Rényi divergence inequalities via interpolation, with applications to
generalised entropic uncertainty relations University of Technology Sydney
Faculty of Engineering and Information Technology Master of Science (Research)
in Computing Sciences Dr. Marco Tomamichel Dr. Christopher Ferrie January 2021

###### Abstract

We investigate quantum Rényi entropic quantities, specifically those derived
from ‘sandwiched’ divergence. This divergence is one of several proposed Rényi
generalisations of the quantum relative entropy. We may define Rényi
generalisations of the quantum conditional entropy and mutual information in
terms of this divergence, from which they inherit many desirable properties.
However, these quantities lack some of the convenient structure of their
Shannon and von Neumann counterparts. We attempt to bridge this gap by
establishing divergence inequalities for valid combinations of Rényi order
which replicate the chain and decomposition rules of Shannon and von Neumann
entropies. Although weaker in general, these inequalities recover equivalence
when the Rényi parameters tend to one.

To this end we present Rényi mutual information decomposition rules, a new
approach to the Rényi conditional entropy tripartite chain rules and a more
general bipartite comparison. The derivation of these results relies on a
novel complex interpolation approach for general spaces of linear operators.

These new comparisons allow us to employ techniques that until now were only
available for Shannon and von Neumann entropies. We can therefore directly
apply them to the derivation of Rényi entropic uncertainty relations.
Accordingly, we establish a family of Rényi information exclusion relations
and provide further generalisations and improvements to this and other known
relations, including the Rényi bipartite uncertainty relations.

Acknowledgments

This thesis could not have happened without the consistent and invaluable
support of Marco.

The inspiration for the interpolation technique was supplied by Dr. Salman
Beigi, whose contributions in discussions about the nature of Pisier norms
were more than illuminating.

Lucie, for her patience.

This is a CONVENTIONAL THESIS. Published work arising from research during the
candidature of the degree has been included in an appendix.

DECLARATION OF PUBLICATIONS INCLUDED IN THE THESIS Title : Decomposition Rules
for Quantum Rényi Mutual Information with an Application to Information
Exclusion Relations Parts included : Subsections IV.A and IV.B Authors :
Alexander McKinlay, Marco Tomamichel Contributions : AM derived results and
developed the method of the proofs in consultation with MT. Status : Published
Journal : Journal of Mathematical Physics, vol. 61, no. 7, p. 072202, 2020
Address : https://doi.org/10.1063/1.5143862

###### Contents

  1. Introduction
  2. Notation and nomenclature
  3. 1 Background and current literature
    1. 1.1 Probability theory
      1. 1.1.1 Events and outcomes
      2. 1.1.2 Random variables
      3. 1.1.3 Probability and related quantities
    2. 1.2 Banach spaces of operators
      1. 1.2.1 Matrices as operators
      2. 1.2.2 Operator decompositions
      3. 1.2.3 The Schatten operator norm
    3. 1.3 Quantum mechanics and information
      1. 1.3.1 The state of a quantum system
      2. 1.3.2 State vectors and probability density matrices
      3. 1.3.3 Composite systems
      4. 1.3.4 Measurement
    4. 1.4 Entropy
      1. 1.4.1 Classical Entropy
      2. 1.4.2 Quantum entropies
      3. 1.4.3 Rényi entropy
      4. 1.4.4 Quantum Rényi entropy
    5. 1.5 Rényi divergence and related quantities
      1. 1.5.1 Generalised Rényi entropic quantities
      2. 1.5.2 Properties of the Rényi divergence
      3. 1.5.3 Dupuis’ chain rules
    6. 1.6 Entropic uncertainty relations
      1. 1.6.1 Commutator formulation
      2. 1.6.2 Entropic formulation
      3. 1.6.3 Improvements and extensions
  4. 2 Interpolation framework
    1. 2.1 The super-operator $\Gamma_{\sigma,\tau}$
      1. 2.1.1 Unbalanced weighted norms
      2. 2.1.2 Rényi Divergence in terms of weighted norms
    2. 2.2 Pisier’s norms
      1. 2.2.1 Two-part norms
      2. 2.2.2 The case of positive operators
      3. 2.2.3 Weighted two-part norms
      4. 2.2.4 Three-part norms
      5. 2.2.5 Adapting the general formulation
    3. 2.3 Three-line theorem for unbalanced weighted norms
  5. 3 Rényi entropy divergence inequalities
    1. 3.1 Main results
    2. 3.2 Proofs
      1. 3.2.1 Applying the three-line theorem
      2. 3.2.2 Bipartite divergence inequalities
      3. 3.2.3 The tripartite chain rules
  6. 4 Generalised Rényi divergence uncertainty relations
    1. 4.1 Bipartite conditional entropy relations
    2. 4.2 Information exclusion relations
    3. 4.3 Proofs
      1. 4.3.1 Proofs of the bipartite conditional entropy relations
      2. 4.3.2 Proofs of the Rényi information exclusion relations
  7. 5 Discussion
    1. 5.1 Immediate observations
      1. 5.1.1 Specialising to classical Rényi entropies
      2. 5.1.2 Monotonicity in $\alpha$
    2. 5.2 Advantages and drawbacks
      1. 5.2.1 Place within the broader theory
      2. 5.2.2 Applicability
    3. 5.3 Future work
      1. 5.3.1 Numerical simulations
      2. 5.3.2 Multiple measurements
      3. 5.3.3 POVMs and tripartite uncertainty relations
      4. 5.3.4 Adapting Pisier norms
  8. A An alternative proof of the decomposition rules
    1. A.1 Operator-vector correspondence
    2. A.2 Identities in terms of operator-vector correspondence
    3. A.3 Interpolating for valid choices of $\theta$
    4. A.4 Consolidation and analysis
  9. B Other useful results
    1. B.1 Background results
    2. B.2 Ancillary results

###### List of Tables

  1. 0.1 Overview of notation
  2. 3.1 Some particular choices satisfying $\alpha\beta\gamma-2\beta\gamma-\alpha+\beta+\gamma=0$

## Introduction

This thesis contributes some concrete results in the form of inequalities for
quantum Rényi entropies and a direct application to determining Rényi entropy
uncertainty relations. However, these results are a consequence of a more
general desire to explore the mathematical structure and place of Rényi
entropies in the broader theory of quantum information.

######

Entropy is a concept that was developed in the study of thermodynamics to
quantify the disorder of a system. The statistical understanding of entropy
was formalised by Boltzmann [1], where it was viewed as a property of
thermodynamic systems. The quantities we investigate stem from the information
theoretic concept of entropy, characterised by the Shannon entropy $H(X)$ of a
random variable [2]. Although having other uses within information theory,
entropy is prominently known as a measure of uncertainty (or spread) of a
given random variable, quantifying the average information which would be
gained (or uncertainty removed) from the observation of that variable.

Associated with the Shannon entropy of a classical random variable $X$ are
several entropic quantities which each have their own uses and
interpretations. The joint entropy $H(XY)$ describes the entropy of a
multivariate random variable and, as for joint probability, is affected by
whether or not the variables are dependent. Conditional entropy $H(X|Y)$
quantifies the average uncertainty of $X$ given that $Y$ has been observed.
This quantity is well-known and understood, being a fundamental tool in
cryptography, statistical analysis and as a general measure of uncertainty.
Mutual information $I(X:Y)$ quantifies the level of correlation between $X$
and $Y$ – the difference between the total of the individual entropies of $X$
and $Y$ and the joint entropy of $XY$. Most prominently, Shannon [2]
established that the capacity of any discrete memoryless communication channel
is given by the maximal mutual information between the channel’s input and
output. Beyond its original use in information theory, it has found many other
applications in information processing from such a wide range as machine
learning (see, e.g., [3, 4, 5]) and computational linguistics (see, e.g.,
[6]).

The classical Rényi entropies $H_{\alpha}(X)$ [7] generalise the Shannon
entropy. They form a family of entropies parametrised by their order,
$\alpha>0$, that produce measurements of uncertainty that give more or less
weight to events with high or low information content. More specifically, they
weigh the surprisal of outcomes differently depending on the order. We recover
the Shannon entropy when $\alpha=1$ which, naturally, is one of the more
useful orders of Rényi entropy, along with min-entropy – when $\alpha=\infty$,
and the collision entropy – when $\alpha=2$. Formal definitions are given in
Section 1.4.

We consider also the quantum generalisation of Rényi entropies. It is prudent
therefore to initially cover the quantum analogue of Shannon entropy, the von
Neumann entropy $H(\rho)$. This entropy of a probability density matrix $\rho$
is well understood and induces quantum versions of the related entropic
quantities that are well behaved and reflect our physical understanding of
quantum systems.

The quantum conditional entropy has so far found uses in quantum information
such as in the decoupling theorem [8] and more generally as an entanglement
witness and as a measure of uncertainty (see [9, 10]). We also have a quantum
generalisation of the mutual information which is compatible with von Neumann
entropy and quantum conditional entropy. Quantum mutual information has
analogous applications in quantum information, for example characterising the
capacity of classical to quantum channels [11, 12, 13] and the quantum channel
capacity under entanglement assistance [14, 15, 16, 17]. It has also found
applications in other areas of quantum physics, for example as an entanglement
and correlation measure (see, e.g., [18]) and to quantify Heisenberg’s
uncertainty principle (see [19] and [20] for a review on related work).

From this point we will mostly consider quantum entropies and hence omit the
‘quantum’ and rather specify that an entropy is classical if necessary.

Certain equivalences of entropic quantities are known as chain or
decomposition rules. Aside from highlighting the relationships between the
quantities, these equivalences can be considered as a method to define them.
Indeed, given the relatively simple definitions of the von Neumann entropy and
joint entropy we find the conditional entropy _chain rule_ , for $\rho_{AB}$
on the system $AB$ with marginal $\rho_{B}$ on $B$,

$H(A|B)_{\rho}=H(\rho_{AB})-H(\rho_{B}).$ (0.1)

This concept can be extended to the tripartite conditional entropy chain rule:
for a state $\rho_{ABC}$ on the system $ABC$,

$H(AB|C)_{\rho}=H(A|BC)_{\rho}+H(A|C)_{\rho}=H(B|AC)_{\rho}-H(B|C)_{\rho}.$
(0.2)

Similarly, mutual information can be described as equivalent to an expression
of conceptually simpler entropies, i.e. we define

$I(A:B)_{\rho}=H(\rho_{A})+H(\rho_{B})-H(\rho_{AB})=H(\rho_{A})-H(A|B)_{\rho}=H(\rho_{B})-H(B|A)_{\rho}.$
(0.3)

These equivalences are interpreted as _decomposition rules_ , expressing the
mutual information in terms of the von Neumann entropies of the different
marginals of the joint state on the systems $A$ and $B$. By appealing to the
intuition that entropy measures uncertainty in a quantum system, they reveal
that mutual information measures the uncertainty in $A$ that is due to the
lack of knowledge of $B$, and vice versa.

Another important and equivalent pair of definitions for conditional entropy
and mutual information is given in terms of quantum relative entropy [21],
namely as optimisations of the relative entropy between the joint state
$\rho_{AB}$ and particular product states of the marginals:

$\displaystyle H(A|B)_{\rho}$
$\displaystyle=-\min_{\sigma_{B}}D(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\sigma_{B})=-D(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\rho_{B})\quad\text{and}$
(0.4) $\displaystyle I(A:B)_{\rho}$
$\displaystyle=\min_{\sigma_{A},\sigma_{B}}D(\rho_{AB}\|\sigma_{A}\otimes\sigma_{B})=\min_{\sigma_{B}}D(\rho_{AB}\|\rho_{A}\otimes\sigma_{B})=D(\rho_{AB}\|\rho_{A}\otimes\rho_{B})\,,$
(0.5)

where in the expressions on the right we observed that the minima are achieved
for the marginals $\rho_{A}$ and $\rho_{B}$ of the joint state $\rho_{AB}$.
These expressions reveal a fundamental property of these quantities that is
not evident from the chain and decomposition rules, namely the _data-
processing inequality_. Specifically, this property entails that the
conditional entropy is monotonically non-decreasing and the mutual information
non-increasing under any local processing of information on $A$ and $B$. Its
satisfaction directly follows from the monotonicity under quantum channels of
the underlying relative entropy and the above equivalences. This property is
crucial for many applications relating to the physical interpretations of
these quantities, since it corresponds to our intuition that correlations
cannot be created by acting on only one of the constituent parts (or by acting
on them independently).

On the other hand, Rényi entropic quantities which maintain useful properties
are not as straight-forward to derive and involve a significant increase in
complexity. That said, a natural method of deriving suitable Rényi
generalisations of the above quantities is to replace the relative entropy in
Eqs. (0.4) and (0.5) with a (quantum) Rényi relative entropy which naturally
satisfies the data-processing inequality. It is important to note here that,
in general, the equivalences in Eqs. (0.2)-(0.5) no longer hold in the case of
Rényi entropic quantities defined via this method.

Various generalisations of the concepts of conditional entropy and mutual
information to one-parameter families of operationally significant measures
have been proposed. Possible candidates for a Rényi conditional entropy have
been put forward, generalising both the quantum conditional entropy and
various classical Rényi entropies (see [22, 23, 24] for more detailed
treatments). Similarly, for mutual information we have propositions in both
the classical (see, e.g., [25, 26], for recent discussions) and the quantum
setting (see, e.g., [27]). We call such measures (quantum) Rényi conditional
entropy or mutual information respectively, if they satisfy the data-
processing inequality.

We are essentially looking for quantities which reproduce the relationships
between the classical entropic quantities, have operational significance,
reflect the physical situation and are mathematically convenient. Of these
propositions for such definitions, quantities derived from the so-called
‘sandwiched’ divergence [28, 29] have in recent years surfaced as some of the
most suitable. This parametrised quantum relative entropy, which we from this
point refer to as simply ‘Rényi divergence’, meets the above criteria and has
found important applications within quantum information. This thesis
essentially provides new results for Rényi divergence based quantum entropic
quantities.

Rényi divergence generalises the quantum relative entropy from which we derive
the quantum entropic quantities detailed above. It has many desirable
properties as a measure of uncertainty and its structure is amenable to
formulating inequalities. As an analogue to the quantum relative entropy, it
induces definitions of the quantum Rényi conditional entropy [28] and mutual
information [27]. These definitions take the form

$\displaystyle
H_{\alpha}(A|B)_{\rho}:=-\min_{\sigma_{B}}{D}_{\alpha}(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\sigma_{B})\quad\text{and}\quad
I_{\alpha}(A\;;\>\\!B)_{\rho}:=\min_{\sigma_{B}}{D}_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\sigma_{B})\,,$
(0.6)

where ${D}_{\alpha}(\cdot\|\cdot)$ denotes the Rényi divergence and
$\alpha\in\left[\frac{1}{2},\infty\right)$. Note that, given this definition,
we are lacking an obvious way to decompose the Rényi conditional entropy and
mutual information into Rényi entropies of the state’s marginals.

However, these quantities generalise the conditional entropy and mutual
information respectively, which are recovered by setting $\alpha=1$. These
quantities and similar constructions are widely used in the study of strong
converses and the analysis of channel coding problems (see, e.g., [30, 31, 32,
29, 33, 34]) and Rényi mutual information has found direct operational
interpretation in classical and quantum hypothesis testing [27, 26].

Uncertainty relations are important in any field that involves signals and
waves, but have found particular significance in quantum mechanics. These
relations are often put in terms of commutators and standard deviations, for
example the Robertson relation (see Section 1.6.1), which is a generalisation
of the famous Heisenberg Uncertainty Principle [35]. Entropic uncertainty
relations, on the other hand, take a more information theoretic approach,
expressing uncertainty in terms of its entropy.

There are two canonical entropic uncertainty relations, the more well known
being the Maassen-Uffink relation [36] which indicates that the total
uncertainty of measuring in two different bases is bounded below by a constant
depending on the compatibility of the measurements. Even though it is usually
expressed in terms of Shannon entropies, the general form of the theorem
encompasses a family of relations in terms of classical Rényi entropy. There
are some current well-known extensions and improvement of this relation which
we cover in Section 1.6.3.

By extending the Shannon version of the Maassen-Uffink relation to conditional
entropies we can then derive the Hall information exclusion principle [19].
This result is an entropic uncertainty relation in terms of mutual
information, describing the level of correlation between parts of a quantum
system and some memory containing information about the system. This relation,
dual to Heisenberg’s uncertainty principle, gives upper bounds on the total
amount of correlation between a state measured in either one of two
incompatible bases and some classical memory with information about how the
initial state was prepared, i.e. if there is a high level of correlation
between one measurement of the state and the memory, then there is a
proportionally low level of correlation for the other measurement, depending
on the compatibility of measurements. Hall’s original relation has been
improved upon, with bounds stronger than that found in the ‘Maassen-Uffink’
type relations.

The derivation of these relations and their improvements rely on the chain and
decomposition rules of quantum entropies, motivating our investigation of
feasible Rényi generalisations. We defer the details and formal statements to
Section 1.6.

The applications of entropic uncertainty relations and related concepts
include quantum randomness, quantum cryptography and entanglement witnessing,
see [20] for a more extensive review.

######

We now outline the core results of this thesis, i.e. establishing Rényi
versions of the chain and decomposition rules in Eqs. (0.2) and (0.3). So far
the main example of such an extension is Dupuis’ chain rules [37], which
generalise the tripartite chain rule to conditional entropies based on Rényi
divergence.

In particular, we provide decomposition rules for Rényi mutual information
which can then be applied to determine Rényi information exclusion relations.
These rules take the form

$\displaystyle I_{\alpha}(A\;;\>\\!B)_{\rho}\geq
H_{\beta}(\rho_{A})-H_{\gamma}(A|B)_{\rho}\quad\textrm{and}\quad
I_{\alpha}(A\;;\>\\!B)_{\rho}\leq
H_{\bar{\beta}}(\rho_{A})-H_{\bar{\gamma}}(A|B)_{\rho}$ (0.7)

for suitable choices of Rényi orders $\beta,\bar{\beta},\gamma$ and
$\bar{\gamma}$. The formal result is presented in Theorem 3.3. The two
inequalities above reduce to the equality in Eq. (0.3) when we take all the
parameters to $1$.

We employ a novel interpolation approach which additionally yields a general
bipartite divergence inequality and an alternative derivation of the
tripartite chain rules. For completeness, we include a previously published
proof of the decomposition rules in Appendix A which employs a similar method
to that used by Dupuis in his demonstration of the chain rules. Interestingly,
these different approaches produce slightly varied valid ranges for the Rényi
parameters, the explanation of this discrepancy is left as an open question.

The principle mechanism in demonstrating the desired Rényi divergence
inequalities is exploiting their connection to Schatten norms. This is done by
performing complex interpolation on specific linear operator valued functions
which yield our entropic quantities. The main machinery used to this end is an
extension of Beigi’s three-line theorem [38] to spaces of general linear
operators, which we detail in Chapter 4.

