be used to incorporate hardware physics, such as temperature dependence
and system Hamiltonians. We also show that a subset of single-shot Rényi
entropies remain well-defined on quasi-distributions, are fully meaningful in
terms of data processing and can acquire negative values that signal magic. We
find that the mana of a magic state is the measure of divergence of these
Rényi entropies as one approaches the Shannon entropy for Wigner
distributions, and discuss how distillation lower bounds could be obtained in
this setting. This use of majorization for quasi-distributions could find
application in other studies of non-classicality, and raises nontrivial
questions in the context of classical statistical mechanics.

††preprint: APS/123-QED

## Introduction

There has been rapid progress towards the goal of a fault-tolerant quantum
computer campbell_roads_2017 ; cit:raussendorf ; Raussendorf_2013 ;
Nickerson_2014 ; Nikahd_2017 ; chao_2018 ; lin_pieceable_2020 ; Lin_2020 ;
chamberland_2020 ; Bourassa_2021 . Nevertheless, many challenges remain and
there is increasing need for theory to take into account physical limitations
of the hardware involved. The surface code Bravyi_1998 ; Freedman_2001 ;
Dennis_2002 ; Raussendorf_2007  is a leading framework for fault-tolerance
with very high error thresholds. Within this scheme, Clifford unitaries can be
implemented in a robust, fault-tolerant way. However, due to the Eastin-Knill
theorem Eastin_2009 , we also know that it is impossible to have a universal
set of transversal gates, and although Clifford unitaries can be realized
transversally Calderbank_1996 ; Steane_1996 , one needs to find ways around
the Eastin-Knill restriction. This can be achieved by injecting in quantum
states, called magic states, which promote the Clifford group to universal
quantum computing cit:bravyi . The obstacle to this is that these states are
invariably noisy, so protocols involving stabilizer operations must be
employed to purify many copies of the magic states and improve the overall
performance of the induced quantum gates cit:bravyi ; Jones_2013 ;
Ogorman_2017 . A key question then arises about the overhead on purifying many
copies of a magic state into less noisy forms.

To address this, concrete distillation protocols have been developed, such as
the Bravyi-Haah qubit protocol that provides a quadratic reduction in noise
per cycle Bravyi2012 . Such distillation rates have been improved in more
recent works Jones_2013 ; haah2017magic ; Hastings2018 ; Litinski_2019  and
there have been proposals of protocols for qudits of odd dimension
CampbellAnwar_2012 ; Anwar_2012 ; Dawkins_2015 ; Krishna2019 ; cit:prakash  as
well as of protocols within a full architectural analysis
chamberland_very_2020 ; Chamberland2019faulttolerantmagic . There is also
analysis of magic protocols from the perspective of magic monotones, which
provide upper bounds on distillation rates and find application in the
analysis of gate synthesis Campbell_2017 ; Howard_2017 ; Prakash_2018 ;
Seddon_2021 ; leone2021renyi . These frameworks for magic view magic states as
resource states with respect to a natural class of quantum operations that are
considered cheap, or free, such as stabilizer operations Gour_2019 ;
cit:ahmadi ; cit:seddon ; Wang_2019 . Beyond quantum computing, recent work
has also looked at how magic can be used to analyse many-body physics
Sarkar_2020  and conformal field theories White_2021  as well its connection
to contextuality Vega_2017 ; cit:howard2 ; Zurel_2020 ; Okay_2021 ;
schmid2020structure ; Schmid_2021 .

Work towards fault-tolerance is increasingly bridging the gap between abstract
theory and experiment. Extensive work is being done on error mitigation
jones_2012 ; Li_2017 ; Temme_2017 ; Endo_2018 ; McClean_2017 , and the
incorporation of hardware physics into the theoretical models Kandala_2019 ;
holmes_resource_2019 ; Colless_2018 ; song2018quantum ; Bravyi_2021 . For
example, the XZZX code bonilla_ataides_xzzx_2021  is a variant of the surface
code that incorporates noise bias explicitly and has been shown to attain the
hashing bound of random codes Bennett_1996 . In this work, we develop a
framework to analyse magic state distillation protocols where explicit physics
of the system is incorporated in the distillation bounds. This is achieved by
considering how a given magic state protocol transforms a pair of quantum
states – one being a noisy magic state, and the other a stabilizer state that
is distinguished by the physics of the system (for example a state at a
characteristic temperature, or the maximally mixed state) and acts as a
reference state for the protocol. The magic distillation bounds can then be
expressed in terms of the physics of the reference state (e.g. free energy
changes in the case of temperature). Such bounds are of potential interest in
assessing how physical features such as temperature, noise biases or fixed-
point structure associated with restricted gate-sets constrain distillation
protocols Tuckett_2019 ; Aliferis_2008 ; Stephens_2013 ; Li_2015 ;
Babbush_2018 ; Guillaud_2019 ; Fowler_2019 .

The approach we take is most closely aligned with resource theories of magic,
although it differs in key ways. We obtain distillation upper bounds without
the use of monotones, but instead use tools from statistical mechanics and
recent work in single-shot resource theories cit:janzing ; cit:horodecki2013 ;
Brandao_2015 ; Vinjanampathy_2016 ; Goold_2016 ; cit:lostaglio ; cit:gour .
Our analysis relies on the discrete Wigner representation of quantum systems,
in which all states and operations can be described on a discrete phase space
Ferrie_2008 ; Okay_2021 . Crucially, we focus on magic states with negativity
in their Wigner representation, which is known to be a necessary condition for
universality in the state-injection model cit:veitch ; cit:mari ;
cit:gottesman ; cit:knill ; Campbell_2011 . However, taking a statistical
mechanical perspective in this context raises a problem: the standard
Boltzmann entropy is not defined for quasi-distributions. We circumvent this
obstacle by making use of a more fundamental tool – majorization theory
cit:marshall ; Veinott_1971 ; Ruch_1976 . To our knowledge majorization of
quasi-distributions has not been considered in quantum physics before, and
therefore these methods could find application in other studies of non-
classicality in quantum systems Fine_1982 ; Allahverdyan_2018 ; arvidsson_2020
; halpern_2018 ; Lostaglio_2018 ; Levy_2020 .

Figure 1: Magic distillation bounds from majorization of Wigner quasi-
distributions. The above plot shows distillation upper bounds for qutrits
obtained from majorization on unital protocols. The constraints are plotted as
a function of the depolarizing error $\epsilon$ for a noisy Strange state (Eq.
(23)). The bound $R_{\infty}$ follows from a Lorenz curve analysis presented
before Eq. (32), while the bound $R_{10}$ comes from single-shot entropies on
the Wigner quasi-distributions, as discussed in the derivation of Eq. (52).
Numerical bounds $R_{\rm{num}}$ are obtained by using the full set of
majorization constraints. Bounds $R_{\infty}$, $R_{10}$, $R_{\rm{num}}$ all
provide stricter constraints than mana cit:veitch2  and max–thauma
$\theta_{\rm{max}}$ Wang_2020  bounds for noisy Strange states.

We begin by describing how stabilizer operations correspond to stochastic maps
on a discrete phase space, and how majorization can be applied to constrain
magic state transformations. We then consider families of magic protocols with
increasing complexity. The simplest case we consider is for magic protocols
that lead to unital channels. These are a subset of protocols that leave
invariant some distinguished state or have some equilibrium fixed-point
structure. We then consider protocols that are non-equilibrium processes, but
only generate sub-linear correlations that enable a simple description in the
thermodynamic limit before giving entropic analysis that can be applied
generally.

The relations between the sets of protocols considered are as follows:

Unital $\displaystyle\subset\mbox{ local fixed-points}$
$\displaystyle\subset\mbox{ sub-linear correlations}\subset\mbox{ general
protocols.}$

We provide explicit bounds for magic protocols that generate unital channels
(Eq. (32)), as well as bounds that incorporate the temperature and Hamiltonian
of the system (Theorem 7) and discuss extensions to more general scenarios.
These bounds are shown to be tighter than other bounds such as those that come
from mana and thauma (Fig. 1).

We find that the analysis in the presence of negativity displays a range of
features that do not appear in classical statistical mechanics, and leads to a
picture of Wigner negativity in a quantum circuit being described as non-
classical free energy that is processed under stochastic dynamics (Fig. 2).
This is demonstrated by non-monotonic Lorenz curves (Fig. 3), and by single-
shot entropies for general Wigner quasi-distributions that are well-defined,
obey data-processing under stabilizer operations and can take on negative
values (Eq. (45)).

Figure 2: Probing magic protocols with reference states. A magic protocol (top
figure) converts $n$ copies of some noisy magic state $\rho$ with error
parameter $\epsilon$ to $m$ copies of some less noisy state $\rho^{\prime}$
with $\epsilon^{\prime}<\epsilon$. The physics of the protocol can be analysed
by considering how it would transform some distinguished reference stabilizer
state $\tau^{\otimes n}$ (bottom figure). For the case where this reference
state is a thermal state at some temperature $T$ our magic distillation bounds
are functions of $T$, the error parameters $\epsilon,\epsilon^{\prime}$, and
free energies $F,F^{\prime}$. In addition, a quantity $\phi$ (Eq. (115))
relates the computational and energy bases, and corresponds to the degree to
which the Wigner negativity of $\rho$ can be given a sharp energy via the
system Hamiltonian.

We show that the mana of a quantum state arises as a residue in the
$\alpha$–parameter for Rényi entropies, and gives the rate of divergence to
$-\infty$ as we approach the Shannon entropy of a Wigner distribution (Eq.
(47)). Finally, in the Discussion section, we discuss how our approach could
be extended to construct explicit lower bounds on distillation that take into
account physical constraints.

## Results

### Phase space representation of quantum mechanics

Central to our construction is the representation of any quantum state or
quantum operation on a system of dimension $d$ in terms of quasi-probability
representations on a discrete phase space Gross2006 ; Ferrie_2008 . This
construction is a discrete version of Wigner representations in quantum optics
Wigner_1932 ; Vourdas_2004 ; Kenfack_2004 .

Consider a $d$–dimensional quantum system with Hilbert space ${\cal H}_{d}$,
and let $\\{|0\rangle,|1\rangle,\dots,|d-1\rangle\\}$ denote the standard
computational basis, defined over $\mathbb{Z}_{d}=\\{0,1,\dots,d-1\\}$. On
this space, generalised Pauli matrices $X,Z$ can be defined by their
respective roles as shift and phase operators, acting on the basis states as
follows,

$\displaystyle X\ket{k}$ $\displaystyle=\ket{k+1}$ (1) $\displaystyle
Z\ket{k}$ $\displaystyle=\omega^{k}\ket{k}.$ (2)

Here $\omega\coloneqq e^{2\pi i/d}$ is the $d$-th root of unity and addition
is taken modulo $d$. From these we can construct a phase space ${\cal
P}_{d}\coloneqq\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ that provides a complete
representation of the quantum system. Given a point
$\boldsymbol{z}\coloneqq(q,p)$ we define a displacement operator,

$D_{\boldsymbol{z}}\coloneqq\tau^{qp}X^{q}Z^{p},\ $ (3)

where the phase factor $\tau\coloneqq-\omega^{1/2}$ ensures unitarity. We
assume going forward that $d$ is an odd prime Appleby_2005 . The qubit case
$d=2$ is known to have obstacles to using a Wigner representation to quantify
magic in terms of negativity Mermin_1990 ; cit:howard2 , although recent work
Raussendorf_qubits  has shown how to extend the representation to include
qubits by enlarging the phase space. For a composite system with dimension
$d_{\rm tot}=d_{1}\dots d_{n}$ and Hilbert space ${\cal
H}_{d_{1}}\otimes\dots\otimes{\cal H}_{d_{n}}$, we define displacement
operators as

$D_{\boldsymbol{z}}\coloneqq D_{(q_{1},p_{1})}\otimes\dots\otimes
D_{(q_{n},p_{n})},$ (4)

where now we have

$\displaystyle\boldsymbol{z}$
$\displaystyle\coloneqq(q_{1},\dots,q_{n},p_{1},\dots,p_{n})\in{\cal
P}_{d_{1}}\times\dots\times{\cal P}_{d_{n}},$

to denote the phase space point for the composite system. For simplicity, we
assume $n$ copies of a $d$–dimensional system $d_{1}=d_{2}=\cdots=d$, and
therefore we have that $\boldsymbol{z}\in\mathbb{Z}_{d}^{2n}$.

The displacement operators form the Heisenberg-Weyl group Folland_1989 ;
Bengtsson_2006  under matrix multiplication modulo phases,

${\rm{HW}}_{d}^{n}\coloneqq\\{\tau^{k}D_{\boldsymbol{z}}:k\in\mathbb{Z}_{d},\boldsymbol{z}\in{\cal
P}_{d}^{n}\\}.$ (5)

The Clifford group ${\cal C}_{d}^{n}$ is then defined as the set of unitaries
that normalise the Heisenberg-Weyl group Appleby_2005 . We may define the pure
stabilizer states as those states obtained by acting on $|0\rangle$ with
Clifford unitaries Gross2006  and, finally, the full set of stabilizer states
as the convex hull of all pure stabilizers, namely all probabilistic mixtures
of states of the form $U|0\rangle\langle 0|U^{\dagger}$ where $U$ is Clifford.

### Wigner representation for quantum states and quantum operations

In order to provide a complete decomposition of arbitrary quantum states and
quantum operations we now define a basis of Hermitian observables that
transforms covariantly under the action of the Clifford group. At every point
$\boldsymbol{z}\in{\cal P}_{d}$ we define the phase-point operator

$A_{\boldsymbol{z}}\coloneqq\frac{1}{d}\sum_{\boldsymbol{y}\in{\cal
P}_{d}}\omega^{\eta(\boldsymbol{z},\boldsymbol{y})}D_{\boldsymbol{y}},$ (6)

where $\eta(\boldsymbol{z},\boldsymbol{y})$ is the symplectic inner product
between any two points $\boldsymbol{z},\boldsymbol{y}\in{\cal P}_{d}$, and is
given explicitly by

$\eta(\boldsymbol{z},\boldsymbol{y})\coloneqq\boldsymbol{y}^{T}\begin{pmatrix}0&\mathbbm{1}\\\
-\mathbbm{1}&0\\\ \end{pmatrix}\boldsymbol{z},$ (7)

where $0,\mathbbm{1}$ denote the $n\times n$ zero and identity matrices.

The phase-point operators form an orthogonal operator basis with respect to
the Hilbert-Schmidt inner product, as discussed in Supplementary Note 1.
Therefore, any quantum state $\rho$ can be expressed as a linear combination
of them, $\rho=\sum_{\boldsymbol{z}\in{\cal
P}_{d}}W_{\rho}(\boldsymbol{z})A_{\boldsymbol{z}}$, where the coefficients
$W_{\rho}(\boldsymbol{z})$ give the Wigner distribution of state $\rho$,

$W_{\rho}(\boldsymbol{z})\coloneqq\frac{1}{d}{\rm{tr}}[A_{\boldsymbol{z}}\rho].$
(8)

For any quantum state $\rho$, the Wigner distribution
$W_{\rho}(\boldsymbol{z})$ is readily seen to be a $d^{2}$-dimensional quasi-
distribution over ${\cal P}_{d}$. More precisely, $W_{\rho}(\boldsymbol{z})$
is a real-valued function on ${\cal P}_{d}$ with the normalisation property
$\sum_{\boldsymbol{z}}W_{\rho}(\boldsymbol{z})=1$ (see Supplementary Note 1
for details). Moreover, the above Wigner representation has recently been
shown to be the only non-contextual quasi-probability representation of
stabilizer theories in odd prime dimensions schmid2021noncontextual .

Any quantum channel ${\cal E}$ also admits a Wigner representation. If ${\cal
E}$ maps some quantum system $A$ of dimension $d_{A}$ to a quantum system $B$
of dimension $d_{B}$ and ${\cal J}({\cal E})=(id\otimes{\cal
E})(|\phi^{+}\rangle\langle\phi^{+}|)$, with
$|\phi^{+}\rangle=d_{A}^{-1/2}\sum_{k}|kk\rangle$, is its associated Choi
state watrous_2018 , then we can define

$W_{{\cal E}}(\boldsymbol{y}|\boldsymbol{z})\coloneqq d_{A}^{2}W_{{\cal
J}({\cal E})}(\bar{\boldsymbol{z}}\oplus\boldsymbol{y}),$ (9)

where $\bar{\boldsymbol{z}}=(q_{1},\dots,q_{n},-p_{1},\dots,-p_{n})$ can be
viewed as the time-reversed version of $\boldsymbol{z}$ in the discrete phase
space, where momenta are reflected while position coordinates remain
unchanged.

### Magic theories for quantum computation

A number of magic theories exist, where magic states are computational
resource states with respect to a class of quantum operations that are
considered free Gour_2019 . One natural class of free operations are those
obtained from Clifford unitaries, Pauli measurements and the ability to
discard quantum systems. However, there are also other candidates cit:ahmadi ;
cit:seddon ; Wang_2019 .

In any theory of magic, one route to bounding distillation rates is through a
magic monotone. A magic monotone is a real-valued function of any quantum
state $M(\rho)$ that is monotonically non-increasing under the free operations
of the magic theory. More precisely $M(\sigma)\leq M(\rho)$ whenever it is
possible to convert $\rho$ into $\sigma$ using free operations.

One prominent magic monotone is the mana of a state cit:veitch2 , defined as

${\cal M}(\rho)\coloneqq\log(2\hskip 1.0pt{\rm sn}(\rho)+1),$ (10)

where the sum-negativity cit:veitch2  is the sum of the negative components in
$W_{\rho}$,

${\rm
sn}(\rho)\coloneqq\sum\limits_{\boldsymbol{z}:W_{\rho}(\boldsymbol{z})<0}\absolutevalue{W_{\rho}(\boldsymbol{z})}.$
(11)

Using the fact that $W_{\rho}(\boldsymbol{z})$ is a normalized quasi-
distribution, we can also write ${\cal
M}(\rho)=\sum_{\boldsymbol{z}}|W_{\rho}(\boldsymbol{z})|$.

Mana is an additive magic monotone, and the fact that it is non-increasing
under free operations provides a constraint on magic state
interconvertibility.

