# Local Asymptotic Normality and Optimal Estimation of low-rank Quantum
Systems

Samriddha Lahiry 111Corresponding author. Email<EMAIL_ADDRESS>Department
of Statistics and Data Science, Cornell University  Michael Nussbaum 222
Email<EMAIL_ADDRESS>Department of Mathematics, Cornell University

###### Abstract

In classical statistics, a statistical experiment consisting of $n$ i.i.d
observations from d-dimensional multinomial distributions can be well
approximated by a $d-1$ dimensional Gaussian distribution. In a quantum
version of the result it has been shown that a collection of $n$ qudits of
full rank can be well approximated by a quantum system containing a classical
part, which is a $d-1$ dimensional Gaussian distribution, and a quantum part
containing an ensemble of $d(d-1)/2$ shifted thermal states. In this paper, we
obtain a generalization of this result when the qudits are not of full rank.
We show that when the rank of the qudits is $r$, then the limiting experiment
consists of an $r-1$ dimensional Gaussian distribution and an ensemble of both
shifted pure and shifted thermal states. We also outline a two-stage procedure
for the estimation of the low-rank qudit, where we obtain an estimator which
is sharp minimax optimal. For the estimation of a linear functional of the
quantum state, we construct an estimator, analyze the risk and use quantum LAN
to show that our estimator is also optimal in the minimax sense.

††Keywords and phrases: low rank states, quantum LAN, quantum minimax
estimation, functional estimation

##  1 Introduction

Recent breakthroughs in quantum technology, such as quantum computing,
communication, and metrology ([48]) have created renewed interest in the
probabilistic and statistical problems that arise from quantum information
theory. In particular, the age old tools of mathematical statistics have found
their way into the toolbox of scientists working with quantum information.

One of the fundamental problems in quantum statistics is quantum estimation,
i.e. estimation of a quantum state or functionals of a state. State estimation
consists of the following problem: given a collection of identically prepared
states indexed by a parameter, one designs a measurement and then provides an
estimate of the parameter based on the outcome of the measurement. On the
other hand, in the case of functional estimation, one is interested in
measuring some particular property of the state instead of the whole state. In
analogy to the classical decision theoretic approach, one can develop a
quantum decision theoretic framework for such problems where the "best
estimate" also involves optimizing over the measurements. In this direction, a
quantum Cramer-Rao bound has been obtained in [10, 29, 32] for the covariance
matrix of unbiased estimators. Another closely related problem pertains to the
discrimination between multiple quantum states, in which case one can devise
optimal testing procedures. In the asymptotic setup, a quantum Chernoff bound
has been established in [6, 7, 40, 42].

In classical statistics, the first step for obtaining optimal decision
theoretic procedures often consists of approximating complicated experiments
(families of laws, or models) by simpler ones (see [39] for details). In
particular one establishes asymptotic equivalence between i.i.d models indexed
by a local parameter and a Gaussian shift model (with the shift given by the
same local parameter). This approximation is called local asymptotic normality
(LAN) and allows one to construct an estimator from a procedure in the
Gaussian model with similar risk bounds. Since the approximation is only local
in nature, one constructs a preliminary estimate of the parameter first and
then uses LAN in the neighborhood of the estimated value. Global equivalence
has also been shown between many nonparametric estimation problems like
nonparametric regression [11], density estimation [43] and the Gaussian white
noise model. The theory of LAN can also be used to show optimality in the
estimation of functionals by restating the functional estimation problem as a
smooth parametric estimation problem(see [34]). In the quantum setup, quantum
LAN theory for parametric models, established in [25, 26, 35, 50], shows that
a model given by a large collection of identically prepared finite dimensional
states can be approximated by a quantum Gaussian shift model in a local
neighborhood. An extension of quantum LAN theory towards a quantum version of
local asymptotic equivalence in nonparametric models can be found in [13]
where it is shown that an ensemble of pure states in infinite dimensional
Hilbert space can be approximated by coherent states, which constitute a
quantum counterpart of the Gaussian white noise model.

Our contribution in this paper is threefold; as a first result we establish
LAN for low-rank quantum states. We restrict ourselves to finite dimensional
states, in particular to a setup similar to [23, 26, 35]. In [26] the authors
show that a quantum statistical experiment consisting of a large number of
qubits (two dimensional quantum states) with rotational shift can be
approximated by a quantum Gaussian state, specifically a shifted thermal
state, while in [23] the authors improved the result to include diagonal
perturbation of the qubits. They showed that the limiting experiment in this
case contains a classical part which is Gaussian and a quantum part which is a
shifted thermal state. Later in [35] the authors extended the results of the
qubit to qudits (d-dimensional quantum systems) and showed that a large number
of full rank qudits can be well approximated by a quantum experiment
consisting of a $d-1$ dimensional "classical" Gaussian part (corresponding to
the diagonal elements of the original qudit model) and a quantum part which is
a tensor product of $d(d-1)/2$ shifted thermal states (corresponding to the
off-diagonal elements of the original qudit model). The latter paper used
technical results on symmetric and general linear groups but was crucially
dependent on the assumption that the original qudit was of full rank. We
extend this result to the case of rank-r qudits and obtain an interesting
limiting experiment. While the classical part contains an $r-1$ dimensional
classical Gaussian as expected from the degeneracy of the eigenvalues, the
quantum part contains a tensor product of both shifted thermal states
($r(r-1)/2$ many) and shifted pure states ($r(d-r)$ many). We observe that for
the full rank case there are only shifted thermal states in the quantum part
of the limit as in [23] and [35], while for the rank $r=1$ case we only have
shifted pure states (without a classical part). On the other hand, one can
easily modify the proof of local asymptotic equivalence in [13] to show that
the limiting experiment of an ensemble of finite dimensional pure states gives
rise to a tensor product of $d-1$ shifted pure states (without a classical
part) in the limit. Our result for rank $1$ qudits matches this result. Thus
we obtain a unifying picture that suitably generalizes previous results on
finite dimensional quantum LAN.

Secondly, we address the question of optimal estimation of low-rank qudits
using the LAN result. Recently there has been considerable interest in the
study of low-rank quantum models, although in most cases the authors study
them in the high dimensional low-rank setup and obtain finite sample bounds in
contrast to our asymptotic setup. In the framework of low-rank quantum state
tomography, the problem is treated in [16, 36, 37] where the authors consider
a trace regression model with the observables being random and then establish
minimax bounds in the estimation of the low-rank state. A similar problem has
been studied in [15] but with a sparsity assumption different from the low-
rank structure. The noiseless case (also called quantum compressed sensing)
has been studied in [18, 22, 41] and similar methods of quantum tomography
(using a small number of randomized measurements to reconstruct a low rank
state) have been considered in [1, 2]. A spectral thresholding and a rank
penalized method are used for the low-rank quantum state tomography problem in
[12] and [5] respectively while [27] uses a projected least square approach. A
comparative study of these approaches can be found in [3]. However, most of
these methods are only optimal in the rate sense and fail to obtain sharp
constants. We employ the methods of [12] and [27] to obtain a preliminary
estimator which is both rank consistent and lies close to the original state
and then use our LAN result to obtain the minimax risk up to sharp constants.
The optimal minimax risk for the full rank case is obtained in [24]; we
compare our results for the rank $r$ case and show that it matches with former
for $r=d$.

Finally, in the spirit of [34, 38], we use LAN to address the question of
optimal estimation of linear functionals of low-rank quantum states. While
property estimation is a well-established concept in quantum information
theory (for example see [4] for estimation of von-Neumann entropy), the
optimal procedures for estimation of a linear functional of a quantum state
are less well known. In this direction, the quantum analog of the concept of
least favorable sub-family is treated in [47]. We construct an estimator of
the linear functional and use the concept of a least favorable sub-family and
LAN to show optimality (up to sharp constants) of our estimator. It should be
noted that while [47] gives a pointwise optimality result, we establish a
minimax result; the classical analog of the latter can be found in [38].

###  1.1 Outline of results

The paper is organized as follows. In Section 2, we review the basic quantum
mechanical concepts of states, measurements, observables, and quantum
channels. Section 3 reviews classical parametric LAN and quantum LAN for the
full rank case as obtained in [35]. It also contains our main theorem (Theorem
3.3) where we establish LAN for the low-rank model.

Section 4 describes the Bayes estimator in the Gaussian case from [30, 31],
which will be useful in our construction of the optimal estimator and also in
establishing the minimax lower bound. Analogous to the classical case we also
observe a shrinkage phenomenon for this Bayes estimator.

In section 5 we construct the optimal estimator for rank $r$ qudits. First, in
Theorem 5.1, using a part of the sample, we obtain an estimator in which lies
close to the original state and has rank $r$ with high probability. We then
use the quantum channel on the remaining sample to transfer the estimation
problem of qudits to the estimation of parameters in a limiting Gaussian model
which contains a classical Gaussian and a tensor product of both shifted and
pure thermal states in its quantum part. An asymptotic minimax result for the
limiting Gaussian model is given in Theorem 5.2. To prove this theorem we use
the result from Section 4 to get a lower bound of the minimax risk by the
Bayes risk and then use a covariant measurement to give a matching upper
bound. Finally, we observe that the Hilbert-Schmidt norm between the two
qudits is locally quadratic approximately and then transfer the risk from the
Gaussian model to get the optimal risk in the low-rank qudit model in Theorem
5.3.

Estimation of a linear functional is treated in Section 6. We construct an
estimate of the functional in question using the appropriate observable to
give an upper bound. The lower bound is given by constructing a least
parametric subfamily and then using LAN to obtain a lower bound in the
limiting Gaussian model. The latter is indexed by a one-dimensional parameter
and we use a Bayesian result from [31] to give the lower bound.

Some representation theoretic tools are needed for proving Theorem 3.3 and
they are reviewed in Appendix A. The Bayesian result for a one dimensional
parameter is discussed in Appendix B. Proofs of the main theorems are given in
Appendix C while the proofs of the more technical lemmas are deferred to
Appendix D.

###  1.2 Notation

In physics, the vectors of a Hilbert space $\mathcal{H}$ (assumed separable)
are written as "ket” $|v\rangle$, $v^{*}$ as "bra” $\langle v|$ and the inner
product of two vectors as the "bra-ket” $\langle u|v\rangle\in\mathbb{C}$
which is linear with respect to the right entry and anti-linear with respect
to the left entry. Similarly, $M:=|u\rangle\langle v|$ is the rank one
operator acting as $M:|w\rangle\mapsto M|w\rangle=\langle
v|w\rangle|u\rangle$. For an operator $A$ the expression $\langle u|Av\rangle$
will sometimes be denoted as $\langle u|A|v\rangle$. The space of bounded
linear operators on $\mathcal{H}$ is denoted by $\mathcal{L}(\mathcal{H})$. Of
particular interest are the following two subspaces of
$\mathcal{L}(\mathcal{H})$.

  1. 1.

$\mathcal{T}_{1}(\mathcal{H})\subset\mathcal{L}(\mathcal{H})$ \- the trace
class defined by
$\mathcal{T}_{1}(\mathcal{H})=\\{A:\mathcal{H}\rightarrow\mathcal{H}:\mathrm{Tr}(A^{*}A)^{1/2}<\infty\\}$.
Operators in $\mathcal{T}_{1}(\mathcal{H})$ are equipped with the norm
$\mathrm{Tr}(A^{*}A)^{1/2}$.

  2. 2.

$\mathcal{T}_{2}(\mathcal{H})\subset\mathcal{L}(\mathcal{H})$ \- the Hilbert
Schmidt operators defined by
$\mathcal{T}_{2}(\mathcal{H})=\\{A:\mathcal{H}\rightarrow\mathcal{H}:\mathrm{Tr}(A^{*}A)<\infty\\}$.
Operators in $\mathcal{T}_{2}(\mathcal{H})$ are equipped with the norm
$(\mathrm{Tr}(A^{*}A))^{1/2}$. The class $\mathcal{T}_{2}(\mathcal{H})$ is a
Hilbert space with respect to the inner product $(A,B):=\mathrm{Tr}(A^{*}B)$.

It is well known that
$\mathcal{T}_{1}(\mathcal{H})\subset\mathcal{T}_{2}(\mathcal{H})$. For any
Hilbert space, the usual norm will be denoted by $||.||$ and the identity
operator on that space by $\mathbf{1}$ where the particular space will be
understood from the context. We will denote by $||\mu-\nu||_{\mathrm{TV}}$ the
total variation norm between two measures $\mu$ and $\nu$. By $a\vee b$ and
$a\wedge b$ we will denote $\max(a,b)$ and $\min(a,b)$ respectively and
$a_{+}$ will be used to denote $a\vee 0$. By $\lfloor a\rfloor$ and $\lceil
a\rceil$, we will denote the largest integer less than or equal to $a$ and the
smallest integer greater than or equal to $a$ respectively. We will use the
notation $a_{n}\asymp b_{n}$ whenever
$c<\liminf_{n}(a_{n}/b_{n})\leq\limsup_{n}(a_{n}/b_{n})<C$ for some constants
$c,C>0$. Throughout the paper, $c$ and $C$ will denote arbitrary constants.

##  2 Quantum Mechanics Preliminaries

###  2.1 States, Measurements, and Observables

A state of a quantum system is described by a self-adjoint trace class
operator $\rho$ on a complex Hilbert space $\mathcal{H}$, which is positive
$(\rho\geq 0$) and normalized to $\mathrm{Tr}\left(\rho\right)=1$ (a density
operator). A state is called pure if it is of the form
$\rho=|\psi\rangle\langle\psi|$, otherwise it is called a mixed state. We
denote the set of states by $\mathcal{S}(\mathcal{H})$.

Data on a quantum system are obtained from observables which are self-adjoint
operators $S$ in the Hilbert space $\mathcal{H}$. If $S$ has spectral
decomposition $S=\sum_{j}\lambda_{j}\Pi_{j}$ where $\Pi_{j}$s are projectors,
then a measurement generates a discrete random variable $X_{S}$ taking values
in the set of eigenvalues $\left\\{\lambda_{1},\lambda_{2},\ldots\right\\}$
with probabilities $p_{j}=\mathrm{Tr}(\rho\cdot\Pi_{j})$. The expectation of
$X_{S}$ under the state $\rho$ is then given by the Born-von Neumann
postulate:

$E_{\rho}X_{S}=\sum_{j}\lambda_{j}\mathrm{Tr}\left(\rho\Pi_{j}\right)=\mathrm{Tr}\left(\rho
S\right).$ (1)

In quantum mechanics, one needs generalized versions of the above definitions
of observables and measurements because the spectral decomposition of self-
adjoint operators in the form of a weighted sum of projectors may fail to hold
when the Hilbert space is infinite dimensional. If a measurement has outcomes
in a measurable space $(\Omega,\mathfrak{B})$, it is determined by a positive
operator-valued measure.

######  Definition 1.

A positive operator valued measure (POVM) is a map
$M:\mathfrak{B}\to\mathcal{L}(\mathcal{H})$ having the following properties

  * 1)

positivity: $M(B)\geq 0$ for all events $B\in\mathfrak{B}$ (hence M(B) is
self-adjoint)

  * 2)

$\sigma$-additivity: $M(\cup_{i}B_{i})=\sum_{i}{M}(B_{i})$ for any countable
set of mutually disjoint events $B_{i}$ (here the convergence is in the weak
operator topology of $\mathcal{L}(\mathcal{H})$)

  * 3)

normalization: $M(\Omega)=\mathbf{1}$.

If the operators $M(B)$ are also orthogonal projections, i.e. $M(A)^{2}=M(A)$
and $M(B)M(A)=0$ when $A\cap B=\emptyset$, then it is called a simple
measurement. The collection of projectors $\\{\Pi_{j}\\}$ in the spectral
decomposition $S=\sum_{j}\lambda_{j}\Pi_{j}$ is an example of a simple
measurement. The outcome of the measurement has probability distribution

$P_{\rho}(B)=\mathrm{Tr}(\rho M(B)),\qquad B\in\mathfrak{B}.$ (2)

The spectral theorem shows that any self-adjoint operator
$S:\mathcal{H}\rightarrow\mathcal{H}$ can be diagonalized as follows:

$S=\int_{\sigma(S)}xM(dx)$ (3)

where $\sigma(S)$ is the spectrum of $S$ and $M$ is a POVM, also called
spectral measure associated with the operator $S$. When $S$ is an observable
with a continuous spectrum, it generates a continuous random variable $X_{S}$
with probabilities given by (2). Also, it easily follows that

$E[X_{S}]=\mathrm{Tr}(S\rho).$

The expected value of an observable $S$ is often denoted as $\langle
S\rangle$, when the state dependence is not explicitly shown. There are POVMs
(called generalized measurements) where the orthogonality does not hold, but
these can be extended to a POVM in a larger Hilbert space where the extended
version is orthogonal. Let $\mathit{POVM}(\Omega,\mathcal{H})$ be the set of
POVMs with values in $\mathcal{L}(\mathcal{H})$ and outcome space $\Omega$ and
let $\mathcal{H}_{0}$ be another Hilbert space with a density operator
$\rho_{0}$. Then any simple measurement $M^{\prime}$ in
$\mathit{POVM}(\Omega,\mathcal{H}\otimes\mathcal{H}_{0})$ induces a
measurement $M$ in $\mathit{POVM}(\Omega,\mathcal{H})$ which is determined by

$\mathrm{Tr}(\rho
M(B))=\mathrm{Tr}((\rho\otimes\rho_{0})M^{\prime}(B)),B\in\mathfrak{B}$

for all states $\rho$ on $\mathcal{H}$. The pair $(\mathcal{H}_{0},\rho_{0})$
is called an ancilla and it is known that (cf. [32], Section 2.5) given any
measurement $M$ in $\mathcal{H}$, there exists an ancilla
$(\mathcal{H}_{0},\rho_{0})$ and a simple measurement $M^{\prime}$ such that
the above equation holds. The triple $(\mathcal{H}_{0},\rho_{0},M^{\prime})$
is called a realization of the measurement $M$ and the notion of adding an
ancilla before taking simple measurements is called quantum randomization in
[8].

In many cases, it is convenient to perform a measurement after "changing" the
state of the original system by interacting with other systems. The maps
describing such transformations are called quantum channels.

######  Definition 2.

A quantum channel between systems with Hilbert spaces $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$ is a mapping $\ T$ which assigns to every state $\rho$ on
$\mathcal{H}_{1}$ the state $T(\rho)$ on $\mathcal{H}_{2}$ given by

$T(\rho)=\sum_{i=1}^{\infty}{K}_{i}\rho K_{i}^{\ast}$ (4)

where $\left\\{{K}_{i}\right\\}$ are bounded operators
$K_{i}:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ such that
$\sum_{i=1}^{\infty}K_{i}^{\ast}K_{i}=\mathbf{1}$ (the series converging in
the strong operator topology of $\mathcal{L}(\mathcal{H}))$.