The choices we make in these interpolations are not determined by mere luck or
simply arrive out of the void, rather they are inspired and motivated by the
norms on general $L_{p}$-spaces developed by Pisier [39]. These ‘Pisier norms’
have already found utility in quantum information (see [40, 41]) due to how
they relate to Rényi divergence when constrained to finite spaces. In Chapter
2 we discuss some observations of their properties in the context of Rényi
entropies and how they influence the derivation of the divergence
inequalities.

As a direct application we are able to establish Rényi generalisations of the
Maassen-Uffink and Hall relations, and make further improvements on the
bounds. Some uncertainty relations of this genre have already been treated,
notably the bipartite uncertainty relation found in [42], which provides a
quantum Rényi extension to an already improved version of the Maassen-Uffink
relation found in [43].

We are able to improve the bipartite uncertainty relation by deriving state-
dependent and state-independent bounds analogous to those determined in [44].
These new bounds are determined by the other Rényi orders present in the
relation and we may recover known relations with particular choices of
parameters. We are then able to establish a new family of information
exclusion relations which provide a direct extension of the Hall relation to
Rényi entropies. This family of relations can then be further improved, again
with an analogous method to that used in [44]. Interestingly, the resulting
tighter upper bounds are not order-dependent, unlike those for the improved
Rényi bipartite uncertainty relations.

######

The structure of the thesis is as follows:

In Chapter 1 we cover the mathematical fundamentals, including a brief
treatment of probability theory (mainly for the purposes of notation) and an
overview of linear operators and Schatten norms. The remainder of this chapter
is dedicated to the formal definitions of Rényi divergence and its related
quantities, with a summary of some useful properties.

We explore Pisier norms and interpolation in Chapter 2. This involves
introducing a family of super-operators and associated norms, which help
inform some observations about certain specifications of Pisier norms. We also
establish the three-line theorem for spaces of linear operators, integral to
the proofs of the subsequent chapter.

Chapter 3 details the Rényi divergence inequalities and their proofs,
including the generalised bipartite divergence inequalities, the decomposition
rules, the tripartite chain rules and some other comparisons which directly
follow. This chapter also includes some specifications of the new
interpolation result which facilitate the derivation of the main inequalities.

The improved Rényi uncertainty relations are then established formally in
Chapter 4. We first cover the bipartite uncertainty relations, starting with a
generalised version of the result from [42], then move to a version with an
improved, state-independent bound. We conclude the main results with a Rényi
information exclusion relation and its improved version.

We then provide a brief discussion, detailing some possible future work, such
as further generalisations and applications. We also examine how these results
may fit into the broader theory, notably how it could aid in establishing
generalisations of the chain and decomposition rules for smooth Rényi
entropies [45, 46] and how it could inform a clearer definition of the Rényi
conditional mutual information (see [47]), both currently open questions in
quantum information.

## Notation and nomenclature

We use standard notation as summarised in the following table:

Table 0.1: Overview of notation Symbol |  Meaning
---|---
$\mathrm{supp}(X)$ $[\mathrm{supp}(\rho)]$ |  The support of a random variable [probability density matrix], i.e. the values of $x$ such that $P(X=x)\neq 0$ [the vectors such that $\rho\left|\psi\right\rangle\neq 0$].
$|X|$ $[|\mathrm{supp}(\rho)|]$ |  For a random variable $X$, the cardinality of the support of $P(X=x)$. [For a probability density matrix $\rho$, the dimension of $\mathrm{supp}(\rho)$.]
$\langle X\rangle$ |  The expected value of $X$, $\sum_{x}xP(X=x)$.
$\log$ |  The logarithm to base 2, we also take the convention that $0\log 0=0$.
$A,B,AB$ |  Quantum system or subsystems
$\mathcal{H}_{A},\mathcal{H}_{B}$ $[\mathcal{H}_{AB}]$ |  The Hilbert spaces of states corresponding to the quantum systems $A$ and $B$. [$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$]
$\mathcal{L}(A,B)$ $[\mathcal{L}(A)]$ |  Set of linear operators from $\mathcal{H}_{A}$ to $\mathcal{H}_{B}$ [$\mathcal{H}_{A}$].
$\mathcal{P}(A)$ $[\mathcal{P}^{*}(A)]$ |  The set of positive semi-definite [strictly positive] operators from $\mathcal{H}_{A}$ to $\mathcal{H}_{A}$.
$\mathcal{D}(A)$ $[\mathcal{D}^{*}(A)]$ |  The subset of $\mathcal{P}(A)$ $[\mathcal{P}^{*}(A)]$ of operators with unit trace.
$\mathcal{U}(A)$ $[\mathcal{U}(A,B)]$ |  The set of unitary operators in $\mathcal{L}(A)$ [isometries in $\mathcal{L}(A,B)$].
$\mathrm{CPTP}(A,B)$ |  The set of completely-positive trace-preserving operator maps from $\mathcal{L}(A)$ to $\mathcal{L}(B)$.
$\operatorname{\mathrm{tr}}_{A}(\cdot)$ |  The partial trace over $A$, $\operatorname{\mathrm{tr}}_{A}(K_{A}\otimes L_{B})=\operatorname{\mathrm{tr}}(K_{A})L_{B}$.
$\sigma\gg\rho$ |  $\sigma$ ‘dominates’ $\rho$, i.e the kernel of $\sigma$ is contained in the kernel of $\rho$.
$\sigma\perp\rho$ $[\sigma\not\perp\rho]$ |  $\sigma$ and $\rho$ are [not] perpendicular, i.e. the span of $\sigma$ and $\rho$ have [non-]empty intersection.
$\operatorname{\mathbbm{1}}_{A}\in\mathcal{L}(A)$ |  The identity map on $\mathcal{H}_{A}$.
$\mathrm{Op}_{A\rightarrow B}(\cdot)$ |  The operator mapping from $A$ to $B$. For basis vectors $\left|e_{i}\right\rangle\in\mathcal{H}_{A},\left|f_{j}\right\rangle\in\mathcal{H}_{B}$, $\mathrm{Op}_{A\rightarrow B}:\mathcal{H}_{AB}\rightarrow\mathcal{L}(AB),\left|e_{i}\right\rangle\otimes\left|f_{j}\right\rangle\mapsto\left|f_{j}\right\rangle\\!\\!\left\langle e_{i}\right|$.
$\|\cdot\|_{p}$ |  The $p$-Schatten norm, $\|X\|_{p}=\operatorname{\mathrm{tr}}\left[(X^{\dagger}X)^{\frac{p}{2}}\right]^{\frac{1}{p}}$. This is not a norm for $p<1$.
$[\|\cdot\|^{*}_{p}]$ |  [The dual $p$-norm - the norm induced on the space of functionals on the space equipped with $\|\cdot\|_{p}$.]
$\Re(z)$ |  The real part of the complex number $z$.
$\lambda_{\max}\left[M\right]$ |  The maximum eigenvalue of $M$.
$[X,Y]$ |  The commutator: for $X,Y\in\mathcal{L}(A)$, $[X,Y]=XY-YX$.

##  Chapter 1 Background and current literature

###  1.1 Probability theory

Before we discuss entropy, we must establish some definitions and notations
for probability.

####  1.1.1 Events and outcomes

Here we consider only discrete random variables and, accordingly, a set of
outcomes is a discrete set which describes the observable results of an
experiment. An event, on the other hand, describes a subset of possible
outcomes. For example, when rolling a standard die, the numbers 1 through 6
are possible outcomes whereas the outcome being an odd number is an event
which includes the outcomes 1, 3 and 5.

Given a set of outcomes $\Sigma$, we consider the probability of an event
$E\subseteq\Sigma$ as a subadditive function from the power set of $\Sigma$ to
the real unit interval, $P:2^{\Sigma}\rightarrow[0,1]$. $P(E)=1$ means that
$E$ is guaranteed to happen and $P(E)=0$ means $E$ will not happen.

Mutually exclusive events are events that cannot occur at the same time. Heads
or tails being the result of a coin toss are mutually exclusive whereas it
raining on the weekend or it reaching $20^{\circ}$ are not mutually exclusive.
We therefore expect that, for mutually exclusive events $E$ and $F$, if
$P(E)=p$ then

$P(F)\leq 1-p\implies P(E)+P(F)\leq 1.$ (1.1)

A complete set of mutually exclusive events encompasses all possible outcomes
of an experiment. If $\\{E_{i}\\}$ is a complete set of mutually exclusive
events we have $\bigcap_{i}E_{i}=\emptyset$ and $\bigcup_{i}E_{i}=\Sigma$.
Moreover,

$\displaystyle P(E_{k})=1-\sum_{i\neq k}P(E_{i})\implies\sum_{i}P(E_{i})=1.$
(1.2)

Eq. (1.2) is known as the completeness relation.

####  1.1.2 Random variables

A random variable $X$ associates the outcomes of an experiment with real
values, so we may consider the set of outcomes
$\Sigma\subseteq\operatorname{\mathbb{R}}$. Accordingly, we have the
probability of an outcome $x\in\Sigma$, $P(X=x)$ and the probability of an
event, $X\in E\subseteq\Sigma$, written $P(X\in E)=\sum_{x\in E}P(X=x)$.
Clearly, if $x_{1}\neq x_{2},\,X=x_{1}\implies X\neq x_{2}$, hence the set
$\\{x_{i}\\}_{i}$ is complete and mutually exclusive, i.e.
$\sum_{i}P(X=x_{i})=1$. A discrete random variable $X$ is described by its
probability mass function $p(x)=P(X=x)$.

An important property of a random variable is its expected value $\langle
X\rangle$. This is a measure of centre which plays a significant role in the
analysis of random variables. We define the expected value as the sum of the
outcome values, weighted by their probabilities: $\langle
X\rangle=\sum_{x\in\Sigma}xp(x)$. Note that in general this does not reproduce
the most likely value, but rather characterises the average of the values
generated by many independent observations of identical random variables.

####  1.1.3 Probability and related quantities

The joint probability of two events $E$ and $F$ is written $P(E,F)$, read “the
probability of $E$ and $F$”. It describes the chance that both $E$ and $F$
will occur simultaneously. In terms of random variables we write
$P(X=x,Y=y)=p(xy)$.

The conditional probability $P(E|F)$ is similar but instead considers the
chance that $E$ occurs given that $F$ occurs. For random variables we write
$P(X=x|Y=y)=p(x|y)$.

For independent random variables we have $p(xy)=p(x)p(y)$, otherwise

$p(xy)=p(x)p(y|x)=p(y)p(x|y)\implies p(x)=\frac{p(xy)}{p(y|x)},\quad
p(x|y)=\frac{p(y|x)p(x)}{p(y)}.$ (1.3)

The equation on the far right is known as Bayes’ Theorem.

This notation will be used for the remainder of this thesis, except for some
cases where it may cause ambiguity.

###  1.2 Banach spaces of operators

The main accepted mathematical model of quantum mechanics and information is
based on complex-valued linear algebra and the analysis of vector spaces. In
this section we cover some of the theory and tools available to us due to this
model. In particular, we detail some results which are useful in the following
sections.

####  1.2.1 Matrices as operators

An operator, in general, is a mapping that takes an element of one space to
the element of another space (possibly the same space). In our context of
linear operators acting on finite-dimensional Hilbert spaces, we may represent
operators with matrices and we use the terms interchangeably when the focus
dictates it.

##### Transformations on operators

For reference we cover some basic manipulations of operators.

  * •

The inverse of a square matrix can be considered as the operator which
reverses the action of the original. Not all matrices are invertible, these
are known as singular or degenerate matrices. Given invertible
$M\in\mathcal{L}(A)$ its inverse, $M^{\textsc{I}}$ is such that if, for
$v\in\mathcal{H}_{A}$,

$Mv=u\implies v=M^{\textsc{I}}u=M^{\textsc{I}}Mv.$ (1.4)

As the above indicates, the inverse relationship is symmetric and
$MM^{\textsc{I}}=M^{\textsc{I}}M=\operatorname{\mathbbm{1}}_{A}$.

  * •

The transpose is essentially a reflection of the matrix across its diagonal.
Given $M=(a_{ij})$, ${M^{\top}=(a_{ji})}$. Many properties and attributes are
preserved under the transpose, the trace for example. If
$M\in\mathcal{L}(A,B)$ then $M^{\top}\in\mathcal{L}(B,A)$.

  * •

The conjugate $\overline{M}$ of complex-valued matrix $M$ is just the matrix
whose entries are the complex conjugates of the entries of $M$.

  * •

The adjoint $M^{\dagger}$ of an operator is the operator such that for the
inner product of the Hilbert space it is acting upon $\langle
u,Mv\rangle=\langle M^{\dagger}u,v\rangle$. In our context, this simplifies to
the conjugate transpose of $M$, i.e. $M^{\dagger}=\overline{M}^{\top}$.

##### Types of operators

We have particular categories of operators each with specific properties. We
work our way from the most general to the least.

  * •

A square matrix is any matrix that has the same number of rows as columns. As
an operator it maps between spaces of the same dimension – any operator that
maps to the same space must at least be square.

  * •

A diagonalisable matrix is a square matrix for which there exists a basis such
that it can be written as a diagonal matrix. The entries of this diagonal
matrix are the eigenvalues of the original.

  * •

A normal matrix is such that $M^{\dagger}M=MM^{\dagger}$. As a result all
normal matrices map to and from the same space.

  * •

A self-adjoint (Hermitian) matrix is such that $M^{\dagger}=M$. A self-adjoint
matrix is thus normal. The eigenvalues of a self-adjoint matrix are real and
its eigenvectors form an orthonormal basis.

  * •

A positive semi-definite matrix is a self-adjoint matrix whose eigenvalues are
greater than or equal to zero. A strictly positive or positive definite matrix
is similar except that its eigenvalues are all greater than zero. We often
shorten positive semi-definite to just positive. If $X\in\mathcal{P}(A)$ is
positive then there exists an $M\in\mathcal{L}(A,B)$ such that
$X=M^{\dagger}M$ and a self-adjoint $Y\in\mathcal{L}(A)$ such that $X=Y^{2}$.

  * •

A density matrix is a trace-1 positive matrix.

  * •

An isometry $U$ is an operator such that
$U^{\dagger}U=\operatorname{\mathbbm{1}}$. The length of a vector is invariant
under isometric operations.

  * •

A unitary operator $U$ is an isometry which is also square. Any evolution of a
quantum state can be represented as a unitary operator since it takes unit
vectors to unit vectors of the same dimension.

##### Functions on operators

Given a function from $\mathbb{C}\rightarrow\mathbb{C}$, we can define an
extension of this function on any normal matrix, depending on its domain.

For a subset $\Omega\subseteq\mathbb{C}$ and $f:\Omega\rightarrow\mathbb{C}$
we define $f:\Omega^{m\times m}\rightarrow\mathbb{C}^{m\times m}$ where, if
the eigenvalues of $M$ are $\\{\lambda_{i}\\}_{i}\subseteq\Omega$, the
eigenvalues of $f(M)$ are $\\{f(\lambda_{i})\\}_{i}$.

Most functions are well defined for normal matrices since we are guaranteed a
unitarily similar diagonalisation (see Eq. (1.6)). However, sometimes we must
restrict our focus for functions with limited natural domains. For example,
the logarithm can only be applied to positive matrices since their eigenvalues
are all real and non-negative.

We often use the function $f(M)=M^{-1}$ to denote the pseudoinverse [48,
Section 5.5.4]. Usually this notation refers to the usual matrix inverse but
we require a more general definition. In our context we take advantage of two
main properties of the pseudoinverse. Firstly, for matrices with full support,
the pseudoinverse coincides with the usual matrix inverse. However, if
$M\in\mathcal{L}(A)$ does not have full support (and is therefore singular),
$M^{-1}M$ is instead a projector onto
$\mathrm{supp}(M)\subset\mathcal{H}_{A}$. This is problematic as we cannot
guarantee a given density operator has full support. This brings us to the
second useful property: there are situations where we maintain the usual
behaviour of the inverse in the trace, i.e. if
$\mathrm{supp}(M)\subset\mathcal{H}_{A}$ then for $X\in\mathcal{L}(A)$ with
$\mathrm{supp}(X)\subseteq\mathrm{supp}(M)$ we have
$\operatorname{\mathrm{tr}}M^{-1}MX=\operatorname{\mathrm{tr}}X$. In other
words, equivalence holds if no information is lost by the action of the
projector $M^{-1}M$.

####  1.2.2 Operator decompositions

As an aid to computation we often make use of operator decompositions. These
observations essentially allow us to decompose certain classes of operators
into combinations of simpler operators which are more mathematically
convenient.

##### The spectral decomposition

Also known as the ‘eigendecomposition’, the spectral decomposition allows us
to reformulate a given matrix in terms of its eigenvalues and eigenvectors:

######  Proposition 1.1.

Any diagonalisable matrix $A$ can be factorised as

$A=P\Lambda P^{-1},$ (1.5)

where $\Lambda$ is a diagonal matrix whose $i$th entry is the $i$th
eigenvalue, $\lambda_{i}$, of $A$ and the $i$th column of $P$ is the
eigenvector corresponding to $\lambda_{i}$. For normal matrices this reduces
to

$X=U\Lambda U^{\dagger},$ (1.6)

where $U$ is unitary.

##### The polar decomposition

The polar decomposition allows us to express any square matrix in a much more
convenient form, especially in the context of Schatten norms (see the next
section).

######  Proposition 1.2.

Any operator $M\in\mathcal{L}(A)$ can be decomposed into the product of a
positive operator and a unitary operator i.e. for $P,Q\in\mathcal{P}(A)$ and
$U\in\mathcal{U}(A)$

$M=UP=QU.$ (1.7)

The above are known respectively as the left and right polar decompositions.

##### The singular value decomposition

We may also decompose non-square matrices in a similar way to the spectral
decomposition.

######  Proposition 1.3.

For any $M\in\mathcal{L}(A,B)$ we may write

$M=UDV$ (1.8)

where $U\in\mathcal{U}(C,B),V\in\mathcal{U}(A,C)$ are isometries and $D$ is a
positive semi-definite diagonal matrix in $\mathcal{L}(C)$ with dimension
equal to the number of non-zero eigenvalues of $M^{\dagger}M$.

The diagonal entries of $D$ are known as the singular values of $M$ which we
denote $\\{s_{i}(M)\\}_{i}$.

Most expositions of this proposition instead present the decomposition in
terms of unitary $U$ and $V$ in $\mathcal{U}(B)$ and $\mathcal{U}(A)$,
respectively, and $D$ as a rectangular diagonal matrix in $\mathcal{L}(A,B)$.
However, for our purposes, the above form is more convenient (See, for
example, Theorem 2.12). Hence we include a brief proof for clarity.

