# Polar Codes for Quantum Reading

Francisco Revson F<EMAIL_ADDRESS>School of Science and
Technology, University of Camerino, I-62032 Camerino, Italy  INFN, Sezione di
Perugia, I-06123 Perugia, Italy  Stefano Mancini  School of Science and
Technology, University of Camerino, I-62032 Camerino, Italy  INFN, Sezione di
Perugia, I-06123 Perugia, Italy

###### Abstract

Quantum reading provides a general framework where to formulate the
statistical discrimination of quantum channels. Several paths have been taken
for such a problem. However, there is much to be done in the avenue of
optimizing channel discrimination using classical codes. At least two open
questions can be pointed to: how to construct low complexity encoding schemes
that are interesting for channel discrimination and, more importantly, how to
develop capacity-achieving protocols. The aim of this paper is to present a
solution to these questions using polar codes. Firstly, we characterize the
rate and reliability of the channels under polar encoding. We also show that
the error probability of the scheme proposed decays exponentially with respect
to the code length. Lastly, an analysis of the optimal quantum states to be
used as probes is given.

Keywords: Quantum Reading, Polar Codes, Capacity-Achieving Protocols.

##  1 Introduction

Quantum hypothesis testing aims to identify strategies in order to
statistically discriminate quantum states or processes. The former is called
quantum state discrimination and has been largely analyzed in the literature
(starting from Refs. [14, 16]). The latter, under the name of quantum channel
discrimination, has been recently addressed [8, 2, 11, 30, 29, 36, 10, 13,
12]. In its basic formulation, one has to identify a quantum channel selected
from a set accordingly to a probability measure. This should be done by using
suitable input state and output measurement. As such, it is a double-
optimization problem and hence results in a daunting task. Performance are
usually quantified in terms of minimum error probability and recently bounds
on it were found for general strategies [24, 19]. Although theoretically
subtle, quantum channel discrimination is interesting for practical
applications. For instance, it is at the basis of the decoding procedure of
two-way quantum cryptography [26] where the secret information is encoded in a
Gaussian ensemble of phase-space displacements. It also appears in the quantum
illumination of targets [20, 35], where the sensing of a remote low-reflective
object in a bright thermal environment corresponds to the binary
discrimination between a lossy channel (presence of target) and a depolarizing
channel (absence of target). Following this line, quantum channel
discrimination can be reformulated in the framework of quantum reading [23].
There, the data storage corresponds to a process of channel encoding, where
information is recorded into a cell by storing a quantum channel picked from a
given ensemble. Then readout corresponds to the process of channel decoding,
which is equivalent to discriminating between the various channels in the
ensemble. In such a setting, using a quantum resource, such as entanglement,
was shown to overcome any classical strategy based on mixtures of coherent
states, hence the name of “quantum” reading.

Efficient paths to quantum reading can be also envisaged by the use of coding
techniques [25, 9]. There are at least two possible approaches: classical
coding on the quantum memory cell labels and quantum coding on the probe
states used in the readout. This paper takes the former one. Similar to
channel coding, where redundancy is added to protect the information to be
transmitted, we use classical codes to encode the information to be recorded
in the memory cell. The goal is to decrease the error probability in
discriminating the channels in the ensemble. There are several classical codes
that one could use for quantum reading, depending on the goal in mind. In this
work, we are interested in information-theoretically provable codes that could
be used to attain high information rates and low error probability. Therefore,
suitable codes can be derived from the family of polar codes.

Creating capacity-achieving codes has been a challenge since the development
of Classical Information Theory. Even more difficult was to show there exist
theoretically-proven such codes. Fortunately, developing the idea of
information combining, Arikan was able to show that for binary memoryless
symmetric (BMS) channels such codes exist [3]. They are called polar codes.
The major achievement of Arikan’s paper is to show a clever way to combine
information, which leads to new synthetic channels manifesting a polarization
phenomenon. They are commonly divided into two groups named “good” and “bad”
channel222The reason for such names is due to the capacity of these new
channels been close to its maximum value and close to zero, respectively..
Furthermore, the fractions of good and bad channels are related to the
capacity of the original channel in consideration. Encoding and decoding
schemes take advantage of these properties to attain low complexity. It is
worth mention that polar code construction highly depends on the channel into
consideration.

Several papers extend or apply the idea of polarization in a diversity of
channels and areas [32, 4, 31, 18]. In particular, polar codes have also been
constructed to classical-quantum channels [37], and quantum channels [38, 27].
However, this is not the case when one considers quantum reading, which is the
goal of this paper. The distribution of the channels to be discriminated can
be symmetric or asymmetric. Asymmetric quantum reading can be seen as a
generalization of the former quantum reading and this formulation can be
justified by the measurement strategy implemented in the decoding process or
energy constrain imposed over the channels. Since these two conditions are
plausible hypothesis to be considered in classical digital memories, which are
the main goal of channel model used in this paper, we will formulate our
results for asymmetric quantum reading and, where it makes necessary, an
adaptation for symmetric quantum reading is given. Therefore, our polar code
will be design to, primarily, asymmetric quantum channels. In the classical
paradigm, this has been done by Sutter, et al. [33], and Honda and Yamamoto
[18]. Some of the ideas used in Ref. [18] are applied in this paper. However,
as we will see, the transition from the classical to quantum paradigm is
subtle and major changes are needed to be done to derive our construction.
This is particularly true for the measurement process implemented in the
decoding part. An adequately definition for rate and reliability is also
needed.

The analysis of quantum memory cell in this paper is twofold. Initially, polar
coding is applied on the labels of the quantum memory cell in order to
decrease the error probability in distinguishing them. In this direction, we
show that the polarization phenomenon can be characterized in quantum memory
cell by studying its composing parts; namely, channel combining and channel
splitting. The former constitutes a systematic approach to combine source bits
so the polarization phenomenon emerges. An encoding matrix, sometimes also
called combining function, describing the process is given in this paper. The
latter is an information-theoretically analysis of the synthesized channels
created in the channel combining part. Initially, the first level encoding
process combining two quantum channels is analyzed. We show that the rate and
reliability of the synthesized channels polarize. This is later used in the
asymptotic analysis. A connection between rate and reliability is also
presented, showing that rate is inversely proportional to reliability. Next,
our first major result is given. It characterizes the asymptotic distribution
of the synthesized channel with respect to the length code. This result is
based on proving the existence, shown in this paper, of a symmetric quantum
channel that can be used to obtain the statistics of the (asymmetric) quantum
channel of interest. Lastly, in this first part, we draw the encoding and
decoding scheme. The strategy used for encoding the frozen bits does not
follow the scheme used in previous works on polar codes for classical-quantum
channels. Additionally, examining the error probability obtained after the
decoding scheme, we see that it decays exponentially with respect to the
length code. The second part addresses the optimization problem of probe
states. As a first-order approximation of classical digital memories, we
consider amplitude damping channels as our channel model. The fundamental
result of this part is showing that the optimal probe states are pure states.
Even though we have taken into consideration just the first level polar
encoding, it seems satisfactory supposing this can be extended to any
$N$-level polar encoding.

This paper is organized as follows. In Section 2 we present previous results
on polar codes for classical-quantum channels, and some important definition
used to characterize quantum memory cells. This section constitutes the
fundamentals that our work in based on. Next, the main results are shown in
Section 3. Encoding and decoding schemes are described and analyzed in detail,
showing that the use of polar coding is also interesting for quantum memory
cells. The following subject treated in this work is the optimization of
states used to probe the channels. This is drawn in Section 4. Lastly, we draw
our conclusions and some final remarks in Section 5.

###  1.1 Notation

We denote classical random variables as $X$, $Y$, $U$, whose realizations are
elements of the finite sets $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{U}$,
respectively. The probability distributions are respectively represented by
$p_{X}(x)$, $p_{Y}(y)$, and $p_{U}(u)$ for the random variables $X$, $Y$, $U$.
In particular, $X$ is always assumed to be discrete and, in some parts of the
text, to be a Bernoulli random variable with values in $\\{0,1\\}$ and
$P_{X}(0)=p$ by using the notation $X\sim\text{Ber}(p)$. For such random
variables, its binary Shannon entropy is defined as
$H(X)=h(p):=-p\log(p)-(1-p)\log(1-p)$. The use of subscript and superscript on
a letter indicates a sequence starting with the element denoted by the
subscript and ending with the element denoted by the superscript, e.g.
$X_{i}^{j}$ (with $j>i$) is the sequence of random variables
$X_{i},X_{i+1},\ldots,X_{j}$. $X^{N}$ simply stands for the sequence
$X_{1},X_{2},\ldots,X_{N}$. A classical memoryless channel is written as
$W\colon\mathcal{X}\rightarrow\mathcal{Y}$ from the input alphabet
$\mathcal{X}$ to the output alphabet $\mathcal{Y}$. We write $W^{N}$ to denote
$N$ uses of the channel $W$; i.e.,
$W^{N}\colon\mathcal{X}^{N}\rightarrow\mathcal{Y}^{N}$ with
$W^{N}(y^{N}|x^{N})=\prod_{i=1}^{N}W(y_{i}|x_{i})$ since $W$ is memoryless.
Quantum systems $A$, $B$, and $C$ correspond to Hilbert spaces
$\mathcal{H}_{A}$, $\mathcal{H}_{B}$, and $\mathcal{H}_{C}$. The notation
$A^{N}:=A_{1}A_{2}\cdots A_{N}$ denotes a joint system consisting of $N$
subsystems, each of which is isomorphic to $\mathcal{H}_{A}$. Let
$\mathcal{L}(\mathcal{H}_{A})$ denote the algebra of bounded linear operators
acting on a Hilbert space $\mathcal{H}_{A}$. The subset
$\mathcal{L}_{+}(\mathcal{H}_{A})$ of $\mathcal{L}(\mathcal{H}_{A})$ denotes
the set of all positive semi-definite operators. A special and important class
of operators in $\mathcal{L}_{+}(\mathcal{H}_{A})$ is the one containing
density operators $\mathcal{D}(\mathcal{H}_{A})$. A density operator
$\rho_{A}\in\mathcal{D}(\mathcal{H}_{A})$ is a positive semi-definite operator
with unit trace, $\text{Tr}\\{\rho_{A}\\}=1$, and represents the state of a
quantum system $A$. Lastly, a quantum channel $\mathcal{W}$ is a linear
completely positive trace-preserving map from $\mathcal{L}(\mathcal{H}_{A})$
to $\mathcal{L}(\mathcal{H}_{B})$.

##  2 Preliminaries

This section is devoted to introducing the main concepts of polar codes and
quantum reading. We begin with a brief overview of binary polar codes and
their desirable attributes. They can be used to achieve the capacity of
discrete memoryless channels (DMC); there are encoding and decoding efficient
schemes; and it is possible to reach error probabilities that decay
exponentially in the square root of the blocklength. In the following, quantum
reading problematic is introduced. The general concept is given, followed by
the channel model adopted for analyzing probe states. In the end, we present
the quantities assumed as rate and reliability.

###  2.1 Classical-Quantum Polar Codes

Let $W\colon\mathcal{X}\rightarrow\mathcal{D}(\mathcal{H}_{B})$ be a binary-
input memoryless classical-quantum channel (cq channel) with input and output
alphabets given by $\mathcal{X}$ and $\mathcal{D}(\mathcal{H}_{B})$,
respectively. Suppose the random variable $X$ is $\text{Ber}(\frac{1}{2})$.
Then the transition probabilities can be derived from the joint input-output
state

$\rho^{XB}=\frac{1}{2}\outerproduct{0}{0}^{X}\otimes\rho_{0}^{B}+\frac{1}{2}\outerproduct{1}{1}^{X}\otimes\rho_{1}^{B},$
(1)

where $\rho_{x}^{B}=W(x)$, for $x=0,1$. Notice that we are using superscripts
to emphasize the system to which each part belongs. Whenever this is clear
from context, they will be omitted. One important characterization can be
given to this cq channel: since the possible outcomes are uniformly
distributed over all possible labels $x\in\mathcal{X}$, we say that the cq
channel $W$ is symmetric. Similarly, a cq channel is called asymmetric when
this does not happen. Though in this subsection we are going to deal with a
symmetric channel, the rest of the text adopts asymmetric channel in order to
provide a more general formulation. As a last comment to be made, we adopt as
input alphabet $\mathcal{X}=\mathbb{Z}_{2}$, and arbitrary finite-dimensional
output alphabet $\mathcal{D}(\mathcal{H}_{B})$ and transition probability.
This choice of input alphabet allows us to operate with their elements; in
particular, we can use XOR operations, or sum$\mod 2$ operations, with the
elements of the input alphabet. An appropriate combination of these operations
leads to channel polarization.