It can be shown that the map $T$ is trace preserving and _completely positive_
, i.e. $\mathrm{Id}_{m}\otimes T$ is positive for all $m\geq 1$, where
$\mathrm{Id}_{m}$ is the identity map on the space of $m$ dimensional
matrices. The simplest example of a quantum channel is a transformation
$\rho\mapsto U\rho U^{*}$, where $U$ is a unitary operator on $\mathcal{H}$.
More generally, if $|\varphi\rangle\in\mathcal{K}$ is a pure state of an
ancillary system, and $V$ is a unitary on $\mathcal{H}\otimes\mathcal{K}$,
then

$\rho\mapsto
T(\rho):=\mathrm{Tr}_{\mathcal{K}}(V(\rho\otimes|\varphi\rangle\langle\varphi|)V^{*})$

is a quantum channel where $\mathrm{Tr}_{\mathcal{K}}$ is the partial trace
over $\mathcal{K}$ (with respect to an orthonormal basis
$\\{|f_{i}\rangle\\}_{i=1}^{\mathrm{dim}\mathcal{K}}$). If we define operators
$K_{i}$ on $\mathcal{H}$ such that
$\langle\psi|K_{i}|\psi^{\prime}\rangle:=\langle\psi\otimes
f_{i}|V|\psi^{\prime}\otimes\varphi\rangle$, then it can be seen that
$T(\rho)$ can be written as in the form given in Definition 2. We define a
dual map $T^{\ast}$ of a quantum channel as follows

$\displaystyle T^{\ast}:\mathit{POVM}(\Omega,\mathcal{H}_{2})$
$\displaystyle\rightarrow\mathit{POVM}(\Omega,\mathcal{H}_{1})$ $\displaystyle
T^{\ast}(M)(B)$ $\displaystyle=\sum_{i=1}^{\infty}K_{i}^{\ast}M(B)K_{i}$

where the $\sum_{i=1}^{\infty}K_{i}^{\ast}M(B)K_{i}$ is a strongly convergent
sum. From the definition, it can be easily verified that $T^{\ast}(M)$ is
indeed an element of $\mathit{POVM}(\Omega,\mathcal{H}_{1})$ (i.e. a POVM
satisfying properties 1,2 and 3 of Definition 1) and that it satisfies the
following duality relation

$\mathrm{Tr}(\rho T^{*}(M)(B))=\mathrm{Tr}(T(\rho)M(B)),\quad\forall
B\in\mathfrak{B}$

and all states $\rho$ on $\mathcal{H}_{1}$ (cf. 29.9 of [44]).

For estimation purposes, we will need to define the following distances
between two quantum states. The _trace-norm_ distance between two states
$\rho_{0},\rho_{1}\in\mathcal{S(\mathcal{H})}$ is given by

$\|\rho_{0}-\rho_{1}\|_{1}:=\mathrm{Tr}(|\rho_{0}-\rho_{1}|)$

where $|\tau|:=\sqrt{\tau^{\ast}\tau}$ denotes the absolute value of $\tau$.
An interpretation of this metric in terms of quantum testing can be found in
[28]. In the special case of pure states, the trace-norm distance is given by

$\||\psi_{0}\rangle\langle\psi_{0}|-|\psi_{1}\rangle\langle\psi_{1}|\|_{1}=2\sqrt{1-|\langle\psi_{0}|\psi_{1}\rangle|^{2}}.$
(5)

Similarly one can define the $L^{2}$ distance between two states induced by
the Hilbert-Schmidt norm as

$\|\rho_{0}-\rho_{1}\|_{2}:=[\mathrm{Tr}((\rho_{0}-\rho_{1})^{*}(\rho_{0}-\rho_{1}))]^{1/2}.$

###  2.2 Qudits under local parametrization

Consider a qudit or $d$-dimensional density matrix i.e. $\rho\in
M_{d}(\mathbb{C})$ (the space of $d\times d$ complex matrices), with $\rho\geq
0$ and $\mathrm{Tr}(\rho)=1$. A natural way to parametrize qudits is to write
it in the form $\mathscr{U}(\zeta)\rho_{0}\mathscr{U}^{*}(\zeta)$ with
$\rho_{0}=$diag$(\mu_{1},\mu_{2},\ldots,\mu_{d})$ with
$\sum_{i=1}^{d}\mu_{i}=1$, where $\mathscr{U}(\zeta)$ are unitary matrices
i.e, elements of the Lie group $SU(d)$ parametrized by
$\zeta\in\mathbb{C}^{d(d-1)/2}$. We describe the unitaries in more detail.
Consider the generators of the Lie algebra $\mathfrak{su}(d)$ (cf. [19]):

$\displaystyle H_{j}$ $\displaystyle=E_{jj}-E_{j+1,j+1}\text{ for }1\leq j\leq
d-1$ $\displaystyle T_{j,k}$ $\displaystyle=iE_{j,k}-iE_{k,j}\text{ for }1\leq
j<k\leq d$ $\displaystyle T_{k,j}$ $\displaystyle=E_{j,k}+E_{kj}\text{ for
}1\leq j<k\leq d$

where $E_{i,j}$ is the matrix with $(i,j)^{th}$ entry equal to $1$, and all
other entries equal to $0$.

Define

$\mathscr{U}(\zeta)=\exp\left[i\left(\sum_{1\leq j<k\leq
d}\frac{Re(\zeta_{j,k})T_{j,k}+Im(\zeta_{j,k})T_{k,j}}{\mu_{j}-\mu_{k}}\right)\right]$
$\mathscr{U}^{r}(\zeta)=\exp\left[i\left(\sum_{\begin{subarray}{c}1\leq j\leq
r\\\ j<k\leq
d\end{subarray}}\frac{Re(\zeta_{j,k})T_{j,k}+Im(\zeta_{j,k})T_{k,j}}{\mu_{j}-\mu_{k}}\right)\right].$

Next, we consider local models by first perturbing the eigenvalues only:

$\rho_{0,u}=diag(\mu_{1}+u_{1},\mu_{2}+u_{2},\ldots,\mu_{d}+u_{d})$

where $\sum u_{i}=0$; that gives a state in a local neighborhood of
$\rho_{0}$. To describe all possible states in the local neighborhood of
$\rho_{0}$, we should also consider rotations using unitaries, i.e.

$\rho_{\vartheta}=\mathscr{U}(\zeta)\rho_{0,u}\mathscr{U}^{*}(\zeta)$ (6)

where $\vartheta=(u,\zeta)$. Similarly we consider a low-rank state in the
local neighborhood of
$\rho_{0,r}=diag(\mu_{1},\mu_{2},\ldots,\mu_{r},,0,\ldots,0)$:

$\displaystyle\rho_{0,u,r}$
$\displaystyle=diag(\mu_{1}+u_{1},\mu_{2}+u_{2},\ldots,\mu_{r}+u_{r},0,\ldots,0)$
(7) $\displaystyle\rho_{\vartheta,r}$
$\displaystyle=\mathscr{U}^{r}(\zeta)\rho_{0,u,r}\mathscr{U}^{r*}(\zeta).$ (8)

Let $\mathfrak{z}_{ij}=\frac{\zeta_{ij}}{\sqrt{\mu_{i}-\mu_{j}}}$. For
notational convenience we will denote $\theta=(u,\mathfrak{z})$ and redefine
the unitaries $\mathscr{U}(\zeta)$ and $\mathscr{U}^{r}(\zeta)$ as
$U(\mathfrak{z})$ and $U^{r}(\mathfrak{z})$ respectively. We will denote the
state by $\rho_{\vartheta}$ ($\rho_{\vartheta,r}$ respectively) or
$\rho_{\theta}$ ($\rho_{\theta,r}$ respectively) depending on whether it is
indexed by $\vartheta$ or $\theta$.

In classical LAN we are interested in the limit for $n$ i.i.d copies of the
experiment with the local parameter lying in an $n^{-1/2}$ neighborhood of $0$
(see section 3 for more details). In the quantum setup, this amounts to
studying the following operators

$\rho^{\theta,n}=\rho^{\otimes
n}_{\theta/\sqrt{n}},\quad\rho^{\theta,r,n}=\rho^{\otimes
n}_{\theta/\sqrt{n},r}.$

###  2.3 Gaussian States, Fock Space, Quantum Gaussian Sequence Model

To obtain Gaussian random variables, in the space
$\mathcal{H}$$=L^{2}\left(\mathbb{R}\right)$ one considers two special
observables $Q,P$ with continuous spectrum:

$\left(Qf\right)\left(x\right)=xf\left(x\right)\text{,
}\left(Pf\right)\left(x\right)=-i\frac{df}{dx}\left(x\right),\;\;\;\;f\in
D\subset L^{2}\left(\mathbb{R}\right)$

(defined on an appropriate domain D) often associated to position ($Q)$ and
momentum ($P$) of a particle. These operators satisfy the Heisenberg
commutation relations

$[Q,P]=i\textbf{1}.$

It can be shown that $Z_{u}:=u_{1}Q+u_{2}P$, $u\in\mathbb{R}^{2}$ are
observables (called the canonical observables). In this context we define the
quantum characteristic function as
$\tilde{W}_{\rho}(u_{1},u_{2})=\mathrm{Tr}(\rho\exp\left(iZ_{u}\right))$. If
the following relation holds

$E_{\rho}\exp\left(iZ_{u}\right)=\mathrm{Tr}(\rho\exp\left(iZ_{u}\right))=\exp\left(iu^{T}\mu-\frac{1}{2}u^{T}\Sigma
u\right)\text{, }u\in\mathbb{R}^{2},$

then $\rho$ is called a Gaussian state with mean $\mu$ and covariance matrix
$\Sigma$. For such quantum Gaussian states in $L^{2}\left(\mathbb{R}\right)$
we adopt a compact notation, resembling the one for the $2$-variate normal
law:

$\rho=\mathbb{N}_{2}\left(\mu,\Sigma\right).$ (9)

To define the simplest Gaussian state, let $\psi_{0}=\sqrt{\varphi_{1/2}}$ be
the square root of the density function of the normal $N\left(0,1/2\right)$
distribution and consider the operator $\rho_{0}$ acting by
$\rho_{0}f=\psi_{0}\left\langle\psi_{0},f\right\rangle$, $\;f\in
L^{2}\left(\mathbb{R}\right)$. Since $\psi_{0}$ is a unit vector in
$L^{2}\left(\mathbb{R}\right)$, the operator $\rho_{0}$ (henceforth called the
vacuum state) is a projection (written
$\rho_{0}=\left|\psi_{0}\right\rangle\left\langle\psi_{0}\right|$ in Dirac
notation) and it can be shown that
$\rho_{0}=\mathbb{N}_{2}\left(0,I_{2}/2\right)$ in the notation described
above.

An important class is the collection of coherent states
$\mathbb{N}_{2}\left(\mu,I_{2}/2\right)$; these are pure states which can be
interpreted as a vacuum shifted by $\mu\in\mathbb{R}^{2}$ (similar to the
Gaussian shift model in classical statistics). Consider the operators
$a^{*}=(Q-iP)/\sqrt{2}$ (the creation operator), $a=(Q+iP)/\sqrt{2}$ (the
annihilation operator) and $N=a^{*}a$ (the number operator). It is well known
that the Hermite basis $\\{|0\rangle,|1\rangle\,\ldots\\}$ forms an eigenbasis
of the number operator, i.e. $N|k\rangle=k|k\rangle$. For any $z\in\mathbb{C}$
define the displacement operator as

$D(z)=\exp(za^{*}-\bar{z}a)$

and the coherent state as

$|G(z)\rangle=D\left(z\right)\left|0\right\rangle=\exp(-|z|^{2}/2)\sum_{k=0}^{\infty}\frac{z^{k}}{\sqrt{k!}}|k\rangle.$
(10)

In the density operator notation this pure Gaussian state is
$|G(z)\rangle\langle G(z)|$. The expectations of the canonical observables $Q$
and $P$ under the state $|G(z)\rangle\langle G(z)|$ are

$\left\langle Q\right\rangle=\sqrt{2}\operatorname{Re}z\text{, }\left\langle
P\right\rangle=\sqrt{2}\operatorname{Im}z$

and the characteristic function of $|G(z)\rangle\langle G(z)|$ is

$\varphi\left(t\right)=\exp\left(i\left(t_{1}\sqrt{2}\operatorname{Re}z+t_{2}\sqrt{2}\operatorname{Im}z\right)-\frac{1}{4}\left(t_{1}^{2}+t_{2}^{2}\right)\right)\text{,
}t\in\mathbb{R}^{2}.$ (11)

The presence of the factor $\sqrt{2}$ motivates us to adopt a modified
notation for the coherent vector: setting
$\mu=\left(\sqrt{2}\operatorname{Re}z,\sqrt{2}\operatorname{Im}z\right)$, we
will write $|G(z)\rangle=\left|\psi_{\mu}\right\rangle$ so that now the
expectations are $\left(\left\langle Q\right\rangle,\left\langle
P\right\rangle\right)=\mu$. The characteristic function
$\varphi\left(t\right)$ is that of $\mathcal{N}_{2}(\mu,I/2)$ and hence in the
notations of (9)

$|G(z)\rangle\langle
G(z)|=|\psi_{\mu}\rangle\langle\psi_{\mu}|=\mathbb{N}_{2}\left(\mu,I/2\right).$
(12)

Other important classes of Gaussian states are thermal states and shifted
thermal states; these are mixed states unlike the vacuum and coherent states.
We define below the thermal state with temperature $\beta$ as

$\phi_{\beta}=(1-e^{-\beta})\sum_{k=0}^{\infty}e^{-k\beta}|k\rangle\langle
k|.$

Shifted thermal states are defined using the shift operator $D(z)$ as follows:

$\phi^{z}_{\beta}=D(z)\phi_{\beta}D^{*}(z).$

One can show that the quantum characteristic function
$\mathrm{Tr}(\phi^{z}_{\beta}\exp(iu_{1}Q+iu_{2}P))$ of the shifted thermal
state is given by

$\mathrm{Tr}(\phi^{z}_{\beta}\exp(iu_{1}Q+iu_{2}P))=\exp(i(u_{1}\sqrt{2}Re(z)+u_{2}\sqrt{2}Im(z))-\frac{\sigma^{2}_{\beta}}{2}(u_{1}^{2}+u_{2}^{2}))$

where $\sigma^{2}_{\beta}=\frac{\coth(\beta/2)}{2}$. Defining
$\mu=(\sqrt{2}Re(z),\sqrt{2}Im(z))$, we write that

$\phi_{\beta}^{z}=\mathbb{N}_{2}(\mu,\sigma^{2}_{\beta}I_{2}).$ (13)

To define a k-mode Gaussian state one considers the space
$\bigotimes_{i=1}^{k}L^{2}(\mathbb{R})$ and identifies the number basis as
follows

$|\mathbf{m}\rangle=\otimes_{1\leq i\leq
k}|m_{i}\rangle,\quad\mathbf{m}=\\{m_{i}\in\mathbb{N}:1\leq i\leq k\\}.$ (14)

We consider the collection of operators $\\{Q_{1},P_{1},\ldots,Q_{k},P_{k}\\}$
where each $Q_{i}$ and $P_{i}$ are position and momentum operators of a
particular mode (i.e. acting on a particular $L^{2}(\mathbb{R})$). These
operators satisfy the joint commutation relation as follows:

$[Q_{i},P_{j}]=i\delta_{i,j}\mathbf{1},\quad[Q_{i},Q_{j}]=0,\quad[P_{i},P_{j}]=0.$
(15)

One can then define the creation and annihilation operators $a_{i}^{*}$ and
$a_{i}$ for each mode and proceed to define the displacement operator as

$D(\mathbf{z})=\exp(\mathbf{z}.\mathbf{a}^{*}-\mathbf{\bar{z}}.\mathbf{a})$

where $\mathbf{z}.\mathbf{a}^{*}=\sum_{i=1}^{k}z_{i}a^{*}_{i}$ and similarly
for $\mathbf{\bar{z}}.\mathbf{a}$. Then we have

$|G(\mathbf{z})\rangle=D\left(\mathbf{z}\right)\left|\mathbf{0}\right\rangle=\exp(-|\mathbf{z}|^{2}/2)\sum_{m_{1},\ldots,m_{k}=0}^{\infty}\prod_{i=1}^{k}\frac{z_{i}^{m_{i}}}{\sqrt{m_{i}!}}|\mathbf{m}\rangle.$
(16)

To describe our limiting model we need the following multimode Fock spaces:

$\displaystyle\mathcal{F}$ $\displaystyle:=\bigotimes_{1\leq i<j\leq
d}L^{2}(\mathbb{R})$ (17) $\displaystyle\mathcal{F}^{r}$
$\displaystyle:=\bigotimes_{1\leq i\leq r,i<j\leq d}L^{2}(\mathbb{R}).$ (18)

Recall the shifted thermal state $\mathbb{N}_{2}(\mu,\sigma^{2}_{\beta}I_{2})$
and that in the complex notations it is denoted by $\phi^{z}_{\beta}$ (see
equation (13)). Similarly the shifted pure state in (12) can be written as
$\phi^{z}_{\infty}$ (noting that the case $\beta=\infty$ corresponds to the
pure case). The limiting model considered in [35] is as follows

$\Phi^{\theta}=\mathcal{N}_{d-1}(u,V_{\mu})\otimes\bigotimes_{1\leq i<j\leq
d}\phi_{\beta_{ij}}^{\mathfrak{z}_{ij}}$ (19)

where $\beta_{ij}=\ln(\mu_{i}/\mu_{j})$ and
$\mathfrak{z}_{ij}=\frac{\zeta_{ij}}{\sqrt{\mu_{i}-\mu_{j}}}$ i.e. the
diagonal perturbation only appears in the classical part which is a $d-1$
multivariate normal experiment with mean $u$ and covariance matrix $V_{\mu}$.
The latter is the covariance matrix of a multinomial random variable with
probabilities $\mu_{i}$, while the rotation perturbations determine the
$d(d-1)/2$ shifted thermal states. Since $\beta_{ij}$ are constants we will
use the abbreviated notation

$\phi^{\mathbf{\mathfrak{z}}}=\bigotimes_{1\leq i<j\leq
d}\phi_{\beta_{ij}}^{\mathfrak{z}_{ij}}\quad\in\mathcal{T}_{1}(\mathcal{F})$
(20)

where $\mathbf{\mathfrak{z}}=(\mathfrak{z}_{ij})_{1\leq i<j\leq d}$ is a
vector in $\mathbb{C}^{d(d-1)/2}$.

We also consider the following model which is a tensor product of both thermal
and pure quantum states along with a classical part which is given by a
multivariate normal random variable.