###### Proof of Proposition 1.3.

Let $M\in\mathcal{L}(A,B)$ be an $n\times m$ matrix. We have that
$M^{\dagger}M$ is positive semi-definite so by Eq. (1.6) there exists a
$\tilde{V}^{\dagger}\in\mathcal{U}(A)$ such that

$\tilde{V}^{\dagger}M^{\dagger}M\tilde{V}=\tilde{D}=\begin{bmatrix}D^{2}&0\\\
0&0\end{bmatrix}$ (1.9)

where $D^{2}\in\mathcal{L}(C)$ is diagonal and positive semi-definite with
$\dim(\mathcal{H}_{C})$ equal to the number of non-zero eigenvalues of
$M^{\dagger}M$. Again by Eq. (1.6), we may write

$\tilde{V}^{\dagger}=\begin{bmatrix}V^{\dagger}&V_{0}^{\dagger}\end{bmatrix}$
(1.10)

where $V^{\dagger}$ and $V_{0}^{\dagger}$ are matrices whose columns are the
eigenvectors of the, respectively, non-zero and zero eigenvalues. In this
sense we can write

$\begin{bmatrix}D^{2}&0\\\
0&0\end{bmatrix}=\begin{bmatrix}VM^{\dagger}MV^{\dagger}&VM^{\dagger}MV_{0}^{\dagger}\\\
V_{0}M^{\dagger}MV^{\dagger}&V_{0}M^{\dagger}MV_{0}^{\dagger}\end{bmatrix},$
(1.11)

hence $D^{2}=VM^{\dagger}MV^{\dagger}$ with $V^{\dagger}\in\mathcal{U}(C,A)$.

Define $U=MV^{\dagger}D^{-1}\in\mathcal{L}(C,B)$, thus

$UDV=MV^{\dagger}D^{-1}DV=MV^{\dagger}V=M.$ (1.12)

Moreover,

$U^{\dagger}U=(MV^{\dagger}D^{-1})^{\dagger}MV^{\dagger}D^{-1}=D^{-1}VM^{\dagger}MV^{\dagger}D^{-1}=D^{-1}D^{2}D^{-1}=\operatorname{\mathbbm{1}}_{B},$
(1.13)

Hence $U\in\mathcal{U}(C,B)$. ∎

##### The Schmidt decomposition

Finally, we have the Schmidt decomposition, considered an extension of the
singular value decomposition to vectors, which allows us to represent any pure
state as a set of orthonormal vectors localised on distinct subsystems.

######  Theorem 1.4.

We may write any vector $v\in\mathcal{H}_{AB}$ in the form

$v=\sum_{i}r_{i}\left(e^{i}_{A}\otimes f^{i}_{B}\right)$ (1.14)

where $\\{e^{i}_{A}\\}_{i}$ and $\\{f^{i}_{B}\\}_{i}$ form orthonormal bases
on their respective subsystems and the Schmidt coefficients $\\{r_{i}\\}_{i}$
are real and non-negative.

For proofs and related material see [49, Section 2.5].

####  1.2.3 The Schatten operator norm

If we introduce a norm on a set of a finite complex-valued linear operators we
form a Banach space. The Schatten norms form a family which generalise some of
the more common norms associated with spaces of operators. Essentially, they
can be considered as a matrix generalisation of the $p$-norms on Lebesgue
spaces.

For $M\in\mathcal{L}(A,B)$, we define

$\left\|M\right\|_{p}=\left[\operatorname{\mathrm{tr}}\left(M^{\dagger}M\right)^{\frac{p}{2}}\right]^{\frac{1}{p}},\quad
p\geq 1.$ (1.15)

Note these norms are well defined for rectangular matrices since
$M^{\dagger}M$ is by definition self-adjoint. In fact, the eigenvalues of
$|M|=\left(M^{\dagger}M\right)^{\frac{1}{2}}$ coincide with the singular
values of $M$ (and $M^{\dagger}$) (see [9, Section 1.1.3]). As such, we may
write the above in the form

$\left\|M\right\|_{p}=\left[\sum_{i=1}^{k}s_{i}\left(M\right)^{p}\right]^{\frac{1}{p}},$
(1.16)

where $k$ is the rank of $M$.

##### Properties

The above definition induces some useful properties:

######  Proposition 1.5.

The Schatten norm is unitarily invariant, i.e. for $M\in\mathcal{L}(A,B)$,
$U\in\mathcal{U}(B)$ and $V\in\mathcal{U}(A)$,

$\|UMV\|_{p}=\|UM\|_{p}=\|MV\|_{p}=\|M\|_{p}.$ (1.17)

###### Proof.

By the singular value decomposition we may write
$UMV=UU^{\prime}DV^{\prime}V$, where $D\in\mathcal{L}(C)$ is a diagonal
matrix, $UU^{\prime}\in\mathcal{U}(C,B)$ and $V^{\prime}V\in\mathcal{U}(A,C)$.
This demonstrates that the singular values of $M$ and $UVM$ coincide, hence by
Eq. (1.16) we have the equivalences in Eq. (1.17). ∎

######  Proposition 1.6.

For $M\in\mathcal{L}(A,B)$,

$\left\|M\right\|_{p}=\left\|M^{\dagger}\right\|_{p}=\left\|\overline{M}\right\|_{p}=\left\|M^{\top}\right\|_{p}=\left\||M|\right\|_{p}.$
(1.18)

###### Proof.

We simply observe that the singular values of $M$ are invariant under these
transformations. ∎

######  Proposition 1.7.

Given $M\in\mathcal{L}(A,B)$ we may write
$\left\|M^{\dagger}M\right\|_{p}=\left\|M\right\|^{2}_{2p}$.

###### Proof.

$\displaystyle\left\|M^{\dagger}M\right\|_{p}$
$\displaystyle=\left[\operatorname{\mathrm{tr}}\left(M^{\dagger}MM^{\dagger}M\right)^{\frac{p}{2}}\right]^{\frac{1}{p}}$
(1.19)
$\displaystyle=\left[\operatorname{\mathrm{tr}}\left(M^{\dagger}M\right)^{\frac{2p}{2}}\right]^{\frac{2}{2p}}$
(1.20) $\displaystyle=\left\|M\right\|^{2}_{2p}.$ (1.21)

∎

We conclude with an observation about the form of positive operators in the
unit ball of a given norm:

######  Proposition 1.8.

For all $\sigma\in\mathcal{D}(A)$ $\left\|\sigma^{\frac{1}{p}}\right\|_{p}=1$.

###### Proof.

This is evident by considering the definition of the Schatten norm and the
fact that ${\operatorname{\mathrm{tr}}(\sigma)=1}$. ∎

##### Examples

The Schatten norms generalise some well known norms on spaces of operators:

  * •

The trace norm is equivalent to $\|\cdot\|_{1}$ and recovers the trace for
self-adjoint operators.

  * •

The Hilbert-Schmidt (or Frobenius) norm is equivalent to $\|\cdot\|_{2}$. This
norm is induced by the inner product $\langle
M,N\rangle=\operatorname{\mathrm{tr}}(M^{\dagger}N)$, hence $\mathcal{L}(A,B)$
equipped with this norm is a Hilbert space.

  * •

The operator norm is equivalent to $\|\cdot\|_{\infty}$, defined as

$\displaystyle\|M\|_{\infty}$
$\displaystyle=\sup_{v\in\mathcal{H}_{A}}\frac{\|Mv\|_{2}}{\|v\|_{2}}$
$\displaystyle\left[\|v\|_{2}=\sqrt{\langle v,v\rangle}\right].$ (1.22)

###  1.3 Quantum mechanics and information

The following follows the material presented in [49].

####  1.3.1 The state of a quantum system

A quantum system can be any physical system, but generally we only consider
systems where quantum effects are significant, such as an electron, a molecule
or the crystalline structure of a material like doped silicon. A quantum
system $A$ is associated with a separable111Having a countable orthonormal
basis. Hilbert space $\mathcal{H}_{A}$ over $\mathbb{C}$. Here, we consider
only finite Hilbert spaces equipped with an inner product

$\displaystyle\langle\cdot,\cdot\rangle$ $\displaystyle:$
$\displaystyle\;\mathcal{H}_{A}\times\mathcal{H}_{A}$
$\displaystyle\longrightarrow$ $\displaystyle\mathbb{C}$ (1.23)
$\displaystyle:$ $\displaystyle\;\langle u,v\rangle$
$\displaystyle\longmapsto$ $\displaystyle\sum_{i}\overline{u}_{i}v_{i},$
(1.24)

where $u_{i}$ and $v_{i}$ are respectively the $i$th elements of $u$ and $v$.
The possible states of the system $A$ are described by the elements of
$\mathcal{H}_{A}$.

We use bra-ket notation to represent quantum states: for
$\psi\in\mathcal{H}_{A}$, its ‘ket’ is written $\left|\psi\right\rangle$ and
it can be considered as a column vector such that for an operator
$M\in\mathcal{L}(A,B)$,

$M\left|\psi\right\rangle=M\psi.$ (1.25)

From this point we use the shorthand
$\left|\psi\right\rangle=\psi\in\mathcal{H}_{A}$. Considering
$\left|\psi\right\rangle$ as a matrix, we define the ‘bra’ of $\psi$ as
$\left\langle\psi\right|=\left|\psi\right\rangle^{\dagger}$.

We find this definition notationally convenient as, considering Eq. (1.24), we
may write ${\left\langle\varphi|\psi\right\rangle=\langle\varphi,\psi\rangle}$
and

$\langle\varphi,A\psi\rangle=\left\langle\varphi|A\psi\right\rangle=\left\langle
A^{\dagger}\varphi|\psi\right\rangle=\left\langle\varphi\right|A\left|\psi\right\rangle.$
(1.26)

####  1.3.2 State vectors and probability density matrices

A finite dimensional state vector can be represented by a complex-valued ket
$\left|\psi\right\rangle$, for which $\left\langle\psi|\psi\right\rangle=1.$
This is to say that in an orthonormal basis
$\\{\left|e_{i}\right\rangle\\}_{i}$ where
$\left|\psi\right\rangle=\sum_{i}a_{i}\left|e_{i}\right\rangle$, the square
sum of the moduli of the complex coefficients, $\sum_{i}|a_{i}|^{2}=1$.

The probability density matrix (or probability operator) $\rho$ of a quantum
system is determined by an ensemble of pure states,
$\\{p_{i},\left|\psi_{i}\right\rangle\\}_{i}$. We define

$\rho=\sum_{i}p_{i}\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|,$
(1.27)

where $\rho$ is a density operator and as such we may express its spectral
decomposition (see section 1.2.2) in terms of bra-ket notation:

$\rho=\sum_{j}\lambda_{j}\left|j\right\rangle\left\langle j\right|,$ (1.28)

where the eigenvectors form an orthonormal eigenbasis and
$\sum_{j}\lambda_{j}=1$.

A system is considered to be in a pure state when its probability density
matrix can be represented by a single state vector, i.e.
$\rho=\left|\psi\right\rangle\left\langle\psi\right|$. This reflects that we
have complete knowledge of the state of the system. Note we must still
consider superposition and the underlying uncertainty when measuring a pure
state in a particular basis. A system that is not in a pure state is said to
be in a mixed state.

If $\rho$ has the spectral decomposition as in Eq. (1.28) then

$\displaystyle\rho^{2}$
$\displaystyle=\sum_{i,j}\lambda_{i}\lambda_{j}\left|i\right\rangle\left\langle
i|j\right\rangle\left\langle j\right|$ (1.29)
$\displaystyle=\sum_{j}\lambda_{j}^{2}\left|j\right\rangle\left\langle
j\right|$ (1.30) $\displaystyle\implies\operatorname{\mathrm{tr}}(\rho^{2})$
$\displaystyle=\sum\lambda_{j}^{2}.$ (1.31)

For all $j$, $0\leq\lambda_{j}\leq 1$, so $\lambda_{j}^{2}\leq\lambda_{j}$,
with equality if and only if $\lambda_{j}$ is the only eigenvalue, i.e.
equality if and only if $\rho$ is a pure state.

This gives us the criterion:

$\operatorname{\mathrm{tr}}(\rho^{2})\leq 1,$ (1.32)

with equality if and only if $\rho$ is a pure state.

####  1.3.3 Composite systems

A composite quantum system is one that is made up of distinct subsystems. We
write that $\rho_{AB}\in\mathcal{D}(AB)$ is a probability density matrix of
the composite system of $A$ and $B$. To recover the state on one of the
subsystems we ‘trace out’ the other subsystem with the partial trace: given a
basis $\\{\left|e_{i}\right\rangle\\}_{i}\subset\mathcal{H}_{B}$ and
$M\in\mathcal{L}(AB)$, we define the marginal of $M$ in $A$:

$\operatorname{\mathrm{tr}}_{B}(M)=\sum_{i}\left\langle
e_{i}\right|M\left|e_{i}\right\rangle=M_{A}\in\mathcal{L}(A).$ (1.33)

Given $\rho_{AB}\in\mathcal{D}(AB)$, the reduced state
$\rho_{A}\in\mathcal{D}(A)$ is the marginal of $\rho_{AB}$ in $A$, i.e.
$\operatorname{\mathrm{tr}}_{B}(\rho_{AB})=\rho_{A}$. We can consider the
reduced state as a description of the state on a subsystem, at least when it
comes to making measurements only on this subsystem [49]. The state of a
composite system may or may not be entangled. A pure state on a composite
system can be decomposed as $\rho_{AB}=\rho_{A}\otimes\rho_{B}$ if and only if
no entanglement exists between the subsystems.

##### Classical-quantum states

It is possible to define joint classical-quantum systems, required in the case
of measurement on a single subsystem (see next section). In order to consider
classical subsystems in the context of density operators we introduce an
auxiliary Hilbert space $\mathcal{H}_{X}$ for the random variable $X$, with
orthonormal basis $\\{\left|x\right\rangle\\}_{x}$ that acts as a ‘classical
register’ [20]. Accordingly, we have

$\rho_{XB}=\sum_{x}p(x)\left|x\right\rangle\\!\\!\left\langle
x\right|\otimes\rho^{x}_{B},$ (1.34)

where $\rho^{x}_{B}$ is the quantum state of the system $B$ conditioned on
$X=x$, i.e. we have

$\operatorname{\mathrm{tr}}_{X}\rho_{XB}=\sum_{x}p(x)\rho^{x}_{B}=\rho_{B}.$
(1.35)

##### Observations about pure states

Given a pure state is represented by a vector in its respective Hilbert space,
we can use the Schmidt decomposition to derive some useful equivalences:

######  Proposition 1.9.

Given the pure state
$\rho_{AB}=\left|\varphi\right\rangle\\!\\!\left\langle\varphi\right|\in\mathcal{D}(AB)$
and the Schmidt decomposition
${\left|\varphi\right\rangle=\sum_{i}r_{i}\left|i\right\rangle_{A}\otimes\left|i\right\rangle_{B}}$,
if we let $X\in\mathcal{L}(A,B)$ such that
$X=\sum_{i}r_{i}\left|i\right\rangle_{B}\left\langle i\right|_{A}$ then

$X^{\dagger}X=\rho_{A}\quad\text{and}\quad XX^{\dagger}=\rho_{B}.$ (1.36)

###### Proof.

We prove the first statement explicitly, then the second follows from a
symmetric argument.

First note that since $\\{\left|i\right\rangle_{A}\\}_{i}$ is an orthonormal
basis for $\mathcal{H}_{A}$ we may write the partial trace over $A$ as

$\displaystyle\rho_{B}$
$\displaystyle=\operatorname{\mathrm{tr}}_{A}(\rho_{AB})$ (1.37)
$\displaystyle=\sum_{k}\left\langle
k\right|_{A}\sum_{i,j}r_{i}r_{j}\left(\left|i\right\rangle_{A}\otimes\left|i\right\rangle_{B}\right)\left(\left\langle
j\right|_{A}\otimes\left\langle j\right|_{B}\right)\left|k\right\rangle_{A}$
(1.38) $\displaystyle=\sum_{i,j,k}r_{i}r_{j}\left(\left\langle
k|i\right\rangle_{A}\otimes\left|i\right\rangle_{B}\right)\left(\left\langle
j|k\right\rangle_{A}\otimes\left\langle j\right|_{B}\right)$ (1.39)
$\displaystyle=\sum_{i}r_{i}^{2}\left|i\right\rangle_{B}\left\langle
i\right|_{B}.$ (1.40)

Moreover,

$\displaystyle XX^{\dagger}$
$\displaystyle=\sum_{i,j}r_{i}r_{j}\left|i\right\rangle_{B}\left\langle
i|j\right\rangle_{A}\left\langle j\right|_{B}$ (1.41)
$\displaystyle=\sum_{i}r_{i}^{2}\left|i\right\rangle_{B}\left\langle
i\right|_{B}.$ (1.42)

∎

######  Proposition 1.10.

Given a pure state $\left|\varphi\right\rangle\in\mathcal{H}_{ABC}$ with
Schmidt decompositions

$\left|\varphi\right\rangle=\sum_{i}r_{i}\left|i\right\rangle_{A}\otimes\left|i\right\rangle_{BC}=\sum_{i}s_{i}\left|i\right\rangle_{AB}\otimes\left|i\right\rangle_{C},$
(1.43)

we have, for $K_{A}\in\mathcal{L}(A)$ and $L_{C}\in\mathcal{L}(C)$,

$\left\|L_{C}\sum_{i}r_{i}\left|i\right\rangle_{BC}\left\langle
i\right|_{A}K_{A}\right\|_{2}=\left\|L_{C}\sum_{i}s_{i}\left|i\right\rangle_{C}\left\langle
i\right|_{AB}K_{A}\right\|_{2}.$ (1.44)

This implies that we are free to choose any valid Schmidt decomposition in the
product of an argument of the Schatten $2$-norm – effectively ‘shifting’ a
subsystem to the other side – given that subsystem is not otherwise
represented in the product.