Channel polarization consists mainly of two parts. The first one is named
channel combining, which describes a method of combining inputs of $N$ cq
channels. The second is channel splitting. This part is an information-
theoretical analysis of new inputs and outputs that the channel combining
produces. These new inputs and outputs generate synthesized channels. With a
careful examination the synthesized channels, it is possible to show that, for
an arbitrarily large number of them, they fit into two sets called good and
bad channels. The statistical behavior of them gives the desirable attributes
of polar codes [3, 5, 18, 37, 38, 27]: they achieve the capacity when used for
transmitting information over a cq channel; they can be encoded efficiently
(with a complexity that is essentially linear in the blocklength); the error
probability of the decoder decays exponentially in the square root of the
blocklength. A descriptive explanation of channel polarization, synthesized
channels, and some attributes of polar codes are given below.

Suppose there are $N$ copies of a cq channel $W$, which we denote by $W^{N}$,
and $N$ realizations $u^{N}$ of a random variable $U$ representing the source.
The general formulation of polar codes consists of applying a composition
function to the input $u^{N}$, traditionally represented by a matrix $G_{N}$,
and using the output of the composition function, defined as $x^{N}$, as the
actual inputs for the channels $W^{N}$. This general procedure is called
channel combining, and a scheme is shown in Fig. 1. Observe that this is the
same scheme used in any linear channel coding; the type of the composition
function is what determines the coding scheme to generate a polar code. For a
particular example of coding scheme and composition function, with $N=2$, see
Fig. 2.

Figure 1: General polar encoding scheme for a classical-quantum channel. The
matrix $G_{N}$ represents the composition function applied on the input
$u^{N}$ and resulting in the vector $x^{N}=u^{N}G_{N}$. The density operator
$\rho_{x^{N}}=\rho_{x_{1}}\otimes\cdots\otimes\rho_{x_{N}}$ is the output
$W^{N}(x^{N})$. Figure 2: Polar encoding scheme for $N=2$. The choice of the
composition function characterizes the encoding scheme to be polar.

Now, we can introduce channel splitting and the polarization phenomenon that
emerges from it. For $i\in\\{1,\ldots,N\\}$, we define the $i$-th synthesized
channel $W_{N}^{(i)}$ with input alphabet $\mathcal{U}$ and output
$\mathcal{D}(\mathcal{H}_{U_{1}^{i-1}}\mathcal{H}_{B^{N}})$ as

$\displaystyle W_{N}^{(i)}(u_{i})$ $\displaystyle=$
$\displaystyle\sum_{u_{1}^{i-1}}\frac{1}{2^{i-1}}\outerproduct{u_{1}^{i-1}}{u_{1}^{i-1}}^{U_{1}^{i-1}}\otimes\overline{\rho}_{u_{1}^{i-1}}^{B^{N}},$
(2) $\displaystyle\overline{\rho}_{u_{1}^{i-1}}^{B^{N}}$ $\displaystyle=$
$\displaystyle\sum_{u_{i+1}^{N}}\frac{1}{2^{N-i}}\rho_{u^{N}G_{N}}^{B^{N}},$
(3)

where $N$ is a power of two and $G_{N}$ is the composition function. This
formulation comes from analyzing a successive decoder acting on the output
channel with the help of a genie. For more details, have a look at the
original formulation by Arikan [3] or its extension to classical-quantum
channels in Ref. [37].

There are two important quantities used to quantify the polarization
phenomenon and error probability decay in polar coding. These quantities are
called rate and reliability. Their definition depends on the channel under
consideration. For a classical-quantum channel $W$, rate is defined as the
mutual information $I(X;B)$ and we denote it as $I(W)$. Reliability is adopted
as the fidelity between the possible channel outputs, i.e., reliability is
given by $F(\rho_{0},\rho_{1})=||\sqrt{\rho_{0}}\sqrt{\rho_{1}}||_{1}$, where
$||A||_{1}$ is the Schatten 1-norm of a operator
$A\in\mathcal{L}(\mathcal{H}^{B})$. It is possible to show that these
quantities are inversely proportional to each other in the sense that when one
has a value close to its maximum, the other has a value close to zero [37].
Some interesting results are obtained studying the rate and reliability over
channels produced in the channel splitting part. In particular, for $N$
sufficiently large, it is shown that the channels $W_{N}^{(i)}$ are divided in
two sets: one set with $I(W_{N}^{(i)})$ close to unit, these channels are
called “good”; and a set with $I(W_{N}^{(i)})$ close to zero, called “bad”
channels. We denote by $\mathcal{A}$ the set with indexes labeling good
channels. Analyzing how the fraction of good and bad channels grow when $N$
goes to infinity, it is shown the polarization phenomenon and a capacity-
achieving coding strategy for polar codes. See the following proposition for a
formal formulation of the exposed ideas.

######  Proposition 1.

[37, Thm. 2, Prop. 4] Let $W$ be a classical-quantum channel. Then the
following is true:

  1. 1.

The channels $\\{W_{N}^{(i)}\\}$ polarize in the sense that, for any
$\delta\in(0,1)$, as $N$ goes to infinity through powers of two, the fraction
of indexes $i\in\\{1,2,\ldots,N\\}$ for which $I(W_{N}^{(i)})\in(1-\delta,1]$
goes to $I(W)$ and the fraction for which $I(W_{N}^{(i)})\in[0,\delta)$ goes
to $1-I(W)$;

  2. 2.

For any choice of parameters $(N,K,\mathcal{A})$ for a classical-quantum polar
code, the probability of error is bounded above by

$P_{e}(N,K,\mathcal{A})\leq
2\sum_{i\in\mathcal{A}}\sqrt{F(W^{(i),0},W^{(i),1})}.$ (4)

In particular, for any fixed $R=K/N<I(W)$ and $\beta<1/2$, block error
probability for polar coding under sequential decoding satisfies

$P_{e}(N,K)=o(2^{-N^{\beta}}),$ (5)

where $o(\cdot)$ is the little-O notation from complexity theory.

We make use of Proposition 1 in the following section to show that polar codes
can be applied in the task of quantum memory cell discrimination.

###  2.2 Quantum Memory Cell

We now formulate the general description of quantum memory cell used in this
paper. A quantum memory cell is defined as the set
$\\{\mathcal{W}^{x}\\}_{x\in\mathcal{X}}$ of quantum channels. For a fixed
$x$, we have

$\displaystyle\mathcal{W}^{x}\colon\mathcal{D}(\mathcal{H}_{B^{\prime}})$
$\displaystyle\rightarrow\mathcal{D}(\mathcal{H}_{B})$ (6) $\displaystyle\rho$
$\displaystyle\mapsto\mathcal{W}^{x}(\rho),$ (7)

where $\mathcal{D}(\mathcal{H}_{B^{\prime}}),\mathcal{D}(\mathcal{H}_{B})$ are
the sets of input and output density states of the quantum channel
$\mathcal{W}^{x}$. We call $x\in\mathcal{X}$ the quantum memory cell index.
Sometimes we denote $\mathcal{W}^{x}$ as $\mathcal{W}_{{B^{\prime}}\rightarrow
B}^{x}$ to highlight the input and output systems of $\mathcal{W}^{x}$. An
important hypothesis is given in here. We are supposing that the distribution
of the random variable $X$ describing the label of the quantum channels is
non-uniform. This can be justified by the measurement strategy implemented in
the decoding process or energy constrain imposing this distribution. Since
these two reasoning are plausible in this paper, we are going to adopt, in
most parts of this paper, that we are dealing with asymmetric quantum reading.
The definition of asymmetric quantum reading goes similarly to the definition
of asymmetric cq channels. A quantum memory cell
$\\{\mathcal{W}^{x}\\}_{x\in\mathcal{X}}$ is called asymmetric if the possible
outcomes are non-uniformly distributed over all possible labels
$x\in\mathcal{X}$. Lastly, notice that minor changes over our results lead to
the applicability of them to symmetric quantum reading.

######  Remark 2.

Before introducing the channel model used for numerical analysis in this
paper, an important point-of-view over quantum memory cell need to be given. A
common approach to quantum memory cell is to study how to discriminate their
elements using different probe states but having the labels fixed. However, in
this paper, we are also interested in improving quantum memory cell
discrimination by working on the channel labels. This optimization is
performed by fixing the probe states used. Along these lines, quantum memory
cell can be seen as a set of classical-quantum channels. Thus, a capacity-
achieving approach using polar codes is possible when one deals with quantum
memory cell elements having indexes obeying some rule. The rule used in this
paper is given by the composition function of polar codes.

For this paper, we will adopt the amplitude damping (AD) channel as our
channel model. This is because AD channel is the finite-dimensional first-
order approximation of bosonic attenuator channel, which is the standard model
for digital memory cell. They can be described in the following manner. Let
$\rho\in\mathcal{D}(\mathcal{H})$ be a single-qubit density state, then the AD
channel can be described by the following Kraus expression:

$\mathcal{W}^{x}(\rho)=A_{0}\rho A_{0}^{\dagger}+A_{1}\rho A_{1}^{\dagger},$
(8)

where

$A_{0}=\begin{pmatrix}1&0\\\ 0&\sqrt{1-x}\end{pmatrix},\quad
A_{1}=\begin{pmatrix}0&\sqrt{x}\\\ 0&0\end{pmatrix}.$ (9)

Observe that the same notation for general quantum memory cell and AD channel
has been used. However, through the paper will be clear when we are talking
about one or the other.

The goal of the following sections is to show that using polar codes it is
possible to attain optimal rate with low reliability. This is possible by
using synthesized channels formulation and connecting its properties with the
original channel under analysis. A method for approaching this is firstly
introducing the joint input-output density state and characterizing the
channel via this density state. Thus, lets $X$ be a random variable with
probability law $p_{X}$. We can write the joint density state describing the
systems $X$ and $B$ as

$\rho^{XB}=p_{X}(0)\outerproduct{0}{0}^{X}\otimes\mathcal{W}^{0}(\rho)+p_{X}(1)\outerproduct{1}{1}^{X}\otimes\mathcal{W}^{1}(\rho).$
(10)

As outlined previously, there are two parameters at the center of polarization
phenomenon: rate and reliability. Rate is defined in this work as the quantum
mutual information between the source $X$ and the output system $B$:

######  Definition 3.

Let $X\sim\text{Ber}(p)$ and $\mathcal{W}_{B^{\prime}\rightarrow B}^{x}(\rho)$
be a quantum memory cell, where $x\in\mathcal{X}$. The rate of $\mathcal{W}$
is defined as $I(\mathcal{W})_{\rho}:=I(X;B)_{\rho}$. A direct computation of
this quantum mutual information shows that

$I(\mathcal{W})_{\rho}=H\Big{(}p\mathcal{W}^{0}(\rho)+(1-p)\mathcal{W}^{1}(\rho)\Big{)}-pH(\mathcal{W}^{0}(\rho))-(1-p)H(\mathcal{W}^{0}(\rho)),$
(11)

where $H(\sigma)$ is the von Neumann entropy for a density operator
$\sigma\in\mathcal{D}(\mathcal{H})$.