$\Phi^{\theta,r}=\mathcal{N}_{r-1}(u,V_{\mu})\otimes\bigotimes_{1\leq i<j\leq
r}\phi_{\beta_{ij}}^{\mathfrak{z}_{ij}}\otimes\bigotimes_{\begin{subarray}{c}1\leq
i\leq r\\\ r+1\leq j\leq
d\end{subarray}}\phi_{\infty}^{\mathfrak{z}_{ij}}=\mathcal{N}_{r-1}(u,V_{\mu})\otimes\phi^{\mathfrak{z},r}.$
(21)

where

$\phi^{\mathbf{\mathfrak{z},r}}=\bigotimes_{1\leq i<j\leq
r}\phi_{\beta_{ij}}^{\mathfrak{z}_{ij}}\otimes\bigotimes_{\begin{subarray}{c}1\leq
i\leq r\\\ r+1\leq j\leq
d\end{subarray}}\phi_{\infty}^{\mathfrak{z}_{ij}}\quad\in\mathcal{T}_{1}(\mathcal{F}^{r}).$

In section 3 we show that the states $\Phi^{\theta,r}$ arise as limiting
models for low-rank qudits.

The classical-quantum limiting state also motivates us to adopt a notation
(similar to [20]) that allows us to describes the commutation relations in the
hybrid system in a compact fashion. Note that for the $k$-mode system we have
defined operators $\\{Q_{i},P_{i}\\}_{i=1}^{k}$ given by (15). In addition to
this, we allow $l$ classical random variables $C_{1},\ldots,C_{l}$ that
commute with each other and with all $(Q_{i},P_{i})$. We can denote the
$m=2k+l$ variables as

$(X_{1},\ldots,X_{m})\equiv(C_{1},\ldots,C_{l},Q_{1},P_{1},\ldots,Q_{k},P_{k})$
(22)

and the commutation relation as

$[X_{i},X_{j}]=iS_{ij}\mathbf{1}$

where $S=0_{l\times l}\oplus{\bigoplus_{i=1}^{2k}}\Omega$ and

$\Omega=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right).$

We can define a state $\varrho=f\otimes\rho$ in the space
$L^{1}(\mathbb{R}^{l})\otimes\mathcal{T}_{1}(\mathcal{F}^{k})$, with $\rho$ a
quantum state and $f$ a probability density. Now define the hybrid
characteristic function of the state as

$E_{\varrho}(e^{i\sum_{i=1}^{m}u_{i}X_{i}}):=\int\mathrm{Tr}(\rho
e^{\sum_{i=l+1}^{2k+l}u_{i}X_{i}})f(y)e^{\sum_{j=1}^{l}u_{i}y_{j}}dy_{1}\ldots
dy_{l}.$ (23)

######  Definition 3.

A hybrid state $\varrho$ is called classical-quantum Gaussian if the
characteristic function has the following form:

$E_{\varrho}(e^{i\sum_{i=1}^{m}u_{i}X_{i}})=e^{iu^{T}\tau-u^{T}\Sigma u/2}$

where $\tau\in\mathbb{R}^{m}$ and $\Sigma$ is a $m\times m$ real covariance
matrix.

With $\nu_{ij}=(\sqrt{2}Re(\mathfrak{z}_{ij}),\sqrt{2}Im(\mathfrak{z}_{ij}))$
each shifted thermal state can be denoted by
$\phi_{\beta_{ij}}^{\mathfrak{z}_{ij}}=\mathbb{N}_{2}(\nu_{ij},\sigma^{2}_{\beta_{ij}}I_{2})$.
Since the classical part in the limiting model is also a Gaussian (given by
$\mathcal{N}_{r-1}(u,V_{\mu})$), we adopt the following alternate notation for
the classical-quantum Gaussian:

$\Phi^{\theta,r}=\mathfrak{N}(\tau,\mathscr{S})$ (24)

where

$\displaystyle\Sigma$ $\displaystyle=V_{\mu}\oplus{\bigoplus_{1\leq i\leq
r,i<j\leq d}}\sigma^{2}_{\beta_{ij}}I_{2}$ $\displaystyle S$
$\displaystyle=0_{r-1\times r-1}\oplus{\bigoplus_{1\leq i\leq r,i<j\leq
d}}\Omega$ $\displaystyle\mathscr{S}$
$\displaystyle=\Sigma+iS,\quad\tau=u\oplus{\bigoplus_{1\leq i\leq r,i<j\leq
d}}\nu_{ij}.$

We use $\mathscr{S}$ instead of $\Sigma$ in $\mathfrak{N}(\tau,\mathscr{S})$
that captures the underlying non-commutative structure via $S$. We note that a
similar notation incorporating the commutation relations into a complex
covariance matrix has been used in [50]. This notation will be useful in
computing a Bayes risk for a one dimensional parameter (see Appendix B) which
will subsequently be used in establishing a minimax lower bound for the
estimation of a linear functional of the state.

###  2.4 Quantum Statistical Inference

In this section we formalize the quantum counterparts of the basic notions of
optimality in classical statistical inference. In classical statistics, an
experiment is defined to be a family of probability measures on a sample space
and denoted by $\mathcal{E}=\\{P_{\theta},\theta\in\Theta\\}$ where $\Theta$
is the parameter space.

######  Definition 4.

A quantum statistical model over a parameter space $\Theta$ consists of a
family of quantum states $\mathcal{Q}=\\{\rho_{\theta}:\,\theta\in\Theta\\}$
on a Hilbert space $\mathcal{H}$, indexed by an unknown parameter
$\theta\in\Theta$.

Inference in quantum models generally involves two steps. In the first step
one performs a measurement on the state $\rho_{\theta}$ and generates data,
while in the second step, one uses standard statistical tools to solve the
specific decision problem using data from the first step. If one performs a
measurement $M$ on the system in state $\rho_{\theta}$, a random outcome is
obtained with distribution $P^{M}_{\theta}(E):=\mathrm{Tr}(\rho_{\theta}M(E))$
(cf. Subsection 2.1). The measurement data is therefore described by the
classical model $\mathcal{P}^{M}:=\\{P^{M}_{\theta}:\,\theta\in\Theta\\}$ and
the estimation problem can be treated using "classical” statistical methods.
However, in many scenarios, the optimal estimators for individual components
of a parameter are incompatible with each other and the optimal joint
estimator for the two components can be entirely different from the optimal
estimators of the individual components.

In the classical setup, a randomized decision function is given by a Markov
kernel $\nu$. If $L(\theta,u)$ is the loss function then the risk is given by

$R(\theta,\nu)=\int\int L(\theta,u)\nu_{x}(du)\mu_{\theta}(dx)=\int
L(\theta,u)\int\nu_{x}(du)\mu_{\theta}(dx)=\int
L(\theta,u)\tilde{\nu}_{\theta}(du)$ (25)

where $\tilde{\nu}_{\theta}(A)=\int\nu_{x}(A)\mu_{\theta}(dx)$.

Section 2.2.4 of [33] discusses the quantum counterpart of this classical
formulation. Let $\rho_{\theta}$ be the quantum state and
$\mu^{M}_{\theta}(B)=Tr(\rho_{\theta}M(B))$ be the probability measure
generated by the POVM $M$. Then the risk is given by

$R(\theta,M)=\int L(\theta,u)\mu^{M}_{\theta}(du).$ (26)

By using the fact that every affine map
$\rho_{\theta}\rightarrow\mu_{\theta}()$ can be associated with a POVM, we see
that $M$ is an analog of the classical randomized decision function $\nu$
given in (25). We can easily define the Bayes and minimax problems for quantum
estimation.

Minimax problem

$\inf_{M}\sup_{\theta\in\Theta}R(\theta,M)=\inf_{M}\sup_{\theta\in\Theta}\int
L(\theta,u)\mu^{M}_{\theta}(du)=\inf_{\hat{m}}\sup_{\theta\in\Theta}E_{\theta}[L(\theta,\hat{m})]$

Bayes problem

$\inf_{M}\int_{\Theta}R(\theta,M)\pi(d\theta)=\inf_{M}\int_{\Theta}\int
L(\theta,u)\mu^{M}_{\theta}(du)\pi(d\theta)=\inf_{\hat{m}}\int
E_{\theta}[L(\theta,\hat{m})]\pi(d\theta)$

The notations
$\inf_{\hat{m}}\sup_{\theta\in\Theta}E_{\theta}[L(\theta,\hat{m})]$ and
$\inf_{\hat{m}}\int E_{\theta}[L(\theta,\hat{m})]\pi(d\theta)$ will be called
condensed notations and will be used henceforth. Note that the infimum is over
all POVM and the notation $\hat{m}$ should not be confused with a
deterministic estimator seen in the classical setup. We will also denote the
Bayes risk as $\inf_{\hat{m}}E[L(\theta,\hat{m})]$ where the expectation is
also taken over the parameter $\theta$.

In classical statistics, a well-known paradigm is using asymptotic equivalence
of experiments to transfer risk bounds from one experiment to another. Suppose
we have two experiments $\mathcal{E}=\\{P_{\theta},\theta\in\Theta\\}$ on a
sample space $(\Omega_{1},\mathcal{A}_{1})$ and
$\mathcal{F}=\\{Q_{\theta},\theta\in\Theta\\}$ on a sample space
$(\Omega_{2},\mathcal{A}_{2})$. Also let the loss function satisfy the
condition $0\leq L(\theta,u)\leq 1$. If there exists a Markov kernel $K$ such
that

$\sup_{\theta\in{\Theta}}||KP_{\theta}-Q_{\theta}||_{TV}\leq\epsilon$

then for any randomized decision function $\mu$ of $\theta$ in the model
$Q_{\theta}$, the randomized decision function $\nu=\mu\circ K$ (composition
of two Markov kernels) satisfies

$R^{1}({\theta,\nu})\leq R^{2}(\theta,\mu)+\epsilon$

where $R^{1}({\theta,\nu})$ and $R^{2}(\theta,\mu)$ are the risks in the
models $\mathcal{E}$ and $\mathcal{F}$ respectively. We discuss the
generalization of this paradigm to the quantum setup and also generalize it to
the case of unbounded loss.

The quantum equivalent of a Markov kernel is the transformation by quantum
channels. The quantum model $\mathcal{Q}$ can be transformed into another
quantum model
$\mathcal{Q}^{\prime}:=\\{\rho_{\theta}^{\prime}:\theta\in\Theta\\}$ on a
Hilbert space $\mathcal{H}^{\prime}$ by applying a quantum channel

$\displaystyle T$ $\displaystyle:$
$\displaystyle\mathcal{T}_{1}(\mathcal{H})\to\mathcal{T}_{1}(\mathcal{H}^{\prime})$
$\displaystyle T$ $\displaystyle:$
$\displaystyle\rho_{\theta}\mapsto\rho_{\theta}^{\prime}.$

In this context, we define the quantum Le Cam distance between two models from
[13].

######  Definition 5.

Let $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ be two quantum models over
$\Theta$. The deficiency of $\mathcal{Q}$ with respect to
$\mathcal{Q}^{\prime}$ is defined by

$\delta\left(\mathcal{Q},\mathcal{Q}^{\prime}\right):=\inf_{T}\sup_{\theta\in\Theta}\|T(\rho_{\theta})-\rho_{\theta}^{\prime}\|_{1}$

where the infimum is taken over all channels $T$. The _Le Cam_ distance
between $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ is defined as

$\Delta\left(\mathcal{Q},\mathcal{Q}^{\prime}\right):=\mathrm{max}\left(\delta\left(\mathcal{Q},\mathcal{Q}^{\prime}\right)\,,\,\delta\left(\mathcal{Q}^{\prime},\mathcal{Q}\right)\right).$
(27)

Its interpretation is that models which are "close” in the Le Cam distance
have similar risk bounds. Now, suppose we have two sequences of quantum models
(or experiments)
$\mathcal{E}^{(n)}=\\{\rho^{(1,n)}_{\theta}:\theta\in\Theta\\}$ and
$\mathcal{F}^{(n)}=\\{\rho^{(2,n)}_{\theta}:\theta\in\Theta\\}$ . Assume that
$\Delta\left(\mathcal{E}^{(n)},\mathcal{F}^{(n)}\right)\rightarrow 0$ with the
associated sequence of Hilbert spaces $\mathcal{H}^{1,n}$ and
$\mathcal{H}^{2,n}$. This implies
$\delta\left(\mathcal{E}^{(n)},\mathcal{F}^{(n)}\right)\rightarrow 0$ and in
particular there exists a sequence of quantum channels $T_{n}$, such that

$||T_{n}(\rho^{(1,n)}_{\theta})-\rho^{(2,n)}_{\theta}||_{1}=o(1).$

Let the loss function also change with $n$ and satisfy the relation $0\leq
L_{n}(\theta,u)\leq c_{n}$. Also, assume that the sequence of quantum channels
$T_{n}$ is such that

$c_{n}\sup_{\theta\in\Theta}||T_{n}(\rho^{(1,n)}_{\theta})-\rho^{(2,n)}_{\theta}||_{1}=o(1).$
(28)

Recall the dual map $T^{*}$ of a quantum channel $T$. It follows that for any
$M\in\mathit{POVM}(\Omega,\mathcal{H}^{2,n})$

$\displaystyle R^{1}_{n}(\theta,T_{n}^{*}(M))=$ $\displaystyle\int
L_{n}(\theta,u)\mathrm{Tr}(\rho_{\theta}^{(1,n)}T_{n}^{*}(M(du)))$
$\displaystyle=$ $\displaystyle\int
L_{n}(\theta,u)\mathrm{Tr}(\rho_{\theta}^{(2,n)}M(du))$ (29)
$\displaystyle+\int
L_{n}(\theta,u)[\mathrm{Tr}(\rho_{\theta}^{(1,n)}T_{n}^{*}(M(du)))-\mathrm{Tr}(\rho_{\theta}^{(2,n)}M(du))]$
$\displaystyle=$ $\displaystyle R^{2}_{n}(\theta,M)+\int
L_{n}(\theta,u)[\mathrm{Tr}(T_{n}(\rho_{\theta}^{(1,n)})(M(du)))-\mathrm{Tr}(\rho_{\theta}^{(2,n)}M(du))]$
$\displaystyle\leq$ $\displaystyle
R^{2}_{n}(\theta,M)+c_{n}||T_{n}(\rho^{(1,n)}_{\theta})-\rho^{(2,n)}_{\theta}||_{1}$
$\displaystyle\leq$ $\displaystyle R^{2}_{n}(\theta,M)+o(1)$ (30)

the term $o(1)$ tending to $0$ uniformly over all $\theta$. Thus we can
compare the risks of the two models $E^{(n)}$ and $F^{(n)}$ if (28) holds.
Note that we have similar relations for minimax risks and Bayes risks, by
taking a supremum over $\Theta$ or integrating with respect to a prior,
respectively, and then taking an infimum over all estimators:

$\displaystyle\inf_{M}\sup_{\theta\in\Theta}R^{1}_{n}(\theta,M)$
$\displaystyle\leq\inf_{M}\sup_{\theta\in\Theta}R^{2}_{n}(\theta,M)+o(1).$
(31) $\displaystyle\inf_{M}\int_{\Theta}R^{1}_{n}(\theta,M)\pi(d\theta)$
$\displaystyle\leq\inf_{M}\int_{\Theta}R^{2}_{n}(\theta,M)\pi(d\theta)+o(1).$
(32)

##  3 Local Asymptotic Normality in low-rank systems

###  3.1 Classical LAN

Consider a collection of i.i.d random variables $\\{X_{1},\ldots,X_{n}\\}$
taking values in a measurable space $(\mathcal{X},\Sigma_{\mathcal{X}})$ with
$X_{i}\sim P_{\theta}$ where $\theta$ belongs to $\Theta$, which is an open
subset of $\mathbb{R}^{d}$. We can consider a local perturbation around a
fixed point $\theta_{0}$ and if we denote $\theta=\theta_{0}+u/\sqrt{n}$ (with
$u$ bounded), then we can represent the aforementioned collection of random
variables by a statistical experiment
$\mathcal{E}_{n}=\\{P_{\theta_{0}+u/\sqrt{n}}^{n},||u||\leq C\\}$ on a sample
space $(\mathcal{X}^{n},\Sigma_{\mathcal{X}}^{n})$ where
$P_{\theta_{0}+u/\sqrt{n}}^{n}$ is an $n$-fold product of
$P_{\theta_{0}+u/\sqrt{n}}$. Under some regularity assumptions (see theorem
below) $\mathcal{E}_{n}$ can be approximated by a Gaussian shift experiment
$\mathcal{F}=\\{N(u,I_{\theta_{0}}^{-1}),||u||\leq C\\}$ where
$I_{\theta_{0}}$ is the Fisher information matrix at $\theta_{0}$. The
following result is well known (cf. [45], Theorem 2.9 along with [46], 79.3).

######  Theorem 3.1.

Assume $\left(\mathcal{X},\Sigma_{\mathcal{X}}\right)$ is a Polish (complete
separable metric) space with its Borel $\sigma$-algebra. Assume further that
(i) the experiment $\mathcal{E}_{n}$ is dominated: $P_{\theta}\ll\mu$,
$\theta\in\Theta$ where $\mu$ is a $\sigma$-finite measure on
$\Sigma_{\mathcal{X}}$,
(ii) the densities $p_{\theta}\left(x\right)=\left(dP_{\theta}/d\mu\right)(x)$
are jointly measurable in $\left(x,\theta\right)$ and differentiable in
quadratic mean at $\theta=\theta_{0}$, i.e. for some measurable function
$\ell_{\theta}:\mathcal{X}\rightarrow\mathbb{R}^{d}$

$\int\left[p_{\theta+u}^{1/2}-p_{\theta}^{1/2}-u^{T}\ell_{\theta}p_{\theta}^{1/2}\right]^{2}d\mu=o\left(\left\|u\right\|^{2}\right)\text{
as }u\rightarrow 0,$

(iii) the Fisher information matrix
$I_{\theta}=4E_{\theta}[\ell_{\theta}\ell_{\theta}^{T}]$ is nonsingular at
$\theta=\theta_{0}$.
Then the experiments $\mathcal{E}_{n}$ and $\mathcal{F}$ are asymptotically
equivalent.

In other words, there exist sequences of Markov kernels $T_{n}$ and $S_{n}$,
such that:

$\displaystyle\lim_{n\rightarrow\infty}\sup_{||u||\leq
C}||T_{n}(P^{n}_{\theta_{0}+u/\sqrt{n}})-N(u,I^{-1}_{\theta_{0}})||_{TV}=0$
$\displaystyle\lim_{n\rightarrow\infty}\sup_{||u||\leq
C}||P^{n}_{\theta_{0}+u/\sqrt{n}}-S_{n}(N(u,I^{-1}_{\theta_{0}}))||_{TV}=0.$

###  3.2 Quantum LAN

Consider the following domain of the local parameters

$\Theta_{n,\beta,\gamma}=\\{(u,\mathfrak{z}):|u_{k}|\leq
n^{\gamma},|\mathfrak{z}_{ij}|\leq n^{\beta},\ \forall 1\leq k\leq d-1,\ 1\leq
i<j\leq d\\}$

for some $\beta>0,\gamma>0$.