###### Proof.

We first rewrite the left-hand side as a trace:

$\displaystyle\left\|L_{C}\sum_{i}r_{i}\left|i\right\rangle_{BC}\left\langle
i\right|_{A}K_{A}\right\|_{2}^{2}$
$\displaystyle=\operatorname{\mathrm{tr}}\left(K_{A}^{\dagger}\sum_{i}r_{i}\left|i\right\rangle_{A}\left\langle
i\right|_{BC}L_{C}^{\dagger}L_{C}\sum_{j}r_{j}\left|j\right\rangle_{BC}\left\langle
j\right|_{A}K_{A}\right)$ (1.45)
$\displaystyle=\operatorname{\mathrm{tr}}\left(\sum_{i,j}r_{i}r_{j}K_{A}^{\dagger}\left|i\right\rangle_{A}\left\langle
i\right|_{BC}L_{C}^{\dagger}L_{C}\left|j\right\rangle_{BC}\left\langle
j\right|_{A}K_{A}\right)$ (1.46) $\displaystyle=\sum_{i,j,k}\left\langle
k\right|_{A}r_{i}r_{j}K_{A}^{\dagger}\left|i\right\rangle_{A}\left\langle
i\right|_{BC}L_{C}^{\dagger}L_{C}\left|j\right\rangle_{BC}\left\langle
j\right|_{A}K_{A}\left|k\right\rangle_{A}$ (1.47)
$\displaystyle=\sum_{i,j,k}r_{i}r_{j}\left\langle
i\right|_{A}K_{A}\left|k\right\rangle_{A}\left\langle
k\right|_{A}K_{A}^{\dagger}\left|j\right\rangle_{A}\left\langle
i\right|_{BC}L_{C}^{\dagger}L_{C}\left|j\right\rangle_{BC}$ (1.48)
$\displaystyle=\sum_{i,j}r_{i}r_{j}\left\langle
i\right|_{A}\otimes\left\langle i\right|_{BC}\left(K_{A}K_{A}^{\dagger}\otimes
L_{C}^{\dagger}L_{C}\right)\left|j\right\rangle_{A}\otimes\left|j\right\rangle_{BC}$
(1.49)
$\displaystyle=\left\langle\varphi\right|\left(K_{A}K_{A}^{\dagger}\otimes
L_{C}^{\dagger}L_{C}\right)\left|\varphi\right\rangle$ (1.50)
$\displaystyle=\sum_{i,j}s_{i}s_{j}\left\langle
i\right|_{AB}\otimes\left\langle i\right|_{C}\left(K_{A}K_{A}^{\dagger}\otimes
L_{C}^{\dagger}L_{C}\right)\left|j\right\rangle_{AB}\otimes\left|j\right\rangle_{C}.$
(1.51)

From this point, reversing the process obtains the right-hand side. ∎

####  1.3.4 Measurement

Uncertainty relations are a characterisation of how much information we can
extract from a system by measurement, hence we require a formal understanding
of what measurement means in the context of quantum information theory.

Although an interesting avenue for further research, this thesis does not
treat general projector operator-valued measurements – or POVMs, instead we
cover the simpler situation of measurements in orthonormal bases, or ONBs for
short.

An ONB $\mathbb{X}$ can be represented by a set of rank 1 projectors:
$\mathbb{X}=\\{\left|x\right\rangle\left\langle x\right|\\}$, whose spans are
mutually orthogonal.

In a real physical situation, if a state is measured in this basis the system
will collapse to one of the pure states described by the basis vectors. The
probability of observing the system in that basis state can be determined by
acting on the state with its associated measurement map:

$\displaystyle\mathcal{M}_{X}(\cdot)$ $\displaystyle:$
$\displaystyle\;\mathcal{L}(A)$ $\displaystyle\longrightarrow$
$\displaystyle\mathcal{L}(X)$ (1.52) $\displaystyle:$
$\displaystyle\;\rho_{A}$ $\displaystyle\longmapsto$
$\displaystyle\sum_{x}\left|x\right\rangle\\!\\!\left\langle
x\right|\rho_{A}\left|x\right\rangle\\!\\!\left\langle
x\right|=\sum_{x}\left\langle
x\right|\rho_{A}\left|x\right\rangle\left|x\right\rangle\\!\\!\left\langle
x\right|,$ (1.53)

$\mathcal{M}_{X}\in\mathrm{CPTP}(A,X)$ and we often denote
$\mathcal{M}_{X}(\rho_{A})=\rho_{X}$.

By the Born Rule, when a pure state $\left|\psi\right\rangle$ is measured in
an ONB with basis vectors $\left\\{\left|x_{i}\right\rangle\right\\}_{i}$, the
square modulus of the coefficients $\lambda_{x}$ of each eigenvector can be
interpreted as the probability $p(k)$ that the system will be observed in the
state $\left|x_{k}\right\rangle$, i.e.

$\displaystyle\left\langle\psi|x_{k}\right\rangle\left\langle
x_{k}|\psi\right\rangle$
$\displaystyle=\sum_{i}|\lambda_{x_{i}}|^{2}\left\langle
x_{i}|x_{k}\right\rangle\left\langle x_{k}|x_{i}\right\rangle$ (1.54)
$\displaystyle=\sum_{i}|\lambda_{x_{i}}|^{2}\delta_{ik}$ (1.55)
$\displaystyle=|\lambda_{k}|^{2}=p(k).$ (1.56)

For a mixed state the probability is instead the weighted sum of the
coefficients for each pure state in the ensemble.

We may also formalise measurement maps on only part of a multipartite system.
In this case we consider $\mathcal{M}^{*}_{X}\in\mathrm{CPTP}(AB,XB)$ noting
that even though it is defined on the larger space it only acts on the
relevant subsystem. With this in mind, the expansion to any additional
subsystem is assumed in a given context and we denote
$\mathcal{M}_{X}(\rho_{AB})=\rho_{XB}$, where it does not cause ambiguity.

The probability that measuring $\rho$ in $\mathbb{X}$ will have the outcome
$x$, i.e. the sum of the joint probabilities that $\rho$ is in the state
$\left|\psi_{i}\right\rangle$ and that it collapses to $\left|x\right\rangle$,
induces the random variable $X$ with probability mass function: $\left\langle
x\right|\rho\left|x\right\rangle=p(x)$, given by the eigenvalues of
$\mathcal{M}_{X}(\rho)$.

###  1.4 Entropy

Entropy itself is, in broad terms, a measurement of chaos. Systems that are
well structured and predictable have low entropy – consider a closed container
of liquid water and air – whereas high entropy systems have little structure
and are hard to predict, such as the same container holding a homogenous
mixture of water vapour and air. Basically, a system that is more “mixed up”
has more entropy.

From an information theoretic perspective, entropy is instead viewed in terms
of random variables. A random variable with a uniform distribution – where
each outcome is equally likely – has high entropy and one with a degenerate
distribution – where there is only one possible outcome – has low entropy. In
terms of the above example, we can consider the distribution of whether or not
a water molecule will be observed at a particular position. In the low
entropy, liquid state we would be guaranteed one way or the other depending on
which side of the surface we measure, but in the high entropy, gaseous state
we would be much less sure of the outcome. This statistical interpretation is
well formalised by the Shannon entropy.

####  1.4.1 Classical Entropy

The Shannon entropy $H(X)$ of a random variable $X$ is the most common, and
arguably the most useful, definition of entropy. It satisfies the following
postulates, desirable for a measurement of uncertainty [50]: If $n=|X|$, then

  1. (a)

$H(X)=H(p(x_{1}),p(x_{2}),...,p(x_{n}))$ is symmetric,

  2. (b)

$H(p(x),1-p(x))$ is continuous for $0<p(x)\leq 1$,

  3. (c)

$H(\underbrace{1/n,1/n,...,1/n}_{\text{$n$ times}})=\log_{2}n$,

  4. (d)

$H(tp(x_{1}),(1-t)p(x_{1}),p(x_{2}),...,p(x_{n}))=H(p(x_{1}),p(x_{2}),...,p(x_{n}))+p(x_{1})H(t,1-t)$,
for $0\leq t\leq 1$.

##### Shannon entropy and surprisal

To determine a rigorous definition of entropy we first consider surprisal [20,
Section III.A.1].

Let’s restrict our focus to a single outcome. The amount of information gained
from observing this outcome is related to its probability – we gain more
information when less likely outcomes occur, i.e. we can consider surprisal to
quantify how surprised one would be if the outcome occurred. Additionally, we
want the information gained from two independent outcomes to be the sum of the
information gained from each outcome, i.e. we want a function $f(P(X=x))=f(x)$
such that

$\displaystyle p(x_{1})<p(x_{2})\implies f(x_{1})>f(x_{2})\text{ and
}f(x,y)=f(p(x)p(y))=f(x)+f(y).$ (1.57)

Any function of the form $-\lambda\log_{2}p(x)$,
$\lambda\in\operatorname{\mathbb{R}}^{+}$ satisfies these conditions, and we
take $\lambda=1$ for simplicity [2], giving us the definition
$f(x):=-\log_{2}p(x)$. [cite]

Since we want to know the expected amount of information gained from observing
the random variable we take the average of the surprisal to obtain the Shannon
entropy:

$H(X)=-\sum_{x}p(x)\log p(x).$ (1.58)

This satisfies the postulates (a)-(d) and is an adequate if not preferable
measurement of uncertainty in many situations.

For any random variable $X$ its Shannon entropy is bounded: $0\leq
H(X)\leq\log|X|$, with equality on the left if and only if the distribution of
$X$ is degenerate and equality on the right if and only if the distribution of
$X$ is uniform on its support.

##### Joint entropy

Joint entropy is the entropy of the random variable described by the joint
probability of two or more random variables. Given two random variables $X$
and $Y$, their joint entropy is defined

$H(XY)=-\sum_{x,y}p(xy)\log p(xy).$ (1.59)

By Bayes’ Theorem (see Eq. (1.3)) and Jensen’s inequality, we have

$H(XY)\leq H(X)+H(Y),$ (1.60)

with equality if and only if $X$ and $Y$ are independent.

##### Conditional entropy

The conditional entropy of $X$ given $Y$ is average the amount of information
left to gain about $X$ once $Y$ has been observed. This is the remaining
uncertainty of $XY$ when given access to the side information provided by $Y$
about $X$. As such

$H(X|Y)=H(XY)-H(Y).$ (1.61)

From Eq. (1.60) we can see that

$H(X|Y)\leq H(X).$ (1.62)

The conditional entropy is equivalently defined [51]

$\displaystyle H(X|Y)$ $\displaystyle=\sum_{y}p(y)H(X|Y=y)$ (1.63)
$\displaystyle=-\sum_{x,y}p(y)p(x|y)\log p(x|y)$ (1.64)
$\displaystyle=-\sum_{x,y}p(xy)\log\frac{p(xy)}{p(y)}.$ (1.65)

Note that Eq. (1.61) is the bivariate form of the more general multivariate
chain rule:

$H(X|YZ)=H(XY|Z)+H(Y|Z).$ (1.66)

##### Mutual information

Mutual information, on the other hand, is the amount of information shared
between random variables, quantifying how well they are correlated. This
notion is conveyed through its definition in terms of the decomposition rules:

$\displaystyle I(X:Y)$ $\displaystyle=H(X)+H(Y)-H(XY)$ (1.67)
$\displaystyle=H(XY)-H(X|Y)-H(Y|X)$ (1.68)
$\displaystyle=H(X)-H(X|Y)=H(Y)-H(Y|X).$ (1.69)

From Eqs. (1.62) and (1.69) we can see $I(X:Y)\geq 0$.

##### Relative entropy

Relative entropy, also know as Kullback-Leibler divergence, is a quantity that
can be used to measure the closeness of two distributions over the same index
set in terms of their entropies.

For $X_{1}$ and $X_{2}$, $P(X_{1}=x)=p_{1}(x),\,P(X_{2}=x)=p_{2}(x)$ we define

$\displaystyle D(X_{1}\|X_{2})=\sum_{x}p_{1}(x)\log\frac{p_{1}(x)}{p_{2}(x)}.$
(1.70)

In the case that $p_{2}(x)$ does not dominate $p_{1}(x)$, i.e. there exists a
$x$ in the index set such that $p_{1}(x)>0$ and $p_{2}(x)=0$, then
$D(X_{1}\|X_{2})=\infty$.

We can express classical entropies in terms of this quantity:

$\displaystyle H(X)$
$\displaystyle=\log|X|-D\left(p(x)\left\|\frac{\\{1\\}_{x}}{|X|}\right.\right)$
(1.71) $\displaystyle=-\sum_{x}p(x)\log{p(x)},$ (1.72) $\displaystyle H(X|Y)$
$\displaystyle=-D(p(xy)\|p(y))$ (1.73)
$\displaystyle=-\sum_{x,y}p(xy)\log\frac{p(xy)}{p(y)}$ (1.74)
$\displaystyle=\sum_{y}p(y)\log{p(y)}-\sum_{x,y}p(xy)\log{p(xy)}$ (1.75)
$\displaystyle=H(XY)-H(Y),$ (1.76) $\displaystyle I(X:Y)$
$\displaystyle=D(p(xy)\|p(x)p(y))$ (1.77)
$\displaystyle=\sum_{x,y}p(xy)\log\frac{p(xy)}{p(x)p(y)}$ (1.78)
$\displaystyle=\sum_{x,y}p(xy)\log p(xy)-\sum_{x}p(x)\log
p(x)-\sum_{y}p(y)\log p(y)$ (1.79) $\displaystyle=H(X)+H(Y)-H(XY)$ (1.80)
$\displaystyle=H(X)-H(X|Y).$ (1.81)

####  1.4.2 Quantum entropies

We want to find applications of entropy in quantum information, so here we
discuss the quantum analogues of the classical entropies covered in the
previous sections. But first, a run-down of the concepts and notation.

##### Von Neumann entropy

Related to the Shannon entropy is its quantum analogue, von Neumann entropy.
Instead of taking a random variable as input, the von Neumann entropy takes
the probability density matrix $\rho$ describing the state of a quantum
system. It is defined similarly to Shannon entropy:

$H(\rho)=-\operatorname{\mathrm{tr}}(\rho\log\rho).$ (1.82)

Fortunately, when $\rho$ is expressed in terms of its spectral decomposition
(see Eq. (1.28)), the above simplifies to the Shannon entropy of the random
variable with probability mass function given by the eigenvalues of $\rho$.
i.e. $H(\rho)=-\sum_{j}\lambda_{j}\log\lambda_{j}$. We can use the von Neumann
entropy as a stepping off point to derive the quantum generalisations of the
classical entropic quantities via the intuitive chain and decomposition rules.
Accordingly, we have the quantum joint entropy:

$H(\rho_{AB})=-\operatorname{\mathrm{tr}}(\rho_{AB}\log\rho_{AB}),$ (1.83)

the quantum conditional entropy:

$H(A|B)=H(\rho_{AB})-H(\rho_{B})$ (1.84)

and the quantum mutual information:

$I(A:B)=H(\rho_{A})+H(\rho_{B})-H(\rho_{AB}).$ (1.85)

Similar to the classical entropies, these relationships coincide with the
definitions established via the quantum relative entropy [21]

$D(\rho\|\sigma)=\operatorname{\mathrm{tr}}\left[\rho(\log\rho-\log\sigma)\right].$
(1.86)

Analogous to the classical case, if $\sigma\not\gg\rho$ then
$D(\rho\|\sigma)=\infty$.

We have for $\rho_{AB}\in\mathcal{D}(AB)$:

$\displaystyle H(\rho_{AB})$
$\displaystyle=\log|\mathrm{supp}(\rho_{AB})|-D\left(\rho_{AB}\left\|\frac{\operatorname{\mathbbm{1}}_{AB}}{|\mathrm{supp}(\rho_{AB})|}\right.\right),$
(1.87) $\displaystyle H(A|B)_{\rho}$
$\displaystyle=-D(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\rho_{B})$
(1.88)
$\displaystyle=-\operatorname{\mathrm{tr}}(\rho_{AB}\log\rho_{AB})+\operatorname{\mathrm{tr}}(\rho_{B}\log\rho_{B})$
(1.89) $\displaystyle=H(\rho_{AB})-H(\rho_{B}),$ (1.90) $\displaystyle
I(A:B)_{\rho}$ $\displaystyle=D(\rho_{AB}\|\rho_{A}\otimes\rho_{B})$ (1.91)
$\displaystyle=\operatorname{\mathrm{tr}}(\rho_{AB}\log\rho_{AB})-\operatorname{\mathrm{tr}}(\rho_{A}\log\rho_{A})-\operatorname{\mathrm{tr}}(\rho_{B}\log\rho_{B})$
(1.92) $\displaystyle=H(\rho_{A})+H(\rho_{B})-H(\rho_{AB})$ (1.93)
$\displaystyle=H(\rho_{A})-H(A|B)_{\rho}.$ (1.94)

In fact we obtain equivalent definitions for the conditional entropy and
mutual information by minimising over one or both subsystems. As this is shown
for conditional entropy in [28], we provide a similar demonstration for mutual
information: if we consider the positive-definiteness of the relative entropy
due to Klein’s inequality [52], and observe that when $\sigma_{B}=\rho_{B}$
then $D(\rho_{B}\|\sigma_{B})=0$, we can write

$\displaystyle I(A:B)_{\rho}$
$\displaystyle=D(\rho_{AB}\|\rho_{A}\otimes\rho_{B})+\inf_{\sigma_{B}\in\mathcal{D}(B)}D(\rho_{B}\|\sigma_{B})$
(1.95)
$\displaystyle=\operatorname{\mathrm{tr}}(\rho_{AB}\log\rho_{AB}-\rho_{AB}\log(\rho_{A}\otimes\rho_{B}))+\operatorname{\mathrm{tr}}(\rho_{B}\log\rho_{B})-\inf_{\sigma_{B}\in\mathcal{D}(B)}\operatorname{\mathrm{tr}}(\rho_{B}\log\sigma_{B})$
(1.96)
$\displaystyle=\inf_{\sigma_{B}\in\mathcal{D}(B)}\operatorname{\mathrm{tr}}(\rho_{AB}\log\rho_{AB}-\rho_{AB}\log(\rho_{A}\otimes\sigma_{B}))$
(1.97)
$\displaystyle=\inf_{\sigma_{B}\in\mathcal{D}(B)}D(\rho_{AB}\|\rho_{A}\otimes\sigma_{B}).$
(1.98)

A similar calculation can be used to show that the equivalence also holds when
minimised over both subsystems, i.e.

$I(A:B)_{\rho}=\inf_{\begin{subarray}{c}\sigma_{A}\in\mathcal{D}(A)\\\
\sigma_{B}\in\mathcal{D}(B)\end{subarray}}D(\rho_{AB}\|\sigma_{A}\otimes\sigma_{B}).$
(1.99)

So far we have been discussing these relative entropy or divergence quantities
only in terms of Shannon or von Neumann entropies. It is important to point
out that this framework allows for a relatively clear and simple definition of
all these quantities but in the case of Rényi entropies things are not so
straight-forward. Nonetheless, armed with these definitions it is much easier
to discuss how to extend this structure to Rényi entropies.

####  1.4.3 Rényi entropy

Rényi entropies were first proposed in [50] as an alternative definition of
entropy, having most of the desired properties as well as allowing the
consideration of information of different order.

The Rényi entropy of order $\alpha\in(0,1)\cup(1,\infty)$ of a classical
random variable $X$, with probability mass function $p(x)$, is defined as

$H_{\alpha}(X)=\frac{1}{1-\alpha}\log\left(\sum_{x}p(x)^{\alpha}\right).$
(1.100)

It generalises the Shannon entropy and serves to weigh outcomes with more or
less likelihood differently depending on the order $\alpha$. The Shannon
entropy is recovered in the limit $\alpha\rightarrow 1$ (for the proof see
Appendix B.1). Rényi’s original derivation is beyond the scope of this thesis,
but it is constructive to note that in his formalism the Shannon entropy
corresponds to a linear case of a more general exponential function where the
order $\alpha$ acts as the exponent.