For reliability of the quantum memory cell $\mathcal{W}$, it is used the
fidelity between the possible output states:

######  Definition 4.

Let $X\sim\text{Ber}(p)$ and $\mathcal{W}_{B^{\prime}\rightarrow B}^{x}(\rho)$
be a quantum memory cell, where $x\in\mathcal{X}$. The reliability of the
quantum memory cell $\mathcal{W}$ is defined as

$Z(\mathcal{W})_{\rho}:=2\sqrt{p(1-p)}F(\mathcal{W}^{0}(\rho),\mathcal{W}^{1}(\rho))=2\sqrt{p(1-p)}||\sqrt{\mathcal{W}^{0}(\rho)}\sqrt{\mathcal{W}^{1}(\rho)}||_{1},$
(12)

where $F(\rho,\sigma)=||\sqrt{\rho}\sqrt{\sigma}||_{1}$ is the fidelity, and
$||A||_{1}$ is the Schatten 1-norm of a operator
$A\in\mathcal{L}(\mathcal{H})$.

The following section describes the polar coding scheme proposed in this paper
and applies the rate and reliability defined above to quantify the goodness of
the codes created.

##  3 Polar Coding Scheme

In this section, we consider the polarization phenomenon induced by a
combining function acting on the quantum memory cells indexes. This approach
is similar to the classical-quantum polarization explained in the previous
section. However, significant refinements in the arguments and proofs are
needed for the results to be valid in our situation.

###  3.1 Channel Polarization

####  3.1.1 Channel Combining

As mentioned before, for a fixed quantum probe state, the elements
$\mathcal{W}^{x}$, for $x\in\mathbb{Z}_{2}$, of a quantum memory cell can be
seen as a classical-quantum channel. Then, without loss of generality, we will
treat them in this form through this section. For an illustrative description
of our coding scheme in conjunction with our hypothesis, see Fig. 3. Let
$\mathcal{W}$ be a classical-quantum channel from which we derive an $N$-fold
classical-quantum channel $\mathcal{W}_{N}$ recursively, where $N=2^{n}$ with
$n\in\mathbb{N}_{0}$. The zeroth level of recursion gives solely the channel
$\mathcal{W}_{1}(u)=\mathcal{W}^{u}(\rho)$, for all $u\in\mathbb{Z}_{2}$. The
first level is a composition of two zeroth level channels; i.e., the
classical-quantum channel $\mathcal{W}_{2}$ is given by

$\displaystyle\mathcal{W}_{2}(u_{1},u_{2})$ $\displaystyle=$
$\displaystyle\mathcal{W}_{1}(u_{1}\oplus u_{2})\otimes\mathcal{W}_{1}(u_{2})$
(13) $\displaystyle=$ $\displaystyle\mathcal{W}^{u_{1}\oplus
u_{2}}(\rho)\otimes\mathcal{W}^{u_{2}}(\rho).$ (14)

This is shown in Fig. 4.

Figure 3: Classical polar coding scheme. Figure 4: Fundamental polar encoding
block

The second level follows from two copies of first level channels. The rule is

$\displaystyle\mathcal{W}_{4}(u_{1},u_{2},u_{3},u_{4})$ $\displaystyle=$
$\displaystyle\mathcal{W}_{2}(u_{1}\oplus u_{2},u_{3}\oplus
u_{4})\otimes\mathcal{W}_{2}(u_{2},u_{4})$ (15) $\displaystyle=$
$\displaystyle\mathcal{W}^{u_{1}\oplus u_{2}\oplus u_{3}\oplus
u_{4}}(\rho)\otimes\mathcal{W}^{u_{3}\oplus
u_{4}}(\rho)\otimes\mathcal{W}^{u_{2}\oplus
u_{4}}(\rho)\otimes\mathcal{W}^{u_{4}}(\rho).$ (16)

We depict this scheme in Fig. 5.

Figure 5: Second level of polar coding scheme.

Following the same procedure, we can derive the $n$-level channels. As
described in [3], the matrix $G_{N}$, which connects the source output $u^{N}$
to the channel input $x^{N}$ by $x^{N}=u^{N}G_{N}$, can be expressed as

$G_{N}=R_{N}F^{\otimes n},$ (17)

with $F^{\otimes n}$ being the $n$-fold Kronecker product of the matrix

$F=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix},$ (18)

and $R_{N}$ is a permutation matrix known as bit-reversal [3]. In particular,
the second level combining function is

$G_{4}=\begin{pmatrix}1&0&0&0\\\ 1&0&1&0\\\ 1&1&0&0\\\ 1&1&1&1\end{pmatrix}.$
(19)

The next step is to characterize the synthesized channels produced by the
action of $G_{N}$. This is the goal of the next subsection.

####  3.1.2 Channel Splitting

First of all, consider two realizations of the channel $\mathcal{W}$. A new
input-output relation can be generated through the channel combining procedure
described before. These channels are called synthesized since they are not
real channels but new point-of-views obtained from the relations created. We
denote this transformation by
$(\mathcal{W},\mathcal{W})\rightarrow(\mathcal{W}^{-},\mathcal{W}^{+})$, where
$\mathcal{W}^{-}$ and $\mathcal{W}^{+}$ are the synthesized channels. Fixing
each input and examining the corresponding outputs, we come to the definition
of the synthesized channels below

$\displaystyle\mathcal{W}^{-}$ $\displaystyle\colon$ $\displaystyle
u_{1}\in\mathbb{Z}_{2}\mapsto\mathcal{W}^{-,u_{1}}(\rho)\in\mathcal{D}(\mathcal{H}_{B_{1}}\otimes\mathcal{H}_{B_{2}}),$
(20a) $\displaystyle\mathcal{W}^{+}$ $\displaystyle\colon$ $\displaystyle
u_{2}\in\mathbb{Z}_{2}\mapsto\mathcal{W}^{+,u_{2}}(\rho)\in\mathcal{D}(\mathcal{H}_{U_{1}}\otimes\mathcal{H}_{B_{1}}\otimes\mathcal{H}_{B_{2}}),$
(20b) with
$\mathcal{W}^{-,u_{1}}(\rho)=\sum_{u_{2}\in\mathbb{Z}_{2}}p_{U}(u_{2})\mathcal{W}^{u_{1}\oplus
u_{2}}(\rho)\otimes\mathcal{W}^{u_{2}}(\rho)$ (20c) and
$\mathcal{W}^{+,u_{2}}(\rho)=\sum_{u_{1}\in\mathbb{Z}_{2}}p_{U}(u_{1})\outerproduct{u_{1}}{u_{1}}\otimes\mathcal{W}^{u_{1}\oplus
u_{2}}(\rho)\otimes\mathcal{W}^{u_{2}}(\rho).$ (20d)

Before presenting channel splitting description for $N$ copies of the channel
$\mathcal{W}$, some properties of the 2-fold case need to be given.

######  Proposition 5.

Consider the transformation
$(\mathcal{W},\mathcal{W})\to(\mathcal{W}^{-},\mathcal{W}^{+})$ for some
channels satisfying 20. Then the following rule holds for the rates:

$\displaystyle I(\mathcal{W}^{-})_{\rho}+I(\mathcal{W}^{+})_{\rho}$
$\displaystyle\leq$ $\displaystyle 2I(\mathcal{W})_{\rho}$ (21) $\displaystyle
I(\mathcal{W}^{+})_{\rho}$ $\displaystyle\geq$ $\displaystyle
I(\mathcal{W})_{\rho}.$ (22)

###### Proof.

For the first statement, it is easy to see that

$\displaystyle I(\mathcal{W}^{-})_{\rho}+I(\mathcal{W}^{+})_{\rho}$
$\displaystyle=$ $\displaystyle I(U_{1};B_{1}B_{2})+I(U_{2};B_{1}B_{2}U_{1})$
(23a) $\displaystyle\overset{(i)}{=}$ $\displaystyle
I(U_{1};B_{1}B_{2})+I(U_{2};B_{1}B_{2}|U_{1})$ (23b)
$\displaystyle\overset{(ii)}{=}$ $\displaystyle I(X_{1}X_{2};B_{1}B_{2})$
(23c) $\displaystyle\overset{(iii)}{=}$ $\displaystyle
I(X_{1};B_{1})+I(X_{2};B_{2})=I(X_{1};B_{1})+I(\mathcal{W}).$ (23d)

The equality in $(i)$ follows from $U_{1}$ and $U_{2}$ being independent.
$(ii)$ is derived from the chain rule for mutual information and the existence
of a one-to-one function from $U_{1},U_{2}$ to $X_{1},X_{2}$. Lastly, the
independence between $X_{1}$ and $X_{2}$ is applied in $(iii)$. Now, notice
that we are dealing with a (possibly non-uniform) Bernoulli random variable
$X_{1}$, which implies $I(X_{1};B_{1})\neq I(\mathcal{W})$ in general. From
$X_{1}=U_{1}+U_{2}$ and $X_{2}=U_{2}$, we have that $X_{1}$ is
$\text{Ber}(p^{2}+(1-p)^{2})$ and $X_{2}$ is $\text{Ber}(p)$. Then, it is
possible to bound $I(X_{2};B_{2})-I(X_{1};B_{1})$ by the following inequality

$\displaystyle I(X_{2};B_{2})-I(X_{1};B_{1})$ $\displaystyle\geq$
$\displaystyle-\log\\{p^{2}+(1-p)^{2}\\}-\log\\{p(1-p)\\}-\log\text{Tr}\\{\sqrt{\mathcal{W}^{0}(\rho)}\sqrt{\mathcal{W}^{1}(\rho)}\\}$
(24)
$\displaystyle-2\sqrt{(1-F(\mathcal{W}^{0}(\rho),\mathcal{W}^{1}(\rho))^{2})2p(1-p)(p^{2}+(1-p)^{2})}.$

We have used the bounds

$I(X_{2};B_{2})\geq-\log\text{Tr}\\{(p_{X_{2}}(0)\sqrt{\mathcal{W}^{0}(\rho)}+p_{X_{2}}(1)\sqrt{\mathcal{W}^{1}(\rho)})^{2}\\},$
(25)

from [17, Prop. 1], and

$\displaystyle I(X_{1};B_{1})$ $\displaystyle\leq$ $\displaystyle
H(\sigma)=h\Big{(}\frac{1}{2}(1-\sqrt{1-4(1-F(\mathcal{W}^{0}(\rho),\mathcal{W}^{1}(\rho))^{2})2p(1-p)(p^{2}+(1-p)^{2})})\Big{)}$
(26a) $\displaystyle\leq$ $\displaystyle
2\sqrt{(1-F(\mathcal{W}^{0}(\rho),\mathcal{W}^{1}(\rho))^{2})2p(1-p)(p^{2}+(1-p)^{2})},$
(26b)

where

$\sigma=\begin{pmatrix}p_{X_{2}}(0)&\sqrt{p_{X_{2}}(0)p_{X_{2}}(1)}F(\mathcal{W}^{0}(\rho),\mathcal{W}^{1}(\rho))\\\
\sqrt{p_{X_{2}}(0)p_{X_{2}}(1)}F(\mathcal{W}^{0}(\rho),\mathcal{W}^{1}(\rho))&p_{X_{2}}(0)\end{pmatrix},$
(27)

from [28, Thm. 3]. Calling the RHS of Eq. 24 by $f(p)$ and analyzing its first
and second derivative, we can conclude that $f(p)\geq 0$ and, thus,
$I(\mathcal{W}^{-})_{\rho}+I(\mathcal{W}^{+})_{\rho}\leq
2I(\mathcal{W})_{\rho}$.

The second statement is derived from

$I(\mathcal{W}^{+})_{\rho}=I(U_{2};B_{1}B_{2}U_{1})\geq
I(U_{2};B_{2})=I(X_{2};B_{2})=I(\mathcal{W}).$ (28)

∎

######  Remark 6.

Proposition 5 shows that the polarization phenomenon can also be derived in
quantum reading problematic. However, it is worth mention the importance of
the probe state used during the process. As explicitly shown in Section 4,
there are probe states that can polarize “faster” than other, in the sense
that
$I(\mathcal{W}^{+})_{\rho}-I(\mathcal{W}^{-})_{\rho}>I(\mathcal{W}^{+})_{\sigma}-I(\mathcal{W}^{-})_{\sigma}$
for some probe states $\rho$ and $\sigma$. Therefore, even though any probe
state generates the polarization phenomenon and, thus, can be used to achieve
a nonzero communication rate with arbitrarily low error probability, it may
not be optimal. There may exist another probe state that produces the same
results but demanding a polar code with lower length.