In this work we develop bounds that apply to any reasonable magic theory. The
central idea is to apply majorization theory to the quasi-distributions that
arise for magic states. In the next section we explain how majorization
relates to the free operations in a theory of magic.

### Stochastic representation of magic protocols

Within the Wigner representation for odd prime dimension, it is well-known
that all positively represented states used in Clifford circuits admit an
efficient classical simulation cit:mari , so negativity is a necessary
resource for universal fault-tolerant quantum computing cit:veitch .
Therefore, the free states in any magic theory are required to be a subset of

${\cal F}\coloneqq\\{\rho:W_{\rho}(\boldsymbol{z})\geq 0\text{ for all
}\boldsymbol{z}\in{\cal P}_{d}\\}.$ (12)

Our focus is on states with negativity, so the particular choice of free
states is not critical for our analysis. The remaining component that defines
any magic theory is the set of free quantum operations. The most basic
assumption we require on free operations is that they send any free state to
another free state.

Any magic state protocol will correspond to a quantum channel ${\cal E}$, so
from Eq. (9) it admits a Wigner representation $W_{{\cal
E}}(\boldsymbol{y}|\boldsymbol{z})$ that acts as a transition matrix mapping
phase space points $\boldsymbol{z}\rightarrow\boldsymbol{y}$. The
representation obeys the relation

$W_{{\cal E}(\rho)}(\boldsymbol{y})=\sum_{\boldsymbol{z}\in{\cal
P}_{d}}W_{{\cal E}}(\boldsymbol{y}|\boldsymbol{z})W_{\rho}(\boldsymbol{z}),$
(13)

for any $\rho$. Since the magic protocol sends free states to free states,
${\cal E}$ sends all positively represented quantum state to other positively
represented quantum states. In such cases, it can be shown Wang_2019  that if
${\cal E}$ is a free operation then the associated Wigner representation
$W_{{\cal E}}(\boldsymbol{y}|\boldsymbol{z})$ must form a stochastic matrix.
In particular, all stabilizer operations correspond to stochastic matrices in
the Wigner representation.

We note however, that not all stochastic maps on the phase space correspond to
valid quantum operations. The reason is that the maps must also respect the
symplectic structure of the phase space, which is an additional non-trivial
constraint.

In what follows, we shall assume that we have a magic theory ${\cal R}=({\cal
F},{\cal O})$ in which the free states ${\cal F}$ are represented by non-
negative Wigner functions, while the free operations ${\cal O}$ are stochastic
maps in the Wigner representation. Our analysis of magic in this phase space
setting makes use of majorization theory, which we describe in the next
section.

### Quantifying disorder without entropies

Majorisation cit:marshall ; Blackwell_1953  is a collection of powerful tools
that has found many applications in quantum information theory Nielsen_1999 ;
cit:cwiklinski ; cit:lostaglio2 ; cit:gour ; cit:gour2 ; Horodecki_2003 ;
Puchala_2013 ; Vallejos_2021 . It describes the disorder of distributions that
undergo stochastic transformations, and in its simplest form defines a pre-
order on probability distributions. Given two distributions
$\boldsymbol{p}=(p_{1},\dots,p_{n})$ and $\boldsymbol{q}=(q_{1},\dots,q_{n})$
over $n$ outcomes, we say that $\boldsymbol{p}$ majorizes $\boldsymbol{q}$,
denoted $\boldsymbol{p}\succ\boldsymbol{q}$, if there exists a bistochastic
map $A=(A_{ij})$ such that $A\boldsymbol{p}=\boldsymbol{q}$, where
bistochastic means that $A_{ij}\geq 0$ and $\sum_{i}A_{ij}=\sum_{j}A_{ij}=1$.
It can be shown cit:marshall  that the condition
$\boldsymbol{p}\succ\boldsymbol{q}$ over probability distributions is
equivalent to $n-1$ inequalities, which can be checked efficiently.

There is a natural generalisation, which is called
$\boldsymbol{d}$–majorization Veinott_1971 , or in the context of
thermodynamics, thermo-majorization cit:horodecki2013 . For a fixed
probability distribution $\boldsymbol{r}=(r_{1},\dots,r_{n})$ with positive
components, we define majorization relative to $\boldsymbol{r}$ as
$\boldsymbol{p}\succ_{\boldsymbol{r}}\boldsymbol{q}$, if and only if there
exists a stochastic map $A$ such that $A\boldsymbol{r}=\boldsymbol{r}$ and
$A\boldsymbol{p}=\boldsymbol{q}$. The original majorization condition between
probability distributions corresponds to the case
$\boldsymbol{r}=(1/n,\dots,1/n)$.

In fact, we can further generalize to relative majorization Blackwell_1953 ;
Ruch_1976 ; ruch_mixing_1978 ; Renes_2016 ; Buscemi_2017 ; Rethinasamy_2020 ,
defined as an ordering between pairs of vectors and write

$(\boldsymbol{p},\boldsymbol{r})\succ(\boldsymbol{q},\boldsymbol{r}^{\prime})$
(14)

if and only if there is a stochastic map $A$ such that
$A\boldsymbol{r}=\boldsymbol{r}^{\prime}$ and
$A\boldsymbol{p}=\boldsymbol{q}$. We retrieve $\boldsymbol{d}$–majorization
when $\boldsymbol{r}=\boldsymbol{r}^{\prime}$.

### Quasi-probability majorization and non-monotonic Lorenz curves

Our analysis applies majorization to magic states at the level of the
associated Wigner distributions. Since these are in general quasi-
distributions, it is important to check how majorization is computed for these
cases and what differences quasi-distributions bring over genuine probability
distributions.

We also make use of the notion of a Lorenz curve of a vector
$\boldsymbol{w}\in\mathbb{R}^{n}$ relative to some other vector
$\boldsymbol{r}\in\mathbb{R}^{n}$. Given a vector $\boldsymbol{w}$ we define
$\boldsymbol{w}^{\downarrow}$ to be the re-arrangement of the components of
$\boldsymbol{w}$ into decreasing order. Given two $n$–component vectors
$\boldsymbol{w}$ and $\boldsymbol{r}$, we first define
$\widetilde{\boldsymbol{w}}=(\widetilde{w}_{i})$, where
$\widetilde{w}_{i}\coloneqq w_{i}/r_{i}$, as the vector of component-wise
ratios between $\boldsymbol{w}$ and $\boldsymbol{r}$. We can now define the
Lorenz curve of $\boldsymbol{w}$ relative to $\boldsymbol{r}$, denoted
$L_{\boldsymbol{w}|\boldsymbol{r}}(x)$, as the piece-wise linear function that
passes through $(0,0)$ and the $n$ points

$(x_{k},L_{\boldsymbol{w}|\boldsymbol{r}}(x_{k}))=\left(\frac{1}{R}\sum_{i=1}^{k}r_{\pi(i)},\sum_{i=1}^{k}w_{\pi(i)}\right),$
(15)

where $R\coloneqq\sum_{i=1}^{n}r_{i}$ and $\pi$ is the permutation on $n$
objects mapping $\widetilde{\boldsymbol{w}}$ to
$\widetilde{\boldsymbol{w}}^{\downarrow}$. The form of this requires that
$\boldsymbol{r}$ has no zero components, which we shall assume without loss of
generality as the rank of a quantum state is not operationally meaningful.

In the usual case where $\boldsymbol{w}$ and $\boldsymbol{r}$ are both
probability distributions the Lorenz curve is defined on the interval $[0,1]$,
and rises monotonically until it reaches the value $1$ at $x=1$. The value
$L_{\boldsymbol{w}|\boldsymbol{r}}(x)=1$ is a global maximum. Moreover, if
$\boldsymbol{w},\boldsymbol{w}^{\prime},\boldsymbol{r},\boldsymbol{r}^{\prime}$
are all valid probability distributions with
$\boldsymbol{r},\boldsymbol{r}^{\prime}$ having positive components, then

$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})\mbox{
if and only if }L_{\boldsymbol{w}|\boldsymbol{r}}(x)\geq
L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(x),$

for all $x\in[0,1]$. This result is shown in ruch_mixing_1978  and we
reproduce it in Supplementary Note 2. It provides a simple way of computing
whether relative majorization holds between pairs of probability
distributions.

However, if $\boldsymbol{w}$ is a quasi-probability distribution with negative
values, and $\boldsymbol{r}$ a regular probability distribution things are
different. Now the Lorenz curve is no longer monotonically increasing, but is
a concave function that breaks through the
$L_{\boldsymbol{w}|\boldsymbol{r}}(x)=1$ barrier at an interior point and
attains some non-trivial maximum $L_{\star}$ above the value $1$, before
decreasing monotonically to $L_{\boldsymbol{w}|\boldsymbol{r}}(x)=1$ at the
end-point $x=1$. See Fig. 3 for examples of non-monotonic Lorenz curves for
quasi-distributions. Within quantum theory, the breaking of this barrier is
associated with the degree of non-classicality.

Figure 3: A ‘heretical’ family of Lorenz curves. Traditionally, Lorenz curves
for distributions are monotone increasing cumulative functions that reach a
maximum value of $1$. In contrast, Lorenz curves for magic states break
through the value of $L(x)=1$ due to the presence of negativity in the
associated quasi-distributions. The above family of curves correspond to
multiple copies of noisy Strange states
$\rho\coloneqq\rho_{\rm{S}}(\epsilon)^{\otimes n}$ (Eq. (23)) for $n=2,4,6$
within ${\cal R}_{\sigma}$, where $\sigma=\mathbbm{1}/3$. Solid lines
represent pure Strange states, while dashed lines represent $\epsilon$-noisy
Strange states with depolarising error $\epsilon=0.1$.

Relative majorization is usually considered for probability distributions, so
we need to verify that the same Lorenz curve conditions apply to quasi-
distributions before proceeding with our analysis. This can be done from first
principles, but a simpler way to see that this holds is to reduce to the
problem of genuine distributions by first ‘masking’ the negativity in the
quasi-distribution $\boldsymbol{w}$ using the reference distribution
$\boldsymbol{r}$ and then applying the conditions for relative majorization.
This negativity masking then gives the following result.

######  Theorem 1.

Let $\boldsymbol{w}\in\mathbb{R}^{n},\boldsymbol{w}^{\prime}\in\mathbb{R}^{m}$
be quasi-distributions, and let
$\boldsymbol{r}\in\mathbb{R}^{n},\boldsymbol{r}^{\prime}\in\mathbb{R}^{m}$ be
probability distributions with non-zero components. Then,
$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})$
if and only if $L_{\boldsymbol{w}|\boldsymbol{r}}(x)\geq
L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(x)$ for all $x\in[0,1]$.

###### Proof.

Since the components of $\boldsymbol{r}$ are strictly positive, there always
exists an $\epsilon>0$ such that
$\boldsymbol{w}_{\epsilon}\coloneqq\epsilon\boldsymbol{w}+(1-\epsilon)\boldsymbol{r}$
is a genuine probability distribution. A similar result holds for
$\boldsymbol{w}^{\prime}$ and $\boldsymbol{r}^{\prime}$ and we choose
$\epsilon$ sufficiently small so that both $\boldsymbol{w}_{\epsilon}$ and
$\boldsymbol{w}_{\epsilon}^{\prime}$ are probability distributions. We now
have that
$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})$
if and only if
$(\boldsymbol{w}_{\epsilon},\boldsymbol{r})\succ(\boldsymbol{w}_{\epsilon}^{\prime},\boldsymbol{r}^{\prime})$.
This equivalence holds because there exists a stochastic map $A$ such that
$A\boldsymbol{w}=\boldsymbol{w}^{\prime}$ and
$A\boldsymbol{r}=\boldsymbol{r}^{\prime}$ if and only if

$A[\epsilon\boldsymbol{w}+(1-\epsilon)\boldsymbol{r}]=\epsilon\boldsymbol{w}^{\prime}+(1-\epsilon)\boldsymbol{r}^{\prime}\mbox{
and }A\boldsymbol{r}=\boldsymbol{r}^{\prime}.$ (16)

In terms of a Lorenz curve condition we have that
$(\boldsymbol{w}_{\epsilon},\boldsymbol{r})\succ(\boldsymbol{w}_{\epsilon}^{\prime},\boldsymbol{r}^{\prime})$
if and only if $L_{\boldsymbol{w}_{\epsilon}|\boldsymbol{r}}(x)\geq
L_{\boldsymbol{w}^{\prime}_{\epsilon}|\boldsymbol{r}^{\prime}}(x)$ for all
$x\in[0,1]$. Additionally, we show in Supplementary Note 2 that the Lorenz
curve for any quasi-distribution obeys the relation

$L_{a\boldsymbol{w}+b\boldsymbol{r}|\boldsymbol{r}}(x)=aL_{\boldsymbol{w}|\boldsymbol{r}}(x)+bx,$
(17)

for any $a>0$ and $b\in\mathbb{R}$. This relation implies that
$L_{\boldsymbol{w}_{\epsilon}|\boldsymbol{r}}(x)\geq
L_{\boldsymbol{w}^{\prime}_{\epsilon}|\boldsymbol{r}^{\prime}}(x)$ if and only
if $\epsilon L_{\boldsymbol{w}|\boldsymbol{r}}(x)+(1-\epsilon)x\geq\epsilon
L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(x)+(1-\epsilon)x$.
Finally, the $(1-\epsilon)x$ terms cancel on both sides and we get the
required relative majorization conditions that
$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})$
if and only if $L_{\boldsymbol{w}|\boldsymbol{r}}(x)\geq
L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(x)$ for all $x\in[0,1]$,
as required. ∎

With this theory in place, we turn to magic distillation protocols.

### Magic distillation bounds with fixed-points

We can now consider how majorization constrains magic distillation rates
starting in this section with an abstract setting based on the fixed-point
structure of the physical operations accessible to us. We show that any magic
theory can always be decomposed in terms of ‘sub-theories’ with a particular
fixed-point structure, which allows the use of $\boldsymbol{d}$–majorization.
Later, we drop the requirement of fixed-points and instead use the full
relative majorization, before giving general entropic constraints.

We begin by defining the following sub-theory of any magic theory ${\cal R}$.

######  Definition 2.

Given a theory of magic ${\cal R}=({\cal F},{\cal O})$, we define the sub-
theory ${\cal R}_{\sigma}=({\cal F},{\cal O}_{\sigma})$, in which the free
operations are

${\cal O}_{\sigma}\coloneqq\\{{\cal E}\in{\cal O}:{\cal E}(\sigma)=\sigma\\},$
(18)

namely those free operations that leave the distinguished state
$\sigma\in{\cal F}$ invariant.

This gives a simple way to break up any theory into smaller, more manageable
parts, and where the union over all sub-theories returns the parent theory, so
we are not discarding any information by breaking up the theory in this way.
We make this precise as follows.

######  Theorem 3.

Let ${\cal R}=({\cal F},{\cal O})$ be a theory of magic. Then a transformation
$\rho_{1}\rightarrow\rho_{2}$ is possible in ${\cal R}$ if and only if the
transformation $\rho_{1}\rightarrow\rho_{2}$ is possible in at least one sub-
theory ${\cal R}_{\sigma}$.

The proof of this is straightforward.

###### Proof.

Suppose the interconversion is possible in a ${\cal R}_{\sigma}$ via some
${\cal E}\in{\cal O}_{\sigma}$. But since ${\cal O}_{\sigma}\subseteq{\cal O}$
it is also possible in ${\cal R}$. Conversely, suppose the transformation is
possible in ${\cal R}$ via some ${\cal E}$ in ${\cal O}$. The free states
${\cal F}$ are a closed, bounded set and moreover the image of ${\cal F}$
under the map ${\cal E}$ is in ${\cal F}$. By the Brouwer fixed-point theorem
cit:brouwer , this mapping must therefore have a fixed point $\sigma\in{\cal
F}$, so ${\cal E}\in{\cal O}_{\sigma}$ and the interconversion is possible in
${\cal R}_{\sigma}$. ∎

We can compare with resource monotones, which are also called resource
measures. A complete set of measures $\\{{\cal M}_{i}\\}_{i}$ is such that
$\rho\rightarrow\sigma$ if and only if ${\cal M}_{i}(\rho)\geq{\cal
M}_{i}(\sigma)$ for all $i$. The above set of sub-theories can therefore be
viewed as a complete set of “co-measures” for the theory, where the parent
resource theory’s complex pre-order of states is mapped to a simpler pre-order
for a particular ${\cal R}_{\sigma}$. In the next section we show that a given
${\cal R}_{\sigma}$ can be approximated by a majorization pre-order, and from
this we compute magic distillation bounds.

### Majorisation of Wigner distributions with fixed-point structure

We consider some $\sigma\in{\cal F}$, and its corresponding ${\cal
R}_{\sigma}$. We are interested in the ability to transform many copies of
some noisy magic state $\rho$ towards a more pure form of magic. The state
$\rho$ is assumed to have negativity in the Wigner representation, so
$W_{\rho}(\boldsymbol{z})<0$ for some regions of $\boldsymbol{z}\in{\cal
P}_{d}$. The state $\sigma$ is assumed to have a Wigner distribution with
$W_{\sigma}(\boldsymbol{z})>0$ for all $\boldsymbol{z}$ in the phase space.
This is justified because a non-full rank state $\sigma$ can be handled as a
limiting case in which we first add an infinitesimal fraction of depolarising
noise $\epsilon(\mathbbm{1}/d)$ and then take $\epsilon\rightarrow 0$.

The free operations within the magic theory ${\cal R}$ are represented by
stochastic maps, and within ${\cal R}_{\sigma}$ by stochastic maps that leave
$W_{\sigma}(\boldsymbol{z})$ invariant. Therefore, a necessary condition for
magic state transformations $\rho_{1}\rightarrow\rho_{2}$ within ${\cal
R}_{\sigma}$ will be that

$W_{\rho_{1}}\succ_{W_{\sigma}}W_{\rho_{2}},$ (19)

or put another way, that the quasi-distribution $W_{\rho_{1}}$ is more ordered
than the quasi-distribution $W_{\rho_{2}}$ relative to $W_{\sigma}$. To
simplify notation, we denote by $L_{\rho|\sigma}(x)$ the Lorenz curve
$L_{W_{\rho}|W_{\sigma}}(x)$, and therefore have that

$\rho_{1}\rightarrow\rho_{2}\mbox{ within }{\cal R}_{\sigma}\mbox{ implies
}L_{\rho_{1}|\sigma}(x)\geq L_{\rho_{2}|\sigma}(x),$ (20)

for all $x\in[0,1]$, which restricts the transformations that are possible.