##### Rényi entropic quantities via relative entropy

There have been various quantities with which we could derive definitions of
Rényi conditional entropy and Rényi mutual information, the foremost being
classical Rényi divergence (Rényi relative entropy) [50]: given two random
variables $X_{1}$ and $X_{2}$ indexed on the same set $\\{x\\}$ with
probability mass functions $p_{1}(x)$ and $p_{2}(x)$ respectively, the Rényi
divergence is defined

$D_{\alpha}(X_{1}\|X_{2})=\frac{1}{\alpha-1}\log\sum_{x}\frac{p_{1}(x)^{\alpha}}{p_{2}(x)^{\alpha-1}}.$
(1.101)

Some specific examples of classical divergences belonging to this family of
divergences are

$\displaystyle D_{0}(X_{1}\|X_{2})$
$\displaystyle=-\log\sum_{x}p_{1}(x)^{0}p_{2}(x)=-\log\sum_{\\{x:p_{1}(x)>0\\}}p_{2}(x),$
(1.102) $\displaystyle D_{1/2}(X_{1}\|X_{2})$
$\displaystyle=-2\sum_{x}\log\sqrt{p_{1}(x)p_{2}(x)}$ (1.103)
$\displaystyle=2D_{B}(X_{1},X_{2})\quad\textit{(twice the Bhattacharyya
distance\leavevmode\nobreak\
\cite[cite]{[\@@bibref{Number}{bhattacharyya}{}{}]})},$ (1.104) $\displaystyle
D_{1}(X_{1}\|X_{2})$ $\displaystyle:=D(X_{1}\|X_{2}),$ (1.105) $\displaystyle
D_{2}(X_{1}\|X_{2})$
$\displaystyle=\log\sum_{x}p_{1}(x)\frac{p_{1}(x)}{p_{2}(x)}=\log\left\langle\frac{p_{1}(x)}{p_{2}(x)}\right\rangle_{X_{1}},$
(1.106) $\displaystyle D_{\infty}(X_{1}\|X_{2})$
$\displaystyle:=\lim_{\alpha\rightarrow\infty}D_{\alpha}(X_{1}\|X_{2})=\log\sup_{x}\frac{p_{1}(x)}{p_{2}(x)}.$
(1.107)

We would expect the Rényi divergence to induce definitions of the the Rényi
entropic quantities in an analogous way to the definition of classical
entropies via relative entropy in Section 1.4.1. That is to say, the Rényi
conditional entropy and mutual information would be defined

$\displaystyle
H_{\alpha}(X|Y)=-D_{\alpha}\left(p_{XY}\|p_{Y}\right)\quad\text{and}\quad
I_{\alpha}(X:Y)=D_{\alpha}\left(p_{XY}\|p_{X}p_{Y}\right).$ (1.108)

Since, in the limit $\alpha\rightarrow 1$, Eq. (1.101) produces the relative
entropy, we expect the above to produce the Shannon and von Neumann quantities
in the same limit.

There are some alternate propositions for conditional Rényi entropy from which
we could also derive a definition of mutual information.

##### Alternate definitions

The main alternative to defining mutual information directly with divergence
is to define it via the conditional entropy. There are some definitions which
have found utility, derived from analogous definitions based on Shannon
entropy or generalisations of other related quantities [54]. For example:

$\displaystyle H_{\alpha}^{\textsc{C}}(X|Y)$
$\displaystyle=\sum_{y}p(y)H_{\alpha}(X|Y=y)$ [55], (1.109) $\displaystyle
H_{\alpha}^{\textsc{JA}}(X|Y)$ $\displaystyle=H_{\alpha}(XY)-H_{\alpha}(Y)$
[56, 57], (1.110) $\displaystyle H_{\alpha}^{\textsc{RW}}(X|Y)$
$\displaystyle=\frac{1}{1-\alpha}\max_{y}\log\sum p(x|y)^{\alpha}$ [58].
(1.111)

Our classical Rényi mutual information could then be defined

$\displaystyle I^{*}_{\alpha}(X:Y)$
$\displaystyle=H_{\alpha}(X)+H_{\alpha}(Y)-H_{\alpha}(XY)$ (1.112)
$\displaystyle=H_{\alpha}(X)-H^{*}_{\alpha}(X|Y)=H_{\alpha}(Y)-H^{*}_{\alpha}(Y|X).$
(1.113)

On closer examination the above does not hold in general. Although similar or
related techniques to derive a Rényi measure of information may find use in
specific applications, this method is not fruitful for our purposes.

####  1.4.4 Quantum Rényi entropy

The quantum Rényi entropy is a quantum generalisation of the Rényi entropy and
is derived in an analogous way to von Neumann entropy: for a probability
density matrix $\rho$ and $\alpha\in(0,1)\cup(1,\infty)$, we define

$H_{\alpha}(\rho)=\frac{1}{1-\alpha}\log\operatorname{\mathrm{tr}}(\rho^{\alpha}).$
(1.114)

There are some particular choices of $\alpha$ which are either mathematically
convenient or reflect specific physical situations.

When $\alpha\rightarrow 1$ we recover the von Neumann entropy:
$H_{1}(\rho):=H(\rho)$. When $\alpha=2$ we have the ‘collision’ entropy:
$H_{2}(\rho)=-\log\operatorname{\mathrm{tr}}(\rho^{2})$, which characterises
the purity of a quantum system (see Eq. (1.32)). The max-entropy could be
naturally defined for $\alpha\rightarrow 0$ but, due to some mathematical
restrictions, we instead use $\alpha=\frac{1}{2}$ (see, for example, Eq.
(1.155)). We have

$\displaystyle H_{0}(\rho)$ $\displaystyle=\log|\mathrm{supp}(\rho)|,$ (1.115)
$\displaystyle H_{\frac{1}{2}}(\rho)=H_{\text{max}}(\rho)$
$\displaystyle=2\log\operatorname{\mathrm{tr}}(\sqrt{\rho}).$ (1.116)

The last quantity, and perhaps the most useful except for $\alpha\rightarrow
1$, is the min-entropy which we obtain when $\alpha\rightarrow\infty$:

$\displaystyle
H_{\infty}(\rho)=H_{\text{min}}(\rho)=-\log\lambda_{\max}[\rho].$ (1.117)

###  1.5 Rényi divergence and related quantities

We now introduce a generalisation of the quantum relative entropy and the
classical Rényi divergence – the ‘sandwiched’ Rényi divergence [28, 29]: for
$\rho\in\mathcal{D}(A),\sigma\in\mathcal{P}(A)$ and
${\alpha\in(0,1)\cup(1,\infty)}$,

$D_{\alpha}(\rho\|\sigma):=\begin{cases}\frac{1}{\alpha-1}\log\operatorname{\mathrm{tr}}\left[\left({\sigma}^{\frac{1-\alpha}{2\alpha}}\rho{\sigma}^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right]\quad&\text{if
}\rho\not\perp\sigma\wedge(\sigma\gg\rho\vee\alpha<1)\\\
\infty\quad&\text{else}\end{cases}.$ (1.118)

From this point we will refer to this quantity as simply ‘Rényi divergence’.

It is prudent at this point to formally introduce the following shorthands
which are used extensively in the remainder of the thesis. We often treat the
relationships $\frac{1}{\alpha}+\frac{1}{\alpha^{\prime}}=1$ and
$\frac{1}{\alpha}+\frac{1}{\hat{\alpha}}=2$. Accordingly, we equate

$\displaystyle\alpha^{\prime}=\frac{\alpha}{\alpha-1}\quad\text{and}\quad\hat{\alpha}=\frac{\alpha}{2\alpha-1}.$
(1.119)

This produces some equivalences that will be useful for later calculations. We
have

$\displaystyle\frac{\alpha^{\prime}}{\alpha}$
$\displaystyle=\frac{\alpha^{\prime}(\alpha^{\prime}-1)}{\alpha^{\prime}}={\alpha^{\prime}-1},$
(1.120) $\displaystyle\frac{\alpha^{\prime}}{\hat{\alpha}}$
$\displaystyle=\frac{\alpha^{\prime}(2\alpha-1)}{\alpha}=\frac{\alpha^{\prime}}{\alpha}\left(\frac{2\alpha^{\prime}}{\alpha^{\prime}-1}-1\right)=\left({\alpha^{\prime}-1}\right)\left(\frac{\alpha^{\prime}+1}{\alpha^{\prime}-1}\right)={\alpha^{\prime}+1}\quad\text{and}$
(1.121) $\displaystyle-\alpha^{\prime}$
$\displaystyle=\frac{\alpha}{1-\alpha}=\left(1-\frac{\hat{\alpha}}{2\hat{\alpha}-1}\right)^{-1}\left(\frac{\hat{\alpha}}{2\hat{\alpha}-1}\right)=\left(\frac{2\hat{\alpha}-1}{\hat{\alpha}-1}\right)\left(\frac{\hat{\alpha}}{2\hat{\alpha}-1}\right)=\frac{\hat{\alpha}}{\hat{\alpha}-1}=\hat{\alpha}^{\prime}.$
(1.122)

Applying a similar technique as in Section 1.4.2, we find that equivalence
does not extend to Rényi divergence but we can still produce rigorous
definitions of the relevant quantum Rényi entropic quantities. The following
notation is adapted from the notation introduced in [22]. We define the
quantum Rényi entropy as

$\displaystyle H_{\alpha}(\rho)$
$\displaystyle=\log|\mathrm{supp}(\rho)|^{\frac{\alpha}{1-\alpha}}-D_{\alpha}\left(\rho\left\|\frac{\operatorname{\mathbbm{1}}}{|\mathrm{supp}(x)|}\right.\right).$
(1.123)

We may also derive quantities that generalise the quantum conditional entropy:

$\displaystyle H^{\downarrow}_{\alpha}(A|B)_{\rho}$
$\displaystyle=-D_{\alpha}(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\rho_{B})\quad\text{and}$
(1.124) $\displaystyle H^{\uparrow}_{\alpha}(A|B)_{\rho}$
$\displaystyle=-\inf_{\sigma_{B}\in\mathcal{D}(B)}D_{\alpha}(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\sigma_{B}).$
(1.125)

The ordering $H^{\downarrow}_{\alpha}(A|B)_{\rho}\leq
H^{\uparrow}_{\alpha}(A|B)_{\rho}$, obvious from the definition, is indicated
by the direction of the superscript arrow. We can safely assume that
$\operatorname{\mathbbm{1}}_{A}\otimes\sigma_{B}\gg\rho_{AB}$, since any
choice of $\sigma_{B}$ where this is not the case would certainly not achieve
the infimum. Similarly, we derive generalisations of the quantum mutual
information, originally proposed in [27]:

$\displaystyle I^{\uparrow}_{\alpha}(A\;;\>\\!B)_{\rho}$
$\displaystyle=\inf_{\sigma_{B}\in\mathcal{D}(B)}D_{\alpha}(\rho_{AB}\|\rho_{A}\otimes\sigma_{B})\quad\text{and}$
(1.126) $\displaystyle I^{\downarrow}_{\alpha}(A:B)_{\rho}$
$\displaystyle=\inf_{\begin{subarray}{c}\sigma_{A}\in\mathcal{D}(A)\\\
\sigma_{B}\in\mathcal{D}(B)\end{subarray}}D_{\alpha}(\rho_{AB}\|\sigma_{A}\otimes\sigma_{B}).$
(1.127)

Again, the superscript arrows indicate the ordering of each version and we
satisfy the support condition as result of the minimisations. The use of ‘;’
in Eq. (1.126) indicates that this quantity is not symmetric between the
systems $A$ and $B$, whereas Eq. (1.127) is.

####  1.5.1 Generalised Rényi entropic quantities

In order to express our results in a more general setting, we make use of the
following notation for the generalised Rényi mutual information [27] and
conditional entropy:

$\displaystyle H_{\alpha}(\rho_{AB}\|\tau_{B})$
$\displaystyle=-D_{\alpha}(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\tau_{B}),$
(1.128) $\displaystyle I_{\alpha}(\rho_{AB}\|\tau_{A})$
$\displaystyle=\inf_{\sigma_{B}\in\mathcal{D}(B)}D_{\alpha}(\rho_{AB}\|\tau_{A}\otimes\sigma_{B}).$
(1.129)

We can readily verify that the above quantities generalise Eqs.
(1.124)-(1.127), i.e.

$\displaystyle H^{\downarrow}_{\alpha}(A|B)_{\rho}$
$\displaystyle=H_{\alpha}(\rho_{AB}\|\rho_{B}),$ $\displaystyle
H^{\uparrow}_{\alpha}(A|B)_{\rho}$
$\displaystyle=\sup_{\sigma_{B}\in\mathcal{D}(B)}H_{\alpha}(\rho_{AB}\|\sigma_{B}),$
(1.130) $\displaystyle I^{\uparrow}_{\alpha}(A\;;\>\\!B)_{\rho}$
$\displaystyle=I_{\alpha}(\rho_{AB}\|\rho_{A}),$ $\displaystyle
I^{\downarrow}_{\alpha}(A:B)_{\rho}$
$\displaystyle=\inf_{\sigma_{A}\in\mathcal{D}(A)}I_{\alpha}(\rho_{AB}\|\sigma_{A}).$
(1.131)

####  1.5.2 Properties of the Rényi divergence

This definition of a quantum Rényi divergence is not the only one proposed,
however, it maintains many useful and desirable properties of the quantities
it generalises for a wider range Rényi orders, whereas other
propositions222The main contender in this context is the Petz quantum Rényi
divergence. It is a simpler generalisation but only satisfies many desired
properties of a divergence for $\alpha\in[0,2]$. See [42] and [59] for more
details. do not. There are a few properties of the Rényi divergence which we
find particularly useful, which we now detail (for a more comprehensive
treatment see [28, 29, 38]).

Firstly, it generalises the quantum relative entropy. Indeed, it is recovered
by taking the limit $\alpha\rightarrow 1$ as in Proposition B.1. It is also
monotonically increasing in $\alpha$ [38], i.e for $\alpha>\beta$,

$D_{\alpha}(\rho\|\sigma)\geq D_{\beta}(\rho\|\sigma).$ (1.132)

It is useful to re-express the Rényi divergence as a Schatten norm of the
arguments dependent on $\alpha$:

$D_{\alpha}\left(\rho\|\sigma\right)=\log\left\|\sigma^{\frac{-1}{2\alpha^{\prime}}}\rho\sigma^{\frac{-1}{2\alpha^{\prime}}}\right\|_{\alpha}^{\alpha^{\prime}}.$
(1.133)

Its close relation to Schatten norms is mathematically convenient and as such
derived quantities exhibit duality relations. For example we have the duality
of the Rényi conditional entropy [38, 28]:

######  Proposition 1.11.

For a pure state $\rho_{ABC}\in\mathcal{D}(ABC)$ and $\alpha>\frac{1}{2}$ such
that $\frac{1}{\alpha}+\frac{1}{\hat{\alpha}}=2$ we have

$\displaystyle
H^{\uparrow}_{\alpha}(A|B)_{\rho}=-H^{\uparrow}_{\hat{\alpha}}(A|C)_{\rho}.$
(1.134)

And the duality of the generalised mutual information [27]:

######  Proposition 1.12.

For a pure state $\rho_{ABC}\in\mathcal{D}(ABC),\tau_{B}\in\mathcal{P}(B)$ and
$\alpha>\frac{1}{2}$ such that $\frac{1}{\alpha}+\frac{1}{\hat{\alpha}}=2$ we
have

$I_{\alpha}(\rho_{AB}\|\tau_{A})=-I_{\hat{\alpha}}(\rho_{AC}\|\tau_{A}^{-1}).$
(1.135)

Additionally, it satisfies the data-processing inequality [38, 60, 28]:

######  Proposition 1.13.

If $\alpha\geq\frac{1}{2}$ and $\Phi\in\mathrm{CPTP}(A,B)$, then for
$\rho\in\mathcal{D}(A),\sigma\in\mathcal{P}(A)$

$D_{\alpha}(\rho\|\sigma)\geq D_{\alpha}(\Phi(\rho)\|\Phi(\sigma)).$ (1.136)

In order to extend certain statements about strictly positive matrices to
positive semi-definite matrices we can observe the continuity of the Rényi
divergence. The following proposition is adapted from [28, Lemma 13] and
ensures the Rényi divergence is continuous even when the rank of $Y$
decreases.

######  Proposition 1.14.

Let $X\in\mathcal{D}(A),Y\in\mathcal{P}(A)$ with $X\neq 0$ and
$\alpha\in(0,1)\cup(1,\infty)$. We have

$D_{\alpha}(X\|Y)=\lim_{\varepsilon\rightarrow
0^{+}}\frac{1}{\alpha-1}\log\operatorname{\mathrm{tr}}\left[\left({(Y+\varepsilon\operatorname{\mathbbm{1}}_{A})}^{\frac{1-\alpha}{2\alpha}}X{(Y+\varepsilon\operatorname{\mathbbm{1}}_{A})}^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right]$
(1.137)

and the limit exists in the weaker sense in which a real valued sequence which
is bounded from below and not bounded from above and which does not have an
accumulation point is considered as being convergent to $+\infty$.

Essentially, this indicates that if we can make a statements for strictly
positive matrices, e.g.
$Y+\varepsilon\operatorname{\mathbbm{1}}_{A}\in\mathcal{P}^{*}(A)$, then we
can extend that statement to positive semi-definite matrices by taking
${Y+\varepsilon\operatorname{\mathbbm{1}}_{A}\rightarrow Y}$ with
$\mathrm{supp}(Y)\subset\mathcal{H}_{A}$. Note however, that in some cases the
quantity will still diverge if $Y\not\gg X$.

These properties naturally extend to any quantity defined using Rényi
divergence – properties which coincide with the mathematical and physical
interpretation of quantum entropies. That is to say, monotonicity in $\alpha$
reflects the expected behaviour of Rényi entropy when weighing more or less
likely outcomes differently and the data-processing inequality reflects that
entropy can only ever increase (or correlation decrease) when information is
processed (on each system independently).

####  1.5.3 Dupuis’ chain rules

One of the major contributions of this work is to formalise and provide Rényi
approximations of the chain and decomposition rules that exist for Shannon and
von Neumann entropies and which often are an important tool in the derivation
of related results. To this end, Dupuis’ chain rule [37] provides a foundation
for the methodology and a proof-of-concept for the more general inequalities
covered in Chapter 3.

######  Theorem 1.15.

Let $\frac{\alpha}{\alpha-1}=\frac{\beta}{\beta-1}+\frac{\gamma}{\gamma-1}$
with $\alpha,\beta,\gamma\in\left(1/2,1\right)\cup\left(1,\infty\right)$. For
$\rho_{ABC}\in\mathcal{D}(ABC)$ and $\tau_{C}\in\mathcal{P}(C)$, if
$(\alpha-1)(\beta-1)(\gamma-1)>0$,

$H_{\alpha}(\rho_{ABC}\|\tau_{C})\geq
H^{\uparrow}_{\beta}(A|BC)_{\rho}+H_{\gamma}(\rho_{BC}\|\tau_{C}).$ (1.138)

Otherwise, if $(\alpha-1)(\beta-1)(\gamma-1)<0$,

$H_{\alpha}(\rho_{ABC}\|\tau_{C})\leq
H^{\uparrow}_{\beta}(A|BC)_{\rho}+H_{\gamma}(\rho_{BC}\|\tau_{C}).$ (1.139)

This result shows that we may establish chain rules for conditional Rényi
entropies for particular choices of Rényi order. Evidently, these chain rules
are weaker than the Shannon and von Neumann versions, but still provide a
rigorous generalisation – note that by pinching the inequalities we recover an
equality when all orders tend to $1$. We provide an alternative proof of this
theorem and a variation on the bipartite case in Chapter 3.