The following proposition shown in Ref. [27] relates the reliability of the
synthesized channels to the reliability of the original one.

######  Proposition 7.

[27, Prop. 4] Consider the transformation
$(\mathcal{W},\mathcal{W})\to(\mathcal{W}^{-},\mathcal{W}^{+})$ for some
channels satisfying Eq. 20. Then the following rule holds for the reliability:

$\displaystyle Z(\mathcal{W}^{+})_{\rho}$ $\displaystyle=$ $\displaystyle
Z(\mathcal{W})_{\rho}^{2}$ (29) $\displaystyle Z(\mathcal{W}^{-})_{\rho}$
$\displaystyle\leq$ $\displaystyle
2Z(\mathcal{W})_{\rho}-Z(\mathcal{W})_{\rho}^{2}.$ (30)

Now, we can extend the splitting analysis to a $N$-fold combining of
$\mathcal{W}$. Wilde and Guha have shown how to extend the previous
characterization of synthesized channels [37]. We are going to follow a
similar path but using a proper description for quantum memory cells. Let
$N=2^{n}$, where $n\in\mathbb{N}_{0}$. First of all, in this general case is
preferable to label the synthesized channels by natural numbers instead of
$\\{+,-\\}^{n}$. Let $i\in\\{1,2,\ldots,N\\}$. The $i$-th synthesized channel
is given by the map

$\mathcal{W}_{N}^{(i)}\colon u_{i}\mapsto\mathcal{W}^{(i),u_{i}}(\rho^{\otimes
N}),$ (31)

where

$\displaystyle\mathcal{W}^{(i),u_{i}}(\rho^{\otimes N})$ $\displaystyle=$
$\displaystyle\sum_{u_{1}^{i-1}\in\mathbb{Z}_{2}^{i-1}}p_{U^{i-1}}(u_{1}^{i-1})\outerproduct{u_{1}^{i-1}}{u_{1}^{i-1}}\otimes\overline{\mathcal{W}}^{u_{1}^{i}}(\rho^{\otimes
N}),$ (32) $\displaystyle\overline{\mathcal{W}}^{u_{1}^{i}}(\rho^{\otimes N})$
$\displaystyle=$
$\displaystyle\sum_{u_{i+1}^{N}\in\mathbb{Z}_{2}^{N-i}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{u^{N}G_{N}}(\rho^{\otimes
N}).$ (33)

From Eq. 32 we see that the description of $\mathcal{W}_{N}^{(i)}$ supposes
the knowledge of the previous input $u_{1}^{i-1}$. For a finite-length
analysis, this hypothesis may hold with the use of a “genie-aided” successive
cancellation decoder similar to Refs. [3, 37]. In the asymptotic analysis,
this is not needed.

Having described channel combining and splitting, and given important
definitions like the meaning of synthesized channels, our next step is to
characterize the behavior of rate, reliability, and error probability in the
asymptotic scenario. See the next subsection.

####  3.1.3 Rate, Reliability, and Error Probability

The first result present in this section describes the connection between rate
and reliability. It shows, as expected, that the rate
$I(\mathcal{W})_{\rho}\to 0$ (or $I(\mathcal{W})_{\rho}\to h(p)$) when the
reliability $Z(\mathcal{W})_{\rho}\to 2\sqrt{p(1-p)}$ (or
$Z(\mathcal{W})_{\rho}\to 0$).

######  Proposition 8.

Let $X\sim\text{Ber}(p)$ and $\mathcal{W}^{x}(\rho)$ be a quantum memory cell,
where $x\in\mathbb{Z}_{2}$. Then the following holds

$\displaystyle I(\mathcal{W})_{\rho}$ $\displaystyle\geq$ $\displaystyle
h(p)-\log(1+Z(\mathcal{W})_{\rho}),$ (34) $\displaystyle
I(\mathcal{W})_{\rho}$ $\displaystyle\leq$
$\displaystyle\sqrt{4p(1-p)-Z(\mathcal{W})_{\rho}^{2}}.$ (35)

###### Proof.

The first inequality follows from [15, Thm. C.1]. For the second one, we need
to use the following inequality derived from Theorem 3 in Ref. [28]:

$I(\mathcal{W})_{\rho}\leq H(\sigma),$ (36)

where

$\sigma=\begin{pmatrix}p&\frac{Z(\mathcal{W})_{\rho}}{2}\\\
\frac{Z(\mathcal{W})_{\rho}}{2}&1-p\end{pmatrix}.$ (37)

Thus

$\displaystyle I(\mathcal{W})_{\rho}$ $\displaystyle\leq$ $\displaystyle
h\Big{(}\frac{1}{2}(1-\sqrt{1-4p(1-p)+Z(\mathcal{W})_{\rho}^{2}})\Big{)}$ (38)
$\displaystyle\leq$ $\displaystyle\sqrt{4p(1-p)-Z(\mathcal{W})_{\rho}^{2}}.$
(39)

∎

Before presenting the next result, we need to introduce a new classical-
quantum symmetric channel. Let $\tilde{U}_{i}\sim\text{Ber}(\frac{1}{2})$,
$U_{j}\sim\text{Ber}(p)$ for all $i,j=1,\ldots,N$. Assume that the output
system of the classical-quantum channel $\tilde{\mathcal{W}}$ is given by
$\mathcal{D}(\mathcal{H}_{\tilde{Z}}\otimes\mathcal{H}_{B})$, where
$\sigma\in\mathcal{D}(\mathcal{H}_{\tilde{Z}}\otimes\mathcal{H}_{B})$ can be
written as

$\sigma=\sum_{\tilde{z}\in\mathbb{Z}_{2}}\outerproduct{\tilde{z}}{\tilde{z}}^{\tilde{Z}}\otimes\rho^{B}.$
(40)

Notice that $\tilde{Z}$ represents the classical input system in the same way
as the system $Z$ associated with the quantum channel $\mathcal{W}$.
Therefore, the associated quantum systems are the same. We keep different
notation here to emphasize that one is associated with a symmetric channel and
the other with an asymmetric channel. Now, we can properly describe the
classical-quantum channel $\tilde{\mathcal{W}}$. Fix
$\rho\in\mathcal{D}(\mathcal{H}_{A})$ and let $u\in\mathbb{Z}_{2}$. Then
$\tilde{\mathcal{W}}$ is the map given by

$\displaystyle\tilde{\mathcal{W}}\colon\tilde{U}$
$\displaystyle\to\mathcal{D}(\mathcal{H}_{\tilde{Z}}\otimes\mathcal{H}_{B})$
$\displaystyle\tilde{u}$
$\displaystyle\mapsto\tilde{\mathcal{W}}^{\tilde{u}}(\rho):=\sum_{u\in\mathcal{U}}p_{U}(u)\outerproduct{\tilde{u}\oplus
u}{\tilde{u}\oplus u}^{\tilde{Z}}\otimes\mathcal{W}^{u}(\rho)^{B}.$ (41)

It is easy to see that $\tilde{\mathcal{W}}$ is a symmetric classical-quantum
channel. Indeed, this follows from the uniform distribution of $\tilde{U}$ and
the construction of the $\mathcal{H}_{\tilde{Z}}$ part. The next step is to
describe the synthesized channels generated in polar coding on
$\tilde{\mathcal{W}}$. Following the characterization given in Eq. 31, we have
that

$\tilde{\mathcal{W}}_{N}^{(i),\tilde{u}_{i}}(\rho^{N})=\sum_{\tilde{u}_{1}^{i-1}}\frac{1}{2^{i-1}}\outerproduct{\tilde{u}_{1}^{i-1}}{\tilde{u}_{1}^{i-1}}\otimes\Big{(}\sum_{u^{N}}p_{U^{N}}(u^{N})\mathcal{W}^{u^{N}G_{N}}(\rho^{N})\otimes\Big{(}\sum_{\tilde{u}_{i+1}^{N}}\frac{1}{2^{N-i}}\outerproduct{(\tilde{u}^{N}\oplus
u^{N})G_{N}}{(\tilde{u}^{N}\oplus u^{N})G_{N}}\Big{)}\Big{)}.$ (42)

With these tools, we show in the next proposition that any asymmetric
classical-quantum channel can be described via a symmetric one. Moreover, in
the following, a relation between the reliabilities of these channels is
provided.

######  Proposition 9.

Let
$\tilde{\mathcal{W}}_{N}^{(i)}\colon\tilde{U}_{i}\rightarrow\mathcal{D}(\mathcal{H}_{\tilde{U}_{1}^{i-1}}\otimes\mathcal{H}_{B^{N}}\otimes\mathcal{H}_{\tilde{Z}^{N}})$,
where $\tilde{U}_{i}\sim\text{Ber}(\frac{1}{2})$ for all $i=1,\ldots,N$. Then,

$\mathcal{W}_{N}^{(i),u_{i}}(\rho^{N})=\text{Tr}_{\tilde{Z}^{N}}\\{\tilde{\mathcal{W}}_{N}^{(i),u_{i}}(\rho^{N})\outerproduct{0}{0}^{Z^{N}}\\}.$
(43)

###### Proof.

Let $\tilde{\rho}^{(i)}$ be the joint input-output density operator of
$\tilde{\mathcal{W}}_{N}^{(i),\tilde{u}_{i}}$. Thus,

$\displaystyle\tilde{\rho}^{(i)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\tilde{u}_{i}\in\mathbb{Z}_{2}}\outerproduct{\tilde{u}_{i}}{\tilde{u}_{i}}\otimes\tilde{\mathcal{W}}_{N}^{(i),\tilde{u}_{i}}(\rho^{N})$
(44a) $\displaystyle=$
$\displaystyle\sum_{\tilde{u}^{N}}\frac{1}{2^{N}}\outerproduct{\tilde{u}_{1}^{i}}{\tilde{u}_{1}^{i}}\otimes\Big{(}\sum_{u^{N}}p_{U^{N}}(u^{N})\mathcal{W}^{u^{N}G_{N}}(\rho^{N})\otimes\outerproduct{(\tilde{u}^{N}\oplus
u^{N})G_{N}}{(\tilde{u}^{N}\oplus u^{N})G_{N}}\Big{)}$ (44b) $\displaystyle=$
$\displaystyle\sum_{u^{N}}p_{U^{N}}(u^{N})\mathcal{W}^{u^{N}G_{N}}(\rho^{N})\otimes\Big{(}\sum_{\tilde{u}^{N}}\frac{1}{2^{N}}\outerproduct{\tilde{u}_{1}^{i}}{\tilde{u}_{1}^{i}}\otimes\outerproduct{(\tilde{u}^{N}\oplus
u^{N})G_{N}}{(\tilde{u}^{N}\oplus u^{N})G_{N}}\Big{)}.$ (44c)

Defining $\tilde{z}^{N}=(\tilde{u}^{N}\oplus u^{N})G_{N}$, it is possible to
rearrange the sums as

$\displaystyle\tilde{\rho}^{(i)}$ $\displaystyle=$
$\displaystyle\sum_{u^{N}}p_{U^{N}}(u^{N})\mathcal{W}^{u^{N}G_{N}}(\rho^{N})\otimes\Big{(}\sum_{\begin{subarray}{c}\tilde{z}^{N}\\\
[u^{N}\oplus\tilde{z}^{N}G_{N}]_{1}^{i}=\tilde{u}_{1}^{i}\end{subarray}}\frac{1}{2^{N}}\outerproduct{[u^{N}\oplus\tilde{z}^{N}G_{N}]_{1}^{i}}{[u^{N}\oplus\tilde{z}^{N}G_{N}]_{1}^{i}}\otimes\outerproduct{\tilde{z}^{N}}{\tilde{z}^{N}}\Big{)}$
(45) $\displaystyle=$
$\displaystyle\sum_{\tilde{z}^{N}}\frac{1}{2^{N}}\outerproduct{\tilde{z}^{N}}{\tilde{z}^{N}}\otimes\Big{(}\sum_{\tilde{u}_{1}^{i}}p_{U_{1}^{i}}(\tilde{u}_{1}^{i}\oplus[\tilde{z}^{N}G_{N}]_{1}^{i})\outerproduct{\tilde{u}_{1}^{i}}{\tilde{u}_{1}^{i}}\otimes\mathcal{W}^{[(\tilde{u}_{1}^{i},0_{i+1}^{N})G_{N}]_{1}^{i}\oplus\tilde{z}_{1}^{i}}(\rho^{i})$
$\displaystyle\otimes$
$\displaystyle\Big{(}\sum_{u_{i+1}^{N}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{[(0_{1}^{i},u_{i+1}^{N})G_{N}]_{i+1}^{N}}(\rho^{N-i})\Big{)}\Big{)}.$