Recall the $d$-dimensional state $\rho_{\theta}$ given in equation (6) and the
corresponding Gaussian state given in (19) indexed by the same local parameter
$\theta$. Note that we have used the alternate notation using $\theta$ instead
of $\vartheta$ (see the discussion after (8)). We consider the following two
experiments:

$\mathcal{Q}_{n}=\\{\rho^{\theta,n}:\theta\in\Theta_{n,\beta,\gamma}\\},\quad\mathcal{R}_{n}=\\{\Phi^{\theta}:\theta\in\Theta_{n,\beta,\gamma}\\}.$

We state the LAN theorem for full rank states (proved in [35]) which shows
that these local models are asymptotically equivalent.

######  Theorem 3.2.

Recall the Fock space described in (17). Then for $0<\gamma<1/4$ and
$0<\beta<1/9$ there exist quantum channels $T_{n}$ and $S_{n}$

$\displaystyle T_{n}:M(\mathbb{C}^{d})^{\otimes n}\rightarrow
L^{1}(\mathbb{R}^{d-1})\otimes\mathcal{T}_{1}(\mathcal{F})$ $\displaystyle
S_{n}:L^{1}(\mathbb{R}^{d-1})\otimes\mathcal{T}_{1}(\mathcal{F})\rightarrow
M(\mathbb{C}^{d})^{\otimes n}$

such that

$\displaystyle\sup_{\theta\in\Theta_{n,\beta,\gamma}}||\Phi^{\theta}-T_{n}(\rho^{\theta,n})||_{1}=O(n^{-\kappa})$
$\displaystyle\sup_{\theta\in\Theta_{n,\beta,\gamma}}||S_{n}(\Phi^{\theta})-\rho^{\theta,n}||_{1}=O(n^{-\kappa})$

for some $\kappa>0$ which depends on $\beta$ and $\gamma$.

Now we are ready to state the low-rank version of the above theorem. Recall
that a qudit of rank $r$ can be parametrized as $\rho_{\theta,r}$ (see
equation (8) and use the alternate notation i.e. $\theta$ instead of
$\vartheta$) and consider the corresponding Gaussian state given in (21)
indexed by a local parameter $\theta$. Define

$\Theta_{n,r,\beta,\gamma}=\\{(u,\mathfrak{z}):|u_{k}|\leq
n^{\gamma},|\mathfrak{z}_{ij}|\leq n^{\beta}\ \forall\ 1\leq k\leq r-1,1\leq
i\leq r,i<j\leq d\\}$

for some $\beta>0,\gamma>0$. We now have the low-rank versions of the earlier
models:

$\mathcal{Q}^{r}_{n}=\\{\rho^{\theta,r,n}:\theta\in\Theta_{n,r,\beta,\gamma}\\}\quad\mathcal{R}^{r}_{n}=\\{\Phi^{\theta,r}:\theta\in\Theta_{n,r,\beta,\gamma}\\}.$

The following theorem shows that the above models are asymptotically
equivalent.

######  Theorem 3.3.

Recall the Fock space described in (18). Then for $0<\gamma<1/4$ and
$0<\beta<1/9$ there exist quantum channels $T^{r}_{n}$ and $S^{r}_{n}$

$\displaystyle T^{r}_{n}:M(\mathbb{C}^{d})^{\otimes n}\rightarrow
L^{1}(\mathbb{R}^{r-1})\otimes\mathcal{T}_{1}(\mathcal{F}^{r})$ $\displaystyle
S^{r}_{n}:L^{1}(\mathbb{R}^{r-1})\otimes\mathcal{T}_{1}(\mathcal{F}^{r})\rightarrow
M(\mathbb{C}^{d})^{\otimes n}$

such that

$\sup_{\theta\in\Theta_{n,r,\beta,\gamma}}||\Phi^{\theta,r}-T^{r}_{n}(\rho^{\theta,r,n})||_{1}=O(n^{-\kappa})$
(33)
$\sup_{\theta\in\Theta_{n,r,\beta,\gamma}}||S^{r}_{n}(\Phi^{\theta,r})-\rho^{\theta,r,n}||_{1}=O(n^{-\kappa})$
(34)

for some $\kappa>0$ which depends on $\beta$ and $\gamma$.

We observe that the classical part corresponds to a low dimensional ($r-1$\-
variate) normal experiment while the quantum part contains two subparts. When
$\mu_{j}$ and $\mu_{k}$ are both positive (i.e. $1\leq j<k\leq r$), we get
shifted thermal states with temperatures given as before
($\beta_{jk}=\ln(\mu_{j}/\mu_{k})$). When $\mu_{j}>0$ and $\mu_{k}=0$, we get
shifted pure states. We can compare our result with other available results in
quantum LAN.

Comparison with other LAN results

  1. 1.

The diagonal case: In the absence of the rotation by unitaries the diagonal
state represents a local multinomial model and the limiting model contains
only the classical Gaussian part, i.e. $\mathcal{N}_{r-1}(u,V_{\mu})$, and we
recover the classical LAN result of Theorem 3.1. A global version of this
approximation (with possibly increasing dimension of the multinomial) was
obtained in [17].

  2. 2.

$d=2,r=1$: In this case, we observe that the limiting model consists of a
single shifted pure Gaussian state and no classical component. This case was
discussed in [26] using a heuristic argument.

  3. 3.

$d=r>2$: The limiting model consists of a $d-1$ dimensional normal
distribution in the classical part and only shifted thermal states in the
quantum part. We recover the full rank case of [35] i.e. Theorem 3.2.

  4. 4.

$d>r=1$: The limiting model consists of $d-1$ shifted pure states and no
classical component. The case when $d=\infty$ and $r=1$ i.e. infinite
dimensional pure states was treated in [13] under a slightly different
parametrization. The limiting model in [13] is quantum white noise, or
equivalently a tensor product of infinitely many shifted pure states. A slight
modification of the proof of Theorem 4.1 in [14] (proving it for finite
dimensional pure states) shows that the limiting model is indeed a tensor
product of $d-1$ shifted pure states and agrees with our current result.

##  4 Measurement of the shift parameter for a Gaussian state

In this section, we consider measurement of the shift parameter $\mu$ of the
model $\rho=\mathbb{N}_{2}(\mu,\sigma^{2}I_{2})$. We describe the particular
generalized measurement (the covariant measurement) that is used to measure
the shift parameter. It can be shown that a Bayes estimator can be constructed
by appropriately "shrinking" the covariant measurement.

An essential fact is that the coherent vectors
$\left\\{\left|\psi_{m}\right\rangle,m\in\mathbb{R}^{2}\right\\}$ form an
"overcomplete system" (if multiplied by a factor $1/\sqrt{2\pi}$), i.e.
fulfill

$\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\left|\psi_{m}\right\rangle\left\langle\psi_{m}\right|dm=\mathbf{1}$
(35)

where $\mathbf{1}$ is the identity operator in the Hilbert space
$\mathcal{H}=L^{2}\left(\mathbb{R}\right)$ (see equation 3.5.45, p. 101 of
[32], with proof after Proposition 3.5.1). A complete orthonormal system
$\left\\{\left|\psi_{m}\right\rangle\right\\}$ is an example of an
overcomplete system (with integration replaced by summation); however, in the
general case the vectors $\psi_{m}$ can be non-orthogonal and linearly
dependent. The system of coherent vectors
$\left\\{\left|\psi_{m}\right\rangle,m\in\mathbb{R}^{2}\right\\}$ generates a
resolution of the identity, i.e. a normalized POVM $M$ on the measurable space
$\left(\mathbb{R}^{2},\mathfrak{B}\right)$ (with $\mathfrak{B}$ being the
Borel sigma-algebra) according to

$M\left(B\right)=\frac{1}{2\pi}\int_{B}\left|\psi_{m}\right\rangle\left\langle\psi_{m}\right|dm\text{,
}B\in\mathfrak{B}\text{.}$ (36)

The POVM $M$ then generates a (generalized) observable $X_{M}$ with values in
$\mathbb{R}^{2}$ which under the state $\rho$ has probability distribution

$P\left(X_{M}\in B\right)=\mathrm{Tr\;}\rho M\left(B\right)\text{,
}B\in\mathfrak{B}\text{.}$ (37)

This is called the canonical covariant measurement in Section 3.6 of [32],
covariance referring to the action of the Weyl unitaries (or the displacement
operators). There also optimality properties are proved, as well as
equivalence to simple measurements on an extended system (corresponding to an
orthogonal resolution of the identity there). When the covariant measurement
is clear from the context we will often write $X_{M}$ as $X$. It can be shown
that when $\rho$ is a shifted pure state i.e.
$\rho=\mathbb{N}_{2}\left(\mu,I_{2}/2\right)$ then,
$X\sim\mathcal{N}_{2}(\mu,I_{2})$ and if $\rho$ is a shifted thermal state
i.e. $\rho=\mathbb{N}_{2}\left(\mu,\sigma^{2}_{\beta}I_{2}\right)$, then
$X\sim\mathcal{N}_{2}\left(\mu,\frac{(2\sigma_{\beta}^{2}+1)}{2}I_{2}\right)$.

An equivalent description can be given as follows; cf. Proposition 3.6.1 of
[32] and also relation (3.18) in [31]. Let
$\mathcal{H}=L^{2}\left(\mathbb{R}\right)$ and let $\mathcal{H}_{0}$ be an
identical Hilbert space with canonical observables $Q_{0}$ and $P_{0}$. In the
tensor product $\mathcal{H\otimes H}_{0}$, consider the operators

$\tilde{Q}=Q\otimes\mathbf{1}_{0}+\mathbf{1}\otimes
Q_{0},\;\tilde{P}=P\otimes\mathbf{1}_{0}-\mathbf{1}\otimes P_{0}$ (38)

where $\mathbf{1}_{0}$ is the identity operator in $\mathcal{H}_{0}$. Let
$\rho$ be the state in $\mathcal{H}$ to be measured and $\rho_{0}$ be an
auxiliary state in $\mathcal{H}_{0}$ to be chosen; then a simple measurement
of $\rho\otimes\rho_{0}$ can be understood as a "randomized" measurement of
$\rho$. These randomized measurements correspond to nonorthogonal resolutions
like (35).

It can be shown that $\tilde{Q}$ and $\tilde{P}$ commute, which means that the
observables $\tilde{Q}$, $\tilde{P}$ are jointly measurable in the system
given by $\mathcal{H\otimes H}_{0}$. The operators $\tilde{Q}$, $\tilde{P}$
are self-adjoint and thus generate jointly distributed real valued random
variables $X_{\tilde{Q}}$, $X_{\tilde{P}}$. We define $\tilde{X}$ as follows

$\tilde{X}=\left(X_{\tilde{Q}},X_{\tilde{P}}\right)$ (39)

and it can be checked that if
$\rho=\left|\psi_{\mu}\right\rangle\left\langle\psi_{\mu}\right|$ and the
auxiliary state $\rho_{0}$ is the vacuum
$\rho_{0}=\left|\psi_{0}\right\rangle\left\langle\psi_{0}\right|$ then the
distribution of $\tilde{X}$ coincides with the distribution of $X$ (obtained
with the covariant measurement), i.e. with
$\mathcal{N}_{2}\left(\mu,I_{2}\right)$. Similarly when
$\rho=\mathbb{N}(\mu,\sigma^{2}_{\beta}I_{2})$ then the distribution of
$\tilde{X}$ coincides with
$\mathcal{N}_{2}\left(\mu,\frac{(2\sigma_{\beta}^{2}+1)}{2}I_{2}\right)$.

Next, for the state $\mathbb{N}(\mu,\sigma^{2}I_{2})$, we consider the problem
of Bayes estimation with quadratic loss of the parameter
$\mu\in\mathbb{R}^{2}$ under a normal prior
$\mu\sim\mathcal{N}_{2}\left(0,\sigma_{0}^{2}I_{2}\right)$.The solution for a
quadratic risk is given in [33], p. 55, with details and proofs in [31].
Consider the loss function

$L\left(\hat{\mu},\mu\right)=\left(\hat{\mu}_{1}-\mu_{1}\right)^{2}+\left(\hat{\mu}_{2}-\mu_{2}\right)^{2}.$

The observable which is optimal in the Bayes sense (along with the equivalent
POVM) is discussed in [30], Section 3. By Proposition 3 there, the optimal
POVM is given by (denoting $m=(x,y)$)

$M_{c}\left(B\right)=\frac{1}{2\pi
c^{2}}\int_{B}\left|\psi_{m/c}\right\rangle\left\langle\psi_{m/c}\right|dm\text{,
}B\in\mathfrak{B}\text{.}$

From (35) and a change of variables, it can be easily verified that

$\frac{1}{2\pi
c^{2}}\int_{\mathbb{R}^{2}}\left|\psi_{m/c}\right\rangle\left\langle\psi_{m/c}\right|dm=\mathbf{1}$

and hence $M_{c}$ is a resolution of identity (the other properties of a POVM
are also easy to check). Here
$c=2\sigma_{0}^{2}/(2\sigma_{0}^{2}+2\sigma^{2}+1)$ and the Bayes risk is
given by (writing $\sigma^{2}$ as $\sigma^{2}_{\beta}$)

$\inf_{M}R(M,\pi)=\frac{2\sigma_{0}^{2}(2\sigma^{2}_{\beta}+1)}{2(\sigma_{0}^{2}+\sigma^{2}_{\beta})+1}.$
(40)

When $\sigma^{2}=1/2$ (i.e. the state is pure), we have
$c=\frac{\sigma_{0}^{2}}{\sigma_{0}^{2}+1}$ and the Bayes risk according to
(17) in [30] thus becomes

$\inf_{M}R(M,\pi)=\frac{2\sigma_{0}^{2}}{\sigma_{0}^{2}+1}.$ (41)

The randomized measurements are then given by

$\tilde{Q}_{c}=c(Q\otimes\mathbf{1}_{0}+\mathbf{1}\otimes
Q_{0}),\quad\tilde{P}_{c}=c(P\otimes\mathbf{1}_{0}-\mathbf{1}\otimes P_{0}),$
(42)

cf. equation (12) of [30]; it can easily be shown that these randomized
measurements and $M_{c}$ generate the same random variables.

From (42) it is clear that
$\tilde{X}_{c}=(X_{\tilde{Q}_{c}},X_{\tilde{P}_{c}})$ satisfies
$\tilde{X}_{c}=c\tilde{X}$ where $\tilde{X}$ is given in (39). As $c<1$, we
note that Bayes estimation in the quantum case exhibits the same shrinkage
phenomenon as witnessed in the classical counterpart.

However we note that the risk in (41) is greater than the Bayes risk in the
classical model $\mathcal{N}_{2}(\mu,I_{2}/2)$ with the same prior, i.e. we
obtain $2\sigma_{0}^{2}/(\sigma^{2}_{0}+1)$ in the quantum case compared to
$2\sigma_{0}^{2}/(2\sigma^{2}_{0}+1)$ in the classical case. A similar
statement holds for the thermal state as well. This inflation of risk is
essentially due to the quantum nature of the data, in particular to the non-
commutativity of the observables $Q$ and $P$ (as a consequence of which the
random variables generated by $Q$ and $P$ separately do not admit a joint
distribution).

##  5 Optimal estimation of the low-rank state

In this section, we outline a two-stage procedure to construct an estimator
which is asymptotical minimax optimal. We split the sample into two parts. The
first part has sample size $\lfloor n^{\delta}\rfloor$ and is used to obtain a
preliminary estimator which lies within a ball of shrinking radius, centered
at the original state, with high probability. Then we use the LAN result,
obtained in the previous section, on the remaining $n-\lfloor
n^{\delta}\rfloor$ observations to construct an estimator which is sharp
minimax optimal.

For notational convenience, we work with $n$ instead of $\lfloor
n^{\delta}\rfloor$ while showing the properties of the preliminary estimator
and replace $n$ with $\lfloor n^{\delta}\rfloor$ in the final computations.

###  5.1 Preliminary estimator

Rank penalized and "physical" estimators have been constructed in [12] in the
context of estimation of the joint quantum state of k two-dimensional systems
(qubits) i.e. in a special case of our setting where $d=2^{k}$. The authors
obtain a least square estimator constructed using Pauli observables and then
use rank penalization and spectral thresholding to obtain rank penalized and
physical estimator, respectively. Not only do these estimators lie within a
ball of radius $n^{-1/2+\epsilon}$ (for $\epsilon<1/2$) centered at the
original state with high probability; they are also rank consistent in the
sense that the ranks of estimated states match the rank of the original state
with high probability.

The general $d$ dimensional case has been considered in [27], where the
authors construct a least squares estimator using a so-called uniform POVM and
project it into the space of density matrices to obtain an estimator which
lies within a ball of radius $n^{-1/2+\epsilon}$ (for arbitrary small
$\epsilon$) centered at the original state with high probability. Since the
authors are interested in the error of estimation in the trace norm, the rank
consistency is not discussed in the paper. We show the least squares estimator
of [27], when processed suitably like the physical estimator in [12], enjoys
the rank consistency property. We note that in order to use LAN we need a
local parametrization around a preliminary estimate of the state; it is
essential that its rank matches that of the original state and hence rank
consistency is crucial. The following theorem (proved in Appendix C)
summarizes the above discussion.