###  1.6 Entropic uncertainty relations

Uncertainty relations manifest in many areas of science, and can be applied to
situations ranging across the accuracy of radar readings (trajectory vs
position), the analysis of sound waves (frequency vs instant) and the
measurement of subatomic particles (position vs momentum).

####  1.6.1 Commutator formulation

In quantum science, uncertainty relations are a way to characterise what are
referred to as ‘incompatible’ observables. Observables in this context are
operators whose action yields something which can be observed about the state
of a quantum system. Incompatibility can be measured by the commutator
$[C,D]$, a function of operators. Only when $[C,D]=0$ are the observables said
to commute, otherwise they are incompatible. Note that observables commute
only if they are simultaneously diagonalisable.

As explained in Section 1.3.4, when an observable $M$ acts on a quantum state
$\left|\psi\right\rangle$, we can recover the probability $P(m)$ that a
certain value $m$ will be measured for that observable. Using this notation we
write the expected value of the observable

$\langle\psi|M|\psi\rangle=\langle M\rangle=\sum_{m}mP(m)$ (1.140)

and its standard deviation $\sigma(M)=\sqrt{\langle M^{2}\rangle-\langle
M\rangle^{2}},$ which we find by examining the expected difference of the
values $m$ from the expected value:

$\displaystyle(\sigma(M))^{2}$ $\displaystyle=\langle(M-\langle
M\rangle)^{2}\rangle$ (1.141) $\displaystyle=\langle M^{2}-2M\langle
M\rangle+\langle M\rangle^{2}\rangle$ (1.142) $\displaystyle=\langle
M^{2}\rangle-\langle M\rangle^{2}.$ (1.143)

Taking the square root will bring things back to the same dimensionality as
the observable. Standard deviation is a measure of how spread out a
distribution is, and as such is connected to the uncertainty in the
measurement of that observable.

The Robertson relation [61] is described by the inequality

$\sigma(C)\sigma(D)\geq\frac{|\langle[C,D]\rangle|}{2}.$ (1.144)

This implies the combined spread of the observables must be greater than a
function of their incompatibility, the right-hand side is always positive and
only equal to zero for observables that commute on the support of
$\left|\psi\right\rangle$. This allows the left-hand side to be a trivial
bound for some situations where the eigenstates of the observables are
orthogonal to $\left|\psi\right\rangle$.

In general, for non-commuting observables, if the outcomes of one observable
can be predicted with a high level of accuracy (small standard deviation) then
the uncertainty in predicting the outcome of the other observable must be
proportionately high (large standard deviation).

Arguably the most famous application of this relation is the Heisenberg
Uncertainty Principle [35] relating position and momentum:

$\sigma(X)\sigma(P)\geq\frac{\hbar}{2},$ (1.145)

where $X$ is the position observable, $P$ is the momentum observable and
$\hbar$ is the reduced Planck’s constant, equal to the expected value of the
their commutator [62, 63].

####  1.6.2 Entropic formulation

To discuss entropic uncertainty relations we consider finite quantum states
represented by probability density matrices measured in a choice of ONB (see
Section 1.3.4).

##### Uncertainty in measuring in two orthonormal bases

Since for each $\rho\in\mathcal{D}(A)$ we can recover the random variables $X$
and $Z$ for two ONBs $\mathbb{X}$ and $\mathbb{Z}$ respectively, we can
examine the Shannon entropies $H(X)_{\rho}$ and $H(Z)_{\rho}$. Note that this
is not the entropy associated with the state $\rho$, or even the post-
measurement state, but rather the entropy associated with the measurement of
$\rho$ in either basis. We then have the canonical entropic uncertainty
relation [36]:

######  Theorem 1.16.

For a mixed state $\rho_{A}\in\mathcal{D}(A)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ we have

$H(X)_{\rho}+H(Z)_{\rho}\geq\log\frac{1}{c}=:q_{\textsc{MU}},\quad
c=\max_{x,z}|\left\langle x|z\right\rangle|^{2}.$ (1.146)

This indicates that the total uncertainty of measuring two different
observables on the same system must be at least as large as a positive
constant depending on the measurement bases but not on the measured state.

We may extend this to a result involving conditional entropy via the following
rationale: consider a generic Shannon entropic relation

$\sum_{n}H(X_{n})\geq q,$ (1.147)

with $q$ state independent. For some classical memory $Y$ containing
information about the preparation of the state being measured, we can show
[20, Section IV.C] that Eq. (1.147) implies $\sum_{n}H(X_{n}|Y)\geq q$. We may
therefore conclude that

$H(X|Y)_{\rho}+H(Z|Y)_{\rho}\geq q_{\textsc{MU}}.$ (1.148)

##### Information exclusion relations

Also known as mutual information uncertainty relations, information exclusion
relations are entropic uncertainty relations expressed in terms of mutual
information.

The canonical version of this type of relation is the Hall relation [19] which
we may derive from Eq. (1.148) with an application of the decomposition rule,
Eq. (1.69):

######  Theorem 1.17.

For a mixed state $\rho_{AB}\in\mathcal{D}(A)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ and $\mathbb{Y}$ on $B$ we have

$\displaystyle I(X:Y)_{\rho}+I(Z:Y)_{\rho}$
$\displaystyle\leq\log(d^{2}c)=:r_{\textsc{H}},$ (1.149)

where $d$ is the dimension of $\mathcal{H}_{A}$.

Note $d$ appears due to both $H(X)$ and $H(Y)$ being bounded above by $\log
d$.

####  1.6.3 Improvements and extensions

Here we detail some of the pre-existing results relevant to the generalised
entropic uncertainty relations in Chapter 4. These take the form of improved
bounds, tighter inequalities, extensions to Rényi entropies or a combination
thereof.

##### Improved bounds

Berta et al. [43] have improved on Eq. (1.148) by considering the conditional
entropy of the state $\rho$, which quantifies its inherent mixedness:

######  Theorem 1.18.

For a mixed state $\rho_{AB}\in\mathcal{D}(AB)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ we have

$H(X|B)_{\rho}+H(Z|B)_{\rho}\geq q_{\textsc{MU}}+H(A|B)_{\rho}.$ (1.150)

With this relation as a starting point, Coles and Piani [44] have derived a
tighter, state-dependent version:

######  Theorem 1.19.

For a mixed state $\rho_{AB}\in\mathcal{D}(AB)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ we have

$H(X|B)_{\rho}+H(Z|B)_{\rho}\geq q(\rho)+H(A|B)_{\rho},$ (1.151)

where
$q(\rho):=\max\\{q(\rho,\mathbb{X},\mathbb{Z}),q(\rho,\mathbb{Z},\mathbb{X})\\}$
and
$q(\rho_{A},\mathbb{A},\mathbb{B}):=-\sum_{a}p(a)\log\left(\max_{b}\left|\left\langle
a|b\right\rangle\right|^{2}\right)$.

This can be weakened slightly to a state-independent version:

######  Theorem 1.20.

For a mixed state $\rho_{AB}\in\mathcal{D}(AB)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ we have

$H(X|B)_{\rho}+H(Z|B)_{\rho}\geq q_{\textsc{CP}}+H(A|B)_{\rho},$ (1.152)

where
$q_{\textsc{CP}}:=\displaystyle{\min_{\sigma\in\mathcal{D}(AB)}q(\sigma)}$.

These relations are then used to produce improved versions of information
exclusion relation in Eq. (1.149). Firstly, they proved the conjecture of
Grudka et al. [64], who proposed the following:

######  Proposition 1.21.

For a mixed state $\rho_{AB}\in\mathcal{D}(AB)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ we have

$\displaystyle I(X:B)_{\rho}+I(Z:B)_{\rho}\leq
r_{\textsc{G}}\quad\text{where}\quad r_{\textsc{G}}:=\log\left(d\sum_{\text{d
largest}}\max_{x,z}|\left\langle x|z\right\rangle|^{2}\right),$ (1.153)

with the sum over the largest $d$ terms of the matrix
$\left[\left|\left\langle x|z\right\rangle\right|^{2}\right]$.

Indeed, this is actually a weaker version of the main result of Coles and
Piani:

######  Theorem 1.22.

For a mixed state $\rho_{AB}\in\mathcal{D}(AB)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ we have

$\displaystyle I(X:B)_{\rho}+I(Z:B)_{\rho}$ $\displaystyle\leq
r_{\textsc{CP}},$ (1.154)

where
$r_{\textsc{CP}}:=\min\left\\{r(\mathbb{X},\mathbb{Z}),r(\mathbb{Z},\mathbb{X})\right\\}$
and $r(\mathbb{A},\mathbb{B}):=\log\left(d\sum_{a}\max_{b}\left|\left\langle
a|b\right\rangle\right|^{2}\right)$.

##### Extensions to Rényi entropies

We now consider the known extensions of these relations to Rényi entropies of
specific order. In fact, to begin we need look no further than the Maassen-
Uffink relation as it was originally proposed [36] in the general form:

######  Theorem 1.23.

For a mixed state $\rho_{A}\in\mathcal{D}(A)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ with $\alpha,\hat{\alpha}\geq\frac{1}{2}$
such that $\frac{1}{\alpha}+\frac{1}{\hat{\alpha}}=2$,

$\displaystyle H_{\alpha}(X)+H_{\hat{\alpha}}(Z)\geq q_{\textsc{MU}}.$ (1.155)

Note for $\alpha,\beta=1$ we recover Eq. (1.148).

This not only establishes a strong precedent for Rényi uncertainty relations
but reveals that they are a fundamental part of the mathematical structure of
this field.

Tomamichel [42] derived the main improvement of this relation, described in
terms of quantum Rényi conditional entropy:

######  Theorem 1.24.

For a mixed state $\rho_{AB}\in\mathcal{D}(AB)$ and measurement bases
$\mathbb{X}$ and $\mathbb{Z}$ on $A$ with
${\frac{\alpha}{\alpha-1}=\frac{\gamma}{\gamma-1}+\frac{\beta}{\beta-1}},$ and
$(\alpha-1)(\beta-1)(\gamma-1)<0$.

$H^{\uparrow}_{\gamma}(X|B)+H^{\uparrow}_{\beta}(Z|B)\geq
q_{\textsc{MU}}+H^{\uparrow}_{\alpha}(A|B).$ (1.156)

This result generalises both Eq. (1.150) and Eq. (1.155), respectively, by
letting all parameters go to $1$ or choosing $B$ trivial and allowing
$\alpha\rightarrow 0$.

Once adequate Rényi entropy chain and decomposition rules are established, we
may adapt the methodology used for the derivations and proofs of these
improved relations to demonstrate the results in Chapter 4.

##  Chapter 2 Interpolation framework

The principle mechanism which we use to compare Rényi divergence of different
order is complex interpolation. There is a well-established and sophisticated
theory associated with this field of mathematics but in our context we do not
need to go into such depth. Our result is essentially an extension of the
Hadamard three-line theorem and, although Hadamard’s result can be derived via
the broader theory, we purposefully restrict our focus. However, in order to
understand the motivation behind the choices of interpolation function it is
prudent to provide an overview of the interpolation structure and associated
norms of Pisier. The statement and proof of the three-line theorem for
unbalanced weighted norms on linear operators are given in the last section of
this chapter and the specialised results that follow are given in Chapter 3,
in the context in which they are used.

###  2.1 The super-operator $\Gamma_{\sigma,\tau}$

For conciseness in the subsequent chapters, we introduce a more general form
of the super-operator notation found in [38]: with
$\sigma\in\mathcal{P}^{*}(B),\tau\in\mathcal{P}^{*}(A)$, define

$\displaystyle\Gamma_{\sigma,\tau}:$ $\displaystyle\,\mathcal{L}(A,B)$
$\displaystyle\longrightarrow$ $\displaystyle\mathcal{L}(A,B)$ (2.1)
$\displaystyle:$ $\displaystyle M$ $\displaystyle\longmapsto$
$\displaystyle\sigma^{\frac{1}{2}}M\tau^{\frac{1}{2}}.$ (2.2)

If $\sigma$ or $\tau$ are instead operators on only part of a tensor product
we still use the above notation when there is no ambiguity, e.g. for
$\sigma_{C}\in\mathcal{P}(C),\tau_{A}\in\mathcal{P}(A)$ and
$M\in\mathcal{L}(AB,CD)$

$\Gamma_{\sigma_{C},\tau_{A}}^{\frac{1}{\lambda}}(M)=\left(\sigma_{C}^{\frac{1}{2\lambda}}\otimes\operatorname{\mathbbm{1}}_{D}\right)M\left(\tau_{A}^{\frac{1}{2\lambda}}\otimes\operatorname{\mathbbm{1}}_{B}\right).$
(2.3)

In Eq. (2.1) we may choose $\mathcal{H}_{B}=\mathcal{H}_{A}$. Moreover, we
write
$\Gamma_{\sigma,\sigma}^{\frac{1}{\lambda}}=\Gamma_{\sigma}^{\frac{1}{\lambda}}$.

####  2.1.1 Unbalanced weighted norms

We define a norm on $\mathcal{L}(A,B)$ in the following way: given
$\sigma\in\mathcal{P}^{*}(B)$ and $\tau\in\mathcal{P}^{*}(A)$, let

$\left\|Y\right\|_{p,(\sigma,\tau)}=\left\|\sigma^{\frac{1}{2p}}Y\tau^{\frac{1}{2p}}\right\|_{p}=\left\|\Gamma_{\sigma,\tau}^{\frac{1}{p}}\left(Y\right)\right\|_{p}.$
(2.4)

It is relatively straight-forward to show this is indeed a norm for $1\leq
p\leq\infty$.

By defining the inner product $\left\langle
Y,X\right\rangle_{\sigma,\tau}=\operatorname{\mathrm{tr}}\left(Y^{\dagger}\sigma^{\frac{1}{2}}X\tau^{\frac{1}{2}}\right)$
we may show the following duality: for $1\leq p\leq\infty$,

$\|X\|_{p,(\sigma,\tau)}=\sup_{\|Y\|_{p^{\prime},(\sigma,\tau)}=1}\left|\left\langle
Y,X\right\rangle_{\sigma,\tau}\right|,$ (2.5)

where $p^{\prime}$ is the Hölder conjugate such that
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$.

###### Proof of Eq. (2.5).

By the definition of the dual norm [65, Section 2.10] we may write

$\displaystyle\|X\|_{p,(\sigma,\tau)}^{*}$
$\displaystyle=\sup_{Y\in\mathcal{L}(A,B)}\frac{\left|\left\langle
Y,X\right\rangle_{\sigma,\tau}\right|}{\left\|Y\right\|_{p,(\sigma,\tau)}}$
(2.6)
$\displaystyle=\sup_{Y\in\mathcal{L}(A,B)}\frac{\left|\operatorname{\mathrm{tr}}\left(Y^{\dagger}\sigma^{\frac{1}{2}}X\tau^{\frac{1}{2}}\right)\right|}{\left\|\sigma^{\frac{1}{2p}}Y\tau^{\frac{1}{2p}}\right\|_{p}}$
(2.7)
$\displaystyle=\sup_{Y\in\mathcal{L}(A,B)}\frac{\left|\operatorname{\mathrm{tr}}\left((\sigma^{\frac{-1}{2p}}Y\tau^{\frac{-1}{2p}})^{\dagger}\sigma^{\frac{1}{2}}X\tau^{\frac{1}{2}}\right)\right|}{\left\|Y\right\|_{p}}$
(2.8)
$\displaystyle=\sup_{Y\in\mathcal{L}(A,B)}\frac{\left|\operatorname{\mathrm{tr}}\left(Y^{\dagger}\sigma^{\frac{-1}{2p}}\sigma^{\frac{1}{2}}X\tau^{\frac{1}{2}}\tau^{\frac{-1}{2p}}\right)\right|}{\left\|Y\right\|_{p}}$
(2.9)
$\displaystyle=\sup_{Y\in\mathcal{L}(A,B)}\frac{\left|\operatorname{\mathrm{tr}}\left(Y^{\dagger}\sigma^{\frac{1}{2p^{\prime}}}X\tau^{\frac{1}{2p^{\prime}}}\right)\right|}{\left\|Y\right\|_{p}}$
(2.10)
$\displaystyle=\left\|\sigma^{\frac{1}{2p^{\prime}}}X\tau^{\frac{1}{2p^{\prime}}}\right\|_{p^{\prime}}$
(2.11) $\displaystyle=\left\|X\right\|_{p^{\prime},(\sigma,\tau)}.$ (2.12)

∎

The above implies that a Hölder type inequality holds for these norms, i.e.

$\left|\left\langle
Y,X\right\rangle_{\sigma,\tau}\right|\leq\|X\|_{p,(\sigma,\tau)}\|Y\|_{p^{\prime},(\sigma,\tau)},\quad
1\leq p,p^{\prime}\leq\infty.$ (2.13)

We must take note that in general these weighted norms are not unitarily
invariant, however we may state a specialised unitary invariance: for $U$ and
$V$ unitary such that $[U,\sigma]=[V,\tau]=0$,

$\|UXV\|_{p,(\sigma,\tau)}=\|UX\|_{p,(\sigma,\tau)}=\|XV\|_{p,(\sigma,\tau)}=\|X\|_{p,(\sigma,\tau)}.$
(2.14)

####  2.1.2 Rényi Divergence in terms of weighted norms

We may establish the following identities for Rényi entropic quantities in
terms of super-operator notation:

######  Lemma 2.1.

For $M\in\mathcal{L}(AB,C)$ such that $M^{\dagger}M=\rho_{AB}$,
$\tau_{B}\in\mathcal{P}^{*}(B)$ and $\alpha\in(0,1)\cup(1,\infty)$:

$\displaystyle H_{\alpha}(\rho_{A})$
$\displaystyle=-\log\sup_{\sigma_{A}\in\mathcal{D}^{*}(A)}\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{1}{\alpha^{\prime}}}(M)\right\|_{2}^{2\alpha^{\prime}},$
(2.15) $\displaystyle H_{\alpha}(\rho_{AB}\|\tau_{B})$
$\displaystyle=-\log\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{\frac{-1}{\alpha^{\prime}}}(M)\right\|_{2\alpha}^{2\alpha^{\prime}},$
(2.16) $\displaystyle I_{\alpha}(\rho_{AB}\|\tau_{B})$
$\displaystyle=\log\inf_{\sigma_{A}\in\mathcal{D}(A)}\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}\otimes\tau_{B}}^{\frac{-1}{\alpha^{\prime}}}(M)\right\|_{2\alpha}^{2\alpha^{\prime}}.$
(2.17)

Before we present the proof of Lemma 2.1 we find it useful to examine a more
general comparison between Rényi divergence and weighted norms.