Now, we can see that

$\text{Tr}_{\tilde{Z}^{N}}\\{\tilde{\rho}^{(i)}\outerproduct{0}{0}^{Z^{N}}\\}=\sum_{u_{1}^{i}}p_{U_{1}^{i}}(u_{1}^{i})\outerproduct{u_{1}^{i}}{u_{1}^{i}}\otimes\Big{(}\sum_{u_{i+1}^{N}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{u^{N}G_{N}}(\rho^{N})\Big{)},$
(46)

which is the joint input-output state of
$\mathcal{W}^{(i),u_{i}}_{N}(\rho^{N})$. ∎

######  Proposition 10.

Let $\mathcal{W}_{N}^{(i)}$ and $\tilde{\mathcal{W}}_{N}^{(i)}$ be the
synthesized quantum channels described in Eq. 31 and Proposition 43,
respectively. Then

$Z(\tilde{\mathcal{W}}_{N}^{(i)})_{\rho}=Z(\mathcal{W}_{N}^{(i)})_{\rho}.$
(47)

###### Proof.

From the definition of reliability for symmetric classical-quantum channels in
Ref. [37] and Eq 45, we have that

$\displaystyle Z(\tilde{\mathcal{W}}_{N}^{(i)})_{\rho}$ $\displaystyle=$
$\displaystyle
F(\tilde{\mathcal{W}}_{N}^{(i),0}(\rho^{N}),\tilde{\mathcal{W}}_{N}^{(i),1}(\rho^{N}))$
(48) $\displaystyle=$
$\displaystyle\sum_{\tilde{z}^{N}}\sum_{\tilde{u}_{1}^{i-1}}\frac{p_{U_{1}^{i-1}}(\tilde{u}_{1}^{i-1}\oplus[\tilde{z}^{N}G_{N}]_{1}^{i-1})\sqrt{p_{U}(0\oplus[\tilde{z}^{N}G_{N}]_{i})p_{U}(1\oplus[\tilde{z}^{N}G_{N}]_{i})}}{2^{N-1}}F(w^{(i)}_{0},w^{(i)}_{1}),$
(49)

where

$\displaystyle w^{(i)}_{0}$ $\displaystyle=$
$\displaystyle\mathcal{W}^{[(\tilde{u}_{1}^{i-1},0_{i}^{N})G_{N}]_{1}^{i}\oplus\tilde{z}_{1}^{i}}(\rho^{i})\sum_{u_{i+1}^{N}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{[(0_{1}^{i},u_{i+1}^{N})G_{N}]_{i+1}^{N}}(\rho^{N-i}),$
(50) $\displaystyle w^{(i)}_{1}$ $\displaystyle=$
$\displaystyle\mathcal{W}^{[(\tilde{u}_{1}^{i-1},1,0_{i+1}^{N})G_{N}]_{1}^{i}\oplus\tilde{z}_{1}^{i}}(\rho^{i})\sum_{u_{i+1}^{N}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{[(0_{1}^{i},u_{i+1}^{N})G_{N}]_{i+1}^{N}}(\rho^{N-i}).$
(51)

Let $z^{N}=\tilde{z}^{N}G_{N}$ and
$u_{1}^{i-1}=\tilde{u}_{1}^{i-1}\oplus[\tilde{z}^{N}G_{N}]_{1}^{i-1}=\tilde{u}_{1}^{i-1}\oplus
z_{1}^{i-1}$. Since the map
$(\tilde{z}_{1}^{N},\tilde{u}_{1}^{i-1})\mapsto(z_{1}^{N},u_{1}^{i-1})$ is a
bijection, we see that

$\displaystyle Z(\tilde{\mathcal{W}}_{N}^{(i)})_{\rho}$ $\displaystyle=$
$\displaystyle\sum_{z^{N}}\sum_{u_{1}^{i-1}}\frac{p_{U_{1}^{i-1}}(u_{1}^{i-1})\sqrt{p_{U}(0)p_{U}(1)}}{2^{N-1}}F(\mathcal{W}^{(u_{1}^{i-1},0)}(\rho^{i})\sum_{u_{i+1}^{N}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{[(0_{1}^{i},u_{i+1}^{N})G_{N}]_{i+1}^{N}}(\rho^{N-i}),$
(52)
$\displaystyle\mathcal{W}^{(u_{1}^{i-1},1)}(\rho^{i})\sum_{u_{i+1}^{N}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{[(0_{1}^{i},u_{i+1}^{N})G_{N}]_{i+1}^{N}}(\rho^{N-i}))$
$\displaystyle=$ $\displaystyle
2\sqrt{p_{U}(0)p_{U}(1)}\sum_{u_{1}^{i-1}}p_{U_{1}^{i-1}}(u_{1}^{i-1})F(\overline{\mathcal{W}}^{(u_{1}^{i-1},0)}(\rho^{\otimes
N}),\overline{\mathcal{W}}^{(u_{1}^{i-1},1)}(\rho^{\otimes N}))$
$\displaystyle=$ $\displaystyle
2\sqrt{p_{U}(0)p_{U}(1)}F(\mathcal{W}^{(i),0}(\rho^{\otimes
N}),\mathcal{W}^{(i),1}(\rho^{\otimes N}))$ $\displaystyle=$ $\displaystyle
Z(\mathcal{W}_{N}^{(i)})_{\rho},$

where $\overline{\mathcal{W}}^{u_{1}^{i}}(\rho^{\otimes N})$ is defined in Eq.
32. ∎

The next theorem shows that the asymptotically fraction of good channels is
equals to the mutual information between the classical system $X$ and the
quantum system $B$.

######  Theorem 11.

Let $\mathcal{W}_{N}^{(i)}$ be the synthesized quantum channels described in
Eq. 31 and $Z(X|Y)$ be the reliability of two Bernoulli random variables $X$
and $Y$; i.e., $Z(X|Y)=2\sum_{y}\sqrt{p_{X,Y}(0,y)p_{X,Y}(1,y)}$. For every
$\beta<1/2$, we have

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{2^{n}}|\\{i\in\mathbb{Z}_{2}^{n}\colon
Z(\mathcal{W}_{N}^{(i)})_{\rho}\leq 2^{-2^{n\beta}}\text{ and
}Z(U_{i}|U_{1}^{i-1})\geq 1-2^{-2^{n\beta}}\\}|$ $\displaystyle=$
$\displaystyle I(X;B)_{\rho},$ (53)
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{2^{n}}|\\{i\in\mathbb{Z}_{2}^{n}\colon
Z(\mathcal{W}_{N}^{(i)})_{\rho}\geq 1-2^{-2^{n\beta}}\text{ and
}Z(U_{i}|U_{1}^{i-1})\leq 2^{-2^{n\beta}}\\}|$ $\displaystyle=$ $\displaystyle
1-I(X;B)_{\rho},$ (54)

###### Proof.

Firstly, notice that

$\displaystyle I(\tilde{\mathcal{W}})$ $\displaystyle=$ $\displaystyle
I(\tilde{X};\tilde{X}\oplus X,B)$ (55) $\displaystyle=$ $\displaystyle
H(\tilde{X}\oplus X,B)-H(\tilde{X}\oplus X,B|\tilde{X})$ $\displaystyle=$
$\displaystyle 1+H(B)-H(X,B)=1-H(X|B).$

Applying Eq. 55 to Proposition 1, we can deduce that

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{2^{n}}|\\{i\in\mathbb{Z}_{2}^{n}\colon
Z(\mathcal{W}_{N}^{(i)})_{\rho}\leq 2^{-2^{n\beta}}\\}|$ $\displaystyle=$
$\displaystyle 1-H(X|B)_{\rho},$ (56a)
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{2^{n}}|\\{i\in\mathbb{Z}_{2}^{n}\colon
Z(\mathcal{W}_{N}^{(i)})_{\rho}\geq 1-2^{-2^{n\beta}}\\}|$ $\displaystyle=$
$\displaystyle H(X|B)_{\rho}.$ (56b)

Additionally, it is possible to derive
$Z(\mathcal{W}_{N}^{(i)})_{\rho}=Z(U_{i}|U_{1}^{i-1})$ and $H(X|B)=H(X)$ if
the output probe state of $\mathcal{W}$ is independent of $X$. Thus, a similar
result to Eq. 56 is derived

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{2^{n}}|\\{i\in\mathbb{Z}_{2}^{n}\colon
Z(U_{i}|U_{1}^{i-1})\leq 2^{-2^{n\beta}}\\}|$ $\displaystyle=$ $\displaystyle
1-H(X),$ (57a)
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{2^{n}}|\\{i\in\mathbb{Z}_{2}^{n}\colon
Z(U_{i}|U_{1}^{i-1})\geq 1-2^{-2^{n\beta}}\\}|$ $\displaystyle=$
$\displaystyle H(X).$ (57b)

Now, define $A,B,C,$ and $D$ as the sets

$\displaystyle A$ $\displaystyle=$ $\displaystyle\\{i\colon
Z(\mathcal{W}_{N}^{(i)})_{\rho}\leq 2^{-2^{n\beta}}\\},$ (58a) $\displaystyle
B$ $\displaystyle=$ $\displaystyle\\{i\colon
Z(\mathcal{W}_{N}^{(i)})_{\rho}\geq 1-2^{-2^{n\beta}}\\},$ (58b)
$\displaystyle C$ $\displaystyle=$ $\displaystyle\\{i\colon
Z(U_{i}|U_{1}^{i-1})\leq 2^{-2^{n\beta}}\\},$ (58c) $\displaystyle D$
$\displaystyle=$ $\displaystyle\\{i\colon Z(U_{i}|U_{1}^{i-1})\geq
1-2^{-2^{n\beta}}\\}.$ (58d)

From Proposition 8 and $H(U_{i}|U_{1}^{i-1},B^{N})\leq H(U_{i}|U_{1}^{i-1})$,
it is possible to see that $B\cap C$ is empty for sufficiently large $n$.
Furthermore,

$\lim_{n\rightarrow\infty}\frac{|A|+|B|}{2^{n}}=\lim_{n\rightarrow\infty}\frac{|C|+|D|}{2^{n}}=1.$
(59)

Therefore, the claim is derived from

$\lim_{n\rightarrow\infty}\frac{|B\cup
C|}{2^{n}}=\lim_{n\rightarrow\infty}\frac{|B|+|C|}{2^{n}}=1-I(X;B)_{\rho}.$
(60)

and

$\lim_{n\rightarrow\infty}\frac{|A\cap
D|}{2^{n}}=1-\lim_{n\rightarrow\infty}\frac{|B\cup C|}{2^{n}}=I(X;B)_{\rho}.$
(61)

∎

As can be seen in the theorem statement and elaborated in the proof, we had to
impose an additional constraint on the reliability of $U_{i}$ given the
previous $U_{1}^{i-1}$ in order to derive our result. This is because we are
dealing with asymmetric quantum reading. For symmetric quantum reading, the
constrain is not needed.

######  Remark 12.

Subsections 3.1.1 and 3.1.2, and Proposition 8 are easily applied to symmetric
quantum channels discrimination, having it as a particular case when
$p_{X}(0)=p_{X}(1)=\frac{1}{2}$. For the result in Theorem 11, there is no
need to introduce the new type of channel and, furthermore, the asymptotic
analysis of the rate of good and bad channels does not impose anything on
$Z(U_{i}|U_{1}^{i-1})$. The symmetric quantum reading treatment of Theorem 11
goes similarly to Section IV of Ref. [37].