######  Theorem 5.1.

Let $\rho$ be a rank $r$ state in $\mathbb{C}^{d}$ with the minimum eigenvalue
$\lambda_{r}>6\epsilon$. Then there exists an estimator $\tilde{\rho}^{n}$ of
$\rho$ such that the following concentration inequality holds:

$P[||\rho-\tilde{\rho}^{n}||^{2}_{2}\geq 25r\epsilon^{2}]\leq
de^{-3n\epsilon^{2}/16d}.$

Further, if $\hat{r}$ is the rank of the state $\tilde{\rho}^{n}$, then

$P[r=\hat{r}]\geq 1-de^{-3n\epsilon^{2}/16d}.$

###  5.2 Optimal risk in the Gaussian model

Here we consider minimax estimation in the limiting Gaussian model given in
(21). Recall that the local parameter $\theta$ has two components: $u$
corresponds to the classical part and $\mathfrak{z}$ corresponds to the
quantum part. First, we define the loss function with respect to which the
risk will be computed:

$\mathcal{L}(\theta,\hat{\theta})=\sum_{i=1}^{r-1}(u_{i}-\hat{u}_{i})^{2}+(\sum_{i=1}^{r-1}(u_{i}-\hat{u}_{i}))^{2}+2\sum_{\begin{subarray}{c}{1\leq
i\leq r}\\\ {i<j\leq
d}\end{subarray}}(\mu_{i}-\mu_{j})|\mathfrak{z}_{ij}-\hat{\mathfrak{z}}_{ij}|^{2}.$
(43)

Although the particular form of loss may seem arbitrary, we will see that the
$L^{2}$ distance between two density matrices is approximately locally
quadratic and exactly given by this loss. The following theorem gives the
asymptotic minimax bound for estimation of $\theta$:

######  Theorem 5.2.

Let $\Phi^{\theta,r}$ be the limiting Gaussian model given in (21), then

$\lim_{n\rightarrow\infty}\inf_{\hat{\theta}}\sup_{\theta\in\Theta_{n,r,\beta,\gamma}}E_{\theta}[\mathcal{L}(\theta,\hat{\theta})]=\lim_{n\rightarrow\infty}\inf_{M}\sup_{\theta\in\Theta_{n,r,\beta,\gamma}}\int\mathcal{L}(\theta,\hat{\theta})\mathrm{Tr}(\Phi^{\theta,r}M(d\hat{\theta}))=\sum_{i=1}^{r}\mu_{i}(1-\mu_{i})+\sum_{\begin{subarray}{c}{1\leq
i\leq r}\\\ {i<j\leq d}\end{subarray}}2\mu_{i}.$

Note that the limiting model is a tensor product of classical and quantum
shifted Gaussian states. To show the upper bound we use a classical estimator
and a POVM for the classical and quantum parts respectively. For the classical
part, we use $Z\sim\mathcal{N}_{r-1}(u,V_{\mu})$, as the estimator $\hat{u}$.
For the quantum part, we note that each component of the tensored quantum
Gaussian state is either a shifted thermal state or a shifted pure state
indexed by $(i,j)$. We will use the covariant measurement $M$ (see Section 4)
on each component and hence the joint POVM will be given by
$\bar{M}=\bigotimes_{\begin{subarray}{c}{1\leq i\leq r}\\\ {i<j\leq
d}\end{subarray}}M$.

We use a Bayes risk to give a lower bound the minimax risk, by setting up
Gaussian priors for the parameters in both the classical and quantum parts.
The Bayes estimator for the classical part is well known and is given by
shrinkage of the Gaussian random variable. For the quantum part, we directly
compute the Bayes risk using equations (40) or (41) depending on whether the
component is a thermal or pure Gaussian state. We again note that the optimal
POVMs for each component are obtained by appropriately shrinking covariant
measurements. We defer the proof to the appendix.

###  5.3 Localization and the optimal risk

Finally, we are ready to compute the exact asymptotics of the minimax risk in
the low-rank qudit model. First, we state a lemma (proved in Appendeix D) that
shows that the $L^{2}$ distance between two states is approximately a
quadratic loss in the local parameters.

######  Lemma 5.3.

Let $\rho_{\theta_{1},n}$ and $\rho_{\theta_{2},n}$ be two states indexed by
the local parameters $\theta_{1}$ and $\theta_{2}$, where
$\theta_{1},\theta_{2}\in\Theta$, then

$||\rho_{\theta^{(1)}/\sqrt{n},r}-\rho_{\theta^{(2)}/\sqrt{n},r}||^{2}_{2}=\frac{1}{n}\sum_{i=1}^{r}(u^{(1)}_{i}-u^{(2)}_{i})^{2}+\frac{2}{n}\sum_{\begin{subarray}{c}{1\leq
i\leq r}\\\ {i<j\leq
d}\end{subarray}}(\mu_{i}-\mu_{j})|\mathfrak{z}^{(1)}_{ij}-\mathfrak{z}^{(2)}_{ij}|^{2}+O\left(\frac{||\theta^{(1)}||^{3},||\theta^{(2)}||^{3}}{n^{3/2}}\right)$

where $u^{(i)}_{r}=1-\sum_{k=1}^{r-1}u^{(i)}_{k}$ for $i=1,2$.

In subsection 5.1 we have used the first part of the sample to construct a
preliminary estimator $\tilde{\rho}_{n}$ which lies within an
$n^{-1/2+\epsilon}$ ball of the original state $\rho$. We can then parametrize
$\rho$ using the local parameter $\theta$ with $\tilde{\rho}_{n}$ as
$\rho_{0}$, i.e we let $\rho=\rho_{\theta}$ and
$\hat{\rho}_{n}=\rho_{\hat{\theta}}$. Now it is sufficient to estimate
$\theta$ using $\hat{\theta}_{n}$. Let $\mu_{1}>\mu_{2}>\ldots>\mu_{r}>0$ be
the eigenvalues of a density matrix $\sigma$ of rank $r$ and
$||\rho-\sigma||_{2}<n^{-1/2+\varepsilon}$. Since (33) holds one can easily
transfer the risk between the i.i.d and Gaussian models invoking (28) and
(30), i.e we have

$\limsup_{n\rightarrow\infty}\sup_{||\rho-\sigma||_{2}<n^{-1/2+\varepsilon}}E_{\rho}n||\rho-\hat{\rho}^{n}||_{2}^{2}\leq\sum_{i=1}^{r}\mu_{i}(1-\mu_{i})+\sum_{\begin{subarray}{c}{1\leq
i\leq r}\\\ {i<j\leq d}\end{subarray}}2\mu_{i}.$ (44)

The last part uses the upper bound of Theorem 5.2 and the continuity of
eigenvalues, i.e the eigenvalues of $\sigma$ and the central state
$\tilde{\rho}^{n}$ are asymptotically equal (see Appendix C for details). For
the lower bound we choose an arbitrary $\sigma$ and parametrize the state and
its estimate by local parameters $\theta$ and $\hat{\theta}$, i.e we let
$\rho=\rho_{\theta}$ and $\hat{\rho}_{n}=\rho_{\hat{\theta}}$. Since (34)
holds, we can transfer the risk between the i.i.d and Gaussian models invoking
(28) and (30) and then using the lower bound of Theorem 5.2 we have

$\liminf_{n\rightarrow\infty}\inf_{\hat{\rho}_{n}}\sup_{||\rho-\sigma||_{2}<n^{-1/2+\varepsilon}}E_{\rho}n||\rho-\hat{\rho}^{n}||_{2}^{2}\geq\sum_{i=1}^{r}\mu_{i}(1-\mu_{i})+\sum_{\begin{subarray}{c}{1\leq
i\leq r}\\\ {i<j\leq d}\end{subarray}}2\mu_{i}.$ (45)

We note that since the dimension $d$ is kept constant the $||.||_{1}$ and
$||.||_{2}$ norms are equivalent and hence we can easily replace the $L^{2}$
neighborhood with a trace norm neighborhood. We define the trace norm
neighborhood as follows:

$\Sigma_{n,\varepsilon}\left(\sigma\right):=\left\\{\rho\text{ state in
}\mathcal{H}:\left\|\rho-\sigma\right\|_{1}\leq
n^{-1/2+\varepsilon}\right\\}.$ (46)

The upper bound and the lower bound can now be summarized in the following
theorem.

######  Theorem 5.4.

Let $\mu_{1}>\mu_{2}>\ldots>\mu_{r}>0$ be the eigenvalues of a density matrix
$\sigma$ of rank $r$. Then there exists an $\varepsilon<1/2$ such that

$\lim_{n\rightarrow\infty}\inf_{\hat{\rho}_{n}}\sup_{\rho\in\Sigma_{n,\varepsilon}\left(\sigma\right)}E_{\rho}n||\rho-\hat{\rho}^{n}||^{2}=\sum_{i=1}^{r}\mu_{i}(1-\mu_{i})+\sum_{\begin{subarray}{c}{1\leq
i\leq r}\\\ {i<j\leq d}\end{subarray}}2\mu_{i}.$ (47)

##  6 Optimal estimation of linear functional of a low-rank state

We first discuss the classical problem of estimation of a linear functional of
a density. Consider i.i.d observations $X_{1},\ldots,X_{n}$ having density $f$
on $[0,1]$, and consider a bounded linear functional

$\Psi\left(f\right)=\int\varphi\left(x\right)f(x)dx$ (48)

where $\varphi$ is a bounded function on [0,1]. The estimator

$\hat{\Psi}_{n}=n^{-1}\sum_{i=1}^{n}\varphi\left(X_{i}\right)$

satisfies the CLT

$n^{1/2}\left(\hat{\Psi}_{n}-\Psi\left(f\right)\right)\leadsto
N\left(0,V_{f}^{2}\right).$

where

$V_{f}^{2}=\mathrm{Var}_{f}\left(\varphi\left(X\right)\right).$ (49)

One can show by LAN type results that $\hat{\Psi}_{n}$ is asymptotically
optimal (see [38]), i.e. $V_{f}^{2}$ is the best possible variance in a local
asymptotic minimax sense. In this section, we discuss an analogous result for
the quantum case.

Suppose $\rho$ is a state in a d-dimensional Hilbert space $\mathcal{H}$ of
rank $r$ and consider the functional

$\Psi\left(\rho\right)=\mathrm{Tr}\;(A\rho)$ (50)

where $A$ is a self-adjoint matrix. We will denote the operator norm of $A$ by
$||A||$. Let $X_{A}$ be the r.v. generated by $A$; then recall that the
probability distribution of $X_{A}$ is given by (2) and we denote the
expectation by $E_{\rho}X_{A}=\left\langle A\right\rangle_{\rho}$. Also the
variance of $X_{A}$ is given by

$\displaystyle\mathrm{Var}_{\rho}\left(X_{A}\right)$
$\displaystyle=\left\langle A^{2}\right\rangle_{\rho}-\left\langle
A\right\rangle_{\rho}^{2}=\left\langle\tilde{A}^{2}\right\rangle_{\rho}=:V_{\rho}^{2}\text{
where}$ (51) $\displaystyle\tilde{A}$ $\displaystyle:=A-\left\langle
A\right\rangle_{\rho}\mathbf{1}.$

Let $\Sigma_{n,\varepsilon}\left(\rho_{0}\right)$ be a shrinking neighborhood
in trace norm around a certain rank $r$ state $\rho_{0}$ defined as in (46).
We aim to show that with a sample $X_{1},\ldots,X_{n}$, the estimator
$\bar{X}_{n}$ attains the asymptotic variance $V_{\rho_{0}}^{2}$ uniformly
over $\rho\in\Sigma_{n,\varepsilon}\left(\rho_{0}\right)$. For the purpose of
proving an asymptotic minimax theorem, consider the expression

$\sup_{\rho\in\Sigma_{n,\varepsilon}\left(\rho_{0}\right)}E_{\rho}\left(n^{1/2}\left(\bar{X}_{n}-\Psi\left(\rho\right)\right)\right)^{2}.$

We will show that this expression converges to $V_{\rho_{0}}^{2}$.

To give a lower bound to the risk, we will have to find, for a $d$-dimensional
center state $\rho_{0}$ and a parametrization of all $d$-dimensional nearby
states, a "least favorable parametric subfamily" so that the information bound
for estimating $\Psi\left(\rho\right)$ along this family coincides with
$V_{\rho_{0}}^{2}$. We consider the following local model indexed by the
single parameter $t$:

$\rho_{t}=\rho_{0}+n^{-1/2}tH\text{, \ \
}t\in\left(-n^{\varepsilon},n^{\varepsilon}\right)$ (52)

where the matrix $H$ is suitably chosen. The problem of estimation of the
functional $\Psi$ can be shown to be equivalent to the estimation of the
parameter $t$ in the limiting model of the form
$\mathfrak{N}(t\tau,\mathscr{S})$ (see (24)) for appropriately chosen $\tau$
and $\mathscr{S}$. We then use a Bayesian result from [31] to establish a
minimax bound for the estimation of $t$ in the limiting model. We have the
following theorem, the proof of which is deferred to Appendix C.

######  Theorem 6.1.

Let $\Psi$ be the functional given in (50); then there exists an
$\varepsilon<1/2$ such that

$\liminf_{n}\inf_{\hat{\Psi}}\sup_{\rho\in\Sigma_{n,\varepsilon}\left(\rho_{0}\right)}E_{\rho}\left(n^{1/2}\left(\hat{\Psi}-\Psi\left(\rho\right)\right)\right)^{2}=V_{\rho_{0}}^{2}$
(53)

where $\inf_{\hat{\Psi}}$ denotes the infimum over all measurements and all
estimators $\hat{\Psi}$.

##  7 Discussion

We have obtained a generalization of the LAN result of [35] in this paper.
Since a local asymptotic equivalence result was proved in [13] for rank 1
states in the infinite dimensional case, a natural question to ask is what is
the limiting model in the case of an ensemble of rank $r$ states if the
Hilbert space is infinite dimensional. We conjecture that the limiting state
will consist of three parts - a classical multivariate normal, tensor product
of $r(r-1)/2$ shifted thermal states and a tensor product of infinitely many
shifted pure states. Although this result is outside the scope of this paper,
we observe that a restriction of the conjectured model to finite dimension
will correspond to our result while a restriction to rank 1 case (in infinite
dimension) will correspond to the result obtained in [13].

Another interesting phenomenon occurs in the estimation of a functional where
the limiting model is a tensor product of classical and quantum states indexed
by a one-dimensional parameter. It is well known that if the loss function
$L(\hat{\mu},\mu)$ in section 4 is replaced by a weighted loss function
$L_{g}(\hat{\mu},\mu)$ given by

$L_{g}\left(\hat{\mu},\mu\right)=g_{1}\left(\hat{\mu}_{1}-\mu_{1}\right)^{2}+g_{2}\left(\hat{\mu}_{2}-\mu_{2}\right)^{2},$

then for $g_{1}=1$ and $g_{2}=0$, the POVM induced by $Q$ suffices for optimal
estimation, i.e. the Bayes risk is obtained by a simple measurement (the
spectral measure of $Q$) and the associated Bayes risk matches the one with
the classical bivariate normal distribution
$\mathcal{N}_{2}(\mu,\sigma^{2}I_{2})$. Analogously the risk in the limiting
model for the functional estimation matches the same for a related classical
model (see the remark after the proof of Theorem 6.1 in Appendix C). This is
in sharp contrast with the risk observed in the measurement of the shift
parameter of the coherent or thermal state ($g_{1}=g_{2}=1$ in the expression
of $L_{g}(\hat{\mu},\mu)$), where we observe an inflation of the risk compared
to the classical case (see the last paragraph of section 4). However, we
emphasize that the classical-quantum Gaussian shift model indexed by a one-
dimensional parameter is not equivalent to a classical model (in terms of Le
Cam equivalence). A discussion of this phenomenon on a similar model can be
found in [26] where the authors argue that although measuring the position
observable results in a classical model whose "classical Fisher information"
matches that of the quantum Fisher information of the original model, optimal
testing procedures in the classical models are suboptimal in the sense that
the associated risk is higher than that of the optimal testing procedure of
the original quantum model. A more in-depth discussion on this in terms of
sufficient statistics and Le Cam equivalence can be found in [25].

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##  Appendix A Representation theoretic tools

Our main tool in proving Theorem 3.3 will be the representation of the special
unitary group $SU(d)$. We first define the representations of the group
$GL(d)$ and $S(n)$ (the group of invertible matrices and permutations
respectively) on $(\mathbb{C}^{d})^{\otimes n}$. Let
$f_{\mathbf{a}}=f_{{\mathbf{a}_{1}}}\otimes\ldots\otimes
f_{{\mathbf{a}_{n}}}$,where $f_{1},\ldots,f_{d}$ are basis elements of
$\mathbb{C}^{d}$ and $\mathbf{a}_{i}\in\\{1,\ldots,d\\}$ (note that
$f_{\mathbf{a}}$’s span $(\mathbb{C}^{d})^{\otimes n}$) and consider the
following actions:

$\displaystyle\pi_{n}(T):f_{\mathbf{a}_{1}}\otimes\ldots\otimes
f_{\mathbf{a}_{n}}$ $\displaystyle\rightarrow
Tf_{\mathbf{a}_{1}}\otimes\ldots\otimes Tf_{\mathbf{a}_{n}},\quad T\in GL(d)$
(54)
$\displaystyle\tilde{\pi}_{n}(\sigma):f_{\mathbf{a}_{1}}\otimes\ldots\otimes
f_{\mathbf{a}_{n}}$ $\displaystyle\rightarrow
f_{\sigma^{-1}(\mathbf{a}_{1})}\otimes\ldots\otimes
f_{\sigma^{-1}(\mathbf{a}_{n})},\quad\sigma\in S(n).$ (55)

It can be shown that the representation space can be decomposed into direct
sum of subspaces each of which is a tensor product of irreducible
representations of $GL(d)$ and $S(n)$. This decomposition is called Schur-Weyl
duality in the literature (for a detailed account see [19, 21]).