######  Lemma 2.2.

Assume $\alpha\in(0,1)\cup(1,\infty)$. With $M\in\mathcal{L}(AB,C)$ such that
$M^{\dagger}M=\rho_{AB}\in\mathcal{D}(AB)$, $\sigma_{A}\in\mathcal{P}(A)$ and
$\tau_{B}\in\mathcal{P}^{*}(B)$, we have

$\log\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{1}{\alpha}-\frac{1}{\lambda}}\left(\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{-1}(M)\right)\right\|_{2\alpha,(\operatorname{\mathbbm{1}}_{C},\tau_{B}^{2})}^{2\alpha^{\prime}}=D_{\alpha}\left(\rho_{AB}\left\|\sigma_{A}^{1-\frac{\alpha^{\prime}}{\lambda^{\prime}}}\otimes\tau_{B}\right.\right).$
(2.18)

###### Proof.

$\displaystyle\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{1}{\alpha}-\frac{1}{\lambda}}(M)\right\|_{2\alpha,(\operatorname{\mathbbm{1}}_{C},\tau_{B}^{2})}^{2}$
$\displaystyle=\left\|\Gamma_{\sigma_{A}}^{\frac{1}{\alpha}-\frac{1}{\lambda}}(\rho_{AB})\right\|_{\alpha,\tau_{B}}$
(2.19)
$\displaystyle=\left\|\left(\sigma_{A}^{\frac{1}{2\alpha}-\frac{1}{2\lambda}}\otimes\tau_{B}^{\frac{1}{2\alpha}}\right)\rho_{AB}\left(\sigma_{A}^{\frac{1}{2\alpha}-\frac{1}{2\lambda}}\otimes\tau_{B}^{\frac{1}{2\alpha}}\right)\right\|_{\alpha}$
(2.20)
$\displaystyle=\left\|\left(\sigma_{A}^{\frac{1}{2\lambda^{\prime}}-\frac{1}{2\alpha^{\prime}}}\otimes\tau_{B}^{\frac{-1}{2\alpha^{\prime}}}\right)\Gamma_{\tau_{B}}(\rho_{AB})\left(\sigma_{A}^{\frac{1}{2\lambda^{\prime}}-\frac{1}{2\alpha^{\prime}}}\otimes\tau_{B}^{\frac{-1}{2\alpha^{\prime}}}\right)\right\|_{\alpha}$
(2.21)
$\displaystyle\implies\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{1}{\alpha}-\frac{1}{\lambda}}\left(\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{-1}(M)\right)\right\|_{2\alpha,(\operatorname{\mathbbm{1}}_{C},\tau_{B}^{2})}^{2}$
$\displaystyle=\left\|\left(\sigma_{A}^{1-\frac{\alpha^{\prime}}{\lambda^{\prime}}}\otimes\tau_{B}\right)^{\frac{-1}{2\alpha^{\prime}}}\rho_{AB}\left(\sigma_{A}^{1-\frac{\alpha^{\prime}}{\lambda^{\prime}}}\otimes\tau_{B}\right)^{\frac{-1}{2\alpha^{\prime}}}\right\|_{\alpha}$
(2.22)
$\displaystyle\implies\log\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{1}{\alpha}-\frac{1}{\lambda}}\left(\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{-1}(M)\right)\right\|_{2\alpha,(\operatorname{\mathbbm{1}}_{C},\tau_{B}^{2})}^{2\alpha^{\prime}}$
$\displaystyle=D_{\alpha}\left(\rho_{AB}\left\|\sigma_{A}^{1-\frac{\alpha^{\prime}}{\lambda^{\prime}}}\otimes\tau_{B}\right.\right).$
(2.23)

∎

We now make use of the above comparison to prove Lemma 2.1.

###### Proof of Lemma 2.1..

By definition we have

$\displaystyle H_{\alpha}(\rho_{A})$
$\displaystyle=-D_{\alpha}(\rho_{A}\|\operatorname{\mathbbm{1}}_{A})$ (2.24)
$\displaystyle=-\log\|\rho_{A}\|_{\alpha}^{\alpha^{\prime}}$ (2.25)
$\displaystyle=-\log\sup_{\sigma_{A}\in\mathcal{D}^{*}(A)}\left[\operatorname{\mathrm{tr}}\left(\rho_{A}\sigma_{A}^{\frac{1}{\alpha^{\prime}}}\right)^{\alpha^{\prime}}\right]$
(2.26)
$\displaystyle=-\log\sup_{\sigma_{A}\in\mathcal{D}^{*}(A)}\left[\operatorname{\mathrm{tr}}\left(\Gamma_{\sigma_{A}}^{\frac{1}{\alpha^{\prime}}}(\rho_{AB})\right)^{\alpha^{\prime}}\right]$
(2.27)
$\displaystyle=-\log\sup_{\sigma_{A}\in\mathcal{D}^{*}(A)}\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{1}{\alpha^{\prime}}}(M)\right\|_{2}^{2\alpha^{\prime}},$
(2.28)

where in the third line we use Lemma B.2. Note also that in the fourth line,
the partial trace on $B$ does not affect operators localised on $A$.

To show Eq. (2.16) we begin with the comparison in Lemma 2.2 and choose
$\lambda=\alpha$ in Eq. (2.18):

$\displaystyle H_{\alpha}(\rho_{AB}\|\tau_{B})$
$\displaystyle=-D_{\alpha}(\rho_{AB}\|\operatorname{\mathbbm{1}}_{A}\otimes\tau_{B})$
(2.29)
$\displaystyle=-\log\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{-1}(M)\right\|_{2\alpha,(\operatorname{\mathbbm{1}}_{C},\tau_{B}^{2})}^{2\alpha^{\prime}}$
(2.30)
$\displaystyle=-\log\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{\frac{1}{\alpha}-1}(M)\right\|_{2\alpha}^{2\alpha^{\prime}}$
(2.31)
$\displaystyle=-\log\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{\frac{-1}{\alpha^{\prime}}}(M)\right\|_{2\alpha}^{2\alpha^{\prime}}.$
(2.32)

Moreover, choosing instead $\lambda=1$ and
$\tau_{B}=\operatorname{\mathbbm{1}}_{B}$,

$\displaystyle H_{\alpha}(\rho_{AB}\|\sigma_{A})$
$\displaystyle=-D_{\alpha}(\rho_{AB}\|\sigma_{A}\otimes\operatorname{\mathbbm{1}}_{B})$
(2.33)
$\displaystyle=-\log\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{-1}{\alpha^{\prime}}}(M)\right\|_{2\alpha}^{2\alpha^{\prime}}.$
(2.34)

After relabelling in the second case we obtain Eq. (2.16). Similarly, to show
Eq. (2.17) we start with Eq. (2.18), choosing $\lambda=1$:

$\displaystyle I_{\alpha}(\rho_{AB}\|\tau_{B})$
$\displaystyle=\log\inf_{\sigma_{A}\in\mathcal{D}(A)}D(\rho_{AB}\|\sigma_{A}\otimes\tau_{B})$
(2.35)
$\displaystyle=\log\inf_{\sigma_{A}\in\mathcal{D}(A)}\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}}^{\frac{-1}{\alpha^{\prime}}}\left(\Gamma_{\operatorname{\mathbbm{1}}_{C},\tau_{B}}^{-1}(M)\right)\right\|_{2\alpha,(\operatorname{\mathbbm{1}}_{C},\tau_{B}^{2})}^{2\alpha^{\prime}}$
(2.36)
$\displaystyle=\log\inf_{\sigma_{A}\in\mathcal{D}(A)}\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}\otimes\tau_{B}}^{\frac{-1}{\alpha^{\prime}}}(M)\right\|_{2\alpha}^{2\alpha^{\prime}}.$
(2.37)

Moreover,

$\displaystyle I_{\alpha}(\rho_{AB}\|\sigma_{A})$
$\displaystyle=\log\inf_{\tau_{B}\in\mathcal{D}(B)}\left\|\Gamma_{\operatorname{\mathbbm{1}}_{C},\sigma_{A}\otimes\tau_{B}}^{\frac{-1}{\alpha^{\prime}}}(M)\right\|_{2\alpha}^{2\alpha^{\prime}}.$
(2.38)

Again, after relabelling, we arrive at Eq. (2.17). ∎

###  2.2 Pisier’s norms

Although not used directly in the main results of this thesis, the non-
commutative norms of Pisier still inform the interpolation structure and were
crucial in understanding the choices which produce the desired comparisons.
Their relation to Rényi divergence and related quantities has already been put
to use in other areas of quantum information [40, 41].

In this section we cover the basic concepts, some insights which further
simplify things in our context of positive operators and, ultimately, how they
factor into the interpolation results of Chapter 3.

####  2.2.1 Two-part norms

Based on the work of Pisier [39] and the subsequent refinements in [40] we can
use complex interpolation to derive the following definition of a non-
commutative norm on the product of complex Hilbert spaces of finite linear
operators:

######  Theorem 2.3.

For $1\leq q\leq p\leq\infty$ there is a unique $r\in[1,\infty]$ such that
$\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$. With $Y\in\mathcal{L}(AB)$ the
identities

$\displaystyle\|Y\|_{(p,q)}\equiv$
$\displaystyle\sup_{M,N\in\mathcal{L}(A)}\frac{\|(M\otimes\operatorname{\mathbbm{1}}_{B})Y(N\otimes\operatorname{\mathbbm{1}}_{B})\|_{q}}{\|M\|_{2r}\|N\|_{2r}}$
(2.39) and $\displaystyle\|Y\|_{(q,p)}\equiv$
$\displaystyle\inf_{\begin{subarray}{c}M,N\in\mathcal{L}(A)\\\
Y=(M\otimes\operatorname{\mathbbm{1}}_{B})Z(N\otimes\operatorname{\mathbbm{1}}_{B})\end{subarray}}\|M\|_{2r}\|N\|_{2r}\|Z\|_{p}$
(2.40)

define a norm.

Although the above expressions are required for the more general case, we may
take advantage of our specific context to re-express them in a more workable
form.

######  Proposition 2.4.

The expressions in Eq. (2.39) and Eq. (2.40) are equivalent to

$\|Y\|_{(p,q)}=\begin{cases}\displaystyle{\sup_{\sigma,\tau\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma,\tau}^{\frac{-1}{p}}(Y)\|_{q,(\sigma,\tau)}&\text{if
}p\geq q\\\
\displaystyle{\inf_{\sigma,\tau\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma,\tau}^{\frac{-1}{p}}(Y)\|_{q,(\sigma,\tau)}&\text{if
}p\leq q\end{cases}.$ (2.41)

###### Proof.

First we note that for all $M,N\in\mathcal{L}(A)$ we have a left and right
polar decomposition, i.e. there exists unitary operators
$U,V\in\mathcal{U}(A)$ and positive semi-definite operators
$P,Q\in\mathcal{P}(A)$ such that $M=UP$ and $N=QV$. Since Schatten norms are
unitarily invariant we have that Eq. (2.39) becomes

$\sup_{P,Q\in\mathcal{P}(A)}\frac{\|(P\otimes\operatorname{\mathbbm{1}}_{B})Y(Q\otimes\operatorname{\mathbbm{1}}_{B})\|_{q}}{\|P\|_{2r}\|Q\|_{2r}}=\sup_{P,Q\in\mathcal{P}(A)}\frac{\left\|\Gamma_{P,Q}^{2}(Y)\right\|_{q}}{\|P\|_{2r}\|Q\|_{2r}}.$
(2.42)

In order to achieve the supremum we may consider only $P$ and $Q$ such that
$\|P\|_{2r},\|Q\|_{2r}<\infty$ and, without loss of generality, we can further
use linearity and the scalability of Schatten norms to reduce $P$ and $Q$ to
operators such that $\|P\|_{2r}=\|Q\|_{2r}=1$, in which case we can identify
them with density operators with full support, i.e.

$\displaystyle\sup_{\sigma,\tau\in\mathcal{D}^{*}(A)}\frac{\left\|\Gamma_{\sigma,\tau}^{\frac{1}{r}}(Y)\right\|_{q}}{\left\|\sigma^{\frac{1}{2r}}\right\|_{2r}\left\|\tau^{\frac{1}{2r}}\right\|_{2r}}=\sup_{\sigma,\tau\in\mathcal{D}^{*}(A)}\left\|\Gamma_{\sigma,\tau}^{\frac{1}{r}}(Y)\right\|_{q}.$
(2.43)

Note $\frac{1}{2r}=\frac{1}{2q}-\frac{1}{2p}$, hence

$\left\|\Gamma_{\sigma,\tau}^{\frac{1}{r}}(Y)\right\|_{q}=\left\|\Gamma_{\sigma,\tau}^{\frac{1}{q}}(\Gamma_{\sigma,\tau}^{\frac{-1}{p}}(Y))\right\|_{q}=\left\|\Gamma_{\sigma,\tau}^{\frac{-1}{p}}(Y)\right\|_{q,(\sigma,\tau)}.$
(2.44)

Similarly, if we start with Eq. (2.40) we arrive at the second case of Eq.
(2.41). ∎

This formulation includes some cases of particular interest.

######  Corollary 2.5.

$\displaystyle\|Y\|_{(\infty,q)}$
$\displaystyle=\sup_{\sigma,\tau\in\mathcal{D}^{*}(A)}\|Y\|_{q,(\sigma,\tau)},$
(2.45) $\displaystyle\|Y\|_{(1,q)}$
$\displaystyle=\inf_{\sigma,\tau\in\mathcal{D}^{*}(A)}\left\|\Gamma_{\sigma,\tau}^{-1}(Y)\right\|_{q,(\sigma,\tau)},$
(2.46) $\displaystyle\|Y\|_{(p,p)}$ $\displaystyle=\|Y\|_{p}.$ (2.47)

###### Proof.

In Eq. (2.41), replace $p$ with the relevant values. ∎

An important feature of these norms is that they extend the log-convexity of
Schatten norms:

######  Theorem 2.6.

For $\theta\in(0,1)$, if $0\leq q_{0}<q_{1}\leq\infty$, $0\leq
p_{0}<p_{1}\leq\infty$ and

$\frac{1}{p_{\theta}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}},\quad\frac{1}{q_{\theta}}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}$
(2.48)

then

$\|Y\|_{(p_{\theta},q_{\theta})}\leq\|Y\|_{(p_{0},q_{0})}^{1-\theta}\|Y\|_{(p_{1},q_{1})}^{\theta}.$
(2.49)

This estimate follows from the original formulation of Pisier [39] for more
general spaces, but showing this in a self-contained way for our purposes has
proven non-trivial. Although it is related to our three-line theorem – Theorem
2.12, and the derived log-convexity results, it is not actually required to
demonstrate them. We include it here as an indication of the potential of the
general framework, of which we are only utilising a specific part.

####  2.2.2 The case of positive operators

If we restrict the focus to positive operators we can show that the
optimisation is achieved for the same operator, i.e.
$\sigma=\tau\in\mathcal{P}^{*}(A)$. This result and some of the subsequent
observations were originally explored in [40]. It follows that we have a
special case of Proposition 2.4:

######  Proposition 2.7.

For $X\in\mathcal{P}(AB)$,

$\|X\|_{(p,q)}=\begin{cases}\displaystyle{\sup_{\sigma\in\mathcal{D}^{*}(A)}}\left\|\Gamma_{\sigma}^{\frac{-1}{p}}(X)\right\|_{q,\sigma}&\text{if
}p\geq q\\\
\displaystyle{\inf_{\sigma\in\mathcal{D}^{*}(A)}}\left\|\Gamma_{\sigma}^{\frac{-1}{p}}(X)\right\|_{q,\sigma}&\text{if
}p\leq q\end{cases}.$ (2.50)

###### Proof.

For $X\in\mathcal{P}(A),M\in\mathcal{L}(A)$, Hölder’s inequality implies

$\displaystyle\|MXN^{\dagger}\|_{p}^{2}$
$\displaystyle=\|(MX^{\frac{1}{2}}X^{\frac{1}{2}}N^{\dagger})\|_{p}^{2}$
(2.51)
$\displaystyle\leq\|MX^{\frac{1}{2}}\|_{2p}^{2}\|X^{\frac{1}{2}}N^{\dagger}\|_{2p}^{2}$
(2.52) $\displaystyle=\|MXM^{\dagger}\|_{p}\|NXN^{\dagger}\|_{p}$ (2.53)
$\displaystyle\|MXN^{\dagger}\|_{p}$
$\displaystyle\leq\sqrt{\|MXM^{\dagger}\|_{p}\|NXN^{\dagger}\|_{p}}$ (2.54)
$\displaystyle\leq\max\\{\|MXM^{\dagger}\|_{p},\|NXN^{\dagger}\|_{p}\\}.$
(2.55)

Moreover, unitary invariance implies $\|MXM^{\dagger}\|_{p}=\||M|X|M|\|_{p}$.

∎

This special case gives rise to the following identity:

######  Proposition 2.8.

Given $X_{AB}\in\mathcal{P}(AB)$, $\|X_{AB}\|_{(p,1)}=\|X_{A}\|_{p}$.