Note that this is not a single numerical constraint, but a family of
constraints. For $n$ copies of a qudit system of dimension $d$ the number of
terms in $W_{\rho}$ is $d^{2n}$, so imposing the Lorenz curve condition
corresponds to exponentially many constraints.

Before computing explicit examples, we state some generic aspects of Lorenz
curves for magic states, which allow us to interpret previous magic monotones
as features of the curves. The first result gives a simple way to see that the
sum-negativity/mana of a magic state is a monotone cit:veitch2 .

######  Theorem 4.

Majorization in any ${\cal R}_{\sigma}$ implies the monotonicity of sum-
negativity/mana.

###### Proof.

The sum-negativity of a magic state $\rho$ can be written as
$sn(\rho)=\frac{1}{2}(\sum_{\boldsymbol{z}}|W_{\rho}(\boldsymbol{z})|-1)$. We
make use of the $L_{1}$-norm form of relative majorization, (see Supplementary
Note 2), which states that
$\boldsymbol{p}\succ_{\boldsymbol{r}}\boldsymbol{q}$ if and only if

$\sum_{k}|p_{k}-r_{k}t|\geq\sum_{k}|q_{k}-r_{k}t|,$ (21)

for all $t\in\mathbb{R}$. Choosing $t=0$ we get the single condition that
$\sum_{k}|p_{k}|\geq|\sum_{k}|q_{k}|$, independent of the choice of
$\boldsymbol{r}$. Applying this to the Wigner quasi-distributions of two
quantum states immediately gives the result. ∎

If we have a magic state $\rho$ that has negativity in its Wigner
representation, then, as discussed, its Lorenz curve over-shoots $1$ and
reaches a non-trivial maximum $L_{\star}$ that depends on the particular
state. There is an simple relation between $L_{\star}$ and sum-
negativity/mana, which is provided by the following result.

######  Theorem 5.

Given a magic state $\rho$, the maximum $L_{\star}$ of its Lorenz curve
$L_{\rho|\sigma}(x)$ is independent of the ${\cal R}_{\sigma}$ and equal to
$1+{\rm sn}(\rho)$. Moreover, the majorization constraint is stronger than
mana in every ${\cal R}_{\sigma}$.

The proof of this is given in Supplementary Note 2. Using the Lorenz curve
perspective, it is also simple to construct magic monotones. For example,
within a given ${\cal R}_{\sigma}$, the area above the horizontal line $y=1$
is a magic monotone.

######  Theorem 6.

Given a magic state $\rho$ and a free state $\sigma$, let ${\cal
A}_{\sigma}(\rho)$ be the area of the region $\\{(x,y):1\leq y\leq
L_{\rho|\sigma}(x)\\}$. Then ${\cal A}_{\sigma}(\rho)$ is a magic monotone for
${\cal R}_{\sigma}$.

###### Proof.

Consider the transformation $\rho_{1}\rightarrow\rho_{2}$ within ${\cal
R}_{\sigma}$. We have that $L_{\rho_{1}|\sigma}(x)$ is never below the curve
$L_{\rho_{2}|\sigma}(x)$, and therefore the region above $L=1$ for $\rho_{2}$
is a subset of the corresponding region for $\rho_{1}$. Thus, ${\cal
A}_{\sigma}(\rho_{1})\geq{\cal A}_{\sigma}(\rho_{2})$, so ${\cal A}_{\sigma}$
is a magic monotone in ${\cal R}_{\sigma}$. ∎

Note though, in contrast to mana, the area monotone is specific to ${\cal
R}_{\sigma}$, and its value will vary as we change $\sigma$. Therefore its
monotonicity depends on the physics of the fixed point and provides a means to
analyse magic distillation only under free operations that leave $\sigma$
invariant. One can also define an area monotone in the more general setting of
relative majorization, where one drops the fixed point emphasis.

### Magic distillation bounds for unital protocols

We now construct magic state distillation bounds for protocols that generate
unital channels, meaning that ${\cal E}(\mathbbm{1}/d)=\mathbbm{1}/d$.

Our approach works for any odd dimension, but for simplicity we consider
qutrit magic states ($d=3$), for which a canonical magic state exists with a
simple Wigner distribution. This is the Strange state $|S\rangle$, which is
given by

$|S\rangle\coloneqq\frac{1}{\sqrt{2}}(|1\rangle+|2\rangle),$ (22)

in the computational basis. Its distribution $W_{|S\rangle\langle
S|}(\boldsymbol{z})$ has a single negative value of $-1/3$ at
$\boldsymbol{z}=(0,0)$ and the positive value $1/6$ at all other points. We
define the $\epsilon$–noisy Strange state as

$\rho_{\rm{S}}(\epsilon)\coloneqq(1-\epsilon)\ket{\rm{S}}\bra{\rm{S}}+\epsilon\frac{1}{3}\mathbbm{1},$
(23)

where $\epsilon$ is the depolarizing error parameter. Any magic state $\rho$
can be processed via Clifford operations cit:prakash ; cit:prakash2  and put
into this canonical form for some $\epsilon\geq 0$.

The Wigner distribution of the single-copy, $\epsilon$–noisy Strange state is
given by

$W_{\rho_{\rm{S}}(\epsilon)}(\boldsymbol{z})=(1-\epsilon)W_{|\rm{S}\rangle\langle\rm{S}|}(\boldsymbol{z})+\epsilon
W_{\frac{1}{3}\mathbbm{1}}(\boldsymbol{z}).$ (24)

with a single negative component

$-v(\epsilon)\coloneqq-\left(\frac{1}{3}-\frac{4}{9}\epsilon\right)$ (25)

located at the origin $\boldsymbol{z}=\boldsymbol{0}$ and positive components

$u(\epsilon)\coloneqq\frac{1}{6}-\frac{1}{18}\epsilon$ (26)

at the 8 phase space points $\boldsymbol{z}\neq\boldsymbol{0}$. We assume that
$\epsilon<3/4$ to ensure the presence of negativity in the Wigner
distribution.

We now consider the problem of purifying $n$ copies of a noisy Strange state
$\rho_{\rm{S}}(\epsilon)^{\otimes n}$ into a smaller number of copies
$n^{\prime}$ of a less noisy Strange state
$\rho_{\rm{S}}(\epsilon^{\prime})^{\otimes n^{\prime}}$, with
$\epsilon^{\prime}<\epsilon$ and $n^{\prime}\leq n$. In particular, we compute
the Lorenz curves for the transformation

$\rho_{\rm{S}}(\epsilon)^{\otimes
n}\longrightarrow\rho_{\rm{S}}(\epsilon^{\prime})^{\otimes n^{\prime}},$ (27)

and use them to bound the magic distillation rate
$R(\epsilon,\epsilon^{\prime})\coloneqq n^{\prime}/n$.

Due to negativity, the ordering of the components of the rescaled Wigner
distribution $W_{\rho_{n}|\sigma_{n}}(\boldsymbol{z})\coloneqq
W_{\rho_{n}}(\boldsymbol{z})/W_{\sigma_{n}}(\boldsymbol{z})$, for
$\rho_{n}=\rho_{S}(\epsilon)^{\otimes n}$, and $\sigma_{n}=\sigma^{\otimes
n}=(\mathbbm{1}/3)^{\otimes n}$, depends on whether $-v(\epsilon)$ is raised
to an even or odd power. Treating the Wigner distributions as vectors, the
component values $w_{i}$ and associated multiplicities $m_{i}$ in the $n$–copy
case are found to be

$\displaystyle m_{i}$ $\displaystyle=8^{i}\binom{n}{i},$ (28) $\displaystyle
w_{i}$ $\displaystyle=u^{i}(-v)^{n-i},$ (29)

where index $i$ runs through $0,\dots,n$, and we assume for simplicity that
$n$ is even and $\epsilon\leq 3/7$, which implies that $v\geq u$. All details
on the analysis are provided in Supplementary Note 3.

The Lorenz curve $L_{\rho_{n}|\sigma_{n}}(x)$ reaches a maximum value of

$\displaystyle L_{\star}$ $\displaystyle\coloneqq
L_{\rho_{n}|\sigma_{n}}(x_{\star})=\sum_{i=0}^{n/2}m_{2i}w_{2i}$
$\displaystyle=\frac{1}{2}+\frac{1}{2}\left(\frac{15-8\epsilon}{9}\right)^{n}>1,$
(30)

which occurs at $x=x_{\star}$ given by

$x_{\star}=\frac{1}{2}+\frac{1}{2}\left(\frac{7}{9}\right)^{n}.$ (31)

Distillation bounds can be obtained from any part of the Lorenz curve by
imposing the constraint that the Lorenz curve of the input state never dips
below the curve of the output. One simple constraint can be obtained by
considering the initial slopes of the two Lorenz curves.

The coordinates of the first point after the origin for
$\rho_{S}(\epsilon)^{\otimes n}$ are given by $(1/9^{n},v(\epsilon)^{n})$,
while for $\rho_{S}(\epsilon^{\prime})^{\otimes n^{\prime}}$ by
$(1/9^{n^{\prime}},v(\epsilon^{\prime})^{n^{\prime}})$, and by requiring that
the initial slope for the input state Lorenz curve is larger than that of the
output state (Proposition 24 in Supplementary Note 2) we find that

$R(\epsilon,\epsilon^{\prime})\leq
R_{\infty}\coloneqq\frac{\log(3-4\epsilon)}{\log(3-4\epsilon^{\prime})}.$ (32)

The choice of denoting the bound by $R_{\infty}$ is explained in the
derivation of Eq. (52) in relation to single-shot entropies. For the limiting
case of pure magic states on the output ($\epsilon^{\prime}=0$), this
simplifies to

$R\leq 1+\frac{\log(1-\frac{4}{3}\epsilon)}{\log 3}.$ (33)

We compare this to other known distillation bounds coming from mana and
thauma. The mana bound can be directly calculated as

$R\leq\frac{{\cal M}(\rho_{\rm{S}}(\epsilon))}{{\cal
M}(\rho_{\rm{S}}(0))}=1+\frac{\log\left(1-\frac{8}{15}\epsilon\right)}{\log\frac{5}{3}}.$
(34)

The max–thauma Wang_2020  is defined as

$\theta_{\rm{max}}(\rho)\coloneqq\log\min{\\{2\hskip 1.0pt{\rm
sn}(V)+1:V\geq\rho\\}}$ (35)

and can be calculated numerically via a semi-definite program. For the noisy
Strange state, the max–thauma bound coincides with the mana bound, and we find
that they are both looser than the majorization bound $R_{\infty}$ as shown in
Fig. 1.

This figure includes numerical estimates of the optimal majorization bounds
$R_{\rm{num}}$ obtained by considering the full Lorenz curve, which show that
the bound $R_{\infty}$ can be further improved. While this is an improvement
on prior results, all known distillation protocols have rates much lower than
these upper bounds. It remains a major open question to determine what are the
best possible rates that can be achieved.

### Temperature-dependent bounds for magic distillation protocols

We now show how relative majorization can incorporate the system’s Hamiltonian
into magic distillation bounds. The way we do this is by considering how the
protocol would disturb a reference equilibrium state $\tau$ to some different
state $\tau^{\prime}$, if the protocol had been applied to this reference
state instead of the actual $n$-copy magic state.

We consider a magic distillation protocol on multiple identical qudits in a
noisy magic state $\rho(\epsilon)$, with noise parameter $\epsilon$, sending

$\rho(\epsilon)^{\otimes n}\longrightarrow{\cal E}(\rho(\epsilon)^{\otimes
n})=\rho(\epsilon^{\prime})^{\otimes m}$ (36)

with $n\gg 1$, $\epsilon^{\prime}<\epsilon$ and ${\cal E}$ denoting the
quantum channel induced by the protocol. We also assume each qudit has a
Hamiltonian $H$ and neglect interaction terms. We choose some temperature
$T=(k\beta)^{-1}$ where $k$ is Boltzmann’s constant and $\beta$ the inverse
temperature. The reference equilibrium state of the $n$ qudits is given by

$\tau^{\otimes n}=\left(\frac{e^{-\beta H}}{{\cal Z}}\right)^{\otimes n}.$
(37)

We may also assume that the reference state $\tau$ is not a magic state, and
has a strictly positive Wigner distribution $W_{\tau}(\boldsymbol{z})$.

A given magic protocol on the $n$ qudits will correspond to a quantum channel
${\cal E}$. We also assume for simplicity that $U_{\pi}{\cal
E}(X)U_{\pi}^{\dagger}={\cal E}(X)$ for all $X$ and any permutation $U_{\pi}$
of the $m$ output subsystems. This is justified because the protocol on the
input magic state results in state $\rho(\epsilon^{\prime})^{\otimes m}$,
which is invariant under permutations. Therefore, we are always allowed to
symmetrize the output ${\cal E}(X)$ by performing a group average over the
permutation group for the output systems without changing the performance of
the distillation protocol on $\rho(\epsilon)^{\otimes n}$. Thus, we can assume
that ${\cal E}$ always outputs a symmetric state in general. This means that
${\cal E}(\tau^{\otimes n})$ is a symmetric state on $m$ subsystems, so by the
quantum de Finetti theorem hudson_locally_1976 ; christandl_2007  we have that
for $m\gg 1$

${\cal E}(\tau)\approx\int d\mu(x)\tau_{x}^{\prime\otimes m},$ (38)

where $d\mu(x)$ is a probability measure over a set $\\{\tau^{\prime}_{x}\\}$
of single qudit states.

### Free energy and sub-linear correlations in the thermodynamic limit

To keep our analysis simple we make the following physical assumption. We
assume that in the asymptotic/thermodynamic limit $n,m\rightarrow\infty$ the
correlations generated on the reference equilibrium state are negligible. This
implies that $d\mu(x)$ is peaked on a particular state $\tau^{\prime}$, and
${\cal E}(\tau^{\otimes n})\approx\tau^{\prime\otimes m}$. This scenario
occurs in the context of traditional thermodynamics, and states that the
output system is well-described by intrinsic variables that do not scale in
$m$, and correlations are sub-linear in $m$. In particular, this allows us to
compute a free energy per qudit of the output state.

Non-trivial correlations in the thermodynamic limit can also be considered,
but leads to a more complex analysis within our majorization framework of
Wigner distributions. In this direction, we highlight recent work in
majorization in which stochastic independence and correlations are analysed.
It has been shown muller_2015  that stochastic independence (no correlations)
of independent distributions can be viewed as a resource in an extension of
catalytic majorization, and leads to a single-shot operational interpretation
of the Shannon entropy muller_2016 ; muller_2019 .

Our bound depends on the von Neumann entropy $S(\rho)$ of a state, and the
free energy at a particular temperature. For a state $\tau=e^{-\beta H}/{\cal
Z}$, the Helmholtz free energy $F$ is given by

$F\coloneqq{\rm{tr}}[H\tau]-\beta^{-1}S(\tau)=-\beta^{-1}\log{\cal Z},$ (39)

which is obtained from the internal energy via a Legendre transform
Pathria_1997 .

The protocol transforms the equilibrium as $\tau^{\otimes n}\rightarrow{\cal
E}(\tau^{\otimes n})=\tau^{\prime\otimes m}$. The protocol does not generate
magic on its own, so we assume that the output state $\tau^{\prime}$ is also a
Wigner-positive state. However, this is generally a non-equilibrium state for
the system. Despite this it is useful to associate an effective Hamiltonian
$H^{\prime}$ to the output state by considering the change $H\rightarrow
H^{\prime}$ such that equilibrium is restored at the reference temperature
$T$. This Hamiltonian is defined by the expression $\tau^{\prime}=e^{-\beta
H^{\prime}}/{\cal Z}^{\prime}$, and has free energy
$F^{\prime}=-\beta^{-1}\log{\cal Z}^{\prime}$.

The magic state protocol is now considered by how it transforms the pair of
quantum states $(\rho^{\otimes n},\tau^{\otimes n})$ and is then constrained
by the relative majorization condition
$(W_{\rho},W_{\tau})\succ(W_{\rho^{\prime}},W_{\tau^{\prime}})$ that holds due
to the protocol being a stochastic map in the Wigner representation.

In Supplementary Note 4 we analyse the Lorenz curves of this condition and
obtain the following magic distillation bound that combines computational
measures $\epsilon,\epsilon^{\prime}$, with terms that depend on the
Hamiltonian and reference temperature $T$ of the physical system. We state the
result for the case of qutrits, but a similar analysis works for general odd-
dimension qudits.

######  Theorem 7.

Consider a magic distillation protocol on qutrits that transforms $n$ copies
of an $\epsilon$–noisy Strange state into $m$ copies of an
$\epsilon^{\prime}$–noisy Strange state, with depolarising errors
$\epsilon^{\prime}\leq\epsilon\leq 3/7$. We also allow pre/post-processing by
local Clifford unitaries.

Let $T=(k\beta)^{-1}$ be any finite temperature for the physical system and
let $H=\sum_{k\in\mathbb{Z}_{3}}E_{k}|E_{k}\rangle\langle E_{k}|$ be the
Hamiltonian of each qutrit subsystem in its eigen-decomposition. Assume that
in the thermodynamic limit ($n,m\gg 1$), the protocol applied to the
equilibrium state $\tau^{\otimes n}=(e^{-\beta H}/{\cal Z})^{\otimes n}$ maps
$\tau^{\otimes n}\longrightarrow\tau^{\prime\otimes m}$, where we write
$\tau^{\prime}=e^{-\beta H^{\prime}}/{\cal Z}^{\prime}$ for some Hermitian
$H^{\prime}$.

