Inegalités entropiques en théorie de l'information quantique

Quantum entropy inequalities
Théo Fradin
August 24, 2021
You should call it ‘entropy’ for two
reasons; first, the function is already
in use in thermodynamics; second,
and most importantly, most people
don’t know what entropy really is,
and if you use ‘entropy’ in an
argument, you will win every time.
John Von Neumann to Claude
Shannon
Summary
This training was supervised by Marius Lemm, Professor at the École Polytech-
nique Fédérale de Lausanne, chair of analysis and mathematical physics. During
this training in quantum information theory I learnt about quantum entropy. After
having worked on the basic notion : definition of the various quantum entropies,
basic inequalities of entropy, strong subadditivity, data processing inequality; I
learnt that strong subadditivity was the most important entropy inequality and
that every other one could be derived from it. Then, I worked on the main tool used
in a new method to find entropy inequalities from [Sut19], the Golden-Thompson
inequality. After that, I studied the application of this method to prove a variant of
the Linden-Winter inequality, a constrained but new inequality. The proof follows
the note [LX]. Eventually, we did some research about generalizing the proof to the
case of approximate quantum Markov chains, without much success. However we
did manage to bring back the original symmetry in the proof.
Contents
1 Introduction 2
1.1 Notations............................................ 2
1.2 Specifics of quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Entropy ............................................. 5
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2 Golden-Thompson inequality 7
3 Quantum Markov chains 15
3.1 QuantumMarkovchains................................. 15
3.2 Approximate quantum Markov chains . . . . . . . . . . . . . . . . . . . . . . . 17
4 Linden-Winter inequality and possible extensions 18
4.1 Linden and Winter’s original inequality . . . . . . . . . . . . . . . . . . . . . . 18
4.2 A new proof for the inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 Thoughtsontheproof .................................. 22
4.4 Symmetry............................................ 25
1 Introduction
In this section we introduce the notations as well as the basic notions used later.
This follows [NC00],[Wil19]and [Sut18].
1.1 Notations
Let A be a quantum system, i.e. an Hermitian space. We will be using the following
notations :
•ρ†: Conjugate transpose of a matrix ρ
•l o g denotes the natural logarithm, either for complex numbers or matrices.
•|ψ〉:ket ψ, vector of A.
•〈ψ|:bra ψ, vector of the dual of A.
•IA: identity map of A.
•t r ,t rA: trace and partial trace over A.
•s up p (ρ): Support of ρ.
•Sp (ρ): Set of Eigenvalues of ρ.
•⊕: Sum between two operators acting on orthogonal spaces.
•
ρ p: p-Schatten norm of the operator ρ.
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•|ρ|:=ρρ†1/2: Absolute value of the matrix ρ.
•L(A): Set of linear operators over A.
•D(A): Set of density operators over A.
•P(A): Set of positive semi-definite operators over A.
•P+(A): Set of positive definite operators over A.
•T P C P (A,B): set of trace preserving completely positive maps from A to B.
The matrices used here are positive, unless specified otherwise.
1.2 Specifics of quantum physics
A quantum system is a Hermitian space A, whose vectors |ψ〉represent its possible
states. The adjoint of |ψ〉is written 〈ψ|. We have, for example : 〈ψ|ψ〉is the scalar
product of ψwith itself, and |ψ〉〈ψ|is an operator acting on A, more precisely it is
the projector over the subspace spanned by |ψ〉.
We can describe multipartite systems with tensor product. Let AB be a bipartite
system, i.e. it is the Hermitian space : A⊗B. A state of AB which can not be written
in the form |ψA〉 ⊗ |ψB〉is said to be entangled.
We can also represent states with density operators:
Definition 1.1 Density operator
Let {(pi,|ψi〉)}be an set of pure states associated with a probability, where
Pipi=1. Then we define :
ρ:=X
i
pi|ψi〉〈ψi|
the density operator of the system.
A density operator which is a projector, that is to say is is of the form |ψ〉〈ψ|is
called a pure state. Otherwise, it is a mixed state. We now need some mathematical
criterion to describe the evolution of a quantum system. Let Nbe an operator that
maps a density matrix ρAto ρBso that ρBis the evolution of ρAover time. Using
physical reasoning (see [Wil19, Section 4.4.1]) we can define quantum channels:
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Definition 1.2 Quantum channels
Let A and B be quantum systems, and N:L(A)→L(B)be a linear map. It is
called a quantum channel if it is a trace preserving completely positive map,
i.e. :
•∀ρA∈L(A),t r (N(ρA)) = t r (ρA)
• For all quantum system R and , IL(R)⊗Nmaps positive semi-definite
operators to positive semi-definite operators.
We can justify these with the following reasoning : We want a density operator to
be mapped to a density operator,as a quantum system is "stable" over time, so N
has to be trace preserving and has to map positive semi-definite operators to pos-
itive semi-definite operators. But suppose we prepare a larger quantum system,
which subsystems are R and A. Then Ncan be extended as IL(R)⊗Nand this has to
be trace-preserving - which it is automatically- and positive semi-definite, which
is the definition of Nbeing a completely positive map.
We will use density operators to describe quantum systems as it is convenient
for describing multipartite systems. The key notion here is the partial trace :
Definition 1.3 Partial trace
Let |a1〉〈a2|⊗|b1〉〈b2|be in A⊗B. We define :
t rA(|a1〉 〈a2|⊗|b1〉 〈b2|):=|b1〉〈b2|t r (|a1〉 〈a2|)
Where the trace on the right side is the usual trace. Then we extend the defi-
nition by linearity.
An important property of the partial trace (as for the regular trace) is cyclicity:
For any matrices ρ1,ρ2,ρ3acting on a space A⊗Bwhere the products are well de-
fined, we have : t rA(ρ1ρ2ρ3) = t rA(ρ3ρ1ρ2).
The partial trace is the operation which allows us to get the density operators of the
subsystems :
Property 1.4 Reduced states
Let A,B be two systems, with ρAB ,ρA,ρBbe the density operators of AB, A
and B respectively. The so-called reduced density operators ρAand ρBare ob-
tained by taking the partial trace :
ρA=t rBρAB
ρB=t rAρAB
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We now introduce Schatten norms, which are the norms we will use for matri-
ces :
Definition 1.5 Schatten norms
Let p≥1 and L be a matrix. We define :
∥L∥p:=(t r |L|p)1/p
Where : |L|:= (L L †)1/2
1.3 Entropy
Definition 1.6 Classical entropy
Let X be a discrete random variable. We introduce the Shannon entropy :
H(X):=−P
x∈X(Ω)
P(X=x)l o g (P(X=x))
Entropy was developed by Shannon as a way to quantify how much information
can be delivered by a source.
Other entropy-related quantities exist, but the classical case is here only to be com-
pared with the quantum case :
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