$(\mathbb{C}^{d})^{\otimes
n}=\bigoplus_{\lambda}\mathcal{H}_{\lambda}\otimes\mathcal{K}_{\lambda}$ (56)

Here the direct sum is finite and each $\lambda$ corresponds to a Young
diagram. A Young diagram is defined by a ordered tuple of integers
$\lambda=(\lambda_{1},\ldots,\lambda_{d})$ with
$\lambda_{1}\geq\ldots\geq\lambda_{d}\geq 0$ and can be represented
graphically by a diagram with $d$ lines each containing $\lambda_{i}$ boxes.
For example a typical Young diagram looks like:

&

with $\lambda=(3,2,2)$. Also the operator $\pi_{n}(T)$ can be decomposed as
follows :

$\displaystyle\pi_{n}(T)=\bigoplus_{\lambda}\pi_{\lambda}(T)\otimes\mathbbm{1}_{\mathcal{K}_{\lambda}}$

i.e the tensor representation of $T\in GL(d)$ can be decomposed into blocks
$\pi_{\lambda}(T)$ acting on the irreducible representation
$\mathcal{H}_{\lambda}$ and these blocks have multiplicities equal to
dimension of $\mathcal{K}_{\lambda}$. Thus we can decompose the tensor product
of the states as follows

$\displaystyle\rho^{\otimes n}_{0,u,n}$
$\displaystyle=\bigoplus_{\lambda}\tilde{\rho}_{\lambda}^{0,u,n}\otimes\mathbbm{1}_{\mathcal{K}_{\lambda}}$
(57) $\displaystyle\rho^{\otimes n}_{0,u,n}$
$\displaystyle=\bigoplus_{\lambda}\left(\rho^{0,u,n}_{\lambda}\otimes\frac{\mathbbm{1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}\right)p_{n}^{\lambda}$
(58) $\displaystyle U(\mathfrak{z}/\sqrt{n})^{\otimes n}\rho^{\otimes
n}_{0,u,n}U^{*}(\mathfrak{z}/\sqrt{n})^{\otimes n}$
$\displaystyle=\bigoplus_{\lambda}\left(U_{\lambda}(\mathfrak{z}/\sqrt{n})\rho^{0,u,n}_{\lambda}U_{\lambda}^{*}(\mathfrak{z}/\sqrt{n})\otimes\frac{\mathbbm{1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}\right)p_{n}^{\lambda}.$
(59)

In equation (57) we have used the usual decomposition of the $\pi_{n}(T)$
while in equation (58) we have adjusted the factors $p_{n}^{\lambda}$ and
$dim(\mathcal{K}_{\lambda})$ (later we give the exact expressions for these
quantities) so that both $\rho^{0,u,r,n}_{\lambda}$ and
$\frac{\mathbbm{1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}$ are
states (i.e. they are normalized). Equation (59) follows from the fact that
$SU(d)$ is a subgroup of $GL(d)$ and its tensor representation has a similar
decomposition. Define

$\Delta^{\mathfrak{z},n}(A)=U(\mathfrak{z}/\sqrt{n})AU^{*}(\mathfrak{z}/\sqrt{n})$

and $\Delta^{\mathfrak{z},n}_{\lambda}$ in a similar fashion. Then the last
decomposition can be written in a compact fashion as

$\rho^{\otimes
n}_{\theta,n}=\bigoplus_{\lambda}\left(\Delta^{\mathfrak{z},n}_{\lambda}(\rho^{0,u,n}_{\lambda})\otimes\frac{\mathbbm{1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}\right)p_{n}^{\lambda}.$

Next we describe the subspaces $\mathcal{H}_{\lambda}$ in detail. Let
$\mathcal{A}(S(n))$ denote the group algebra, i.e. the linear space spanned by
elements of the form $\sum_{\sigma}c_{\sigma}\sigma$ for complex numbers
$c_{\sigma}$. Also define multiplication between its elements in the usual way
i.e

$\displaystyle a=\sum_{\sigma\in S(n)}a_{\sigma}\sigma,\quad b=\sum_{\tau\in
S(n)}b_{\sigma}\sigma$ $\displaystyle ab=\sum_{\sigma\in
S(n)}\left(\sum_{\tau\in S(n)}a_{\sigma\tau^{-1}}b_{\tau}\right)\sigma.$

Let $\lambda$ be a Young diagram with $|\lambda|=\sum_{i}\lambda_{i}=n$. We
define two subgroups $\mathcal{R}_{\lambda}$ and $\mathcal{C}_{\lambda}$ of
$S(n)$ defined as follows

$\displaystyle\mathcal{R}_{\lambda}$ $\displaystyle=\\{\sigma:\sigma\text{
leaves the rows of }\lambda\text{ unaltered }\\}$
$\displaystyle\mathcal{C}_{\lambda}$ $\displaystyle=\\{\sigma:\sigma\text{
leaves the columns of }\lambda\text{ unaltered }.\\}$

Now consider the action of $\tilde{\pi}_{n}(\sigma)$ (henceforth we will
denote this operator as $\sigma$) given in relation (55) and extend it to the
group algebra. Next we define the following operators

$p_{\lambda}=\sum_{\sigma\in R_{\lambda}}\sigma,\quad
q_{\lambda}=\sum_{\sigma\in C_{\lambda}}sgn(\sigma)\sigma,\quad
y_{\lambda}=q_{\lambda}p_{\lambda}.$

It can be shown that the $\\{y_{\lambda}f_{\mathbf{a}}\\}$ spans the subspace
$\mathcal{H}_{\lambda}$. Of course the elements in
$\\{y_{\lambda}f_{\mathbf{a}}\\}$ are not linearly independent. To describe a
basis for $\mathcal{H}_{\lambda}$ we need the concept of Young tableaux.

A Young tableau is a Young diagram filled with integers. A semistandard Young
tableau is a Young diagram filled with numbers from $1$ to $d$ such that the
numbers are non-decreasing in a row and strictly increasing in a column, for
example:

$1$&$1$ $1$ $2$ $2$$3$ $4$ $4$

is a semistandard Young tableau. Now if we fill a Young tableau with the
integers $\mathbf{a}_{1},\ldots,\mathbf{a}_{n}$ along the rows then for every
$f_{\mathbf{a}}$ we get a corrsponding Young tableau $t_{\mathbf{a}}$. As an
example we associate the following vector with the semistandard Young tableau
shown above - $f_{1}\otimes f_{1}\otimes f_{1}\otimes f_{2}\otimes
f_{3}\otimes f_{4}\otimes f_{4}$. Define the set

$\mathfrak{A}=\\{\mathbf{a}:t_{\mathbf{a}}\text{ a semistandard Young tableau
}\\}.$

It can be shown that
$\\{y_{\lambda}f_{\mathbf{a}}\\}_{\mathbf{a}\in\mathfrak{A}}$ forms a basis of
$\mathcal{H}_{\lambda}$.

An alternative notation (also used in [35]) is to use
$\mathbf{m}=\\{m_{i,j}\\}_{1\leq i<j\leq d}$ where $m_{i,j}$ is the number of
$j$’s in row $i$. Note that if $\mathbf{a}\in\mathfrak{A}$, then $\mathbf{m}$
uniquely identifies $\mathbf{a}$. For the example above
$m_{1,2}=m_{2,3}=m_{2,4}=m_{3,4}=1$ and the rest $m_{i,j}$ are $0$.

Now denote
$|\mathbf{m}_{\lambda}\rangle=y_{\lambda}f_{\mathbf{m}}/||y_{\lambda}f_{\mathbf{m}}||$.
Mapping these basis elements of $\mathcal{H}_{\lambda}$ to the basis elements
of the multimode Fock space will be a key element in constructing the channel.

From the definition of $p_{\lambda}$ and $q_{\lambda}$ it can be easily shown
that

$p_{\lambda}^{2}=(\prod_{i=1}^{d}\lambda_{i}!)p_{\lambda},\quad
q_{\lambda}^{2}=(\prod_{i=1}^{d}i^{\lambda_{i}-\lambda_{i+1}}!)q_{\lambda}$
(60)

and consequently

$\langle y_{\lambda}f_{\mathbf{a}}|y_{\lambda}f_{\mathbf{b}}\rangle=\langle
p_{\lambda}f_{\mathbf{a}}|q^{2}_{\lambda}p_{\lambda}f_{\mathbf{b}}\rangle=(\prod_{i=1}^{d}i^{\lambda_{i}-\lambda_{i+1}}!)\langle
p_{\lambda}f_{\mathbf{a}}|y_{\lambda}f_{\mathbf{b}}\rangle.$ (61)

We also consider the properties of the vector $|{\bf 0}_{\lambda}\rangle$
(which is the _highest weight vector_ of the representation
$(\pi_{\lambda},\mathcal{H}_{\lambda})$) corresponding to the semi-standard
Young tableau where all the entries in row $i$ are $i$. An immediate
consequence is that

$p_{\lambda}|f_{\bf 0}\rangle=(\prod_{i=1}^{d}\lambda_{i}!)|f_{\bf 0}\rangle.$
(62)

Moreover $\langle f_{\bf 0}|q_{\lambda}f_{\bf 0}\rangle=1$ since any column
permutation produces a vector orthogonal to $f_{\bf 0}$ and thus the
normalised vector is:

$|{\bf
0}_{\lambda}\rangle=\frac{1}{\prod_{i=1}^{d}\lambda_{i}!\sqrt{i^{\lambda_{i}-\lambda_{i+1}}}}y_{\lambda}|f_{\bf
0}\rangle.$ (63)

The finite-dimensional coherent states are defined as $\pi_{\lambda}(U)|{\bf
0}_{\lambda}\rangle$ for $U\in SU(d)$. Since
$[p_{\lambda},\pi_{\lambda}(U)]=0$ and (62) holds, we get
$p_{\lambda}\pi_{\lambda}(U)|f_{\bf
0}\rangle=(\prod_{i=1}^{d}\lambda_{i}!)\pi_{\lambda}(U)|f_{\bf 0}\rangle$.
Thus we have

$\displaystyle\langle y_{\lambda}f_{\bf m}|y_{\lambda}\pi_{\lambda}(U)f_{\bf
0}\rangle$ $\displaystyle=\langle p_{\lambda}f_{\bf
m}|q_{\lambda}^{2}p_{\lambda}\pi_{\lambda}(U)f_{\bf 0}\rangle$
$\displaystyle=\langle p_{\lambda}f_{\bf
m}|q_{\lambda}^{2}\pi_{\lambda}(U)p_{\lambda}f_{\bf 0}\rangle$
$\displaystyle=\prod_{i=1}^{d}i^{\lambda_{i}-\lambda_{i+1}}\lambda_{i}!\langle
p_{\lambda}f_{\bf m}|q_{\lambda}\pi_{\lambda}(U)f_{\bf 0}\rangle.$ (64)

##  Appendix B Bayes estimation of a one dimensional shift parameter

In this section, we adopt the framework for Gaussian states $\rho$ in [31] to
develop a Bayes result for estimation of $t$ in the model
$\left\\{\mathfrak{N}(t\tau,\mathscr{S}),t\in\mathbb{R}\right\\}$ (cf. (24)) .
In [31] the characteristic function of $\rho$ is given by

$\phi_{\rho}\left(z\right)=\exp\left(i\theta\left(z\right)-\frac{1}{2}\alpha\left(z,z\right)\right)\text{,
}z\in\mathbb{R}^{m}$ (65)

where $\alpha$ is a positive symmetric form on $\mathbb{R}^{m}$ and $\theta$
is a linear function on $\mathbb{R}^{m}$ ([31], III.2.4). Here the $\rho$ is a
state on the Weyl algebra generated by the Weyl unitaries $W\left(z\right)$,
$z\in\mathbb{R}^{m}$, which satisfy the canonical commutation relations

$W\left(z\right)W\left(w\right)=W\left(z+w\right)\exp\left(\frac{i}{2}\Delta\left(z,w\right)\right)$
(66)

([31] , III.1.3) where $\Delta$ is a bilinear antisymmetric (symplectic) form
on $\mathbb{R}^{m}$. The forms $\alpha$ and $\Delta$ must satisfy the
condition

$\frac{1}{4}\Delta\left(z,w\right)^{2}\leq\alpha\left(z,z\right)\alpha\left(w,w\right)\text{,
}z,w\in\mathbb{R}^{m}.$ (67)

This is equivalent to the following condition ([32], Theorem 5.5.1): for any
$r$ and $z_{1},\ldots,z_{r}$ $\in\mathbb{R}^{m}$, the two complex Hermitian
matrices below are positive semidefinite:

$\left(\alpha\left(z_{j},z_{k}\right)\pm\frac{i}{2}\Delta\left(z_{j},z_{k}\right)\right)_{j,k=1}^{r}\geq
0.$ (68)

Suppose $\alpha$ and $\Delta$ are given by $m\times m$ real matrices:

$\alpha\left(z,w\right)=z^{T}\Sigma w\text{ where }\Sigma>0$ (69)

(strictly positive definite) and

$\Delta\left(z,w\right)=z^{T}Dw$

where $D$ is antisymmetric. The symplectic form $\Delta$ is allowed to be
degenerate ([31], beginning of Chap III, §2), which means that $D$ is allowed
to be singular. Indeed $\Delta\equiv 0$, i.e. $D=0$ is included as the
classical case, where $\rho$ is identified with the $m$-variate normal
distribution $\mathcal{N}_{m}\left(\theta,\Sigma\right)$ (here we identified
the linear form $\theta\left(z\right)$ with a vector
$\theta\in\mathbb{R}^{m}$). The framework of an antisymmetric, possibly
singular $D$ now allows to include products of classical and quantum Gaussian
states.

It is easy to see that (68) is equivalent to

$\Sigma\pm\frac{i}{2}D\geq 0$ (70)

i.e. the above complex Hermitian matrices are positive semidefinite.

In [31], III, §2, (2.1) it is further noted that (under some conditions) one
has $W\left(z\right)=\exp\left(iR\left(z\right)\right)$ where
$R\left(z\right)$ is a self-adjoint operator (often called a "field
operator"), and from (66) it follows that

$\left[R\left(z\right),R\left(w\right)\right]=-i\Delta\left(z,w\right)\;\mathbf{1}.$

(the equality holding in appropriate domains). The case $m=2$ corresponds to

$\Delta\left(z,w\right)=z^{T}Dw\text{ where
}D=\Omega:=\left(\begin{array}[c]{cc}0&1\\\ -1&0\end{array}\right)$ (71)

which yields a one-mode Gaussian state. Assume $z=\left(1,0\right)$ and
$w=\left(0,1\right)$. Then $\Delta\left(z,w\right)=1$ and

$\left[R\left(\left(1,0\right)\right),R\left(\left(0,1\right)\right)\right]=-iz^{T}\Omega
w\;\mathbf{1}=-i\;\mathbf{1}$

and identifying $R\left(\left(1,0\right)\right)$ with $P$ and
$R\left(\left(0,1\right)\right)$ with $Q$, we obtain the standard commutation
relations

$\left[Q,P\right]=i\;\mathbf{1.}$

Then, since for field operators the mapping $z\rightarrow R\left(z\right)$ is
real-linear ([44]), we can write $R\left(z\right)=z_{1}P+z_{2}Q$, and the Weyl
unitaries take the form

$W\left(z\right)=\exp\left(i\left(z_{1}P+z_{2}Q\right)\right).$

It is well know that the characteristic function of the vacuum state in this
setting should be

$\phi_{\rho}\left(z\right)=\exp\left(-\frac{\left\|z\right\|^{2}}{4}\right)\text{,
}z\in\mathbb{R}^{2}.$

Comparing with (65), we see that
$\alpha\left(z,z\right)=\left\|z\right\|^{2}/2$ and hence the covariance
matrix $\Sigma\ $in (70) should be

$\Sigma=\frac{1}{2}I_{2}\text{.}$

We can check condition (70) for this $\Sigma$ and $D=\Omega$: considering the
equivalent condition (67), we have to verify

$\left(\frac{1}{2}z^{T}\Omega w\right)^{2}\leq z^{T}\Sigma z\;w^{T}\Sigma w$

or

$\left(z^{T}\Omega w\right)^{2}\leq\left\|z\right\|^{2}\left\|w\right\|^{2}.$

Since $\Omega^{T}\Omega=I_{2}$, we have $\left\|\Omega
z\right\|^{2}=\left\|z\right\|^{2}$, and setting $u=\Omega z$, the above
inequality is equivalent to

$\left(z^{T}u\right)^{2}\leq\left\|z\right\|^{2}\left\|u\right\|^{2},$

i.e. the Cauchy-Schwarz inequality, which becomes an equality for $u=z$. Hence
(70) is fulfilled for $\Sigma=I_{2}/2$, and for multiples of the unit matrix,
$1/2$ is the minimal factor so that $\Sigma$ can be a covariance matrix. This
agrees with the well-known minimum uncertainty property of the vacuum state;
the thermal states are then those with covariance matrix $\gamma I_{2}$ where
$\gamma>1/2$.

It is now obvious that an $n$-mode zero mean Gaussian state which is a product
of $n$ thermal (or vacuum) states is captured by the framework (65) as
follows: $m=2n$,

$\alpha\left(z,z\right)=z^{T}\Sigma z\text{ where
}\Sigma={\textstyle\bigoplus\nolimits_{j=1}^{n}}\gamma_{j}I_{2}\text{, }$

where $\gamma_{j}\geq 1/2$, $j=1,\ldots,n$ and

$\Delta\left(z,w\right)=z^{T}Dw\text{ where
}D={\textstyle\bigoplus\nolimits_{j=1}^{n}}\Omega$

with $\Omega$ from (71).

Since the form $\Delta$ is allowed to be degenerate, we can also consider the
product of the above with a classical $d$-variate normal distribution
$\mathcal{N}_{d}\left(\mu,V\right)$ by setting $m=2n+d$,

$\displaystyle\alpha\left(z,z\right)$ $\displaystyle=z^{T}\Sigma z\text{ where
}\Sigma=V\oplus{\textstyle\bigoplus\nolimits_{j=1}^{n}}\gamma_{j}I_{2}\text{,}$
(72) $\displaystyle\text{ }\Delta\left(z,w\right)$
$\displaystyle=z^{T}Dw\text{ where }D=0_{d\times
d}\oplus{\textstyle\bigoplus\nolimits_{j=1}^{n}}\Omega.$

A non-zero mean $\theta\in\mathbb{R}^{m}$ is simply accounted for by setting
$\theta\left(z\right)=\theta^{T}z$ in (65).

In [31], III.5 estimation of a one-dimensional shift parameter $t$ is
considered, which has a normal prior distribution with zero mean and variance
$b$. The family of Gaussian states is given by (65) where
$\theta\left(z\right)$ is a linear form on $\mathbb{R}^{m}$ given by
$\theta\left(z\right)=t\theta_{0}\left(z\right)$, $\theta_{0}\left(z\right)$
being a fixed linear form and $t\in\mathbb{R}$. Assume the linear form
$\theta_{0}\left(z\right)$ is given by a vector $\tau$ such that
$\theta_{0}\left(z\right)=\tau^{T}z$, then the family of Gaussian states will
have shift parameters $t\tau$, $t\in\mathbb{R}$ and fixed covariance function
$\alpha\left(z,w\right).$ Recall that according to (69) we have
$\alpha\left(z,w\right)=z^{T}\Sigma w$ where $\Sigma>0$; thus, in this setup
the characteristic function in (65) matches the characteristic function of
$\mathfrak{N}(t\tau,\mathscr{S}),t\in\mathbb{R}.$ Comparing (72) and (24), we
note that $S=D$ when $d=r-1$ and $n=r(r-1)/2+r(d-r)$.

In [31] the linear form $\theta_{0}\left(z\right)$ is represented by a vector
in a different way. First it is to be noted that in this section in [31] the
quadratic form $\alpha$ has acquired a different meaning from (65): the
$\alpha$ there is now written $\alpha_{0}$, and $\alpha$ is now defined as
(see (4.1) in III.4)

$\alpha\left(z,w\right):=\alpha_{0}\left(z,w\right)+\beta\left(z,w\right)$

where $\beta$ is the covariance function of the prior distribution, defined as

$\beta\left(z,w\right)=\int\theta\left(z\right)\theta\left(w\right)\pi\left(d\theta\right)$

where $\pi$ is the prior distribution and $d\theta$ represents integration
over linear forms (i.e. over vectors in $\mathbb{R}^{m}$). In the present
case, the prior distribution is given by $\theta\left(z\right)=t\tau^{T}z$
where $\tau$ is fixed and $t\sim N\left(0,b\right)$. Hence

$\beta\left(z,w\right)=E\left(tz^{T}\tau\right)\left(t\tau^{T}w\right)=Et^{2}\left(z^{T}\tau\tau^{T}w\right)=z^{T}\left(b\tau\tau^{T}\right)w.$

So we have a matrix representation of the quadratic form $\alpha$:

$\alpha\left(z,w\right)=z^{T}\Lambda w\text{, }\Lambda=\Sigma+b\tau\tau^{T}$

where $\Lambda>0$ since $\Sigma>0$.