###  3.2 Polar Coding and Decoding Scheme

The encoding protocol of polar codes consists of setting up the labels for
information bits and frozen bits. The former label is denoted by $\mathcal{A}$
and the later by $\mathcal{A}^{c}$. Thus, we use bits
$u_{\mathcal{A}}=\\{u_{i}\\}_{i\in\mathcal{A}}$ to transmit information. On
the frozen bits, they can be fixed for the whole transmission or can depend on
the previous $u_{1}^{i-1}$ bits. For asymmetric channels, as our case, the
latter strategy is more suitable. See Ref. [18] for more explanations.

The construction of a codeword is done as follows. The source generates a
uniform sequence $u_{1}^{|\mathcal{A}|}$. Next, the encoder determines the
value $u_{i}$, $i\in\mathcal{A}^{c}$, of the frozen bits in the ascending
order by $u_{i}=\lambda_{i}(u_{1}^{i-1})$, where $\lambda_{i}$ is a function
from $\\{0,1\\}^{i-1}$ to $\\{0,1\\}$. Putting the sequence
$u_{1}^{|\mathcal{A}|}$ in the information bits $u_{\mathcal{A}}$ and the
frozen bits in the remaining coordinates, we have the vector $u^{N}$. Now, the
codeword is given by $x^{N}=u^{N}G_{N}$. It is clear that the code rate is
$R=|\mathcal{A}|/N$. It remains to describe how the set $\mathcal{A}$ is
determined and which functions $\lambda_{i}$ do we use.

Determining the set of information bits is crucial for the performance of a
polar code. Firstly, for a sufficiently large code length $N$, we choose the
cardinality of $\mathcal{A}$ so that the code rate $R<I(\mathcal{W})_{\rho}$.
Next, we have to choose coordinates to constitute the set $\mathcal{A}$. A
common approach is to select the coordinates with smallest reliabilities.
Formally, this selection goes as follows. Let $\mathcal{A}$ be the set that
for any $j\in\mathcal{A}^{c}$ we have $Z(\mathcal{W}_{N}^{(j)})\geq
Z(\mathcal{W}_{N}^{(i)})$ for all $i\in\mathcal{A}$. Additionally, because the
construction under consideration is for asymmetric channels, we also impose
$Z(U_{i}|U_{1}^{i-1})$, for all $i\in\mathcal{A}$, to be large when compared
with the elements in $\mathcal{A}^{c}$. Theorem 11 makes use of this constrain
to characterize the asymptotic rates in polar coding. If any of these two
constrains is not satisfied, we decrease the cardinality of $\mathcal{A}$ down
to when they are satisfied.

The functions $\lambda_{i}$ used in this paper is such that optimize the
probability of frozen bit output. Let
$\Lambda_{\mathcal{A}^{c}}=\\{\Lambda_{i}\\}_{i\in\mathcal{A}^{c}}$ be random
variables which are independent of each other and input and output systems,
and satisfy

$p_{\Lambda_{i}}[\Lambda_{i}(u_{1}^{i-1})]=p_{U_{i}|U_{1}^{i-1}}(1|u_{1}^{i-1}),$
(62)

for all $u_{1}^{i-1}\in\\{0,1\\}^{i-1}$. Then, $\lambda_{i}$ is a realization
of the random variable $\Lambda_{i}$. There are practical methods to generate
in practice the functions $\lambda_{i}$ using pseudorandom number generators
[18], but there is no need to address it in this paper.

Considering symmetric channel discrimination, a similar encoding scheme can be
proposed. First of all, the algorithm for choosing the frozen bits is exactly
the one presented here without the dependence of the previous bits. Thus, the
encoding map is a realization of the random variable $\tilde{\Lambda}_{i}$,
where

$p_{\Lambda_{i}}[\Lambda_{i}=i]=\frac{1}{2},$ (63)

for $i=1,2$. Secondly, the proposed strategy for defining the set
$\mathcal{A}$ is applicable for symmetric channels discrimination. We only
need to drop the constrain on $Z(U_{i}|U_{1}^{i-1})$. Approximation techniques
in symmetric polar codes [34] can compose the decoding scheme in order to
derive a faster encoding scheme.

Now we describe the decoding process. Suppose the sequential decoder has
obtained, up to this moment, the vector $\hat{u}_{1}^{i-1}$ and plans to
obtain $\hat{u}_{i}$. Then the decoding process divides into two cases. For
the coordinates in $\mathcal{A}^{c}$, we apply the inverse encoding function
depending on the previous coordinates. Namely, we employ

$\hat{u}_{i}=\lambda_{i}^{-1}(\hat{u}_{1}^{i-1}).$ (64)

Notice that no measurement is implemented in this step and, for sufficiently
large $N$, the error probability is arbitrarily low. See Theorem 11 for the
proof. Next, we deal with the information bits. Quantum successive
cancellation decoder is used in here. First of all, this makes our decoding
strategy and analysis totally different from any decoding method used for
classical polar codes. Secondly, because of the constructive approach adopted
in here, with measurements created for our specific task, the polar decoding
strategy differs from previous works on quantum polar codes.

For characterizing the error probability decay of the polar codes constructed,
we firstly show the existence of “pretty good measurements” design to decode
quantum memory cell and having desirable error probabilities in Proposition
66.

######  Proposition 13.

Let $\mathcal{W}\colon
x\in\mathcal{X}\rightarrow\rho_{x}^{UB}\in\mathcal{D}(\mathcal{H}_{UB})$ be a
cq channel such that

$\rho_{x}^{UB}=\sum_{u\in\mathcal{U}}p_{U}(u)\outerproduct{u}{u}^{U}\otimes\rho_{x,u}^{B},$
(65)

where $X$ is a discrete random variable, and $\\{\ket{u}\\}_{u\in\mathcal{U}}$
is an orthonormal basis of a finite-dimensional Hilbert space. Then, there
exists a POVM $\\{\Lambda_{x}^{UB}\\}_{x\in\mathcal{X}}$ satisfying

$1-\sum_{x\in\mathcal{X}}p_{X}(x)\text{Tr}\\{\Lambda_{x}\rho_{x}\\}<\frac{1}{2}Z(\mathcal{W}).$
(66)

###### Proof.

To derive our claim, we need to invoke a result from Barnum and Knill [7]. Let
$p_{U}(u)$ be the probability of a system to be found in the state $\rho_{u}$,
for $u\in\mathcal{U}$, then there exists a POVM
$\\{\Lambda_{u}^{*}\\}_{u\in\mathcal{U}}$ such that the average success
probability is lower bounded as

$P_{succ}=\sum_{u\in\mathcal{U}}p_{U}(u)\text{Tr}\\{\rho_{u}^{\otimes
M}\Lambda_{u}^{*}\\}\geq 1-\sum_{u\neq
v}\sqrt{p_{U}(u)p_{U}(v)}F(\rho_{u}^{M},\rho_{v}^{M}).$ (67)

In particular, there exists a POVM $\\{\Lambda_{u}\\}_{u\in\mathcal{U}}$ such
that the probability of error follows the following inequality

$P_{err}=1-\sum_{u\in\mathcal{U}}p_{U}(u)\text{Tr}\\{\rho_{u}\Lambda_{u}\\}\leq\sum_{u\neq
v}\sqrt{p_{U}(u)p_{U}(v)}F(\rho_{u},\rho_{v}).$ (68)

Now, we return to our proof. Assume that $\mathcal{W}^{u}\colon
x\rightarrow\rho_{x,u}$ is a cq channel, where $x\in\mathcal{X}$ is a
realization of a discrete random variable $X$. The result in Eq. 68 says that
there exist POVMs $\\{\Lambda_{x,u}\\}_{x\in\mathcal{X}}$ satisfying

$1-\sum_{x\in\mathcal{X}}p_{X}(x)\text{Tr}\\{\rho_{x,u}\Lambda_{x,u}\\}<\frac{1}{2}Z(\mathcal{W}^{u}).$
(69)

Defining, for every $u\in\mathcal{U}$, the POVM

$\Lambda_{x}:=\sum_{u\in\mathcal{U}}\Lambda_{x,u}\otimes\outerproduct{u}{u},$
(70)

we have that

$\displaystyle 1$
$\displaystyle-\sum_{x\in\mathcal{X}}p_{X}(x)\text{Tr}\\{\Lambda_{x}\rho_{x}\\}\overset{(i)}{=}1-\sum_{x\in\mathcal{X}}p_{X}(x)\sum_{u\in\mathcal{U}}p_{U}(u)\text{Tr}\\{\Lambda_{x,u}\rho_{x,u}\\}$
(71) $\displaystyle=$
$\displaystyle\sum_{u\in\mathcal{U}}p_{U}(u)\sum_{x\in\mathcal{X}}p_{X}(x)\Big{(}1-\text{Tr}\\{\Lambda_{x,u}\rho_{x,u}\\}\Big{)}$
(72) $\displaystyle\overset{(ii)}{<}$
$\displaystyle\sum_{u\in\mathcal{U}}p_{U}(u)\frac{1}{2}Z(\mathcal{W}^{u})$
(73) $\displaystyle\overset{(iii)}{=}$
$\displaystyle\sum_{\begin{subarray}{c}x,x^{\prime}\in\mathcal{X}\\\ x\neq
x^{\prime}\end{subarray}}\sqrt{p_{X}(x)p_{X}(x^{\prime})}F\Big{(}\sum_{u\in\mathcal{U}}p_{U}(u)\outerproduct{u}{u}\otimes\rho_{x,u},\sum_{u\in\mathcal{U}}p_{U}(u)\outerproduct{u}{u}\otimes\rho_{x^{\prime},u}\Big{)}$
(74) $\displaystyle\overset{(iv)}{=}$
$\displaystyle\sum_{\begin{subarray}{c}x,x^{\prime}\in\mathcal{X}\\\ x\neq
x^{\prime}\end{subarray}}\sqrt{p_{X}(x)p_{X}(x^{\prime})}F\Big{(}\rho_{x},\rho_{x^{\prime}}\Big{)}$
(75) $\displaystyle=$ $\displaystyle\frac{1}{2}Z(\mathcal{W}).$ (76)

The equality in $(i)$ follows from the definitions of $\rho_{x}$ and
$\Lambda_{x}$ in Eq. 65 and 70, respectively. The inequality in Eq. 69 is used
to obtain $(ii)$. The item $(iii)$ is just the computation of the reliability
for $\mathcal{W}^{u}$. Lastly, we use in $(iv)$ the fact that
$F(\sum_{x}p_{X}(x)\outerproduct{x}{x}\otimes\sigma_{x},\sum_{x}q_{X}(x)\outerproduct{x}{x}\otimes\tau_{x})=\sum_{x}\sqrt{p_{X}(x)q_{X}(x)}F(\sigma_{x},\tau_{x})$.
∎

Analyzing a quantum successive cancellation decoder employing the measurements
in Proposition 66 needs the use of a quantum union bound. Lemma 14 introduces
the one used in this paper. As it is shown in the proof of Theorem 15, the
inequality in Lemma 14 allows us to derive a bound for the error probability
in accordance to what is expected using polar codes.

######  Lemma 14.

[21, Lemma 4.1] Let $\rho$ be a positive semi-definite operator acting on a
separable Hilbert space $\mathcal{H}_{B}$, let $\\{\Lambda_{i}\\}_{i=1}^{L}$
denote a set of positive semi-definite operators such that
$0\leq\Lambda_{i}\leq I$ for all $i\in\\{1,\ldots,L\\}$, and let $c>0$. Then
the following quantum union bound holds

$\displaystyle\text{Tr}\\{\rho\\}-\text{Tr}\\{\Pi_{\Lambda_{L}}\cdots\Pi_{\Lambda_{1}}(\rho\otimes\outerproduct{\overline{0}}{\overline{0}}_{P^{L}})\Pi_{\Lambda_{1}}\cdots\Pi_{\Lambda_{L}}\\}\leq(1+c)\text{Tr}\\{(I-\Lambda_{L})\rho\\}$
$\displaystyle+(2+c+c^{-1})\sum_{i=2}^{L-1}\text{Tr}\\{(I-\Lambda_{i})\rho\\}+(2+c^{-1})\text{Tr}\\{(I-\Lambda_{1})\rho\\},$
(77)

where
$\ket{\overline{0}_{P_{L}}}\equiv\ket{0_{P_{1}}}\otimes\cdots\otimes\ket{0_{P_{L}}}$
is an auxiliary state of $L$ qubit probe systems and $\Pi_{\Lambda_{i}}$ is a
projector defined as $\Pi_{\Lambda_{i}}:=U_{i}^{\dagger}P_{i}U_{i}$ for some
unitary $U_{i}$ and projector $P_{i}$ such that
$\text{Tr}\\{\Pi_{\Lambda_{i}}(\rho\otimes\outerproduct{\overline{0}}{\overline{0}}_{P^{L}})\\}=\text{Tr}\\{\Lambda_{i}\rho\\}$.
In particular, for any set of positive semi-definite operators, there exists a
set of projectors satisfying Eq. 77.