###### Proof.

Using the duality of Schatten norms we can write

$\displaystyle\|X_{AB}\|_{(p,1)}$
$\displaystyle=\sup_{\sigma\in\mathcal{D}^{*}(A)}\left\|\Gamma_{\sigma}^{\frac{-1}{p}}(X_{AB})\right\|_{1,\sigma}$
(2.56)
$\displaystyle=\sup_{\sigma\in\mathcal{D}^{*}(A)}{\|\sigma^{\frac{1}{2p^{\prime}}}\operatorname{\mathrm{tr}}_{B}(X_{AB})\sigma^{\frac{1}{2p^{\prime}}}\|_{1}}$
(2.57)
$\displaystyle=\sup_{\sigma\in\mathcal{D}^{*}(A)}{\operatorname{\mathrm{tr}}X_{A}\sigma^{\frac{1}{p^{\prime}}}}$
(2.58) $\displaystyle=\|X_{A}\|_{p}.$ (2.59)

∎

####  2.2.3 Weighted two-part norms

Based on the definition of weighted norms in section 2.1.1, we may define a
weighted version of the two-part norm on $\mathcal{L}(A,B)$, with
$\sigma\in\mathcal{P}^{*}(B)$ and $\tau\in\mathcal{P}^{*}(A)$:

$\|X\|_{p,q,(\sigma,\tau)}=\|\Gamma_{\sigma,\tau}^{\frac{1}{q}}(X)\|_{(p,q)}.$
(2.60)

This definition may not be particularly useful in general but does allow for
the following identity:

######  Proposition 2.9.

For $Y\in\mathcal{L}(AB)$ and $\sigma_{B},\tau_{B}\in\mathcal{P}^{*}(B)$ we
have

$\|Y\|_{p,q,(\sigma_{B},\tau_{B})}=\begin{cases}\displaystyle{\sup_{\sigma_{A},\tau_{A}\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma_{A},\tau_{A}}^{\frac{-1}{p}}(Y)\|_{q,(\sigma_{A}\otimes\sigma_{B},\tau_{A}\otimes\tau_{B})}&\text{if
}p\geq q\\\
\displaystyle{\inf_{\sigma_{A},\tau_{A}\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma_{A},\tau_{A}}^{\frac{-1}{p}}(Y)\|_{q,(\sigma_{A}\otimes\sigma_{B},\tau_{A}\otimes\tau_{B})}&\text{if
}p\leq q\end{cases}.$ (2.61)

###### Proof.

$\displaystyle\left\|\Gamma_{\sigma_{A},\tau_{A}}^{\frac{-1}{p}}\left(\Gamma_{\sigma_{B},\tau_{B}}^{\frac{1}{q}}(Y)\right)\right\|_{q,(\sigma_{A},\tau_{A})}$
$\displaystyle=\left\|\left(\sigma_{A}\otimes\sigma_{B}\right)^{\frac{1}{2q}}\left(\sigma_{A}\otimes\operatorname{\mathbbm{1}}_{B}\right)^{\frac{-1}{2p}}Y\left(\sigma_{A}\otimes\operatorname{\mathbbm{1}}_{B}\right)^{\frac{-1}{2p}}\left(\tau_{A}\otimes\tau_{B}\right)^{\frac{1}{2q}}\right\|_{q}$
(2.62)
$\displaystyle=\left\|\Gamma_{\sigma_{A},\tau_{A}}^{\frac{-1}{p}}(Y)\right\|_{q,(\sigma_{A}\otimes\sigma_{B},\tau_{A}\otimes\tau_{B})}.$
(2.63)

Moreover
$\|\Gamma_{\sigma_{B},\tau_{B}}^{\frac{1}{q}}(Y)\|_{(p,q)}=\|Y\|_{p,q,(\sigma_{B},\tau_{B})}$,
therefore, taking the infimum or supremum for the relevant comparisons between
$p$ and $q$, we arrive at Eq. (2.61). ∎

This identity also has a specific form for positive operators:

######  Corollary 2.10.

With $1\leq q,p\leq\infty$, $X\in\mathcal{P}(AB)$ and
$\sigma_{B}\in\mathcal{P}^{*}(B)$,

$\|X\|_{(p,q),\sigma_{B}}=\begin{cases}\displaystyle{\sup_{\sigma_{A}\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma_{A}}^{\frac{-1}{p}}(X)\|_{q,\sigma_{A}\otimes\sigma_{B}}&\text{if
}p\geq q\\\
\displaystyle{\inf_{\sigma_{A}\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma_{A}}^{\frac{-1}{p}}(X)\|_{q,\sigma_{A}\otimes\sigma_{B}}&\text{if
}p\leq q\end{cases}.$ (2.64)

###### Proof.

This is clear by replacing $Y$ in Eq. (2.61) with $X\in\mathcal{P}(AB)$. ∎

####  2.2.4 Three-part norms

The framework of Pisier [39] also allows for a natural extension of the two-
part norm. We define a three-part norm on tripartite systems in an analogous
way: for $Y\in\mathcal{L}(ABC)$

$\|Y\|_{(p,q,s)}=\begin{cases}\displaystyle{\sup_{\sigma_{A},\tau_{A}\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma_{A},\tau_{A}}^{\frac{1}{q}-\frac{1}{p}}(Y)\|_{(q,s;AB)}&\text{if
}p\geq q\\\
\displaystyle{\inf_{\sigma_{A},\tau_{A}\in\mathcal{D}^{*}(A)}}\|\Gamma_{\sigma_{A},\tau_{A}}^{\frac{1}{q}-\frac{1}{p}}(Y)\|_{(q,s;AB)}&\text{if
}p\leq q\end{cases}.$ (2.65)

In the above we use the notation $\|Y\|_{(p,q;A)}$ to indicate over which
parts of the space the optimisation occurs. This is an important distinction
as in general, with $Y\in\mathcal{L}(ABC)$ and, for instance, $p\leq q$:

$\displaystyle\|Y\|_{(p,q;A)}$
$\displaystyle=\inf_{\sigma_{A},\tau_{A}\in\mathcal{D}^{*}(A)}\|\Gamma_{\sigma_{A},\tau_{A}}^{\frac{-1}{p}}(Y)\|_{q,(\sigma_{A},\tau_{A})}$
(2.66) $\displaystyle\neq$ $\displaystyle\|Y\|_{(p,q;AB)}$
$\displaystyle=\inf_{\sigma_{AB},\tau_{AB}\in\mathcal{D}^{*}(AB)}\|\Gamma_{\sigma_{AB},\tau_{AB}}^{\frac{-1}{p}}(Y)\|_{q,(\sigma_{AB},\tau_{AB})}.$
(2.67)

This definition allows for a reduction similar to Eq. (2.8), originally given
in [40]:

######  Proposition 2.11.

For $X_{ABC}\in\mathcal{P}(ABC)$,

$\|X_{ABC}\|_{(p,q,1)}=\|X_{AB}\|_{(p,q)}.$ (2.68)

###### Proof.

By Eq. (2.8), with $\sigma_{A}\in\mathcal{P}^{*}(A)$

$\displaystyle\|\Gamma_{\sigma_{A}}^{\frac{1}{q}-\frac{1}{p}}(X_{ABC})\|_{(q,1;AB)}$
$\displaystyle=\|\Gamma_{\sigma_{A}}^{\frac{1}{q}-\frac{1}{p}}(X_{AB})\|_{q}$
(2.69)
$\displaystyle=\|\Gamma_{\sigma_{A}}^{\frac{-1}{p}}(X_{AB})\|_{q,\sigma_{A}}.$
(2.70)

Choosing $p\geq q$ and optimising over $\sigma_{A}\in\mathcal{D}(A)$ on both
sides we obtain

$\displaystyle\sup_{\sigma_{A}\in\mathcal{D}^{*}(A)}\|\Gamma_{\sigma_{A}}^{\frac{1}{q}-\frac{1}{p}}(X_{ABC})\|_{(q,1;AB)}$
$\displaystyle=\sup_{\sigma_{A}\in\mathcal{D}^{*}(A)}\|\Gamma_{\sigma_{A}}^{\frac{-1}{p}}(X_{AB})\|_{q,\sigma_{A}},$
(2.71)

which is equivalent to Eq. (2.68). Choosing $p\leq q$ we instead take the
infimum and arrive again at Eq. (2.68). ∎

####  2.2.5 Adapting the general formulation

In the remainder of this thesis, we essentially use the above quantities but
ignore the optimisations. Overall, the optimisations prove problematic and
unnecessary to reproduce the convenient interpolation structure, i.e. we work
with expressions of the form:

$\left\|\Gamma_{\sigma_{A}}^{\frac{-1}{p}}(X)\right\|_{q,\sigma_{A}\otimes\sigma_{B}}\quad\text{and}\quad\left\|\Gamma_{\tau_{AB}}^{\frac{1}{s}-\frac{1}{q}}\left(\Gamma_{\sigma_{A}}^{\frac{1}{q}-\frac{1}{p}}(X)\right)\right\|_{s}.$
(2.72)

Evidently, these expressions do not define a unique value determined by $p$
and $q$ (and $s$) but rather encompass a family of norm-like expressions which
are free in the first (and second) parameter.

When we remove the optimisation, the reductions in the previous sections
relate closely to situations which produce particular Rényi entropic
quantities. Moreover, the choice of parameters for the relevant comparisons
almost uncannily reproduces the expected form of the chain and decomposition
rules. We may even take advantage of the structure to derive general
divergence inequalities which prove indispensable in establishing the improved
versions of the uncertainty relations and information exclusion relations
covered in Chapter 4.

###  2.3 Three-line theorem for unbalanced weighted norms

This result and its specifications provide the main mechanism by which we
compare divergences of different order. We present an extension of the three-
line theorem from [38] to spaces equipped with unbalanced weighted norms.

######  Theorem 2.12.

We denote $S:=\\{z\in\mathbb{C}:0\leq\Re(z)\leq 1\\}$, the complex strip. Let
$F:S\rightarrow\mathcal{L}(A,B)$ be a bounded map that is holomorphic on the
interior of $S$ and continuous on the boundary. Define

$\mathsf{M}_{k}=\sup_{t\in\operatorname{\mathbb{R}}}\|F(k+\operatorname{\mathbf{i}}t)\|_{p_{k},(\sigma,\tau)},$
(2.73)

then for $1\leq p_{\theta},p_{0},p_{1}\leq\infty$ such that
$\frac{1}{p_{\theta}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$,

$\|F(\theta)\|_{p_{\theta},(\sigma,\tau)}\leq\mathsf{M}_{0}^{1-\theta}\mathsf{M}_{1}^{\theta}.$
(2.74)

This is not only applicable to norms on spaces of square matrices but also on
spaces of non-square matrices – a notion we take advantage of to consider
Rényi orders less than one. Basically, since we are in essence dealing with
positive operators we can instead consider their decomposition into the
product of another operator and its adjoint.

The proof of Theorem 2.12 relies on Hadamard’s three-line theorem, which we
now include for reference.

######  Theorem 2.13 (Hadamard’s three-line [66]).

Let $f:S\rightarrow\mathbb{C}$ be a bounded function that is holomorphic on
the interior of $S$ and continuous on its boundary. For $k\in\\{0,1\\}$, let

$\mathsf{M}_{k}=\sup_{t\in\operatorname{\mathbb{R}}}|f(k+\operatorname{\mathbf{i}}t)|.$
(2.75)

Then for every $\theta\in[0,1]$ we have

$\left|f(\theta)\right|\leq\mathsf{M}_{0}^{1-\theta}\mathsf{M}_{1}^{\theta}.$
(2.76)

The proof of Theorem 2.12 closely follows the structure of the proof of
Theorem 2 in [38], with some adjustments to allow for general spaces of linear
operators.

###### Proof of Theorem 2.12.

Let $X\in\mathcal{L}(A,B)$ be such that
$\|X\|_{p^{\prime}_{\theta},(\sigma,\tau)}=1$ and
$\|F(\theta)\|_{p_{\theta},(\sigma,\tau)}=\left|\left\langle
X^{\dagger},F(\theta)\right\rangle_{\sigma,\tau}\right|$. We have

$\displaystyle\|X\|_{p^{\prime}_{\theta},(\sigma,\tau)}=\left\|\Gamma_{\sigma,\tau}^{\frac{1}{p^{\prime}_{\theta}}}(X)\right\|_{p^{\prime}_{\theta}}=1,$
(2.77)

hence the singular value decomposition (see Proposition 1.3) of
$\Gamma_{\sigma,\tau}^{\frac{1}{p^{\prime}_{\theta}}}(X)$ has the form
${\Gamma_{\sigma,\tau}^{\frac{1}{p^{\prime}_{\theta}}}(X)=UD^{\frac{1}{p^{\prime}_{\theta}}}V}$,
where $U\in\mathcal{U}(C,B),V\in\mathcal{U}(A,C)$ are isometries and $D$ is a
square diagonal matrix in $\mathcal{L}(C)$ whose singular values are real,
positive and sum to $1$. Define

$X(z)=\Gamma_{\sigma,\tau}^{-\left(\frac{1-z}{p^{\prime}_{0}}+\frac{z}{p^{\prime}_{1}}\right)}\left(UD^{\left(\frac{1-z}{p^{\prime}_{0}}+\frac{z}{p^{\prime}_{1}}\right)}V\right)=\sigma^{-\left(\frac{1-z}{2p^{\prime}_{0}}+\frac{z}{2p^{\prime}_{1}}\right)}\left(UD^{\left(\frac{1-z}{p^{\prime}_{0}}+\frac{z}{p^{\prime}_{1}}\right)}V\right)\tau^{-\left(\frac{1-z}{2p^{\prime}_{0}}+\frac{z}{2p^{\prime}_{1}}\right)}.$
(2.78)

Observe that the map $z\mapsto X(z)$ is holomorphic and
$\frac{1}{p_{\theta}^{\prime}}=\frac{1-\theta}{p_{0}^{\prime}}+\frac{\theta}{p_{1}^{\prime}}$,
therefore

$\displaystyle
X(\theta)=\Gamma_{\sigma,\tau}^{-\frac{1}{p^{\prime}_{\theta}}}\left(UD^{\frac{1}{p^{\prime}_{\theta}}}V\right)=\Gamma_{\sigma,\tau}^{-\frac{1}{p^{\prime}_{\theta}}}\left(\Gamma_{\sigma,\tau}^{\frac{1}{p^{\prime}_{\theta}}}(X)\right)=X.$
(2.79)

Now define $g(z)=\left\langle
X^{\dagger}(z),F(z)\right\rangle_{\sigma,\tau}=\operatorname{\mathrm{tr}}\left(X(z)\sigma^{\frac{1}{2}}F(z)\tau^{\frac{1}{2}}\right)$.
Hence $g$ satisfies the requirements of Hadamard’s three-line theorem (Theorem
2.13), and we may write

$\displaystyle\|F(\theta)\|_{p_{\theta},(\sigma,\tau)}$
$\displaystyle=\left|\left\langle
X^{\dagger},F(\theta)\right\rangle_{\sigma,\tau}\right|$ (2.80)
$\displaystyle=\left|\operatorname{\mathrm{tr}}\left(X(\theta)\sigma^{\frac{1}{2}}F(\theta)\tau^{\frac{1}{2}}\right)\right|$
(2.81) $\displaystyle=|g(\theta)|$ (2.82)
$\displaystyle\leq\sup_{t\in\operatorname{\mathbb{R}}}|g(\operatorname{\mathbf{i}}t)|^{1-\theta}\sup_{t\in\operatorname{\mathbb{R}}}|g(1+\operatorname{\mathbf{i}}t)|^{\theta}$
(2.83) $\displaystyle=\sup_{t\in\operatorname{\mathbb{R}}}\left|\left\langle
X^{\dagger}(\operatorname{\mathbf{i}}t),F(\operatorname{\mathbf{i}}t)\right\rangle_{\sigma,\tau}\right|^{1-\theta}\sup_{t\in\operatorname{\mathbb{R}}}\left|\left\langle
X^{\dagger}(1+\operatorname{\mathbf{i}}t),F(1+\operatorname{\mathbf{i}}t)\right\rangle_{\sigma,\tau}\right|^{\theta}$
(2.84)
$\displaystyle\leq\sup_{t\in\operatorname{\mathbb{R}}}\left(\|X(\operatorname{\mathbf{i}}t)\|_{p^{\prime}_{0},(\sigma,\tau)}\|F(\operatorname{\mathbf{i}}t)\|_{p_{0},(\sigma,\tau)}\right)^{1-\theta}\sup_{t\in\operatorname{\mathbb{R}}}\left(\|X(1+\operatorname{\mathbf{i}}t)\|_{p^{\prime}_{1},(\sigma,\tau)}\|F(1+\operatorname{\mathbf{i}}t)\|_{p_{1},(\sigma,\tau)}\right)^{\theta},$
(2.85)

where in the last line we used the Hölder type inequality, Eq. (2.13).

By definition,
$\|X(\operatorname{\mathbf{i}}t)\|_{p^{\prime}_{0},(\sigma,\tau)}=\left\|\Gamma_{\sigma,\tau}^{\frac{1}{p^{\prime}_{0}}}(X(\operatorname{\mathbf{i}}t))\right\|_{p^{\prime}_{0}}$
and

$\displaystyle\Gamma_{\sigma,\tau}^{\frac{1}{p^{\prime}_{0}}}(X(\operatorname{\mathbf{i}}t))$
$\displaystyle=\Gamma_{\sigma,\tau}^{\frac{1}{p^{\prime}_{0}}}\left(\Gamma_{\sigma,\tau}^{-\left(\frac{1-\operatorname{\mathbf{i}}t}{p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{p^{\prime}_{1}}\right)}\left(UD^{\left(\frac{1-\operatorname{\mathbf{i}}t}{p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{p^{\prime}_{1}}\right)}V\right)\right)$
(2.86)
$\displaystyle=\Gamma_{\sigma,\tau}^{-\left(\frac{-\operatorname{\mathbf{i}}t}{p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{p^{\prime}_{1}}\right)}\left(UD^{\left(\frac{1-\operatorname{\mathbf{i}}t}{p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{p^{\prime}_{1}}\right)}V\right)$
(2.87)
$\displaystyle=\sigma^{-\left(\frac{-\operatorname{\mathbf{i}}t}{2p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{2p^{\prime}_{1}}\right)}UD^{\left(\frac{-\operatorname{\mathbf{i}}t}{p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{p^{\prime}_{1}}\right)}D^{\frac{1}{p^{\prime}_{0}}}V\tau^{-\left(\frac{-\operatorname{\mathbf{i}}t}{2p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{2p^{\prime}_{1}}\right)}.$
(2.88)

Observe
$\sigma^{-\left(\frac{-\operatorname{\mathbf{i}}t}{2p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{2p^{\prime}_{1}}\right)}UD^{\left(\frac{-\operatorname{\mathbf{i}}t}{p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{p^{\prime}_{1}}\right)}$
and
$V\tau^{-\left(\frac{-\operatorname{\mathbf{i}}t}{2p^{\prime}_{0}}+\frac{\operatorname{\mathbf{i}}t}{2p^{\prime}_{1}}\right)}$
are isometries, thus
$\|X(\operatorname{\mathbf{i}}t)\|_{p^{\prime}_{0},(\sigma,\tau)}=1$. In a
similar way we can show
$\|X(1+\operatorname{\mathbf{i}}t)\|_{p^{\prime}_{1},(\sigma,\tau)}=1$.

Hence Eq. (2.85) becomes

$\displaystyle\|F(\theta)\|_{p_{\theta},(\sigma,\tau)}\leq\sup_{t\in\operatorname{\mathbb{R}}}\left(\|F(\operatorname{\mathbf{i}}t)\|_{p_{0},(\sigma,\tau)}\right)^{1-\theta}\sup_{t\in\operatorname{\mathbb{R}}}\left(\|F(1+\operatorname{\mathbf{i}}t)\|_{p_{1},(\sigma,\tau)}\right)^{\theta}.$
(2.89)

∎

We defer the introduction of the refinements of this theorem to Chapter 3,
where they can be viewed with the results in which they are used.

##  Chapter 3 Rényi entropy divergence inequalities

This chapter covers Rényi extensions of the chain rule, decomposition rule and
more general divergence equivalences in the form of divergence inequalities.
The first section details the inequalities and the relationships between them
while the technique of interpolation and more detailed proofs are found in the
next section.

As it is closely related to these results, we also include a previously
published approach to the proof of the decomposition rule in Appendix A. This