Then the asymptotic magic distillation rate $R=m/n$ is bounded as

$R\leq\dfrac{\log\big(1-\frac{4}{3}\epsilon\big{missing})+\beta(\phi-F)}{\log\big(1-\frac{4}{3}\epsilon^{\prime}\big{missing})+\beta(\phi^{\prime}-F^{\prime})},$
(40)

where $F$ is the free energy of $\tau$, and

$\phi=-\beta^{-1}\log\zeta$ (41)

with $\zeta$ given by the expressions

$\displaystyle\zeta$
$\displaystyle=\sum_{k\in\mathbb{Z}_{3}}\alpha_{k}e^{-\beta E_{k}},$ (42)
$\displaystyle\alpha_{k}$ $\displaystyle=\langle
E_{k}|A_{\boldsymbol{z}_{\star}}|E_{k}\rangle,$ (43)

and $W_{\tau}(\boldsymbol{z})$ attaining a minimum at
$\boldsymbol{z}=\boldsymbol{z}_{\star}$. The primed variables are defined
similarly for the output system.

The proof of this is provided in Supplementary Note 4, and follows a similar
line to the unital protocol bound.

The above bound depends on:

  * •

Quantum computational measures $\epsilon,\epsilon^{\prime}$,

  * •

Thermodynamic quantities $\beta,F,F^{\prime}$,

  * •

Intermediate terms $\phi,\phi^{\prime}$.

The intermediate terms specify how the energy eigenbasis of the system
$\\{|E_{k}\rangle\\}$ relates to the computational stabilizer basis
$\\{|k\rangle\\}$. In particular, the term $\phi$ quantifies to what degree a
sharp energy value can be associated to the negativity in the Wigner
representation. Its form is similar to $F$, and so $\phi$ can loosely speaking
be viewed as a ‘magic free energy’ term.

The coefficients $\alpha_{k}$ may be negative for some $k$, when the
Hamiltonian has non-stabilizer eigenstates, but the quantity $\phi$ is always
well-defined, since the function $\zeta$ is always positive for $\tau$ in the
interior of ${\cal F}$. The quantity $\phi$ can diverge if the state $\tau$
acquired zero Wigner components, which occurs for $\tau$ on the boundary of
the set of Wigner-positive states, and is not defined outside of ${\cal F}$.
Finally, if $H$ has a stabilizer eigenbasis then $\phi=E_{i}$ for some $i$,
independent of the temperature $T$. Details of this are provided in
Supplementary Note 4.

### Technical aspects and special cases of free energy bounds

Similar to the unital protocol bounds, the above result is based on only part
of the Lorenz curves and so can certainly be tightened with further analysis.
The primary role of the pre/post-processing by Clifford unitaries in Theorem 7
is to simplify the form of the bound, and allow it to be expressed in terms of
the free energy per particle in a way that does not depend on the parameters
$\epsilon,\epsilon^{\prime},\beta$ in a complex form. In Supplementary Note 4,
we give bounds in which one does not include these Clifford changes of basis,
thus having a more non-trivial dependency on the parameters.

Figure 4: Temperature-dependent bounds for magic distillation. Shown are two
contour plots of the bound on $R(\epsilon,\beta)$ for the case
$H=H^{\prime}=\sum_{k\in\mathbb{Z}_{3}}k\outerproduct{k}{k}$ and
$\epsilon^{\prime}=0$, where $\beta$ is the inverse temperature and $\epsilon$
is the depolarising error of the input magic state. The top figure (a) does
not use any changes of Clifford basis, and the form of the bound depends on
both the error parameter and temperature. The curved dashed line is
$\epsilon_{\star}(\beta)$ and given by Eq. (86) in Supplementary Note 4 and
$\beta_{\star}=(kT_{\star})^{-1}$ is given by $E_{2}-\phi=kT_{\star}\ln 2$. In
the bottom figure (b) Clifford processing is used resulting in a smoother
bound. In both figures the $\beta=0$ line correspond to the unital bounds.

The analysis makes other simplifying assumptions that could easily be dropped,
at the price of more complex expressions. We could perform similar analysis
for general qudits, and different choices of magic states, for example. It
might also be of interest to consider other choices of reference states $\tau$
that are more appropriate to the hardware physics, for example for photonic
set-ups bombin2021interleaving .

The assumption which is non-trivial is that we neglect correlations in the
reference state in the thermodynamic limit. However, for more general
scenarios one could make use of variational tools such as the Bogoliubov
inequality bogolyubov_1966  for approximating the free energy of a system via
product states, to obtain similar bounds.

The simplest special case to consider is where the Hamiltonian $H$ is diagonal
in the computational basis, and $H=H^{\prime}$, implying that the protocol
leaves the reference equilibrium state unchanged, hence it corresponds to a
Gibbs-preserving map faist_2015 . For the limiting case of
$\epsilon^{\prime}\rightarrow 0$ we obtain Fig. 4 which is a contour plot of
the bound on $R$ as a function of inverse temperature $\beta$ and the
depolarising error $\epsilon$ for the noisy input magic states. In this figure
we show both the bounds with Clifford basis changes and without them. In the
more general case, $\Delta F\coloneqq F^{\prime}-F\neq 0$, so the protocol,
when applied to the reference state, adds/extracts free energy from the
system. We demonstrate change of the bounds with respect to a varying output
Hamiltonian in Fig. 5.

Figure 5: Variation of distillation bounds with Hamiltonian. We illustrate
variation of the bound in Eq. (114) with respect to the system Hamiltonian. We
fix $(\epsilon,\epsilon^{\prime},\beta)=(0.1,0.0,0.2)$ and look at a family of
Hamiltonians for the qutrit systems, given by $H=A_{\boldsymbol{0}}$ and
$H^{\prime}=(1-p-q)A_{\boldsymbol{0}}+pA_{(1,2)}+q{\rm{diag}}(0,1,2)$, with
varying parameters $p,q$, and where $A_{\boldsymbol{0}}$ and $A_{(1,2)}$ are
phase-point operators.

### Extension of single-shot entropies to quasi-distributions

We have considered Wigner distributions of magic states within a statistical
mechanical setting in which it was argued that magic can be viewed as a non-
classical form of free energy. This was done at the level of majorization
constraints that appear due to stabilizer operations being described by
stochastic maps in the Wigner representation. In this representation, non-
equilibrium states with substantial free energy have Lorenz curves that
deviate noticeably from the line $y=x$. We also found that when considered in
a distillation protocol, magic could also be linked to physical free energies
in a non-trivial way. We can therefore ask: is it possible to make a more
direct link between magic and free energy or entropy?

One obstacle to linking with macroscopic, equilibrium free energy is that the
Wigner distribution is generally a quasi-distribution, so the entropy
$H(\boldsymbol{w})=-\sum_{i}w_{i}\log w_{i}$ of the Wigner distribution is not
defined outside the set ${\cal F}$, let alone have a physically meaningful
interpretation.

However, it turns out that the analysis for the temperature-dependent bound
given by Eq. (114) implicitly made use of the single-shot Rényi divergence
renyi_1960  $D_{\infty}(\boldsymbol{p}\hskip 1.0pt||\hskip
1.0pt\boldsymbol{r})$, which is given by

$D_{\infty}(\boldsymbol{p}\hskip 1.0pt||\hskip
1.0pt\boldsymbol{r})=\log\max_{i}\frac{p_{i}}{r_{i}}.$ (44)

It is clear that this does remain well-defined on quasi-distributions, and
from the majorization relation also has a clear physical meaning, and in
Supplementary Note 4, we show that this divergence has the following
properties that are derived via relative majorization.

######  Theorem 8.

Let $\tau$ be in the interior of ${\cal F}$. Then
$D_{\infty}(W_{\rho}||W_{\tau})$ is well-defined for all $\rho$, and the
following hold:

  1. 1.

$D_{\infty}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})\geq 0$ for all quantum
states $\rho$.

  2. 2.

$D_{\infty}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})=0$ if and only if
$\rho=\tau$.

  3. 3.

$D_{\infty}(W_{\rho^{\otimes 2n}}\hskip 1.0pt||\hskip 1.0ptW_{\tau^{\otimes
2n}})=nD_{\infty}(W_{\rho^{\otimes 2}}\hskip 1.0pt||\hskip
1.0ptW_{\tau^{\otimes 2}})$ for any $n\in\mathbb{N}$.

  4. 4.

$D_{\infty}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})\geq
D_{\infty}(W_{{\cal E}(\rho)}\hskip 1.0pt||\hskip 1.0ptW_{{\cal E}(\tau)})$
for any free operation ${\cal E}$ such that ${\cal E}(\tau)$ is in the
interior of ${\cal F}$.

The reason for the power of $2$ in the third property is that the maximization
is sensitive to the presence of negativity in the Wigner distribution.

We can go further, and show that in fact a range of Rényi entropies remain
both well-defined and meaningful in the Wigner representation. The
$\alpha$–Rényi entropy $H_{\alpha}(\boldsymbol{p})$ is defined as renyi_1960

$H_{\alpha}(\boldsymbol{p})\coloneqq\frac{1}{1-\alpha}\log\sum_{i}p_{i}^{\alpha}.$
(45)

To extend to quasi-distributions we must ensure the following: (a) the entropy
is mathematically well-defined on quasi-distributions, and (b) obeys
monotonicity under the majorization pre-order. In Supplementary Note 5 we
prove the following.

######  Theorem 9.

If $\alpha=\frac{2a}{2b-1}$ for positive integers $a,b$ with $a\geq b$, then
$H_{\alpha}(\boldsymbol{w})$ is well-defined on the set of quasi-
distributions, and if $\boldsymbol{w}\succ\boldsymbol{w}^{\prime}$ for two
quasi-distributions $\boldsymbol{w},\boldsymbol{w}^{\prime}$ then
$H_{\alpha}(\boldsymbol{w})\leq H_{\alpha}(\boldsymbol{w}^{\prime})$.

This monotonicity provides a meaningful extension of entropies to quasi-
distributions. The unusual form of the $\alpha$ parameter is required so that
the entropy function is well-defined and real-valued for all $\boldsymbol{w}$.
The allowed values of $\alpha$ are dense in $\alpha>1$, however, while Rényi
entropies on probabilities can also be used for $0<\alpha<1$, it turns out
that this parameter regime is no longer monotonic in terms of the majorization
pre-order for quasi-distributions (see Supplementary Note 5 for details).

We can prove the following result, which establishes the equivalence between
Wigner negativity and the existence of negative Rényi entropy.

######  Theorem 10.

A quantum state $\rho$ has Wigner negativity if and only if
$H_{\alpha}(W_{\rho})<0$ for some $\alpha=\frac{2a}{2b-1}$, with positive
integers $a\geq b$.

Therefore, the statistical mechanical description of stabilizer operations
provides a setting in which negative entropy is fully meaningful, and
quantifies the fact that magic states are more ordered than a perfectly sharp,
deterministic classical state $\boldsymbol{w}=(1,0,\dots,0)$ with zero
entropy. We note that prior work has shown that negative conditional entropy
rio_thermodynamic_2011  arises in the context of quantum correlations, but is
unrelated to the present negative entropy.

The $\alpha\rightarrow 1$ limit of the Rényi entropy diverges if negativity is
present, however it diverges in a well-defined way. We can write the entropy
as

$H_{1+\epsilon}(W_{\rho})=-\epsilon^{-1}\log\sum_{\boldsymbol{z}}|W_{\rho}(\boldsymbol{z})|^{1+\epsilon}$
(46)

with $\epsilon\rightarrow 0^{+}$ through a sequence of rational values. We
recall that mana can be written as ${\cal
M}(\rho)=\log\sum_{\boldsymbol{z}}|W_{\rho}(\boldsymbol{z})|$, so we obtain a
second perspective on mana:

${\cal M}(\rho)=-\lim_{\epsilon\rightarrow 0^{+}}\epsilon
H_{1+\epsilon}(W_{\rho}).$ (47)

Therefore, the mana of a state $\rho$ is minus the residue of the pole at
$\alpha=1$ for $H_{\alpha}(W_{\rho})$, and quantifies the divergence of the
Rényi-entropy as we approach the limiting Shannon entropy $H(W_{\rho})$.

We also note that the Rényi entropy can be described as a $q$–deformation of
the Shannon entropy baez2011renyi , and that $H_{\alpha}(\boldsymbol{p})$ can
be related to the $q$–derivative of free energy. For this, the Rényi parameter
becomes a temperature term $\alpha=\frac{T_{0}}{T}$ where $T_{0}$ is a
reference temperature and derivatives are considered via the limit
$T\rightarrow T_{0}$. Therefore, if we try to push this to the present
setting, the presence of Wigner negativity would correspond to a divergence in
the first derivative of an effective free energy as $T\rightarrow T_{0}$. It
is interesting to speculate whether this non-classicality could be interpreted
in terms of a phase transition domb2000phase .

### Rényi divergences and general distillation bounds

Every entropy can be obtained from a relative entropy Gour_2020  so we now
state the extensions for Rényi divergences of Wigner distributions. The
$\alpha$–Rényi divergence $D_{\alpha}(\boldsymbol{p}||\boldsymbol{r})$ is
defined as

$D_{\alpha}(\boldsymbol{p}\hskip 1.0pt||\hskip
1.0pt\boldsymbol{r})\coloneqq\frac{1}{\alpha-1}\log\sum_{i}p_{i}^{\alpha}r_{i}^{1-\alpha},$
(48)

where $\boldsymbol{p}$ can be a quasi-distribution and $\boldsymbol{r}$ a
probability distribution with positive components. In Supplementary Note 5, we
also show that the divergence satisfies the following.

######  Theorem 11.

Let $\tau$ be in the interior of ${\cal F}$. If $\alpha=\frac{2a}{2b-1}$ for
positive integers $a,b$ with $a\geq b$, then the $\alpha$-Rényi divergence
$D_{\alpha}(W_{\rho}||W_{\tau})$ is well-defined for all states $\rho$, and
the following hold:

  1. 1.

$D_{\alpha}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})\geq 0$ for all quantum
states $\rho$.

  2. 2.

$D_{\alpha}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})=0$ if and only if
$\rho=\tau$.

  3. 3.

$D_{\alpha}(W_{\rho^{\otimes n}}\hskip 1.0pt||\hskip 1.0ptW_{\tau^{\otimes
n}})=nD_{\alpha}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})$ for any
$n\in\mathbb{N}$.

  4. 4.

$D_{\alpha}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})\geq
D_{\alpha}(W_{{\cal E}(\rho)}\hskip 1.0pt||\hskip 1.0ptW_{{\cal E}(\tau)})$
for any free operation ${\cal E}$ such that ${\cal E}(\tau)$ is in the
interior of ${\cal F}$.

We can now apply this to obtain a family of magic distillation bounds based on
Rényi divergences.

######  Theorem 12.

Consider a general magic state distillation protocol on odd prime dimension
qudits, that converts a magic state $\rho^{\otimes n}\longrightarrow{\cal
E}(\rho^{\otimes n})=\rho^{\prime\otimes m}$ and let $\tau$ be any full-rank
stabilizer reference state on a qudit. Then, the distillation rate $R\coloneqq
m/n$ is upper bounded as

$R\leq\frac{D_{\alpha}(W_{\rho}\hskip 1.0pt||\hskip 1.0ptW_{\tau})}{\hbox
to0.0pt{\raisebox{-0.86108pt}{$\widetilde{\phantom{D}}$}\hss}D_{\alpha}(\rho^{\prime},\tau^{\prime})},$
(49)

where $\alpha=\frac{2a}{2b-1}$ for any positive integers $a,b$ with $a\geq b$
and the average divergence per qudit

$\widetilde{D}_{\alpha}(\rho^{\prime},\tau^{\prime})\coloneqq\frac{1}{m}D_{\alpha}(W_{\rho^{\prime\otimes
m}}\hskip 1.0pt||\hskip 1.0ptW_{\tau^{\prime}_{m}}),$ (50)

between the output magic state $\rho^{\prime\otimes m}$ and
$\tau^{\prime}_{m}={\cal E}(\tau^{\otimes n})$.

The proof is given in Supplementary Note 5.

While this bound is abstract in its present form, it does suggest future
applications by viewing $\widetilde{D}_{\alpha}(\rho^{\prime},\tau^{\prime})$
within the context of hypothesis testing tomamichel2013 , as a measure of
distinguishability between $\rho^{\prime\otimes m}$ and the $m$ qudit state
$\tau^{\prime}_{m}$. However, in order to properly link with hypothesis-
testing one would have to first extend such results to quasi-distributions.

If correlations in $\tau^{\prime}_{m}$ between subsystems can be neglected,
then $\tau^{\prime}_{m}=\tau^{\prime\otimes m}$ for some qudit state
$\tau^{\prime}$ and

$\widetilde{D}_{\alpha}(\rho^{\prime},\tau^{\prime})=D_{\alpha}(W_{\rho^{\prime}}\hskip
1.0pt||\hskip 1.0ptW_{\tau^{\prime}}),$ (51)

which gives a generalized version of Theorem 7, with a term such as
$D_{\alpha}(W_{\rho}||W_{\tau})$ that behaves essentially as an $\alpha$–free
energy difference Brandao_2015  in the Wigner representation.

We can illustrate a basic application of the above bound for magic protocols
that give unital maps. In this case it is easy to see that the above bound
reduces to the simple form

$R\leq R_{\alpha}\coloneqq\frac{2\log d-H_{\alpha}(W_{\rho})}{2\log
d-H_{\alpha}(W_{\rho^{\prime}})}.$ (52)

In particular for the case of noisy Strange states on qutrits and
$\epsilon^{\prime}=0$, this bound becomes

$R\leq R_{\alpha}=\frac{2(1-\alpha)\log
3-\log\left[8(\frac{1}{6}-\frac{\epsilon}{18})^{\alpha}+(-\frac{1}{3}+\frac{4\epsilon}{9})^{\alpha}\right]}{2(1-\alpha)\log
3+\alpha\log 6-\log(8+2^{\alpha})}.$ (53)

This upper bound can be numerically minimized over $\alpha$, for any
$\epsilon$. The case $R_{10}$ is shown in Fig. 1 and numerical evidence shows
it is close to the optimal bound from majorization.