An important attribute of polar codes is the error probability decay when the
length of the code grows. We show in Theorem 15 that error probability decays
exponentially fast in the code length. This result motivates the use of polar
codes in practical protocols devoted to discriminate quantum memory cell.

######  Theorem 15.

Let $\\{\mathcal{W}^{x}\\}_{x\in\mathcal{X}}$ be a quantum memory cell and
$\gamma\in\mathbb{R}$ a positive constant. Then there exists a polar code with
parameters $(N,K,\mathcal{A})$ such that the error probability is bounded
above by

$P_{e}(N,K,\mathcal{A})\leq\gamma\sum_{i\in\mathcal{A}}Z_{\rho}(\mathcal{W}^{(i)}).$
(78)

In particular, for any fixed $R=K/N<I(W)$ and $\beta<1/2$, block error
probability for polar coding under sequential decoding satisfies

$P_{e}(N,K)=o(2^{-N^{\beta}}),$ (79)

where $o(\cdot)$ is the little-O notation from complexity theory.

###### Proof.

First of all, suppose we have access to an auxiliary system
$\ket{\overline{0}_{P_{N}}}=\ket{0_{P_{1}}}\otimes\cdots\otimes\ket{0_{P_{N}}}$.
Lemma 14 guarantees the existence of projective measurements obtained by
extending the pretty good measurements of Proposition 66. We denote the
projective and pretty good measurements by the sets
$\\{\Pi_{\Lambda_{i}}\\}_{i=1}^{N}$ and $\\{\Lambda_{i}\\}_{i=1}^{N}$,
respectively. Then, using these projective measurements on a quantum
successive cancellation strategy for the information bits, a bound for the
error probability is obtained as follows

$\displaystyle P_{e}(N,K,\mathcal{A})$ $\displaystyle=$ $\displaystyle
1-\sum_{u^{N}}p_{U^{N}}(u^{N})\text{Tr}\\{\Pi_{\Lambda_{u^{N}}}\mathcal{W}^{u^{N}}(\rho^{N})\otimes\outerproduct{\overline{0}_{P_{N}}}{\overline{0}_{P_{N}}}\\}$
(80) $\displaystyle=$ $\displaystyle
1-\sum_{u^{N}}p_{U^{N}}(u^{N})\text{Tr}\\{\Pi_{\Lambda_{u_{1}^{N-1}u_{N}}}^{B^{N}}\cdots\Pi_{\Lambda_{u_{1}}}^{B^{N}}\mathcal{W}^{u^{N}}(\rho^{N})\otimes\outerproduct{\overline{0}_{P_{N}}}{\overline{0}_{P_{N}}}\Pi_{\Lambda_{u_{1}}}^{B^{N}}\cdots\Pi_{\Lambda_{u_{1}^{N-1}u_{N}}}^{B^{N}}\\}$
$\displaystyle\leq$
$\displaystyle\sum_{u^{N}}p_{U^{N}}(u^{N})\Big{(}(1+c)\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}^{N-1}u_{N}}^{B^{N}})\mathcal{W}^{u^{N}}(\rho^{N})\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c+c^{-1})\sum_{i=2}^{N-1}\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}^{i-1}u_{i}}^{B^{N}})\mathcal{W}^{u^{N}}(\rho^{N})\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c^{-1})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}}^{B^{N}})\mathcal{W}^{u^{N}}(\rho^{N})\Big{\\}}\Big{)},$
(82)

where the second inequality is obtained from the definition of a quantum
successive cancellation decoder and the inequality from Lemma 14. Distributing
and rearranging the sums, the following equality holds

RHS of Eq. 82 $\displaystyle=$
$\displaystyle(1+c)\sum_{u^{N}}p_{U^{N}}(u^{N})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}^{N-1}u_{N}}^{B^{N}})\mathcal{W}^{u^{N}}(\rho^{N})\Big{\\}}$
(83) $\displaystyle+$
$\displaystyle(2+c+c^{-1})\sum_{i=2}^{N-1}p_{U}(u_{i})\sum_{u_{1}^{i-1}}p_{U_{1}^{i-1}}(u_{1}^{i-1})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}^{i-1}u_{i}}^{B^{N}})\sum_{u_{1}^{i-1}}p_{U_{i+1}^{N}}(u_{i+1}^{N})\mathcal{W}^{u^{N}}(\rho^{N})\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c^{-1})\sum_{u_{1}}p_{U}(u_{1})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}}^{B^{N}})\sum_{u_{2}^{i-1}}p_{U_{2}^{N}}(u_{2}^{N})\mathcal{W}^{u^{N}}(\rho^{N})\Big{\\}}.$

Continuing,

RHS of Eq. 83 $\displaystyle=$
$\displaystyle(1+c)\sum_{u^{N}}p_{U^{N}}(u^{N})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}^{N-1}u_{N}}^{B^{N}})\overline{\mathcal{W}}^{u_{1}^{N}}(\rho^{N})\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c+c^{-1})\sum_{i=2}^{N-1}p_{U}(u_{i})\sum_{u_{1}^{i-1}}p_{U_{1}^{i-1}}(u_{1}^{i-1})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}^{i-1}u_{i}}^{B^{N}})\overline{\mathcal{W}}^{u_{1}^{i}}(\rho^{N})\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c^{-1})\sum_{u_{1}}p_{U}(u_{1})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}}^{B^{N}})\overline{\mathcal{W}}^{u_{1}}(\rho^{N})\Big{\\}}$
$\displaystyle=$
$\displaystyle(1+c)\sum_{u^{N}}p_{U^{N}}(u^{N})\text{Tr}\Big{\\{}\Big{(}\sum_{u_{1}^{N-1}}\outerproduct{u_{1}^{N-1}}{u_{1}^{N-1}}\otimes(I-\Lambda_{u_{1}^{N-1}u_{N}}^{B^{N}})\Big{)}$
$\displaystyle\Big{(}$
$\displaystyle\sum_{u_{1}^{N-1}}p_{U_{1}^{N-1}}(u_{1}^{N-1})\outerproduct{u_{1}^{N-1}}{u_{1}^{N-1}}\otimes\overline{\mathcal{W}}^{u_{1}^{N}}(\rho^{N})\Big{)}\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c+c^{-1})\sum_{i=2}^{N-1}p_{U}(u_{i})\text{Tr}\Big{\\{}\Big{(}\sum_{u_{1}^{i-1}}\outerproduct{u_{1}^{i-1}}{u_{1}^{i-1}}\otimes(I-\Lambda_{u_{1}^{i-1}u_{i}}^{B^{N}})\Big{)}$
$\displaystyle\Big{(}$
$\displaystyle\sum_{u_{1}^{i-1}}p_{U_{1}^{i-1}}(u_{1}^{i-1})\outerproduct{u_{1}^{i-1}}{u_{1}^{i-1}}\otimes\overline{\mathcal{W}}^{u_{1}^{i}}(\rho^{N})\Big{)}\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c^{-1})\sum_{u_{1{}}}p_{U}(u_{1})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}}^{B^{N}})\overline{\mathcal{W}}^{u_{1}}(\rho^{N})\Big{\\}},$
(85)

where the first equality follows from Eq. 33. For the second equality, we have
used the fact that [37]

$\sum_{x}p_{X}(x)\text{Tr}\\{A_{x}\rho_{x}\\}=\text{Tr}\Big{\\{}\Big{(}\sum_{x}\outerproduct{x}{x}\otimes
A_{x}\Big{)}\Big{(}\sum_{x^{\prime}}p_{X}(x)\outerproduct{x^{\prime}}{x^{\prime}}\otimes\rho_{x^{\prime}}\Big{)}\Big{\\}}.$
(86)

Now, from the definition of synthesized channels present in Eq. 32 the next
equality is produced

RHS of Eq. 85 $\displaystyle=$
$\displaystyle(1+c)\sum_{u^{N}}p_{U^{N}}(u^{N})\text{Tr}\Big{\\{}\Big{(}\sum_{u_{1}^{N-1}}\outerproduct{u_{1}^{N-1}}{u_{1}^{N-1}}\otimes(I-\Lambda_{u_{1}^{N-1}u_{N}}^{B^{N}})\Big{)}\mathcal{W}^{(N),u_{N}}(\rho^{N})\Big{\\}}$
(87) $\displaystyle+$
$\displaystyle(2+c+c^{-1})\sum_{i=2}^{N-1}p_{U}(u_{i})\text{Tr}\Big{\\{}\Big{(}\sum_{u_{1}^{i-1}}\outerproduct{u_{1}^{i-1}}{u_{1}^{i-1}}\otimes(I-\Lambda_{u_{1}^{i-1}u_{i}}^{B^{N}})\Big{)}\mathcal{W}^{(i),u_{i}}(\rho^{N})\Big{\\}}$
$\displaystyle+$
$\displaystyle(2+c^{-1})\sum_{u_{1}}p_{U}(u_{1})\text{Tr}\Big{\\{}(I-\Lambda_{u_{1}}^{B^{N}})\mathcal{W}^{(1),u_{1}}(\rho^{N})\Big{\\}}.$

Now, the number of terms in the previous sum can be reduced by means of a
simple observation. Since the frozen bits are obtained by using a classical
function, there is no need to make a measurement on the frozen bits position;
i.e., $\Lambda_{u_{1}^{i-1}u_{i}}^{B^{N}}=I$ for any $i\in\mathcal{A}^{c}$.
This leads to

RHS of Eq. 87 $\displaystyle<$
$\displaystyle\frac{(1+c+c^{-1})}{2}\sum_{i\in\mathcal{A}}Z_{\rho}(\mathcal{W}^{(i)})=o(2^{-N^{\beta}}),$
(88)

where the definition of reliability for synthesized channels have been
applied. The fact that this sum equals to $o(2^{-N^{\beta}})$ is a consequence
of Proposition 1. ∎

When one considers this decoding scheme for symmetric quantum reading, the
same result as in Theorem 15 is obtained. Thus, the strategy used is the same,
and there is only the need to properly design the pretty good measurement for
the symmetric channels under consideration.

##  4 Probe State Analysis

We are going to analyze how reliability and rate depend on the probe state
used. The comparison will be between pure and entangled probe states. In both
cases, a parameter of the probe states is maximized to give the optimal value
of the quantity under consideration.

First of all, we adopt

$\ket{\psi}=\sqrt{1-q}\ket{0}+e^{-i\phi}\sqrt{q}\ket{1}$ (89)

as the pure probe state, where $q\in[0,1]$ and $\phi\in[0,2\pi)$. Because of
the symmetric action of the AD channel with respect to the $z$-axis in the
Bloch sphere representation, we can assume, without loss of generality, that
$\phi=0$. Secondly, the entangled state used as probe is given by

$\ket{\Psi^{xz}}=(Z^{z}X^{x}\otimes I)\ket{\Phi},$ (90)

where $z,x\in\mathbb{Z}_{2}$, and
$\ket{\Phi}=\sqrt{1-q}\ket{00}+\sqrt{q}\ket{11}$, with $q\in[0,1]$. Notice
that for $q=1/2$, we have all four Bell states. Using these pure and entangled
states, we can compare their goodness for quantum reading assisted by polar
codes. This comparison is made by optimizing the quantity under consideration
with respect to $q$ and $q$.