Now the vector $\theta_{0}\in\mathbb{R}^{m}$ to represent the linear form
$\theta_{0}\left(z\right)$ is chosen such that

$\theta_{0}\left(z\right)=\alpha\left(\theta_{0},z\right).$

The relation to $\tau$ is given by

$\displaystyle\tau^{T}z$
$\displaystyle=\theta_{0}\left(z\right)=\theta_{0}^{T}\Lambda z\text{,
}z\in\mathbb{R}^{m},$ $\displaystyle\theta_{0}$
$\displaystyle=\Lambda^{-1}\tau.$

The loss function for estimation of $t$ is $g\left(\hat{t}-t\right)$ for some
$g>0$; we can set $g=1$ here. Then in [31] an optimal measurement for
estimating the parameter $t$ is found, and the Bayes risk is given by

$E\left(\hat{t}-t\right)^{2}=\frac{ba_{0}}{a_{0}+b}$ (73)

(after relation (5.3) of [31]), where $a_{0}=a-b$ and
$a=\alpha\left(\theta_{0},\theta_{0}\right)^{-1}$. Note that we have used the
condensed notation while stating the Bayes risk (see the discussion after the
statement of Bayes problem in subsection 2.4). In order to clarify the
dependence of the Bayes risk on the parameters $\tau$ and $\Sigma$ of the
Gaussian shift model, note that

$\displaystyle a^{-1}$
$\displaystyle=\alpha\left(\theta_{0},\theta_{0}\right)=\tau^{T}\Lambda^{-1}\Lambda\Lambda^{-1}\tau=\tau^{T}\Lambda^{-1}\tau$
$\displaystyle=\tau^{T}\left(\Sigma+b\tau\tau^{T}\right)^{-1}\tau.$

Define $v=\Sigma^{-1/2}\tau$, then

$\tau^{T}\left(\Sigma+b\tau\tau^{T}\right)^{-1}\tau=v^{T}\left(I_{m}+bvv^{T}\right)^{-1}v.$

Now defining a unit vector $e=v/\left\|v\right\|$ we have

$\displaystyle v^{T}\left(I_{m}+bvv^{T}\right)^{-1}v$
$\displaystyle=\left\|v\right\|^{2}e^{T}\left(I_{m}+\left\|v\right\|^{2}bee^{T}\right)^{-1}e$
$\displaystyle=\left\|v\right\|^{2}\left(1+\left\|v\right\|^{2}b\right)^{-1}.$

Consequently

$\displaystyle a$
$\displaystyle=\left\|v\right\|^{-2}\left(1+\left\|v\right\|^{2}b\right)=\left\|v\right\|^{-2}+b,$
$\displaystyle a_{0}$
$\displaystyle=a-b=\left\|v\right\|^{-2}=\left(\tau^{T}\Sigma^{-1}\tau\right)^{-1}.$

The Bayes risk (73) now takes the form

$E\left(\hat{t}-t\right)^{2}=\frac{\left(\tau^{T}\Sigma^{-1}\tau\right)^{-1}b}{\left(\tau^{T}\Sigma^{-1}\tau\right)^{-1}+b}.$
(74)

##  Appendix C Proof of the main theorems

###  C.1 Integral representation of Gaussian states

Recall the displacement operator $D(z)$ (and the coherent state
$D(z)|0\rangle$) defined in equation (10) and $D\left(\mathbf{z}\right)$ (and
the multimode coherent state
$D\left(\mathbf{z}\right)\left|\mathbf{0}\right\rangle$) in equation (16). We
adopt the notation

$|z\rangle=D(z)|0\rangle,\quad|\mathbf{z}\rangle=D\left(\mathbf{z}\right)\left|\mathbf{0}\right\rangle$

which will be the used in the following integral representation and subsequent
proofs.

An alternate representation of the thermal state is given by

$\phi_{\beta}=\frac{e^{\beta}-1}{\pi}\int\exp{-(e^{\beta}-1)|z|^{2}})|z\rangle\langle
z|dz$

i.e. a Gaussian mixture of $|z\rangle\langle z|$’s with the mixture
distribution having mean $0$ and covariance matrix
$\frac{1}{2(e^{\beta}-1)}I_{2}$. Similarly we have the following integral
representation for shifted thermal states:

$\phi^{z_{0}}_{\beta}=D(z_{0})\phi_{\beta}D^{*}(z_{0})=\frac{e^{\beta}-1}{\pi}\int\exp{-(e^{\beta}-1)|z|^{2}})D(z_{0})|z\rangle\langle
z|D^{*}(z_{0})dz.$

Generalizing to the multimodal case, the tensor product of shifted thermal
states given in (20) can be written as:

$\phi^{\mathbf{\mathfrak{z}}}=\prod_{1\leq i<j\leq
d}\frac{e^{\beta_{ij}}-1}{\pi}\int\exp\left(-\sum_{1\leq i<j\leq
d}\frac{e^{\beta_{ij}}-1}{\pi}|z_{ij}|^{2}\right)|\mathfrak{z}+\mathbf{z}\rangle\langle\mathfrak{z}+\mathbf{z}|d\mathbf{z}.$

Similarly we have the following integral representation of the tensor product
of shifted thermal states and shifted pure states given in (2.3):

$\displaystyle\phi^{\mathbf{\mathfrak{z},r}}$ $\displaystyle=\bigotimes_{1\leq
i<j\leq
r}\phi_{\beta_{ij}}^{\mathfrak{z}_{ij}}\otimes\bigotimes_{\begin{subarray}{c}1\leq
i\leq r\\\ r+1\leq j\leq
d\end{subarray}}\phi_{\infty}^{\mathfrak{z}_{ij}}\quad\in\mathcal{T}_{1}(\mathcal{F}^{r})$
$\displaystyle=\left(\prod_{1\leq i<j\leq
r}\frac{e^{\beta_{ij}}-1}{\pi}\int\exp\left(-\sum_{1\leq i<j\leq
r}\frac{e^{\beta_{ij}}-1}{\pi}|z_{ij}|^{2}\right)|\mathfrak{z}_{1}+\mathbf{z}\rangle\langle\mathfrak{z}_{1}+\mathbf{z}|d\mathbf{z}\right)\otimes|\mathbf{\mathfrak{z}_{2}}\rangle\langle\mathbf{\mathfrak{z}_{2}}|.$
(75)

Here $\mathfrak{z}_{1}=(\mathfrak{z}_{ij})_{1\leq i<j\leq r}$ is a vector in
$\mathbb{C}^{r(r-1)/2}$ and $\mathfrak{z}_{2}=(\mathfrak{z}_{ij})_{1\leq i\leq
r,r+1\leq j\leq d}$ is a vector in $\mathbb{C}^{r(d-r)}$. Note that we have
used the notation $|\mathfrak{z}_{1}+\mathbf{z}\rangle$ to denote the vector
as an element of $\mathcal{F}^{r,1}$ and $|\mathfrak{z}_{2}\rangle$ as element
of $\mathcal{F}^{r,2}$, where

$\mathcal{F}^{r,1}:=\bigotimes_{1\leq i<j\leq
r}L^{2}(\mathbb{R}),\quad\mathcal{F}^{r,2}:=\bigotimes_{\begin{subarray}{c}1\leq
i\leq r\\\ r+1\leq j\leq d\end{subarray}}L^{2}(\mathbb{R}).$

Henceforth we will use this notation without mentioning the underlying spaces
$\mathcal{F}^{r,1}$ and $\mathcal{F}^{r,2}$ when there is no chance of
confusion. Following (14) we will define the number basis for
$\mathcal{F}^{r},\mathcal{F}^{r,1},\mathcal{F}^{r,2}$ as
$|\mathbf{m}\rangle,|\mathbf{m}_{1}\rangle$ and $|\mathbf{m}_{2}\rangle$,
respectively. In particular, we have

$\displaystyle|\mathbf{m}\rangle$ $\displaystyle=\otimes_{1\leq i\leq
r,i<j\leq d}|m_{ij}\rangle,\quad\mathbf{m}=\\{m_{ij}\in\mathbb{N}:1\leq i\leq
r,i<j\leq d\\}$ $\displaystyle|\mathbf{m}_{1}\rangle$
$\displaystyle=\otimes_{1\leq i<j\leq
r}|m_{ij}\rangle,\quad\mathbf{m}_{1}=\\{m_{ij}\in\mathbb{N}:1\leq i<j\leq
r\\}$ $\displaystyle|\mathbf{m}_{2}\rangle$ $\displaystyle=\otimes_{1\leq
i\leq r,r+1\leq j\leq
d}|m_{ij}\rangle,\quad\mathbf{m}_{2}=\\{m_{ij}\in\mathbb{N}:1\leq i\leq
r,r+1\leq j\leq d\\}.$

###### Proof of Theorem 3.3.

Construction of the channel $T^{r}_{n}$

We will follow [35] to contsruct the channel. However we first state the
redundancy of some representations in the low dimensional case. We will see
that it is because of these redundancies that the mixture of pure and thermal
states arise in the limit. In particular when stated for the low dimensional
case some of the representations (those with Young diagram containing more
than $r$ rows) will vanish from the sum (57) or (59). Further the $m_{i,j}$s
with $1\leq i<j\leq r$ will determine the central state and the rotation,
while the $m_{i,j}$s with $1\leq i\leq r$ and $r+1\leq j\leq d$ will
contribute only to rotation. The former will correspond to the thermal states
in the limit while the latter will give rise to pure states.

We start by noting that
$f_{\mathbf{a}}=f_{{\mathbf{a}_{1}}}\otimes\ldots\otimes f_{{\mathbf{a}_{n}}}$
is an eigenvector of $\rho^{\otimes n}_{0,u,n}$ with eigenvalues
$\prod_{i=1}^{d}(\mu^{u,n}_{i})^{m_{i}}$, where
$\mu^{u,n}_{i}=\mu_{i}+n^{-1/2}u_{i}$ and $m_{i}$ is the number of times $i$
occur in $(\mathbf{a}_{1},\ldots,\mathbf{a}_{n})$. Since ${\mu^{u,n}_{i}}=0$
for $i>r$ we have

$\rho^{\otimes
n}_{0,u,r,n}|f_{\mathbf{a}}\rangle=\prod_{i=1}^{r}(\mu^{u,n}_{i})^{m_{i}}|f_{\mathbf{a}}\rangle$

if $\mathbf{a}_{j}\in\\{1,2,\ldots,r\\}$ for all $j=1,2,\ldots,n$ and the RHS
of the above equation is $0$ if $\exists\text{ }j$ with $\mathbf{a}_{j}>r$.

Note that $y_{\lambda}f_{\mathbf{a}}$ is a linear combination of tensor
products each containing the same multiplicities (i.e. the set $\\{m_{i}\\}$
is same since $\sigma$ just permutes the basis vectors) and that each tensor
product has the same eigenvalue. Thus $y_{\lambda}f_{\mathbf{a}}$ being a
linear combination of eigenvectors (all of them having same eigenvalue), is
itself an eignevector with the same eigenvalue. Thus we have

$\rho^{\otimes
n}_{0,u,r,n}y_{\lambda}|f_{\mathbf{a}}\rangle=\prod_{i=1}^{r}(\mu^{u,n}_{i})^{m_{i}}y_{\lambda}|f_{\mathbf{a}}\rangle$

if $\mathbf{a}_{j}\in\\{1,2,\ldots,r\\}$ for all $j=1,2,\ldots,n$ and the RHS
of the above equation is $0$ if $\exists\text{ }j$ with $\mathbf{a}_{j}>r$.

Recall that $\\{y_{\lambda}f_{\mathbf{a}}\\}_{\mathbf{a}\in\mathfrak{A}}$
spans $\mathcal{H}_{\lambda}$. Now consider a semistandard tableau with more
than $r$ rows or that $\lambda_{r+1}>0$. By the definition, the entries of a
column are strictly increasing and hence the $(r+1,1)$ entry of the tableau is
greater than $r$. Consequently there exists $j$ such that $\mathbf{a}_{j}>r$
and hence $y_{\lambda}f_{\mathbf{a}}$ belongs to the null space of
$\rho^{\otimes n}_{0,u,r,n}$.

Let $\Lambda_{1}$ be the collection of Young diagrams with number of row less
than equal to $r$ i.e. $\Lambda_{1}=\\{\lambda:\lambda_{k}=0\text{ for
}k>r\\}$. Consider the alternate notation given in the last paragraph Appendix
A where one identifies $\mathbf{a}$ with $\mathbf{m}$ and denotes the
normalized basis elements as $|\mathbf{m}_{\lambda}\rangle$. If
$\lambda\in\Lambda_{1}$, $\mathbf{m}=\\{m_{i,j}\\}_{1\leq i\leq r,i<j\leq d}$
since $i\leq r$ for these $\lambda$s. Let $\mathcal{H}(\\{m_{i}\\})=\text{span
}\\{|\mathbf{l}_{\lambda}\rangle:l_{i}=m_{i}\\}$ i.e the span of the vectors
$l_{\lambda}$ with the same multiplicity $\\{m_{i}\\}$. Note that for each
$\mathbf{m}$ we have a set $\\{m_{i}\\}$ but different $\mathbf{m}$ can give
rise to the same $\\{m_{i}\\}$. We now have a modified version of (57):

$\rho^{\otimes
n}_{0,u,r,n}=\bigoplus_{\lambda\in\Lambda_{1}}\tilde{\rho}_{\lambda}^{0,u,r,n}\otimes\mathbbm{1}_{\mathcal{K}_{\lambda}}$
(76)

with

$\tilde{\rho}_{\lambda}^{0,u,r,n}=\sum_{\\{m_{i}\\}}\prod_{i=1}^{r}(\mu^{u,n}_{i})^{\lambda_{i}}\prod_{j=i+1}^{r}\left(\frac{\mu^{u,n}_{j}}{\mu^{u,n}_{i}}\right)^{m_{i,j}}\sum_{m\in\mathcal{M}^{\lambda}_{r}\cap\\{m_{i}\\}}|\bar{\mathbf{m}}_{\lambda}\rangle\langle\bar{\mathbf{m}}_{\lambda}|$

where $\mathcal{M}^{\lambda}_{r}=\\{m:m_{i,j}=0\text{ for }i>r\text{ or
}j>r\\}$ and $|\bar{\mathbf{m}}_{\lambda}\rangle$s are orthogonal vectors
which spans the space $\mathcal{H}(\\{m_{i}\\})$. We would like to write the
above decomposition in terms of
$|\mathbf{m}_{\lambda}\rangle\langle\mathbf{m}_{\lambda}|$ but since they are
not orthogonal we have to use
$|\bar{\mathbf{m}}_{\lambda}\rangle\langle\bar{\mathbf{m}}_{\lambda}|$
instead. Later we will prove that we can write the same decomposition in terms
of $|\mathbf{m}_{\lambda}\rangle\langle\mathbf{m}_{\lambda}|$ and an error
term that goes to $0$. Note that although $|\mathbf{m}_{\lambda}\rangle$ is a
basis element for all $m$ comes from a semistandard tableau,
$|\mathbf{m}_{\lambda}\rangle$ still belongs to the null space of
$\rho^{\otimes n}_{0,u,n}$ if $m_{i,j}>0$ for some $j>r$.

We normalize $\tilde{\rho}_{\lambda}^{0,u,r,n}$ so that

$\rho^{0,u,r,n}_{\lambda}=\frac{1}{N^{u,r,n}_{\lambda}}\sum_{\\{m_{i}\\}}\prod_{i=1}^{r}(\mu^{u,n}_{i})^{\lambda_{i}}\prod_{j=i+1}^{r}\left(\frac{\mu^{u,n}_{j}}{\mu^{u,n}_{i}}\right)^{m_{i,j}}\sum_{m\in\mathcal{M}^{\lambda}_{r}\cap\\{m_{i}\\}}|\bar{\mathbf{m}}_{\lambda}\rangle\langle\bar{\mathbf{m}}_{\lambda}|$
(77)

where

$N^{u,r,n}_{\lambda}=\sum_{\mathbf{m}\in\mathcal{M}^{\lambda}_{r}}\prod_{i=1}^{r}(\mu^{u,n}_{i})^{\lambda_{i}}\prod_{j=i+1}^{r}\left(\frac{\mu^{u,n}_{j}}{\mu^{u,n}_{i}}\right)^{m_{i,j}}.$

Finally we define
$p_{\lambda}^{u,r,n}=N^{u,r,n}_{\lambda}.\dim(\mathcal{K}_{\lambda})$ (the
dependence of $p_{\lambda}^{u,r,n}$ on $\theta$ coming only from $u$) and we
obtain the modified version of (58):

$\rho^{\otimes
n}_{0,u,r,n}=\bigoplus_{\lambda\in\Lambda_{1}}\left(\rho^{0,u,r,n}_{\lambda}\otimes\frac{\mathbbm{1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}\right)p_{\lambda}^{u,r,n}.$

Since $\mathrm{Tr}{\rho^{\otimes n}_{0,u,r,n}}=1$, taking trace on both sides
of the above equation we obtain
$\sum_{\lambda\in\Lambda_{1}}p_{\lambda}^{u,r,n}=1$. We also observe that
because of the normalization both $\rho^{0,u,r,n}_{\lambda}$ and
$\frac{\mathbbm{1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}$ are
states and so is their tensor product. We have decomposed the tensor state
$\rho^{\otimes n}_{0,u,n}$ into blocks (each having trace 1 and thus a state)
multiplied by a weight. All these weights are positive and add upto $1$ and we
can interpret this as decomposition into mixture of smaller dimensional states
(original state was $dn$ dimensional). Similarly the tensored unitaries
$U(\mathfrak{z}/\sqrt{n})^{\otimes n}$ are decomposed into block matrices.

Defining

$\Delta_{\lambda}^{\mathfrak{z},r,n}(A)=U_{\lambda}^{r}(\mathfrak{z}/\sqrt{n})AU_{\lambda}^{r*}(\mathfrak{z}/\sqrt{n}),$

we obtain the following equivalent of (59) in this modified setting

$\rho^{\otimes
n}_{\theta,r,n}=\bigoplus_{\lambda\in\Lambda_{1}}\left(\Delta^{\mathfrak{z},r,n}_{\lambda}(\rho^{0,u,r,n}_{\lambda})\otimes\frac{\mathbbm{1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}\right)p_{\lambda}^{u,r,n}.$

The channel that we construct will contain two parts - the classical part maps
the weights or probabilities of the block to a normal distribution via a
classical channel or a Markov kernel. The quantum part involves mapping the
basis of each representation to the Fock basis of multimode state and the
action of unitaries under this channel corresponds to action of the
displacement operators. We now give a brief sketch of the channel mentioned
above, the exact details of the terms involved deferred to later sections.