## Discussion

We have described how relative majorization can be used to establish upper
bounds on magic distillation protocols that take into account additional
physics of the system. Our bounds exploited relatively simple aspects of the
Lorenz curves of the quasi-distributions, so it would be of interest to
sharpen these bounds and obtain a better handle on the Lorenz curve structure
in the $n\rightarrow\infty$ limit. It would also be of interest to analyse
what features of single-shot entropies can be extended to quasi-distributions
in a sensible form renner_phd ; tomamichel2013 . This raises interesting
questions as we no longer have a notion of typicality and the central limit
theorem does not apply. That said, for special states such as the Strange
state, the asymptotic behaviour is relatively simple, so exact asymptotics for
this should be possible. In Supplementary Notes 2, 3 and 4 we give additional
results that may be of use for later work.

Because Theorem 3 provides a completeness statement for majorization, we can
ask if this approach could additionally provide lower bounds for distillation
protocols. In contrast to the upper bounds, it is now essential to include the
symplectic constraints on the phase space in the majorization relations. In
principle the resultant majorization (and relative majorization) constraints
should provide exact specifications on what can be obtained via free
operations. Interestingly, this route does not require the use of error
correcting codes, but instead is built on group-theoretic features of the
phase space. The Clifford group action on a quantum system corresponds to the
action of the affine symplectic group $G\coloneqq
SL(2,\mathbb{Z}_{d})\ltimes\mathbb{Z}_{d}^{2}$ on the discrete phase space
Appleby_2005 ; Bengtsson_2006 . Therefore, any convex mixture of Clifford
unitaries will correspond to a convex mixture of these group actions. The
Hardy-Littlewood theorem hardy_1952  tells us that majorization is obtained
from convex mixtures of arbitrary permutations, a result that has also been
generalized to convex mixtures of elements of a general group $G$.
$G$–majorization has been studied in the classical literature and a range of
results are known about it giovagnoli_1985 ; steerneman_1990 ;
giovagnoli_cyclic_1996 ; eaton_1977 . One can therefore consider sub-groups of
the affine symplectic group and analyse the resulting majorization/relative
majorization that follows from its action on quasi-distributions. However if
the sub-group considered is too simple then it may lead to trivial
distillation rates. An example of this is when we restrict to the set of Weyl
covariant channels fukuda2006 , which are represented by a convex mixture of
displacements on the phase space. Initial work by the authors have shown that
the majorization conditions for Weyl covariant channels can be solved exactly
in terms of discrete Fourier transforms, however it is found that the
resultant distillation rates are trivial.

One notable structure that could be exploited for $G$–majorization is when $G$
is a finite reflection group. For this, the $G$–majorization pre-order is
guaranteed to be described by a finite list of conditions giovagnoli_1985 ,
just as is the case for the relative majorization ordering. Therefore one
route to concrete lower bounds on distillation rates is to consider stochastic
maps obtained by reflection sub-groups of
$SL(2,\mathbb{Z}_{d})\ltimes\mathbb{Z}_{2}$, or other simple sub-groups, for
which the majorization relation reduces to a finite set of conditions. Beyond
this, another promising route is to formulate the majorization conditions for
a sub-set of operations as a semi-definite program and then attempt to
estimate realisable distillation rates by exploiting dualities and appropriate
coarse-grainings. This approach has recently been applied by one of the
authors to estimate the set of quantum states accessible by covariant channels
in the resource theory of asymmetry alexander2021  where one allows
depolarizing noise that is then minimized. It would be of interest to see if a
similar approach could be followed for magic state distillation.

Beyond magic state distillation, the topic of $G$–majorization has been
extensively studied, but to our knowledge there has not been work on relative
$G$–majorization. This would correspond to transformations that are not
unital. Physically, this regime would correspond to a form of thermo-
majorization obtained from looking at the action of the Clifford group at a
micro-canonical level, and then reducing to a small subsystem Pathria_1997 .
While this seems like a painful thing to consider, there is motivation for
this beyond the aim of magic protocols: in the case of classical statistical
mechanics on a phase space this is precisely the situation, albeit in the
continuum limit. Statistical mechanics of actual systems obey Hamiltonian
dynamics, thus they automatically respect a symplectic form Arnold_2000 ;
Pathria_1997 . Therefore, the pre-order of statistical mechanical states with
respect to phase space dynamics preserving a Gibbs state must correspond to a
symplectic majorization condition. Of course, technical features arise in the
continuum limit when considering distributions on an unbounded phase space,
but this could be remedied by either considering a compact phase space (e.g.
for a particle on a ring) or by first studying the discrete case. Such
scenarios also arise in the Quantum Hall Effect Klitzing_1980 , which is
another regime where these techniques could be of use.

Lastly, we note that it would be of interest to consider the possibility of
applying the approach taken here to other scenarios in which one wishes to
distinguish classical from non-classical behaviour based on quasi-probability
representations Ferrie_2008 ; barnett_1997 ; Allahverdyan_2018 ;
arvidsson_2020 ; halpern_2018 ; Lostaglio_2018 ; Levy_2020 .

## Methods

### Mathematical methods used for proofs

In order to derive our results we make use of three broad subfields.

The first one is the phase space representation of quantum systems as
described in the first section of our Results. Generally, there are various
possible representations Ferrie_2008 , but we make use of the distinguished
Wigner representation Gross2006  for odd-dimensional systems.

We then develop a theory of majorization cit:marshall ; Veinott_1971 ;
Ruch_1976  for quasi-distributions so as to impose a partial order between
magic states.

Finally, we extend the theory of classical Rényi entropies and divergences
renyi_1960  for probability distributions to quasi-distributions and show that
they possess sufficient properties for the derivation of our entropic bounds.

Precise definitions of all the above concepts and explanations of their
incorporation to the study of magic distillation are provided in the Results
section as the concepts become relevant.

## Data availability

No datasets were generated or analysed during the current study.

## Code availability

The code used to produce the figures in this work is available from the
corresponding author upon reasonable request.

## Acknowledgements

We thank Earl Campbell, Matteo Lostaglio, Philippe Faist and Nicole Yunger
Halpern for helpful comments and discussions. NK is supported by the EPSRC
Centre for Doctoral Training in Controlled Quantum Dynamics. DJ is supported
by the Royal Society and a University Academic Fellowship.

## Competing interests

The authors declare no competing interests.

## Author contributions

DJ proposed the initial idea. Derivations were done by NK. Analysis and
writing of the manuscript were jointly done by NK and DJ.