Lets start considering pure probe states. Substituting Eq. 89 into Eq. 8, it
is possible to see that

$\displaystyle\mathcal{W}^{u_{0}}(\outerproduct{\psi}{\psi})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1-(1-u_{0})q&\sqrt{(1-u_{0})q(q-1)}\\\
\sqrt{(1-u_{0})q(q-1)}&(1-u_{0})q\end{pmatrix},$ (91)
$\displaystyle\mathcal{W}^{u_{1}}(\outerproduct{\psi}{\psi})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1-(1-u_{1})q&\sqrt{(1-u_{1})q(q-1)}\\\
\sqrt{(1-u_{1})q(q-1)}&(1-u_{1})q\end{pmatrix}.$ (92)

Now, the minimal value of reliability with respect to $q$ can be computed. We
are going to study a normalized version of reliability given by

$\overline{Z}(\mathcal{W})_{\outerproduct{\psi}{\psi}}=\max_{q}\frac{Z(\mathcal{W})_{\outerproduct{\psi}{\psi}}}{2\sqrt{p(p-1)}}=\max_{q}F(\mathcal{W}^{u_{0}}(\outerproduct{\psi}{\psi}),\mathcal{W}^{u_{1}}(\outerproduct{\psi}{\psi})).$
(93)

For the current probe state, it is shown in Fig. 6 the minimal values of
$\overline{Z}(\mathcal{W})_{\outerproduct{\psi}{\psi}}$ and the corresponding
$q$. An important fact can be seen in Fig. 6, the optimal pure probe state is
given by the excited state $\ket{1}$.

(a) Minimal value of fidelity w.r.t. $q$.

(b) Distribution of $q$ giving the minimal value of fidelity.

Figure 6: Analysis of fidelity for the pure probe state.

Considering mixed probe states, the same analysis can be done. There is a
difference now on how to probe the channel. Since we have a bipartite
entangled state, it can be used to probe the channel once or twice. The
channel output in both of these situations is given by

$\displaystyle\textstyle\mathcal{W}^{u_{i}}\otimes\text{id}_{A_{2}}(\outerproduct{\Psi^{xz}}{\Psi^{xz}})$
$\displaystyle=$
$\displaystyle\Big{[}\Big{(}(1-q)\delta_{0x}\outerproduct{0}{0}_{A_{2}}+q\delta_{1x}\outerproduct{1}{1}_{A_{2}}\Big{)}$
(94) $\displaystyle+$ $\displaystyle
u_{i}\Big{(}(1-q)\delta_{1x}\outerproduct{0}{0}_{A_{2}}+q\delta_{0x}\outerproduct{1}{1}_{A_{2}}\Big{)}\Big{]}\outerproduct{0}{0}_{A_{1}}$
$\displaystyle+$
$\displaystyle(\delta_{0z}-\delta_{1z})\sqrt{(1-u_{i})q(1-q)}\Big{[}\delta_{0x}\outerproduct{0}{1}_{A_{2}}+\delta_{1x}\outerproduct{1}{0}_{A_{2}}\Big{]}\outerproduct{0}{1}_{A_{1}}$
$\displaystyle+$
$\displaystyle(\delta_{0z}-\delta_{1z})\sqrt{(1-u_{i})q(1-q)}\Big{[}\delta_{1x}\outerproduct{0}{1}_{A_{2}}+\delta_{0x}\outerproduct{1}{0}_{A_{2}}\Big{]}\outerproduct{1}{0}_{A_{1}}$
$\displaystyle+$
$\displaystyle(1-u_{i})\Big{[}(1-q)\delta_{1x}\outerproduct{0}{0}_{A_{2}}+q\delta_{0x}\outerproduct{1}{1}_{A_{2}}\Big{]}\outerproduct{1}{1}_{A_{1}},$

where $i=0,1$ and $\delta_{ab}$ is the Kronecker delta, and

$\displaystyle\textstyle\mathcal{W}^{u_{0}}\otimes\mathcal{W}^{u_{1}}(\outerproduct{\Psi^{xz}}{\Psi^{xz}})$
$\displaystyle=$
$\displaystyle[(1-q)\delta_{0x}+u_{0}(1-q)\delta_{1x}]\outerproduct{0}{0}_{A_{1}}\otimes\outerproduct{0}{0}_{A_{2}}$
(95) $\displaystyle+$ $\displaystyle
q(u_{0}\delta_{0x}+\delta_{1x})\outerproduct{0}{0}_{A_{1}}\otimes[u_{1}\outerproduct{0}{0}_{A_{2}}+(1-u_{1})\outerproduct{1}{1}_{A_{2}}]$
$\displaystyle+$
$\displaystyle(-1)^{z}\sqrt{(1-u_{0})(1-u_{1})q(1-q)}[\delta_{0x}\outerproduct{0}{1}_{A_{1}}+\delta_{1x}\outerproduct{1}{0}_{A_{1}}]\otimes\outerproduct{0}{1}_{A_{2}}$
$\displaystyle+$
$\displaystyle(-1)^{z}\sqrt{(1-u_{0})(1-u_{1})q(1-q)}[\delta_{0x}\outerproduct{1}{0}_{A_{1}}+\delta_{1x}\outerproduct{0}{1}_{A_{1}}]\otimes\outerproduct{1}{0}_{A_{2}}$
$\displaystyle+$
$\displaystyle(1-u_{0})q\delta_{0x}\outerproduct{1}{1}_{A_{1}}\otimes[u_{1}\outerproduct{0}{0}_{A_{2}}+(1-u_{1})\outerproduct{1}{1}_{A_{2}}]$
$\displaystyle+$
$\displaystyle(1-u_{0})(1-q)\delta_{1x}\outerproduct{1}{1}_{A_{1}}\otimes\outerproduct{0}{0}_{A_{2}}.$

For each value of $x,z$, we minimize the fidelity with respect to $q$
considering these two cases, which we call half-probed and full-probed cases.
The first one corresponds to Eq. 94 and the second to Eq. 95. Our first
description is over the half-probed case. See Figs. 7 and 8 below. Notice that
we have shown just the instance $z=0$, since it gives the same results as for
$z=1$.

(a) Minimal value of fidelity w.r.t. $q$.

(b) Distribution of $q$ giving the minimal value of fidelity.

Figure 7: Analysis of fidelity for the half-probed state $\ket{\Psi^{1,0}}$

(a) Minimal value of fidelity w.r.t. $q$.

(b) Distribution of $q$ giving the minimal value of fidelity.

Figure 8: Analysis of fidelity for the half-probed state $\ket{\Psi^{0,0}}$.
Figure 9: Fidelity of pure state minus fidelity of half-probed state
$\ket{\Psi^{0,0}}$.

Two conclusions can be obtained from Figs. 7 and 8. Firstly, the shape of the
fidelity of half-probed state $\ket{\Psi^{10}}$ is exactly the same as for the
pure probe state case. This is clear from the fact that the value of $q$
giving maximal fidelity is $q=0$, which turns the probe state to be $\ket{1}$.
As the second point, we see in Figs. 7 and 8 that half-probing with the state
$\ket{\Psi^{00}}$ can give some improvement in the channel discrimination
task. This improvement is obtained where $q\neq 1$. However, the difference is
not so significant, making the use of $\ket{\Psi^{00}}$ justified only if
entanglement cost is negligible.

The subsequent analysis is over a full-probed state. In opposition to what has
been shown in the half-probed case, there is no improvement in using full-
probed $\ket{\Psi^{00}}$. Fig. 10 shows that $q=1$ gives the minimal value of
fidelity in this case. This leads to the fidelity having values equal to the
square of the pure probe state. Thus, optimizing full-probed strategy using
$\ket{\Psi^{00}}$ leads to the same strategy for pure probe state. Lastly, we
consider full-probed strategy using $\ket{\Psi^{10}}$. We have $q=0$ giving
the optimal value of fidelity. A consequence of this is that now we probe the
quantum channel twice using the state $\ket{0}$ and $\ket{1}$ in each round.
Since for pure probe, we are using the state $\ket{1}$, then the round probing
with $\ket{0}$ contribute degrading the fidelity. Therefore, the optimal
fidelity obtained using full-probed $\ket{\Psi^{10}}$ is much higher than
using full-probed $\ket{\Psi^{00}}$, as can be seen in Fig. 12.

(a) Minimal value of fidelity w.r.t. $q$.

(b) Distribution of $q$ giving the minimal value of fidelity.

Figure 10: Analysis of fidelity for the full-probed state $\ket{\Psi^{0,0}}$.

(a) Minimal value of fidelity w.r.t. $q$.

(b) Distribution of $q$ giving the minimal value of fidelity.

Figure 11: Analysis of fidelity for the full-probed state $\ket{\Psi^{1,0}}$
Figure 12: Fidelity of full-probed state $\ket{\Psi^{1,0}}$ minus fidelity of
full-probed state $\ket{\Psi^{0,0}}$.

The previous results can be extended to the mutual information between the
classical system $X$ and the quantum channel output system $B$. In Proposition
8 we have seen that rate and reliability are connected in a way that when the
rate is close to its maximum, then the reliability is close to its minimum,
and vice versa. Thus, we can conclude the following. For channels with values
of $u_{0}$ and $u_{1}$ close, the mutual information will be close to zero. On
the other hand, when one of the $u_{i}$, $i=0,1$, is close to unit and the
other is close to zero, the mutual information attains its maximum value
possible. Furthermore, probing the quantum memory cell using the half-probed
$\ket{\Psi^{00}}$ strategy is again the optimal procedure for quantum reading.

##  5 Final Remarks

We have demonstrated a new polar coding scheme for quantum memory cell
discrimination. To achieve this goal, we had to introduce new definitions on
the rate and reliability of a quantum channel. The polarization phenomenon
produced by channel combining and analyzed in the channel splitting part has
been shown for these two quantities. This was established due to an inequality
connecting both of them. In the channel splitting part, we have also
introduced the synthesized channels created by the polar coding. It had been
shown that when the number of channels is arbitrarily large, the set of
synthesized channels can be divided in two groups, good and bad channels.
Additionally, the fraction of such channels is related to the mutual
information of the original quantum channel in consideration. This result has
motivated us to construct an optimal encoding scheme for quantum reading. A
decoding scheme was introduced and analyzed, as well. Using an existence proof
of pretty good measurements given in this paper and a previous quantum union
bound from the literature, we have shown that our decoding scheme has error
probability that decays exponentially fast with respect to the code length. At
the end, optimizations over probe states are investigated, leading to the
conclusion of half-probed states as the best choice, in general, of probe
states to be used.

This paper has given some future investigation topics. A question that can be
stated is how to apply polar coding to more general classical and quantum
systems. First of all, we can extend the binary discrimination to a $d$-ary
discrimination task. The mathematical equivalence of this is considering a
classical system with a larger alphabet. Some classical polar codes have been
proposed in the literature and, with the proper adjustments, applied here.
Secondly, we can consider the set of quantum memory cells to be composed of
generalized amplitude damping channel. This class of channel can be seen as a
second-order approximation of classical digital memory, where the model takes
the environment temperature into consideration. In this open question, the
task would be to find the optimal probe state for channel discrimination.
Third, and more important, how can one construct efficient polar codes for
Gaussian bosonic channel discrimination. This still a research topic even for
classical Gaussian channel. Gaussian bosonic channel is the ultimate goal as
quantum channel model for classical digital memories. Because this class of
channels is defined over continuous variables, attacking it needs to be two-
fold. The polar code scheme has to take into account the channel to prove the
polarization phenomenon. Additionally, an optimization over the probe states
is necessary under energy constrain. These two approaches are not independent,
which makes the task even more difficult. Lastly, we do not know if there
exists a provable optimal probe state in the contexts mentioned. Uniqueness
also needs to be verified.

##  6 Acknowledgments

The authors acknowledge the financial support of the Future and Emerging
Technologies (FET) programme, within the Horizon-2020 Programme of the
European Commission, under the FET-Open grant agreement QUARTET, number
862644.

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