Define

$T^{r}_{n}(\rho^{\otimes
n}_{\theta,r,n})=\bigoplus_{\lambda\in\Lambda_{1}}(p_{\lambda}^{u,r,n}\circ\tau^{r,n}_{\lambda})\otimes
T^{r}_{\lambda}(\Delta^{\mathfrak{z},r,n}_{\lambda}(\rho^{0,u,r,n}_{\lambda}))$

where $\tau^{r,n}_{\lambda}$ is a Markov kernel defined below and
$T^{r}_{\lambda}$ is an map of the from $V^{r}_{\lambda}(.)V^{r*}_{\lambda}$
from $M(\mathcal{H}_{\lambda})$ to the multimode Fock space. The explicit from
of the maps $V^{r}_{\lambda}$ will be discussed later. We define
$\tau^{r,n}_{\lambda}$ as follows:

$\tau^{r,n}_{\lambda}(A)=n^{(r-1)/2}\mu(A\cap A_{\lambda,r,n})$ (78)

where both $A$ and $A_{\lambda,r,n}$ are sets in $\mathbb{R}^{r-1}$ and $\mu$
is the Lebesgue measure in $\mathbb{R}^{r-1}$. Here

$A_{\lambda,r,n}=\\{x\in\mathbb{R}^{r-1}:|n^{1/2}x_{i}+n\mu_{i}-\lambda_{i}|\leq
1/2,1\leq i\leq r-1\\}.$

Also define $b_{\lambda}^{u,r,n}=p_{\lambda}^{u,r,n}\circ\tau^{r,n}_{\lambda}$
and
$\Delta^{\mathfrak{z},n}_{\lambda}(\rho^{0,u,r,n}_{\lambda})=\rho_{\lambda}^{\theta,r,n}$.
In this setup we have to show that

$\sup_{\Theta_{n,r,\beta,\gamma}}||\bigoplus_{\lambda\in{\Lambda_{1}}}b_{\lambda}^{u,r,n}\otimes
T^{r}_{\lambda}(\Delta^{\mathfrak{z},r,n}_{\lambda}(\rho^{0,u,r,n}_{\lambda}))-\mathcal{N}_{r-1}(u,V_{\mu})\otimes\phi^{\mathfrak{z},r}||_{1}\rightarrow
0.$

Define the following set of Young diagrams

$\Lambda_{n,\alpha}:=\\{\lambda:|\lambda_{i}-n\mu_{i}|\leq n^{\alpha},1\leq
i\leq r\\},\quad\text{ for }\alpha>1/2.$ (79)

For convenience of notation we will denote $\Lambda_{n,\alpha}$ by
$\Lambda_{2}$ and $\Theta_{n,r,\beta,\gamma}$ by $\Theta$. For the
displacement operator $D(\mathfrak{z})$ we will denote the action
$D(\mathfrak{z})(.)D^{*}(\mathfrak{z})$ by $D^{\mathfrak{z}}$. Finally we note
that $D(\mathbf{\mathfrak{z}})(\phi^{0,r})=\phi^{\mathbf{\mathfrak{z}},r}$. By
triangle inequality one obtains

$\displaystyle||\bigoplus_{\lambda\in{\Lambda_{1}}}b_{\lambda}^{u,r,n}\otimes
T^{r}_{\lambda}(\rho^{\theta,r,n}_{\lambda})-\mathcal{N}_{r-1}(u,V_{\mu})\otimes\phi^{\mathbf{\mathfrak{z},r}}||_{1}$
$\displaystyle\leq$
$\displaystyle\sum_{\lambda\in\Lambda_{1}}||T^{r}_{\lambda}(\rho^{\theta,r,n}_{\lambda})-\phi^{\mathbf{\mathfrak{z},r}}||_{1}||b_{\lambda}^{u,r,n}||_{1}+||\sum_{\lambda\in\Lambda_{1}}b_{\lambda}^{u,r,n}-\mathcal{N}_{r-1}(u,V_{\mu})||_{1}$
$\displaystyle\leq$
$\displaystyle||\sum_{\lambda\in\Lambda_{1}}b_{\lambda}^{u,r,n}-\mathcal{N}_{r-1}(u,V_{\mu})||_{1}+2\sum_{\lambda\in\Lambda_{1}\cap\Lambda^{c}_{2}}p_{\lambda}^{u,r,n}+\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||T^{r}_{\lambda}(\rho_{\lambda}^{\theta,r,,n})-\phi^{\mathbf{\mathfrak{z},r}}||_{1}$
$\displaystyle\leq$
$\displaystyle||\sum_{\lambda\in\Lambda_{1}}b_{\lambda}^{u,r,n}-\mathcal{N}_{r-1}(u,V_{\mu})||_{1}+2\sum_{\lambda\in\Lambda_{1}\cap\Lambda^{c}_{2}}p_{\lambda}^{u,r,n}+\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}(\phi^{0,r})-D^{{\mathfrak{z}}}(\phi^{0,r})||_{1}$
$\displaystyle+\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}T^{r}_{\lambda}(\rho^{0,u,r,n}_{\lambda})-T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}(\phi^{0,r})||_{1}$
$\displaystyle\leq$
$\displaystyle||\sum_{\lambda\in\Lambda_{1}}b_{\lambda}^{u,r,n}-\mathcal{N}_{r-1}(u,V_{\mu})||_{1}+2\sum_{\lambda\in\Lambda_{1}\cap\Lambda^{c}_{2}}p_{\lambda}^{u,r,n}+\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||(T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}-D^{{\mathfrak{z}}})(\phi^{0,r})||_{1}$
$\displaystyle+\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||T^{r}_{\lambda}(\rho^{0,u,r,n}_{\lambda})-\phi^{0,r}||_{1}$

since $T^{r}_{\lambda}$ and $\Delta_{\lambda}^{\mathfrak{z},r,n}$ are
isometries. Using the integral form of $\phi^{0,r}$ (we set $\mathfrak{z}=0$
in (75)), we obtain

$\displaystyle||(T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}-D^{{\mathfrak{z}}})(\phi^{0,r})||_{1}$
$\displaystyle\leq\int
f(\mathbf{z})||(T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}-D^{{\mathfrak{z}}})(|\mathbf{z}\rangle\langle\mathbf{z}|\otimes|\mathbf{0}_{2}\rangle\langle\mathbf{0}_{2}|)||_{1}d\mathbf{z}$
$\displaystyle\leq\int_{||\mathbf{z}||>n^{\beta}}2f(\mathbf{z})d\mathbf{z}+\sup_{||\mathbf{z}||\leq
n^{\beta}}||(T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}D^{z,0}-D^{(\mathfrak{z}_{1}+z,\mathfrak{z}_{2})})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$

where

$f(\mathbf{z})=\prod_{1\leq i<j\leq
r}\frac{e^{\beta_{ij}}-1}{\pi}\exp\left(-\sum_{1\leq i<j\leq
r}\frac{e^{\beta_{ij}}-1}{\pi}|z_{ij}|^{2}\right)$

and

$\int_{||\mathbf{z}||>n^{\beta}}f(\mathbf{z})d\mathbf{z}=c_{1}\exp(-c_{2}n^{2\beta})$

by usual concentration of a Gaussian. The second term can be written as:

$\displaystyle||(T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}D^{\mathbf{z},0}-D^{\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2}})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$
$\displaystyle\leq$
$\displaystyle||(T^{r}_{\lambda}\Delta_{\lambda}^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2}),r,n}T^{r*}_{\lambda}-D^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2})})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}+||(T^{r}_{\lambda}\Delta_{\lambda}^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2}),r,n}T^{r*}_{\lambda}-T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}\Delta_{\lambda}^{\mathbf{z},r,n}T^{r*}_{\lambda})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$
$\displaystyle+||(T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}T^{r*}_{\lambda}D^{\mathbf{z},0}-T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z},r,n}\Delta_{\lambda}^{\mathbf{z},r,n}T^{r*}_{\lambda})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$
$\displaystyle\leq$
$\displaystyle||(T^{r}_{\lambda}\Delta_{\lambda}^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2}),r,n}T^{r*}_{\lambda}-D^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2})})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}+||(\Delta_{\lambda}^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2}),r,n}-\Delta_{\lambda}^{\mathfrak{z},r,n}\Delta_{\lambda}^{\mathbf{z},r,n})T^{r*}_{\lambda}(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$
$\displaystyle+||(T^{r*}_{\lambda}D^{\mathbf{z},0}-\Delta_{\lambda}^{\mathbf{z},r,n}T^{r*}_{\lambda})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$
$\displaystyle\leq$
$\displaystyle||(T^{r}_{\lambda}\Delta_{\lambda}^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2}),r,n}T^{r*}_{\lambda}-D^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2})})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}+||(\Delta_{\lambda}^{(\mathfrak{z}_{1}+\mathbf{z},\mathfrak{z}_{2}),r,n}-\Delta_{\lambda}^{\mathfrak{z},n}\Delta_{\lambda}^{\mathbf{z},r,n})T^{r*}_{\lambda}(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$
$\displaystyle+||(D^{\mathbf{z},0}-T^{r}_{\lambda}\Delta_{\lambda}^{\mathbf{z},r,n}T^{r*}_{\lambda})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}$

In the above equations $\Delta_{\lambda}^{\mathbf{z},r,n}$ is understood to be
$\Delta_{\lambda}^{\mathfrak{z},r,n}$, where $\mathfrak{z}=(\mathbf{z},0)$.
The entire chain of inequalities can be summarized as follows:

$\displaystyle\sup_{\theta\in\Theta}||T_{n}(\rho^{\otimes
n}_{\theta,n})-\mathcal{N}_{r-1}(u,V_{\mu})\otimes\phi^{\mathfrak{z},r}||_{1}$
$\displaystyle\leq$
$\displaystyle\sup_{\theta\in\Theta}||\sum_{\lambda\in\Lambda_{1}}b_{\lambda}^{u,r,n}-\mathcal{N}_{r-1}(u,V_{\mu})||_{1}\text{
(Approximation error of classical channel) }$
$\displaystyle+\sup_{\theta\in\Theta}2\sum_{\Lambda_{1}\cap\Lambda^{c}_{2}}p_{\lambda}^{u,r,n}\text{
(probability of extreme representations) }$
$\displaystyle+\sup_{\theta\in\Theta}\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||T^{r}_{\lambda}(\rho^{0,u,r,n}_{\lambda})-\phi^{0,r}||_{1}\text{
(comparison of unshifted states) }$
$\displaystyle+\sup_{\theta\in\Theta,||\mathbf{z}||\leq
n^{\beta}}\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||(T^{r}_{\lambda}\Delta_{\lambda}^{(\mathfrak{z}+\mathbf{z}),r,n}T^{r*}_{\lambda}-D^{\mathfrak{z}+\mathbf{z}})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}\text{
(comparing shift operators) }$
$\displaystyle+\sup_{\theta\in\Theta,||\mathbf{z}||\leq
n^{\beta}}\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||(D^{\mathbf{z},0}-T^{r}_{\lambda}\Delta_{\lambda}^{\mathbf{z},r,n}T^{r*}_{\lambda})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}\text{
(comparing shift operators) }$
$\displaystyle+\sup_{\theta\in\Theta,||\mathbf{z}||\leq
n^{\beta}}\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||(\Delta_{\lambda}^{(\mathfrak{z}+\mathbf{z}),r,n}-\Delta_{\lambda}^{\mathfrak{z},r,n}\Delta_{\lambda}^{\mathbf{z},r,n})T^{r*}_{\lambda}(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}\text{
(noncomm. of fin. dim. shifts) }$
$\displaystyle+\int_{||\mathbf{z}||>n^{\beta}}f(\mathbf{z})d\mathbf{z}\text{
(concentration term) }.$

The following lemmas deals with each of the above terms separately:

######  Lemma C.1.

. For any given $\epsilon>0$,

$\sup_{\theta\in\Theta}||\sum_{\lambda\in\Lambda_{1}}b_{\lambda}^{u,r,n}-\mathcal{N}_{r-1}(u,V_{\mu})||_{1}=O(n^{-1/4+\epsilon},n^{-1/2+\gamma}).$

######  Lemma C.2.

Recall that $\Lambda_{2}=\Lambda_{n,\alpha}$. Then for $\alpha-\gamma-1/2>0$
we have

$\sum_{\lambda\in\Lambda_{1}\cap\Lambda^{c}_{2}}p_{\lambda}^{u,r,n}=O(n^{r^{2}}\exp(-n^{2\alpha-1}/2)).$

######  Lemma C.3.

For $0<\eta<2/9$ we have

$\sup_{\theta\in\Theta}\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||T^{r}_{\lambda}(\rho^{0,u,r,n}_{\lambda})-\phi^{0,r}||_{1}=O(n^{-1/2+\gamma+\eta},n^{9\eta-2/24})$

######  Lemma C.4.

Let $\epsilon>0$ be such that $2\beta+\eta\leq\eta<2/9$, then we have

$\sup_{\theta\in\Theta,||\mathbf{z}||\leq
n^{\beta}}\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||(T^{r}_{\lambda}\Delta_{\lambda}^{\mathfrak{z}+\mathbf{z},r,n}T^{r*}_{\lambda}-D^{\mathfrak{z}+\mathbf{z}})(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}=R(n)$

with

$R^{2}(n)=O\left(n^{(9\eta-2)/12},n^{-1+2\beta+\eta},n^{-1/2+3\beta+2\epsilon},n^{-1+\alpha+2\beta},n^{-1+\alpha+\eta},n^{-1+3\eta},n^{-\beta}\right)$

######  Lemma C.5.

Assume the condition given in Lemma C.4, then the following result holds:

$\sup_{\theta\in\Theta,||\mathbf{z}||\leq
n^{\beta}}\sup_{\lambda\in\Lambda_{1}\cap\Lambda_{2}}||(\Delta_{\lambda}^{\mathfrak{z}+\mathbf{z},r,n}-\Delta_{\lambda}^{\mathfrak{z},r,n}\Delta_{\lambda}^{\mathbf{z},r,n})T^{r*}_{\lambda}(|\mathbf{0}\rangle\langle\mathbf{0}|)||_{1}=R(n)$

Combining these lemmas we obtain

######  Lemma C.6.

For $0<\gamma<1/4$ and $0<\beta<1/9$ there exists $\kappa>0$ such that

$\sup_{\theta\in\Theta_{n,r,\beta,\gamma}}||\Phi^{\theta,r}-T^{r}_{n}(\rho^{\theta,r,n})||_{1}=O(n^{-\kappa})$

###### Proof.

For $\epsilon>0$, $1/2+\gamma<\alpha<1/2$, $\eta<2/9$ and
$0<\beta<(\eta-\epsilon)/2$ we have

$\sup_{\theta\in\Theta_{n,r,\beta,\gamma}}||\Phi^{\theta,r}-T^{r}_{n}(\rho^{\theta,r,n})||_{1}=O\left(n^{-1/4+\epsilon}+n^{-1/2+\gamma}+n^{-1/2+\gamma+\eta}+n^{9\eta-2/24}\right.\\\
\left.+n^{-1/4+3\beta/2+\epsilon}+n^{-1/2+\alpha/2+\eta/2}+n^{-1/2+3\eta/2}+n^{-\beta/2}\right)$
(80)

using Lemmas C.1 through C.5. Note that we have omitted the error term in
Lemma C.2 since it decays expontially upto a polynomial term and hence is
smaller than the rest of the errors. Also we have used the relations
$n^{-1/2+\beta+\eta/2}=O(n^{-1/2+3\eta/2})$ and
$n^{-1/2+\alpha/2+\beta}=O(n^{-1/2+\alpha/2+\eta/2})$ to gather the error
terms. Now we can easily choose $\epsilon,\alpha$ and $\eta$ suitably so that
the RHS of (80) is $O(n^{-\kappa})$ for some $\kappa>0$. ∎

Construction of the reverse channel $S^{r}_{n}$

We follow the proof given in [35] except that in our case we have a restricted
class of Young diagrams. Let $\sigma^{r,n}$ be defined by

$\sigma^{r,n}:x\in\mathbb{R}^{r-1}\mapsto\delta_{\lambda_{x}}$ (81)

where $\lambda_{x}$ is the Young diagram such that
$\sum_{i=1}^{r}\lambda_{i}=n$, and $|n^{1/2}x_{i}+n\mu_{i}-\lambda_{i}|<1/2$,
for $1\leq i\leq r-1$. If no such diagram exists, we set $\lambda_{x}$ to any
admissible value, for example $(n,0,\dots,0)$. One can easily show that with
(78), $\sigma^{r,n}\circ\tau^{r,n}\circ\sigma^{r,n}=\sigma^{r,n}$ and for any
probability distribution on the $\lambda$ which is in the image of
$\sigma^{r,n}$ we have $\sigma^{r,n}\circ\tau^{r,n}(p^{u,r,n})=p^{u,r,n}$. One
then shows (see lemma 6.8 of [35]) that for any $\epsilon$, we have

$\sup_{\|u\|\leq
n^{\gamma}}\left\|\sigma^{n}\mathcal{N}_{r-1}(u,V_{\mu})-p^{u,r,n}\right\|_{1}=O\left(n^{-1/2+\epsilon},n^{-1/4+\gamma}\right).$
(82)

The channel $S^{r}_{n}$ (which is essentially the pseudo-inverse of
$T^{r}_{n}$) is given as follows. Given a sample from the probability
distribution $\mathcal{N}_{r-1}(u,V_{\mu})$, we use the Markov kernel
$\sigma^{r,n}$ to produce a Young diagram $\lambda$. Conditional on $\lambda$
we send the quantum part through the channel

$S^{r}_{\lambda}:\phi\mapsto\tilde{S}^{r}_{\lambda}(\phi)\otimes\frac{{\bf
1}_{\mathcal{K}_{\lambda}}}{\dim(\mathcal{K}_{\lambda})}$

with

$\tilde{S}^{r}_{\lambda}:\phi\mapsto
T^{r*}_{\lambda}\phi+(1-\mathrm{Tr}(T^{r*}_{\lambda}(\phi)))|{\bf
0}_{\lambda}\rangle\langle{\bf 0}_{\lambda}|.$

For any density operator $\rho_{\lambda}$ on the block $\lambda$, the operator
$\tilde{S}_{\lambda}$ reverts the action of $T^{r}_{\lambda}$:

$\displaystyle\tilde{S}^{r}_{\lambda}T^{r}_{\lambda}(\rho_{\lambda})$
$\displaystyle=T^{r*}_{\lambda}T^{r}_{\lambda}(\rho_{\lambda})+(1-\mathrm{Tr}(T^{r*}_{\lambda}T^{r}_{\lambda}(\rho_{\lambda})))|{\bf
0}_{\lambda}\rangle\langle{\bf 0}_{\lambda}|$