## References

  * (1) Campbell, E. T., Terhal, B. M. & Vuillot, C.  Roads towards fault-tolerant universal quantum computation.  _Nature_ 549, 172–179 (2017).  https://www.nature.com/articles/nature23460.
  * (2) Raussendorf , R. & Briegel, H. J.  A one-way quantum computer.  _Phys. Rev. Lett._ 86, 5188–5191 (2001).  https://link.aps.org/doi/10.1103/PhysRevLett.86.5188.
  * (3) Raussendorf, R., Browne, D. E. & Briegel, H. J.  Measurement-based quantum computation on cluster states.  _Phys. Rev. A_ 68, 022312 (2003).  https://link.aps.org/doi/10.1103/PhysRevA.68.022312.
  * (4) Nickerson, N. H., Fitzsimons, J. F. & Benjamin, S. C.  Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links.  _Phys. Rev. X_ 4, 041041 (2014).  https://link.aps.org/doi/10.1103/PhysRevX.4.041041.
  * (5) Nikahd, E., Sedighi, M. & Saheb Zamani, M.  Nonuniform code concatenation for universal fault-tolerant quantum computing.  _Phys. Rev. A_ 96, 032337 (2017).  https://link.aps.org/doi/10.1103/PhysRevA.96.032337.
  * (6) Chao, R. & Reichardt, B. W.  Fault-tolerant quantum computation with few qubits.  _npj Quantum Inf._ 4, 1–8 (2018).  https://www.nature.com/articles/s41534-018-0085-z.
  * (7) Lin, C., Yang, G., Luo, Q. & Li, X.  Pieceable fault tolerant conversion between 5-qubit code and 7-CSS code.  _Quantum Inf. Process._ 19, 243 (2020).  https://doi.org/10.1007/s11128-020-02740-3.
  * (8) Lin, C. & Yang, G.  Concatenated pieceable fault-tolerant scheme for universal quantum computation.  _Phys. Rev. A_ 102, 052415 (2020).  https://link.aps.org/doi/10.1103/PhysRevA.102.052415.
  * (9) Chamberland, C. _et al._ Building a fault-tolerant quantum computer using concatenated cat codes.  _PRX Quantum_ 3, 010329 (2022).  https://link.aps.org/doi/10.1103/PRXQuantum.3.010329.
  * (10) Bourassa, J. E. _et al._ Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer.  _Quantum_ 5, 392 (2021).  https://doi.org/10.22331/q-2021-02-04-392.
  * (11) Bravyi, S. B. & Kitaev, A. Y.  Quantum codes on a lattice with boundary (1998).  https://arxiv.org/abs/quant-ph/9811052.
  * (12) Freedman, M. H.  Quantum computation and the localization of modular functors.  _Found. Comut. Math._ 1, 183–204 (2001).  https://doi.org/10.1007/s102080010006.
  * (13) Dennis, E., Kitaev, A., Landahl, A. & Preskill, J.  Topological quantum memory.  _J. Math. Phys._ 43, 4452–4505 (2002).  https://aip.scitation.org/doi/10.1063/1.1499754.
  * (14) Raussendorf, R. & Harrington, J.  Fault-tolerant quantum computation with high threshold in two dimensions.  _Phys. Rev. Lett._ 98, 190504 (2007).  https://link.aps.org/doi/10.1103/PhysRevLett.98.190504.
  * (15) Eastin, B. & Knill, E.  Restrictions on transversal encoded quantum gate sets.  _Phys. Rev. Lett._ 102, 110502 (2009).  https://link.aps.org/doi/10.1103/PhysRevLett.102.110502.
  * (16) Calderbank, A. R. & Shor, P. W.  Good quantum error-correcting codes exist.  _Phys. Rev. A_ 54, 1098–1105 (1996).  https://link.aps.org/doi/10.1103/PhysRevA.54.1098.
  * (17) Steane, A.  Multiple-particle interference and quantum error correction.  _Proc. Math. Phys. Eng. Sci._ 452, 2551–2577 (1996).  https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1996.0136.
  * (18) Bravyi , S. & Kitaev, A.  Universal quantum computation with ideal clifford gates and noisy ancillas.  _Phys. Rev. A_ 71, 022316 (2005).  https://link.aps.org/doi/10.1103/PhysRevA.71.022316.
  * (19) Jones, C.  Multilevel distillation of magic states for quantum computing.  _Phys. Rev. A_ 87, 042305 (2013).  https://link.aps.org/doi/10.1103/PhysRevA.87.042305.
  * (20) O’Gorman, J. & Campbell, E. T.  Quantum computation with realistic magic-state factories.  _Phys. Rev. A_ 95, 032338 (2017).  https://link.aps.org/doi/10.1103/PhysRevA.95.032338.
  * (21) Bravyi, S. & Haah, J.  Magic-state distillation with low overhead.  _Phys. Rev. A_ 86, 052329 (2012).  https://link.aps.org/doi/10.1103/PhysRevA.86.052329.
  * (22) Haah, J., Hastings, M. B., Poulin, D. & Wecker, D.  Magic state distillation with low space overhead and optimal asymptotic input count.  _Quantum_ 1, 31 (2017).  https://doi.org/10.22331/q-2017-10-03-31.
  * (23) Hastings, M. B. & Haah, J.  Distillation with sublogarithmic overhead.  _Phys. Rev. Lett._ 120, 050504 (2018).  https://link.aps.org/doi/10.1103/PhysRevLett.120.050504.
  * (24) Litinski, D.  Magic State Distillation: Not as Costly as You Think.  _Quantum_ 3, 205 (2019).  https://doi.org/10.22331/q-2019-12-02-205.
  * (25) Campbell, E. T., Anwar, H. & Browne, D. E.  Magic-state distillation in all prime dimensions using quantum reed-muller codes.  _Phys. Rev. X_ 2, 041021 (2012).  https://link.aps.org/doi/10.1103/PhysRevX.2.041021.
  * (26) Anwar, H., Campbell, E. T. & Browne, D. E.  Qutrit magic state distillation.  _New J. Phys._ 14, 063006 (2012).  https://doi.org/10.1088/1367-2630/14/6/063006.
  * (27) Dawkins, H. & Howard, M.  Qutrit magic state distillation tight in some directions.  _Phys. Rev. Lett._ 115, 030501 (2015).  https://link.aps.org/doi/10.1103/PhysRevLett.115.030501.
  * (28) Krishna, A. & Tillich, J.-P.  Towards low overhead magic state distillation.  _Phys. Rev. Lett._ 123, 070507 (2019).  https://link.aps.org/doi/10.1103/PhysRevLett.123.070507.
  * (29) Prakash, S.  Magic state distillation with the ternary golay code.  _Proc. Math. Phys. Eng. Sci._ 476, 20200187 (2020).  https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2020.0187.
  * (30) Chamberland, C. & Noh, K.  Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits.  _npj Quantum Inf._ 6, 91 (2020).  https://doi.org/10.1038/s41534-020-00319-5.
  * (31) Chamberland, C. & Cross, A. W.  Fault-tolerant magic state preparation with flag qubits.  _Quantum_ 3, 143 (2019).  https://doi.org/10.22331/q-2019-05-20-143.
  * (32) Campbell, E. T. & Howard, M.  Unifying gate synthesis and magic state distillation.  _Phys. Rev. Lett._ 118, 060501 (2017).  https://link.aps.org/doi/10.1103/PhysRevLett.118.060501.
  * (33) Campbell, E. T. & Howard, M.  Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost.  _Phys. Rev. A_ 95, 022316 (2017).  https://link.aps.org/doi/10.1103/PhysRevA.95.022316.
  * (34) Prakash, S., Jain, A., Kapur, B. & Seth, S.  Normal form for single-qutrit clifford+$t$ operators and synthesis of single-qutrit gates.  _Phys. Rev. A_ 98, 032304 (2018).  https://link.aps.org/doi/10.1103/PhysRevA.98.032304.
  * (35) Seddon, J. R., Regula, B., Pashayan, H., Ouyang, Y. & Campbell, E. T.  Quantifying quantum speedups: Improved classical simulation from tighter magic monotones.  _PRX Quantum_ 2, 010345 (2021).  https://link.aps.org/doi/10.1103/PRXQuantum.2.010345.
  * (36) Leone, L., Oliviero, S. F. E. & Hamma, A.  Rényi entropy of magic.  _Phys. Rev. Lett._ 128, 050402 (2022).  https://link.aps.org/doi/10.1103/PhysRevLett.128.050402.
  * (37) Chitambar, E. & Gour, G.  Quantum resource theories.  _Rev. Mod. Phys._ 91, 025001 (2019).  https://link.aps.org/doi/10.1103/RevModPhys.91.025001.
  * (38) Ahmadi, M., Dang, H. B., Gour, G. & Sanders, B. C.  Quantification and manipulation of magic states.  _Phys. Rev. A_ 97, 062332 (2018).  https://link.aps.org/doi/10.1103/PhysRevA.97.062332.
  * (39) Seddon, J. R. & Campbell, E. T.  Quantifying magic for multi-qubit operations.  _Proc. Math. Phys. Eng. Sci._ 475, 20190251 (2019).  https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2019.0251.
  * (40) Wang, X., Wilde, M. M. & Su, Y.  Quantifying the magic of quantum channels.  _New J. Phys._ 21, 103002 (2019).  https://doi.org/10.1088/1367-2630/ab451d.
  * (41) Sarkar, S., Mukhopadhyay, C. & Bayat, A.  Characterization of an operational quantum resource in a critical many-body system.  _New J. Phys._ 22, 083077 (2020).  https://doi.org/10.1088/1367-2630/aba919.
  * (42) White, C. D., Cao, C. & Swingle, B.  Conformal field theories are magical.  _Phys. Rev. B_ 103, 075145 (2021).  https://link.aps.org/doi/10.1103/PhysRevB.103.075145.
  * (43) Bermejo-Vega, J., Delfosse, N., Browne, D. E., Okay, C. & Raussendorf, R.  Contextuality as a resource for models of quantum computation with qubits.  _Phys. Rev. Lett._ 119, 120505 (2017).  https://link.aps.org/doi/10.1103/PhysRevLett.119.120505.
  * (44) Howard, M., Wallman, J., Veitch, V. & Emerson, J.  Contextuality supplies the ‘magic’ for quantum computation.  _Nature_ 510, 351–355 (2014).  https://doi.org/10.1038/nature13460.
  * (45) Zurel, M., Okay, C. & Raussendorf, R.  Hidden variable model for universal quantum computation with magic states on qubits.  _Phys. Rev. Lett._ 125, 260404 (2020).  https://link.aps.org/doi/10.1103/PhysRevLett.125.260404.
  * (46) Okay, C., Zurel, M. & Raussendorf, R.  On the extremal points of the $\Lambda$-polytopes and classical simulation of quantum computation with magic states (2021).  https://arxiv.org/abs/2104.05822.
  * (47) Schmid, D., Selby, J. H., Pusey, M. F. & Spekkens, R. W.  A structure theorem for generalized-noncontextual ontological models (2020).  https://arxiv.org/abs/2005.07161.
  * (48) Schmid, D., Selby, J. H., Wolfe, E., Kunjwal, R. & Spekkens, R. W.  Characterization of noncontextuality in the framework of generalized probabilistic theories.  _PRX Quantum_ 2, 010331 (2021).  https://link.aps.org/doi/10.1103/PRXQuantum.2.010331.
  * (49) Jones, N. C. _et al._ Layered architecture for quantum computing.  _Phys. Rev. X_ 2, 031007 (2012).  https://link.aps.org/doi/10.1103/PhysRevX.2.031007.
  * (50) Li, Y. & Benjamin, S. C.  Efficient variational quantum simulator incorporating active error minimization.  _Phys. Rev. X_ 7, 021050 (2017).  https://link.aps.org/doi/10.1103/PhysRevX.7.021050.
  * (51) Temme, K., Bravyi, S. & Gambetta, J. M.  Error mitigation for short-depth quantum circuits.  _Phys. Rev. Lett._ 119, 180509 (2017).  https://link.aps.org/doi/10.1103/PhysRevLett.119.180509.
  * (52) Endo, S., Benjamin, S. C. & Li, Y.  Practical quantum error mitigation for near-future applications.  _Phys. Rev. X_ 8, 031027 (2018).  https://link.aps.org/doi/10.1103/PhysRevX.8.031027.
  * (53) McClean, J. R., Kimchi-Schwartz, M. E., Carter, J. & de Jong, W. A.  Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states.  _Phys. Rev. A_ 95, 042308 (2017).  https://link.aps.org/doi/10.1103/PhysRevA.95.042308.
  * (54) Kandala, A. _et al._ Error mitigation extends the computational reach of a noisy quantum processor.  _Nature_ 567, 491–495 (2019).  https://www.nature.com/articles/s41586-019-1040-7.
  * (55) Holmes, A. _et al._ Resource optimized quantum architectures for surface code implementations of magic-state distillation.  _Microprocess. Microsyst._ 67, 56–70 (2019).  https://www.sciencedirect.com/science/article/pii/S0141933118303314.
  * (56) Colless, J. I. _et al._ Computation of molecular spectra on a quantum processor with an error-resilient algorithm.  _Phys. Rev. X_ 8, 011021 (2018).  https://link.aps.org/doi/10.1103/PhysRevX.8.011021.
  * (57) Song, C. _et al._ Quantum computation with universal error mitigation on superconducting quantum processor.  _Sci. Adv._ 5, eaaw5686 (2018).  https://www.science.org/doi/10.1126/sciadv.aaw5686.
  * (58) Bravyi, S., Sheldon, S., Kandala, A., Mckay, D. C. & Gambetta, J. M.  Mitigating measurement errors in multiqubit experiments.  _Phys. Rev. A_ 103, 042605 (2021).  https://link.aps.org/doi/10.1103/PhysRevA.103.042605.
  * (59) Bonilla Ataides, J. P., Tuckett, D. K., Bartlett, S. D., Flammia, S. T. & Brown, B. J.  The XZZX surface code.  _Nat. Commun._ 12, 2172 (2021).  https://www.nature.com/articles/s41467-021-22274-1.
  * (60) Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K.  Mixed-state entanglement and quantum error correction.  _Phys. Rev. A_ 54, 3824–3851 (1996).  https://link.aps.org/doi/10.1103/PhysRevA.54.3824.
  * (61) Tuckett, D. K. _et al._ Tailoring surface codes for highly biased noise.  _Phys. Rev. X_ 9, 041031 (2019).  https://link.aps.org/doi/10.1103/PhysRevX.9.041031.
  * (62) Aliferis, P. & Preskill, J.  Fault-tolerant quantum computation against biased noise.  _Phys. Rev. A_ 78, 052331 (2008).  https://link.aps.org/doi/10.1103/PhysRevA.78.052331.
  * (63) Stephens, A. M., Munro, W. J. & Nemoto, K.  High-threshold topological quantum error correction against biased noise.  _Phys. Rev. A_ 88, 060301 (2013).  https://link.aps.org/doi/10.1103/PhysRevA.88.060301.
  * (64) Li, Y.  A magic state’s fidelity can be superior to the operations that created it.  _New J. Phys._ 17, 023037 (2015).  https://doi.org/10.1088/1367-2630/17/2/023037.
  * (65) Babbush, R. _et al._ Encoding electronic spectra in quantum circuits with linear t complexity.  _Phys. Rev. X_ 8, 041015 (2018).  https://link.aps.org/doi/10.1103/PhysRevX.8.041015.
  * (66) Guillaud, J. & Mirrahimi, M.  Repetition cat qubits for fault-tolerant quantum computation.  _Phys. Rev. X_ 9, 041053 (2019).  https://link.aps.org/doi/10.1103/PhysRevX.9.041053.
  * (67) Fowler, A. G. & Gidney, C.  Low overhead quantum computation using lattice surgery (2019).  https://arxiv.org/abs/1808.06709.
  * (68) Janzing, D., Wocjan, P., Zeier, R., Geiss, R. & Beth, T.  Thermodynamic cost of reliability and low temperatures: Tightening landauer’s principle and the second law.  _Int. J. Theor. Phys._ 39, 2717–2753 (2000).  https://doi.org/10.1023/A:1026422630734.
  * (69) Horodecki, M. & Oppenheim, J.  Fundamental limitations for quantum and nanoscale thermodynamics.  _Nat. Commun._ 4, 2059 (2013).  https://doi.org/10.1038/ncomms3059.
  * (70) Brandão, F., Horodecki, M., Ng, N., Oppenheim, J. & Wehner, S.  The second laws of quantum thermodynamics.  _PNAS_ 112, 3275–3279 (2015).  https://www.pnas.org/content/112/11/3275.
  * (71) Vinjanampathy, S. & Anders, J.  Quantum thermodynamics.  _Contemp. Phys._ 57, 545–579 (2016).  https://doi.org/10.1080/00107514.2016.1201896.
  * (72) Goold, J., Huber, M., Riera, A., del Rio, L. & Skrzypczyk, P.  The role of quantum information in thermodynamics—a topical review.  _J. Phys. A Math._ 49, 143001 (2016).  https://doi.org/10.1088/1751-8113/49/14/143001.
  * (73) Lostaglio, M.  An introductory review of the resource theory approach to thermodynamics.  _Reports on Progress in Physics_ 82, 114001 (2019).  https://doi.org/10.1088/1361-6633/ab46e5.
  * (74) Gour, G., Jennings, D., Buscemi, F., Duan, R. & Marvian, I.  Quantum majorization and a complete set of entropic conditions for quantum thermodynamics 9, 5352 (2018).  https://doi.org/10.1038/s41467-018-06261-7.
  * (75) Ferrie, C. & Emerson, J.  Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations.  _J. Phys. A Math._ 41, 352001 (2008).  https://doi.org/10.1088/1751-8113/41/35/352001.
  * (76) Veitch, V., Ferrie, C., Gross, D. & Emerson, J.  Negative quasi-probability as a resource for quantum computation.  _New J. Phys._ 14, 113011 (2012).  https://doi.org/10.1088/1367-2630/14/11/113011.
  * (77) Mari , A. & Eisert, J.  Positive Wigner functions render classical simulation of quantum computation efficient.  _Phys. Rev. Lett._ 109, 230503 (2012).  https://link.aps.org/doi/10.1103/PhysRevLett.109.230503.
  * (78) Gottesman, D.  Stabilizer codes and quantum error correction.  Ph.D. thesis, California Institute of Technology (1997).  https://arxiv.org/abs/quant-ph/9705052.
  * (79) Knill, E.  Quantum computing with realistically noisy devices.  _Nature_ 434, 39–44 (2005).  https://doi.org/10.1038/nature03350.
  * (80) Campbell, E. T.  Catalysis and activation of magic states in fault-tolerant architectures.  _Phys. Rev. A_ 83, 032317 (2011).  https://link.aps.org/doi/10.1103/PhysRevA.83.032317.
  * (81) Marshall, A. W., Olkin, I. & Arnold, B. C.  Inequalities: Theory of Majorization and Its Applications (Springer, 2011).
  * (82) Veinott, A. F.  Least d-majorized network flows with inventory and statistical applications.  _Management Science_ 17, 547–567 (1971).  https://doi.org/10.1287/mnsc.17.9.547.
  * (83) Ruch, E. & Mead, A.  The principle of increasing mixing character and some of its consequences.  _Theoretica chimica acta_ 41, 95–117 (1976).  https://doi.org/10.1007/BF01178071.
  * (84) Fine, A.  Hidden variables, joint probability, and the bell inequalities.  _Phys. Rev. Lett._ 48, 291–295 (1982).  https://link.aps.org/doi/10.1103/PhysRevLett.48.291.
  * (85) Allahverdyan, A. E. & Danageozian, A.  Excluding joint probabilities from quantum theory.  _Phys. Rev. A_ 97, 030102 (2018).  https://link.aps.org/doi/10.1103/PhysRevA.97.030102.
  * (86) Arvidsson-Shukur, D. R. M. _et al._ Quantum advantage in postselected metrology.  _Nat. Commun._ 11, 3775 (2020).  https://www.nature.com/articles/s41467-020-17559-w.
  * (87) Yunger Halpern, N., Swingle, B. & Dressel, J.  Quasiprobability behind the out-of-time-ordered correlator.  _Phys. Rev. A_ 97, 042105 (2018).  https://link.aps.org/doi/10.1103/PhysRevA.97.042105.
  * (88) Lostaglio, M.  Quantum fluctuation theorems, contextuality, and work quasiprobabilities.  _Phys. Rev. Lett._ 120, 040602 (2018).  https://link.aps.org/doi/10.1103/PhysRevLett.120.040602.
  * (89) Levy, A. & Lostaglio, M.  Quasiprobability distribution for heat fluctuations in the quantum regime.  _PRX Quantum_ 1, 010309 (2020).  https://link.aps.org/doi/10.1103/PRXQuantum.1.010309.
  * (90) Veitch, V., Mousavian, S. A. H., Gottesman, D. & Emerson, J.  The resource theory of stabilizer quantum computation.  _New J. Phys._ 16, 013009 (2014).  https://doi.org/10.1088/1367-2630/16/1/013009.
  * (91) Wang, X., Wilde, M. M. & Su, Y.  Efficiently computable bounds for magic state distillation.  _Phys. Rev. Lett._ 124, 090505 (2020).  https://link.aps.org/doi/10.1103/PhysRevLett.124.090505.
  * (92) Gross, D.  Hudson’s theorem for finite-dimensional quantum systems.  _J. Math. Phys._ 47, 122107 (2006).  https://aip.scitation.org/doi/10.1063/1.2393152.
  * (93) Wigner, E.  On the quantum correction for thermodynamic equilibrium.  _Phys. Rev._ 40, 749–759 (1932).  https://link.aps.org/doi/10.1103/PhysRev.40.749.
  * (94) Vourdas, A.  Quantum systems with finite hilbert space.  _Reports on Progress in Physics_ 67, 267–320 (2004).  https://doi.org/10.1088/0034-4885/67/3/r03.
  * (95) Kenfack, A. & Życzkowski, K.  Negativity of the Wigner function as an indicator of non-classicality.  _J. opt. B, Quantum semiclass. opt._ 6, 396–404 (2004).  https://doi.org/10.1088/1464-4266/6/10/003.
  * (96) Appleby, D. M.  Symmetric informationally complete–positive operator valued measures and the extended clifford group.  _J. Math. Phys._ 46, 052107 (2005).  https://doi.org/10.1063/1.1896384.
  * (97) Mermin, N. D.  Simple unified form for the major no-hidden-variables theorems.  _Phys. Rev. Lett._ 65, 3373–3376 (1990).  https://link.aps.org/doi/10.1103/PhysRevLett.65.3373.
  * (98) Raussendorf, R., Bermejo-Vega, J., Tyhurst, E., Okay, C. & Zurel, M.  Phase-space-simulation method for quantum computation with magic states on qubits.  _Phys. Rev. A_ 101, 012350 (2020).  https://link.aps.org/doi/10.1103/PhysRevA.101.012350.
  * (99) Folland, G. B.  Harmonic Analysis in Phase Space (Princeton University Press, 1989).  https://www.jstor.org/stable/j.ctt1b9rzs2.
  * (100) Bengtsson, I. & Życzkowski, K.  Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, 2006).
  * (101) Schmid, D., Du, H., Selby, J. H. & Pusey, M. F.  The only noncontextual model of the stabilizer subtheory is Gross’s (2021).  https://arxiv.org/abs/2101.06263.
  * (102) Watrous, J.  The Theory of Quantum Information (Cambridge University Press, 2018).
  * (103) Blackwell, D.  Equivalent Comparisons of Experiments.  _The Annals of Mathematical Statistics_ 24, 265 – 272 (1953).  https://doi.org/10.1214/aoms/1177729032.
  * (104) Nielsen, M. A.  Conditions for a class of entanglement transformations.  _Phys. Rev. Lett._ 83, 436–439 (1999).  https://link.aps.org/doi/10.1103/PhysRevLett.83.436.
  * (105) Ćwikliński, P., Studziński, M., Horodecki, M. & Oppenheim, J.  Limitations on the evolution of quantum coherences: Towards fully quantum second laws of thermodynamics.  _Phys. Rev. Lett._ 115, 210403 (2015).  https://link.aps.org/doi/10.1103/PhysRevLett.115.210403.
  * (106) Lostaglio, M., Jennings, D. & Rudolph, T.  Description of quantum coherence in thermodynamic processes requires constraints beyond free energy.  _Nat. Commun._ 6, 6383 (2015).  https://doi.org/10.1038/ncomms7383.
  * (107) Gour, G., Müller, M. P., Narasimhachar, V., Spekkens, R. W. & Yunger Halpern, N.  The resource theory of informational nonequilibrium in thermodynamics.  _Physics Reports_ 583, 1–58 (2015).  https://doi.org/10.1016/j.physrep.2015.04.003.
  * (108) Horodecki, M., Horodecki, P. & Oppenheim, J.  Reversible transformations from pure to mixed states and the unique measure of information.  _Phys. Rev. A_ 67, 062104 (2003).  https://link.aps.org/doi/10.1103/PhysRevA.67.062104.
  * (109) Puchała, Z., Rudnicki, Ł. & Życzkowski, K.  Majorization entropic uncertainty relations.  _J. Phys. A Math._ 46, 272002 (2013).  https://doi.org/10.1088/1751-8113/46/27/272002.
  * (110) Vallejos, R. O., de Melo, F. & Carlo, G. G.  Principle of majorization: Application to random quantum circuits.  _Phys. Rev. A_ 104, 012602 (2021).  https://link.aps.org/doi/10.1103/PhysRevA.104.012602.
  * (111) Ruch, E., Schranner, R. & Seligman, T. H.  The mixing distance.  _J. Chem. Phys._ 69, 386–392 (1978).  https://aip.scitation.org/doi/10.1063/1.436364.
  * (112) Renes, J. M.  Relative submajorization and its use in quantum resource theories.  _J. Math. Phys._ 57, 122202 (2016).  https://aip.scitation.org/doi/10.1063/1.4972295.
  * (113) Buscemi, F. & Gour, G.  Quantum relative lorenz curves.  _Phys. Rev. A_ 95, 012110 (2017).  https://link.aps.org/doi/10.1103/PhysRevA.95.012110.
  * (114) Rethinasamy, S. & Wilde, M. M.  Relative entropy and catalytic relative majorization.  _Phys. Rev. Research_ 2, 033455 (2020).  https://link.aps.org/doi/10.1103/PhysRevResearch.2.033455.
  * (115) Brouwer, L. E. J.  Über abbildung von mannigfaltigkeiten.  _Mathematische Annalen_ 71 (1911).  https://doi.org/10.1007/BF01456931.
  * (116) Jain, A. & Prakash, S.  Qutrit and ququint magic states.  _Phys. Rev. A_ 102, 042409 (2020).  https://link.aps.org/doi/10.1103/PhysRevA.102.042409.
  * (117) Hudson, R. L. & Moody, G. R.  Locally normal symmetric states and an analogue of de finetti’s theorem.  _Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete_ 33, 343–351 (1976).  https://doi.org/10.1007/BF00534784.
  * (118) Christandl, M., König, R., Mitchison, G. & Renner, R.  One-and-a-half quantum de finetti theorems.  _Commun. Math. Phys._ 273, 473–498 (2007).  https://doi.org/10.1007/s00220-007-0189-3.
  * (119) Lostaglio, M., Müller, M. P. & Pastena, M.  Stochastic independence as a resource in small-scale thermodynamics.  _Phys. Rev. Lett._ 115, 150402 (2015).  https://link.aps.org/doi/10.1103/PhysRevLett.115.150402.
  * (120) Müller, M. P. & Pastena, M.  A generalization of majorization that characterizes shannon entropy.  _IEEE Trans. Inf. Theory_ 62, 1711–1720 (2016).  https://ieeexplore.ieee.org/document/7404001.
  * (121) Boes, P., Eisert, J., Gallego, R., Müller, M. P. & Wilming, H.  Von neumann entropy from unitarity.  _Phys. Rev. Lett._ 122, 210402 (2019).  https://link.aps.org/doi/10.1103/PhysRevLett.122.210402.
  * (122) Pathria, R. K. & Beale, P. D.  Statistical Mechanics (Elsevier, 2011).  https://www.sciencedirect.com/book/9780123821881/statistical-mechanics.
  * (123) Bombin, H. _et al._ Interleaving: Modular architectures for fault-tolerant photonic quantum computing (2021).  https://arxiv.org/abs/2103.08612.
  * (124) Bogolubov, J., N. N.  On model dynamical systems in statistical mechanics.  _Physica_ 32, 933–944 (1966).  https://doi.org/10.1016/0031-8914(66)90024-3.
  * (125) Faist, P., Oppenheim, J. & Renner, R.  Gibbs-preserving maps outperform thermal operations in the quantum regime.  _New J. Phys._ 17, 043003 (2015).  https://iopscience.iop.org/article/10.1088/1367-2630/17/4/043003.
  * (126) Rényi, A.  On measures of entropy and information.  _Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability_ 1, 547–561 (1960).  https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s4_v1_article-27.pdf.
  * (127) Rio, L. d., Åberg, J., Renner, R., Dahlsten, O. & Vedral, V.  The thermodynamic meaning of negative entropy.  _Nature_ 474, 61–63 (2011).  https://www.nature.com/articles/nature10123.
  * (128) Baez, J. C.  Rényi entropy and free energy (2011).  https://arxiv.org/abs/1102.2098.
  * (129) Domb, C.  Phase transitions and critical phenomena (Elsevier, 2000).  https://www.elsevier.com/books/phase-transitions-and-critical-phenomena/domb/978-0-12-220318-3.
  * (130) Gour, G. & Tomamichel, M.  Entropy and relative entropy from information-theoretic principles.  _IEEE Trans. Inf. Theory_ 67, 6313-–6327 (2021).  http://dx.doi.org/10.1109/TIT.2021.3078337.
  * (131) Tomamichel, M.  A Framework for Non-Asymptotic Quantum Information Theory.  Ph.D. thesis, ETH Zurich (2012).  https://arxiv.org/abs/1203.2142.
  * (132) Renner, R.  Security of Quantum Key Distribution.  Ph.D. thesis, Swiss Federal Institute of Technology Zurich (2005).  https://arxiv.org/abs/quant-ph/0512258.
  * (133) Hardy, G. _et al._ Inequalities (Cambridge University Press, 1952).  https://books.google.gr/books?id=t1RCSP8YKt8C.
  * (134) Giovagnoli, A. & Wynn, H. P.  G-majorization with applications to matrix orderings.  _Linear Algebra Appl._ 67, 111–135 (1985).  https://www.sciencedirect.com/science/article/pii/0024379585901909.
  * (135) Steerneman, A. G. M.  G-majorization, group-induced cone orderings, and reflection groups.  _Linear Algebra Appl._ 127, 107–119 (1990).  https://www.sciencedirect.com/science/article/pii/002437959090338D.
  * (136) Giovagnoli, A. & Wynn, H. P.  Cyclic majorization and smoothing operators.  _Linear Algebra Appl._ 239, 215–225 (1996).  https://www.sciencedirect.com/science/article/pii/S0024379596900130.
  * (137) Eaton, M. L. & Perlman, M. D.  Reflection Groups, Generalized Schur Functions, and the Geometry of Majorization.  _The Annals of Probability_ 5, 829 – 860 (1977).  https://doi.org/10.1214/aop/1176995655.
  * (138) Fukuda, M. & Holevo, A. S.  On weyl-covariant channels (2006).  http://arxiv.org/abs/quant-ph/0510148.
  * (139) Alexander, R., Gvirtz-Chen, S. & Jennings, D.  Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system (2021).  https://arxiv.org/abs/2107.14181.
  * (140) Arnold, V. I.  Symplectic geometry and topology.  _J. Math. Phys._ 41, 3307–3343 (2000).  https://doi.org/10.1063/1.533315.
  * (141) Klitzing, K. v., Dorda, G. & Pepper, M.  New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance.  _Phys. Rev. Lett._ 45, 494–497 (1980).  https://link.aps.org/doi/10.1103/PhysRevLett.45.494.
  * (142) Barnett, S. M. & Radmore, P. M.  Methods in Theoretical Quantum Optics.  Oxford science publications (Clarendon Press, 1997).  https://books.google.gr/books?id=Hp5z09gPmC8C.

## Supplementary Note 1: Properties of Wigner representations

Here we present basic properties of the phase-point operators and the Wigner
distribution that are used throughout the main text.

######  Proposition 13.

For any dimension $d$, the phase-point operators satisfy:

  1. (i)

Hermiticity and unitarity:
$A_{\boldsymbol{z}}^{\dagger}=A_{\boldsymbol{z}}=A_{\boldsymbol{z}}^{-1}$;

  2. (ii)

Closure under transposition: $A_{(q,p)}^{T}=A_{(q,-p)}$;

  3. (iii)

Unit trace for odd $d$: ${\rm{tr}}[A_{\boldsymbol{z}}]=1$;

  4. (iv)

Completeness relation: $\sum_{\boldsymbol{z}\in{\cal
P}_{d}}A_{\boldsymbol{z}}=d\mathbbm{1}$;

  5. (i)

Orthogonality:
${\rm{tr}}[A_{\boldsymbol{z}}^{\dagger}A_{\boldsymbol{y}}]=d\delta_{\boldsymbol{z},\boldsymbol{y}}$.

All properties follow from the definition in Eq. (6) in the main text along
with properties of the displacement operators $D_{\boldsymbol{z}}$ and can be
found in the literature, e.g. app:veitch ; app:Vourdas_2004 ; app:Gross2006

######  Proposition 14.

The state Wigner distribution is

  1. (i)

Real valued: $W_{\rho}(\boldsymbol{z})\in\mathbb{R}^{d^{2}}$;

  2. (ii)

Normalized: $\sum_{\boldsymbol{z}\in{\cal P}_{d}}W_{\rho}(\boldsymbol{z})=1$;

  3. (iii)

Bounded: $\absolutevalue{W_{\rho}(\boldsymbol{z})}\leq\frac{1}{d}$.

  4. (iv)

Additive under mixing:
$W_{p\rho_{1}+(1-p)\rho_{2}}(\boldsymbol{z})=pW_{\rho_{1}}(\boldsymbol{z})+(1-p)W_{\rho_{2}}(\boldsymbol{z})$;

  5. (v)

Multiplicative under tensor products:
$W_{\rho_{A}\otimes\rho_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})=W_{\rho_{A}}(\boldsymbol{z}_{A})W_{\rho_{B}}(\boldsymbol{z}_{B})$.

###### Proof.

Proof of all properties can be found in the literature app:veitch ;
app:Vourdas_2004 ; app:Gross2006 ; app:Wang_2019  except for property (iii)
which we prove here.

Let $\\{\lambda_{i}\\}_{i\in\mathbb{Z}_{d}}$ be the (non-negative) eigenvalues
of $\rho$, summing to 1. Let
$\\{\alpha_{\boldsymbol{z},i}\\}_{i\in\mathbb{Z}_{d}}$ be the eigenvalues of
$A_{\boldsymbol{z}}$. For any
$\boldsymbol{z},\alpha_{\boldsymbol{z},i}\in\\{-1,1\\}$, due to the
hermiticity and unitarity of the phase-point operators. Then,

$\displaystyle\absolutevalue{W_{\rho}(\boldsymbol{z})}$
$\displaystyle=\frac{1}{d}\absolutevalue{{\rm{tr}}[A_{\boldsymbol{z}}\rho]}\leq\frac{1}{d}\absolutevalue{\sum_{i}\alpha_{\boldsymbol{z},i}\lambda_{i}}$
$\displaystyle\leq\frac{1}{d}\sum_{i}\lambda_{i}=\frac{1}{d}.$ (54)

The first inequality follows from Theorem 1 of app:mirsky  for the trace of
complex matrices, while the second is the Cauchy-Schwarz inequality. ∎

######  Proposition 15.

The Wigner distribution of a quantum channel ${\cal E}:{\cal B}({\cal
H}_{d_{A}})\rightarrow{\cal B}({\cal H}_{d_{B}})$ is

  1. (i)

Real-valued: $W_{{\cal
E}}(\boldsymbol{z}|\boldsymbol{y})\in\mathbb{R}^{d_{A}^{2}}\times\mathbb{R}^{d_{B}^{2}}$;

  2. (ii)

Normalized: $\sum_{\boldsymbol{y}\in{\cal
P}_{d_{\scalebox{0.9}{$\scriptscriptstyle B$}}}}W_{{\cal
E}}(\boldsymbol{z}|\boldsymbol{y})=1$
for any $\boldsymbol{z}\in{\cal P}_{d_{A}}$;

  3. (iii)

Bounded: $\absolutevalue{W_{{\cal
E}}(\boldsymbol{z}|\boldsymbol{y})}\leq\frac{d_{A}}{d_{B}}$;

  4. (iv)

$W_{{\cal E}(\rho)}(\boldsymbol{z})=\sum\limits_{\boldsymbol{y}\in{\cal
P}_{d_{A}}}W_{{\cal
E}}(\boldsymbol{z}|\boldsymbol{y})W_{\rho}(\boldsymbol{y})$ for any
$\boldsymbol{z}\in{\cal P}_{d_{B}}$.

###### Proof.

Proof of all properties are provided in app:Wang_2019  except for property
(iii) which is a direct consequence of the definition of $W_{{\cal E}}$ and
property (iii) in Proposition 14. ∎

Within the context of the Wigner representation, we define the rescaled quasi-
distribution

$W_{\rho|\tau}(\boldsymbol{z})\coloneqq\frac{W_{\rho}(\boldsymbol{z})}{W_{\tau}(\boldsymbol{z})},$
(55)

which is well-defined if $\tau$ is a full-rank stabilizer state.

The following result shows that the rescaled Lorenz curves behave in a natural
way under tensor products.

######  Proposition 16.

Let $\tau_{A},\tau_{B}$ be full rank stabilizer states on systems $A$ and $B$,
and let $\rho_{A},\rho_{B}$ be arbitrary states on $A,B$. Then, the rescaled
quasi-distribution obeys

$W_{\rho_{A}\otimes\rho_{B}|\tau_{A}\otimes\tau_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})=W_{\rho_{A}|\tau_{A}}(\boldsymbol{z}_{A})W_{\rho_{B}|\tau_{B}}(\boldsymbol{z}_{B})$
(56)

###### Proof.

This follows from the multiplicativity of the Wigner distribution,

$\displaystyle
W_{\rho_{A}\otimes\rho_{B}|\tau_{A}\otimes\tau_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})$
$\displaystyle=\frac{W_{\rho_{A}\otimes\rho_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})}{W_{\tau_{A}\otimes\tau_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})}$
$\displaystyle=\frac{W_{\rho_{A}}(\boldsymbol{z}_{A})W_{\rho_{B}}(\boldsymbol{z}_{B})}{W_{\tau_{A}}(\boldsymbol{z}_{A})W_{\tau_{B}}(\boldsymbol{z}_{B})}$
$\displaystyle=W_{\rho_{A}|\tau_{A}}(\boldsymbol{z}_{A})W_{\rho_{B}|\tau_{B}}(\boldsymbol{z}_{B}).$

∎

## Supplementary Note 2: Properties of relative majorization and Lorenz curves

### Relative majorization for quasi-distributions

We prove the following result connecting majorization to relative majorization
app:horodecki2013 ; app:Brandao_2015 ; app:lostaglio .

######  Proposition 17.

Let
$\boldsymbol{w}\in\mathbb{R}^{N},\boldsymbol{w}^{\prime}\in\mathbb{R}^{N^{\prime}}$
be quasi-distributions and
$\boldsymbol{r}\in\mathbb{R}^{N},\boldsymbol{r}^{\prime}\in\mathbb{R}^{N^{\prime}}$
probability distributions with positive rational entries given by
$r_{i}=a_{i}/K$ and $r_{i}^{\prime}=a_{i}^{\prime}/K$ for positive integers
$a_{i},a_{i}^{\prime}$ and
$K=\sum_{i=1}^{N}a_{i}=\sum_{i=1}^{N^{\prime}}a_{i}^{\prime}$. Then,

$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})\mbox{
if and only if
}\Gamma_{\boldsymbol{a}}(\boldsymbol{w})\succ\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{w}^{\prime}),$
(57)

where the embedding map
$\Gamma_{\boldsymbol{a}}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{K}$ is given by

$\Gamma_{\boldsymbol{a}}(\boldsymbol{z})\coloneqq\bigoplus_{i=1}^{N}z_{i}\boldsymbol{\eta}_{a_{i}},$
(58)

with $\boldsymbol{\eta}_{a_{i}}=(1/a_{i},1/a_{i},\dots,1/a_{i})$ the uniform
distribution on $a_{i}$ elements.

###### Proof.

We first note that the map $\Gamma_{\boldsymbol{a}}$ is stochastic and has a
well-defined left-inverse
$\Gamma^{-1}_{\boldsymbol{a}}:\mathbb{R}^{K}\rightarrow\mathbb{R}^{N}$ given
by

$\Gamma^{-1}_{\boldsymbol{a}}(\boldsymbol{x})=\left(\sum_{i=1}^{a_{1}}x_{i},\sum_{i=a_{1}+1}^{a_{1}+a_{2}}x_{i},\dots,\sum_{i=a_{1}+\dots+a_{N-1}+1}^{K}x_{i}\right),$
(59)

that obeys
$(\Gamma^{-1}_{\boldsymbol{a}}\circ\Gamma_{\boldsymbol{a}})(\boldsymbol{z})=\boldsymbol{z}$
for all $\boldsymbol{z}\in\mathbb{R}^{N}$.

The claim is equivalent to the statement that there exists a bistochastic map
$B$ sending $\Gamma_{\boldsymbol{a}}(\boldsymbol{w})$ to
$\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{w}^{\prime})$ if and only if
there exists a stochastic map $A$ sending $\boldsymbol{w}$ to
$\boldsymbol{w}^{\prime}$ and $\boldsymbol{r}$ to $\boldsymbol{r}^{\prime}$.

Suppose there is a stochastic map $A$ such that
$A\boldsymbol{w}=\boldsymbol{w}^{\prime}$ and
$A\boldsymbol{r}=\boldsymbol{r}^{\prime}$. We define
$B:\mathbb{R}^{K}\mapsto\mathbb{R}^{K}$ by
$B\coloneqq\Gamma_{\boldsymbol{a}^{\prime}}\circ
A\circ\Gamma_{\boldsymbol{a}}^{-1}$ so that it is stochastic as a composition
of stochastic maps and it preserves the uniform distribution in
$\mathbb{R}^{K}$, since

$\displaystyle
B(1/K,1/K,\dots,1/K)=\big{(}\Gamma_{\boldsymbol{a}^{\prime}}\circ
A\circ\Gamma_{\boldsymbol{a}}^{-1}\big{)}\big{(}\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{r})\big{)}=$
$\displaystyle\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{r}^{\prime})=(1/K,1/K,\dots,1/K),$
(60)

therefore $B$ is bistochastic. Finally, $B$ maps the embedded distributions as
follows,

$B\Gamma_{\boldsymbol{a}}(\boldsymbol{w})=\big{(}\Gamma_{\boldsymbol{a}^{\prime}}\circ
A\circ\Gamma_{\boldsymbol{a}}^{-1}\big{)}\big{(}\Gamma_{\boldsymbol{a}}(\boldsymbol{w})\big{)}=\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{w}^{\prime}).$
(61)

Conversely, suppose a bistochastic map $B$ exists sending
$\Gamma_{\boldsymbol{a}}(\boldsymbol{w})$ to
$\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{w}^{\prime})$. Again, define
$A:\mathbb{R}^{N}\mapsto\mathbb{R}^{N^{\prime}}$ by
$A\coloneqq\Gamma_{\boldsymbol{a}^{\prime}}^{-1}\circ
B\circ\Gamma_{\boldsymbol{a}}$ so that it is stochastic as a composition of
stochastic maps and
$A\boldsymbol{w}=\big{(}\Gamma_{\boldsymbol{a}^{\prime}}^{-1}\circ
B\circ\Gamma_{\boldsymbol{a}}\big{)}\boldsymbol{w}=\boldsymbol{w}^{\prime}$,
as well as $A\boldsymbol{r}=\boldsymbol{r}^{\prime}$. ∎

We note that the vectors $\boldsymbol{r},\boldsymbol{r}^{\prime}$ with
rational components form a dense subset of the positive probability
distributions, and do not consider further technicalities that have no impact
on actual physical measurements, which always have a finite resolution.

Restricting the above result in the context of proper probability
distributions, we can prove the Lorenz curve condition for probability
distributions. We define a Lorenz curve elbow as a point where the slope of
the curve changes, expressed explicitly in Eq. (15) for Lorenz curve
$L_{\boldsymbol{w}|\boldsymbol{r}}(x)$

######  Proposition 18.

Given probability distributions
$\boldsymbol{w},\boldsymbol{r}\in\mathbb{R}^{N}$ and
$\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime}\in\mathbb{R}^{N^{\prime}}$
with $\boldsymbol{r},\boldsymbol{r}^{\prime}$ having positive components, then

$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})\mbox{
if and only if }L_{\boldsymbol{w}|\boldsymbol{r}}(x)\geq
L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(x),$

for all $x\in[0,1]$.

###### Proof.

In the case of $N=N^{\prime}$ and
$\boldsymbol{r}=\boldsymbol{r}^{\prime}=\boldsymbol{\eta}$, where
$\boldsymbol{\eta}=(1/N,1/N,\dots,1/N)$ is the uniform distribution on $N$
elements, the statement reduces to the Lorenz curve condition
$L_{\boldsymbol{w}}(x)\geq L_{\boldsymbol{w}^{\prime}}(x)$ for all $x$, for
simple majorization which follows immediately from the defining set of
inequalities for majorization app:marshall . Namely, the Lorenz curve
$L_{\boldsymbol{w}}(x)$ for $\boldsymbol{w}$ is obtained from the partial sums
of $\boldsymbol{w}$ sorted in non-increasing order. It is also clear from the
definition that the Lorenz curve of $\boldsymbol{w}$ is given by
$L_{\boldsymbol{w}}(x)=L_{\boldsymbol{w}|\boldsymbol{\eta}}(x)$.

To prove the general statement we reduce the relative majorization Lorenz
curve condition to standard majorization. Using the notation and assumptions
of Proposition 17 for distributions with rational components, we can define
$\Gamma_{\boldsymbol{a}}(\boldsymbol{w}),\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{w}^{\prime})$
and the uniform distribution $\boldsymbol{\eta}=(1/K,1/K,\dots,1/K)$.

The key ingredient in the proof is that the Lorenz curve of $\boldsymbol{w}$
relative to $\boldsymbol{r}$ coincides with the Lorenz curve of
$\Gamma_{\boldsymbol{a}}(\boldsymbol{w})$, namely

$L_{\boldsymbol{w}|\boldsymbol{r}}(x)=L_{\Gamma_{\boldsymbol{a}}(\boldsymbol{w})|\boldsymbol{\eta}}(x)=L_{\Gamma_{\boldsymbol{a}}(\boldsymbol{w})}(x)\mbox{
for all }x\in[0,1].$ (62)

To see this we consider the elbows of $L_{\boldsymbol{w}|\boldsymbol{r}}(x)$,

$(x_{k},L_{\boldsymbol{w}|\boldsymbol{r}}(x_{k}))=\left(\sum_{i=1}^{k}r_{\pi(i)},\sum_{i=1}^{k}w_{\pi(i)}\right),$
(63)

where the permutation $\pi$ sorts $(w_{i}/r_{i})$ in non-increasing order.
Expressing $\Gamma_{\boldsymbol{a}}(\boldsymbol{w})$ as

$\Gamma_{\boldsymbol{a}}(\boldsymbol{w})=\frac{1}{K}\bigoplus_{i=1}^{N}\left(\frac{w_{i}}{r_{i}},\dots,\frac{w_{i}}{r_{i}}\right),$
(64)

where $(w_{i}/r_{i},\dots,w_{i}/r_{i})$ has $a_{i}$ elements, it is clear that
permutation $\pi$ sorts $\Gamma_{\boldsymbol{a}}(\boldsymbol{w})$ in non-
increasing order too. The Lorenz curve elbows
$(y_{k},L_{\Gamma_{\boldsymbol{a}}(\boldsymbol{w})|\boldsymbol{\eta}}(y_{k}))$
occur at

$y_{k}=\sum_{j=1}^{a_{\pi(1)}+\dots+a_{\pi(k)}}\frac{1}{K}=\sum_{i=1}^{k}\frac{a_{\pi(i)}}{K}=x_{k}$
(65)

and take values

$L_{\Gamma_{\boldsymbol{a}}(\boldsymbol{w})|\boldsymbol{\eta}}(y_{k})=\sum_{i=1}^{k}a_{\pi(i)}\frac{1}{K}\frac{w_{\pi(i)}}{r_{\pi(i)}}=L_{\boldsymbol{w}|\boldsymbol{r}}(x_{k}).$
(66)

Therefore, the Lorenz curve of the embedded distribution coincides with the
Lorenz curve in the relative majorization setting.

Finally, we have
$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})$
if and only if
$\Gamma_{\boldsymbol{a}}(\boldsymbol{w})\succ\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{w}^{\prime})$,
which holds if and only if
$L_{\Gamma_{\boldsymbol{a}}(\boldsymbol{w})|\boldsymbol{\eta}}(x)\geq
L_{\Gamma_{\boldsymbol{a}^{\prime}}(\boldsymbol{w}^{\prime})|\boldsymbol{\eta}}(x)$
for all $x\in[0,1]$, which in turn holds if and only if
$L_{\boldsymbol{w}|\boldsymbol{r}}(x)\geq
L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(x),\ x\in[0,1]$, which
concludes the proof. ∎

We now verify useful majorization properties in the context of quasi-
probability distributions, starting from the property that the Lorenz curve is
concave.

######  Proposition 19.

Let $\boldsymbol{w}$ be a quasi-distribution and let $\boldsymbol{r}$ be a
probability distribution with strictly non-zero components. Then
$L_{\boldsymbol{w}|\boldsymbol{r}}(x)$ is a concave function on $[0,1]$.

###### Proof.

Let $x_{\star}$ be the point where $L_{\boldsymbol{w}|\boldsymbol{r}}(x)$
first attains it maximum. Therefore, on $[0,x_{\star}]$ the function rises
monotonically to $L_{\boldsymbol{w}|\boldsymbol{r}}(x_{\star})$, via the sum
of all positive entries of $\boldsymbol{w}$, taken in decreasing order.
Likewise on $[x_{\star},1]$, the function decreases monotonically from its
maximum via the partial sums of the negative entries of $\boldsymbol{w}$ in
decreasing order. Let $f_{1}(x)$ be equal to
$L_{\boldsymbol{w}|\boldsymbol{r}}(x)$ on $[0,x_{\star}]$ and $0$ otherwise.
Also let $f_{2}(x)$ be equal to $L_{\boldsymbol{w}|\boldsymbol{r}}(x)$ on
$[x_{\star},1]$ and and $0$ otherwise. By inspection both $f_{1}$ and $f_{2}$
are concave functions, and
$f_{1}(x)+f_{2}(x)=L_{\boldsymbol{w}|\boldsymbol{r}}(x)$ for all $x\in[0,1]$.
However, the sum of two concave functions is also concave which concludes the
proof. ∎

The following result is used in the main text to extend the Lorenz curve
condition of Proposition 18 into the context of quasi-distributions.

######  Proposition 20.

Let $\boldsymbol{w}$ be a quasi-distribution and let $\boldsymbol{r}$ be a
probability distribution with strictly non-zero components. Then,
$L_{a\boldsymbol{w}+b\boldsymbol{r}|\boldsymbol{r}}(x)=aL_{\boldsymbol{w}|\boldsymbol{r}}(x)+bx$
for any constants $a>0$ and $b\in\mathbb{R}$.

###### Proof.

The Lorenz curve of $a\boldsymbol{w}+b\boldsymbol{r}$ relative to
$\boldsymbol{r}$ passes through $(0,0)$ and the points
$(\sum_{i=1}^{k}{r_{\pi(i)}},\sum_{i=1}^{k}(a\boldsymbol{w}+b\boldsymbol{r})_{\pi(i)})$
where $\pi$ is the permutation that puts $(aw_{i}/r_{i}+b)$ in non-increasing
order. Since $a>0$, the permutation $\pi$ puts $(w_{i}/r_{i})$ in non-
increasing order too. We thus have

$\displaystyle\left(\sum_{i=1}^{k}r_{\pi(i)},\sum_{i=1}^{k}(a\boldsymbol{w}+b\boldsymbol{r})_{\pi(i)}\right)=$
$\displaystyle\left(\sum_{i=1}^{k}r_{\pi(i)},a\sum_{i=1}^{k}w_{\pi(i)}+b\sum_{i=1}^{k}r_{\pi(i)}\right),$

so the value of the Lorenz curve at each potential elbow point
$x_{k}=\sum_{i=1}^{k}r_{\pi(i)}$ is given by

$\displaystyle
L_{a\boldsymbol{w}+b\boldsymbol{r}|\boldsymbol{r}}(x_{k})=aL_{\boldsymbol{w}|\boldsymbol{r}}(x_{k})+bL_{\boldsymbol{r}|\boldsymbol{r}}(x_{k})=$
$\displaystyle aL_{\boldsymbol{w}|\boldsymbol{r}}(x_{k})+bx_{k},$ (67)

so we have
$L_{a\boldsymbol{w}+b\boldsymbol{r}|\boldsymbol{r}}(x)=aL_{\boldsymbol{w}|\boldsymbol{r}}(x)+bx$
for any $x\in[0,1]$ due to linearity. ∎

The following provides equivalent formulations of relative majorization on
quasi-distributions.

######  Proposition 21.

Given quasi-distributions $\boldsymbol{w},\boldsymbol{w}^{\prime}$,
$\boldsymbol{r},\boldsymbol{r}^{\prime}$, such that the components of
$\boldsymbol{r}$ and $\boldsymbol{r}^{\prime}$ are positive, the following
statements are equivalent:

  1. (i)

$\boldsymbol{w}^{\prime}=A\boldsymbol{w}$ and
$\boldsymbol{r}^{\prime}=A\boldsymbol{r}$ for a stochastic map $A$;

  2. (ii)

$L_{\boldsymbol{w}|\boldsymbol{r}}(t)\geq
L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(t)$ for $t\in[0,1)$ and
$L_{\boldsymbol{w}|\boldsymbol{r}}(1)=L_{\boldsymbol{w}^{\prime}|\boldsymbol{r}^{\prime}}(1)$;

  3. (iii)

$\sum\limits_{i=1}^{n}\absolutevalue{w_{i}-r_{i}t}\geq\sum\limits_{i=1}^{n}\absolutevalue{w^{\prime}_{i}-r^{\prime}_{i}t}$
for all $t\in\mathbb{R}$.

###### Proof.

The proofs for these properties on proper probability distributions can be
found in app:marshall ; app:ruch_mixing_1978 ; app:Renes_2016 ;
app:Buscemi_2017  and references therein.

The equivalence between (i) and (ii) on quasi-distributions was proven in
Theorem 1 in the main text.

The equivalence between (i) and (iii) follows from a similar argument where we
mask negativity with a probability distribution. Namely, let $\epsilon>0$ be
such that
$\boldsymbol{w}(\epsilon)\coloneqq\epsilon\boldsymbol{w}+(1-\epsilon)\boldsymbol{r}$
and
$\boldsymbol{w}^{\prime}(\epsilon)\coloneqq\epsilon\boldsymbol{w}+(1-\epsilon)\boldsymbol{r}^{\prime}$
are genuine probability distributions. This is guaranteed by picking a
sufficiently small $\epsilon>0$, since $\boldsymbol{r}$ has positive
components. We now have that
$(\boldsymbol{w},\boldsymbol{r})\succ(\boldsymbol{w}^{\prime},\boldsymbol{r}^{\prime})$
if and only if
$(\boldsymbol{w}(\epsilon),\boldsymbol{r})\succ(\boldsymbol{w}^{\prime}(\epsilon),\boldsymbol{r}^{\prime})$.
Moreover, we have that for all $c\in\mathbb{R}$,
$\sum_{i=1}^{n}\absolutevalue{w(\epsilon)_{i}-r_{i}c}\geq\sum_{i=1}^{n}\absolutevalue{w^{\prime}(\epsilon)_{i}-r^{\prime}_{i}c}$,
which leads to

$\displaystyle\sum\limits_{i=1}^{n}\absolutevalue{w_{i}-r_{i}\frac{\epsilon+c-1}{\epsilon}}\geq\sum\limits_{i=1}^{n}\absolutevalue{w^{\prime}_{i}-r^{\prime}_{i}\frac{\epsilon+c-1}{\epsilon}}$

Replacing $t=(\epsilon+c-1)/\epsilon$, we see that $t$ can attain any real
value for $c\in\mathbb{R}$, so we deduce the required $L_{1}$–norm condition
on quasi-distributions $\boldsymbol{w},\boldsymbol{w}^{\prime}$. ∎

### Component-multiplicity pairs

In general, a $1$–copy $d$–dimensional state $\rho$ is described by its
$d^{2}$–dimensional Wigner distribution $W_{\rho}$. The distribution
$W_{\rho}$ is defined on the phase space, but it can be convenient to re-
express this using vector notation. We discuss this in terms of Wigner
distributions, but there is nothing to prevent the discussion from applying to
rescaled Wigner distributions as well.

To each Wigner distribution $W_{\rho}(\boldsymbol{z})$ we can associate
component-multiplicity pairs $\\{(w_{i},m_{i})\\}$ where the value $w_{i}$
occurs in the distribution $W_{\rho}(\boldsymbol{z})$ with multiplicity
$m_{i}$.

As an example, for the Strange state $\rho_{S}\coloneqq\rho_{S}(\epsilon=0)$
with

$W_{\rho_{S}}=(-1/3,1/6,1/6,1/6,1/6,1/6,1/6,1/6,1/6)$ (68)

we have the component-multiplicity pairs: $\\{(-1/3,1),(1/6,8)\\}$. However,
we might also wish more freedom and not require that the different $w_{i}$
values are all distinct. For example, the component-multiplity pairs
$\\{(-1/3,1),(1/6,2),(1/6,3),(1/6,3)\\}$ also describe $W_{\rho_{S}}$.

This representation is more compact when a Wigner distribution has a lot of
multiplicities, and allows for simple handling of multiple copies via the
following fact.

######  Proposition 22.

Consider two Wigner distributions
$W_{\rho_{A}}(\boldsymbol{z}_{A}),W_{\rho_{B}}(\boldsymbol{z}_{B})$ with
component-multiplicity pairs

$\\{(w_{i},m_{i})\\}\text{ and }\\{(w_{j}^{\prime},m_{j}^{\prime})\\},$ (69)

respectively. Then, $\\{(w_{i}w_{j}^{\prime},m_{i}m_{j}^{\prime})\\}$ gives
component-multiplicity pairs for
$W_{\rho_{A}\otimes\rho_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})$.

###### Proof.

This result is true because all components of
$W_{\rho_{A}\otimes\rho_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})$ are
of the form $w_{i}w_{j}^{\prime}$ and

$\sum_{i}\sum_{j}m_{i}m_{j}^{\prime}=\sum_{i}m_{i}\sum_{j}m_{j}^{\prime}=d_{A}^{2}d_{B}^{2},$

where $d_{A},d_{B}$ are the dimensions of $\rho_{A},\rho_{B}$ respectively,
and so the set $\\{(w_{i}w_{j}^{\prime},m_{i}m_{j}^{\prime})\\}$ contains
exactly the Wigner components of
$W_{\rho_{A}\otimes\rho_{B}}(\boldsymbol{z}_{A}\oplus\boldsymbol{z}_{B})$. ∎

The above result also applies when we consider rescaled Wigner distributions,
crucially due to the multiplicativity result shown in Proposition 16.

In the case of distillation protocols, we are interested in copies of
distributions. For this, we have the following result, which follows from
combinatorics.

######  Proposition 23.

Suppose $W_{\rho}$ has a set of $D$ component-multiplicity pairs
$\\{(w_{i},m_{i})\\}$. Then, $W_{\rho^{\otimes n}}$ has component-multiplicity
pairs $\\{(W_{\boldsymbol{q}},M_{\boldsymbol{q}})\\}$, with index
$\boldsymbol{q}$ running through all vectors $(q_{1},\dots,q_{D})$, where
$q_{1},\dots,q_{D}$ are non-negative integers that sum to $n$, and

$\displaystyle W_{\boldsymbol{q}}$
$\displaystyle=\prod_{i=1}^{D}w_{i}^{q_{i}}$ (70) $\displaystyle
M_{\boldsymbol{q}}$
$\displaystyle=\binom{n}{q_{1},q_{2},\dots,q_{d}}\prod_{i=1}^{D}m_{i}^{q_{i}}.$
(71)

The term outside the product in the expression for $M_{\boldsymbol{q}}$ is the
generalized binomial coefficient,

$\binom{n}{q_{1},q_{2},\dots,q_{d}}\coloneqq\frac{n!}{q_{1}!\dots q_{D}!}.$
(72)

###### Proof.

Denote by $C_{D}^{n}\coloneqq\\{\boldsymbol{k}\\}$ the set of all vectors
$\boldsymbol{k}\coloneqq(k_{1},\dots,k_{D})$ with non-negative integer
components that sum to $n$, i.e.

$0\leq k_{1},\dots,k_{D}\leq n\text{ and }k_{1}+\dots+k_{D}=n.$

We proceed by induction. Assume $n=1$ and let $\boldsymbol{k}_{i}$ be the
vector with its $i$-th component equal to 1 and 0’s elsewhere. The set
$C_{D}^{1}$ consists of all vectors of this form, i.e.

$C_{D}^{1}=\\{\boldsymbol{k}_{i}\\}_{i=1,\dots,D}$

It is also true by direct calculation that

$\left(W_{\boldsymbol{k}_{i}},M_{\boldsymbol{k}_{i}}\right)=(w_{i},m_{i}).$

Therefore, $\\{(W_{\boldsymbol{k}},M_{\boldsymbol{k}})\\}_{\boldsymbol{k}\in
C_{D}^{1}}$ is a complete set of component-multiplicity pairs for $W_{\rho}$.

Assume that $\\{(W_{\boldsymbol{k}},M_{\boldsymbol{k}})\\}_{\boldsymbol{k}\in
C_{D}^{n}}$ as given in Eqs. (70, 71) is a complete set of component-
multiplicity pairs for the $n$–copy distribution $W_{\rho^{\otimes
n}}=W_{\rho}^{\otimes n}$. By construction, the distribution
$W_{\rho}^{\otimes(n+1)}=W_{\rho}^{\otimes n}\otimes W_{\rho}$, so it admits
the complete set of component multiplicity pairs

$\\{(W_{\boldsymbol{k}}w_{i},M_{\boldsymbol{k}}m_{i})\\},\ \boldsymbol{k}\in
C_{D}^{n}\text{ and }i=1,\dots,D.$ (73)

Consider the component sum of distribution $W_{\rho}^{\otimes(n+1)}$,

$\displaystyle\sum_{\boldsymbol{k}\in
C_{D}^{n}}\sum_{i=1}^{D}M_{\boldsymbol{k}}m_{i}W_{\boldsymbol{k}}w_{i}=\sum_{\boldsymbol{k}\in
C_{D}^{n}}M_{\boldsymbol{k}}W_{\boldsymbol{k}}\sum_{i=1}^{D}m_{i}w_{i}=$
$\displaystyle\sum_{\boldsymbol{k}\in C_{D}^{n}}\frac{n!}{k_{1}!\dots
k_{D}!}\prod\limits_{i=1}^{D}{m_{i}}^{k_{i}}{w_{i}}^{k_{i}}\sum_{i=1}^{D}m_{i}w_{i}=$
$\displaystyle\left(\sum_{i=1}^{D}m_{i}w_{i}\right)^{n}\left(\sum_{i=1}^{D}m_{i}w_{i}\right)=\left(\sum_{i=1}^{D}m_{i}w_{i}\right)^{n+1}=$
$\displaystyle\sum_{\boldsymbol{q}\in
C_{D}^{n+1}}M_{\boldsymbol{q}}W_{\boldsymbol{q}},$

where in the last expression, vectors $\boldsymbol{q}=(q_{1},\dots,q_{D})$
have non-negative integer components that sum to $(n+1)$ and

$\displaystyle M_{\boldsymbol{q}}$ $\displaystyle=\frac{(n+1)!}{q_{1}!\dots
q_{D}!}\prod\limits_{i=1}^{D}{m_{i}}^{q_{i}},$ $\displaystyle
W_{\boldsymbol{q}}$ $\displaystyle=\prod\limits_{i=1}^{D}{w_{i}}^{q_{i}}.$

We have used the multinomial theorem to proceed between lines 2-3 and lines
3-4 of the derivation.

We have achieved a regrouping of the distribution components. Every component
$W_{\boldsymbol{q}}$ is of the form $W_{\boldsymbol{k}}w_{i}$ with
$q_{i}=k_{i}+1$ and $q_{j}=k_{j}$ for $j\neq i$ and

$\displaystyle\sum_{\boldsymbol{q}\in C_{D}^{n+1}}M_{\boldsymbol{q}}$
$\displaystyle=\sum_{\boldsymbol{q}\in C_{D}^{n+1}}\frac{(n+1)!}{q_{1}!\dots
q_{D}!}\prod\limits_{i=1}^{D}{m_{i}}^{q_{i}}=\left(\sum_{i=1}^{D}m_{i}\right)^{n+1}$
$\displaystyle=d^{2(n+1)},$

which is the dimension of $W_{\rho}^{\otimes(n+1)}$.

Therefore, $\\{(W_{\boldsymbol{q}},M_{\boldsymbol{q}})\\}_{\boldsymbol{q}\in
C_{D}^{n+1}}$ contains exactly the components of $W_{\rho}^{\otimes n}$,
completing the proof. ∎

Once again, the above result applies on rescaled Wigner distributions as well.

### Results on Lorenz curve constraints