1 Introduction

Classical spin chain: typical subsets, Shannon McMillan

	spin configuration $\in \{0,1{\}}^{\mathit{Z}}$
	state (measure) $\psi$ on $\{0,1\},\psi (0)=p$
	state (measure) ${\Psi }^{(n)}$ on $\{0,1{\}}^{[1,n]}$ product state: $\underset{\text{n times}}{\underbrace{\psi \otimes \psi \cdots \otimes \psi }}$
	$\{{𝒯}_n\subset \{0,1{\}}^{[1,n]}{\}}_{n\in \mathit{N}}\text{typical sequence of subsets}:\iff {\Psi }^{(n)}({𝒯}_n)\to 1$
	Theorem by Shannon McMillan characterizes sequences of typical sets in terms of entropy $s=-plogp-(1-p)log(1-p)$: $$\begin{array}{rll}{\tau }^{(n)}\in {𝒯}_n\iff {\Psi }^{(n)}({\tau }^{(n)})& \sim & 2^{-ns}\\ \left|{𝒯}_n\right|& \sim & 2^{ns}\ll 2^n,\text{if}p\ne \frac{1}{2}\end{array}$$
	the size of the typical sets are exponentially small in $n$, but they can not be smaler than what is given by the theorem by Shannon McMillan

1. Question: (interesting for hypothesis testing)
Given two states $\Psi$ and $\Phi$.
Is there a sequence of $\Psi$-typical sets $\{{𝒯}_n{\}}_{n\in \mathit{N}}$ such that ${\Phi }^{(n)}({𝒯}_n)\to 0$?

Answer yes. But the convergence to zero can not be too fast. The optimal rate of convergence is given by an exponential law, where the exponent is the relative entropy density of the two states:

$${\Phi }^{(n)}({𝒯}_n)\sim 2^{-ns(\Psi ,\Phi )}$$

This is the content of Stein's lemma.

The problem in hypothesis testing:
Which one of the 2 states $\Psi$ and $\Phi$ is the better description of the observed data?

To answer this question, the set of n spin configurations $\{0,1{\}}^{[1,n]}$ is partitioned into the test set $\left\{{𝒯}_n\right\}$ and its complement, such that $\Psi ({𝒯}_n)\sim 1,\Phi ({𝒯}_n)\sim 0$. The closer $\{\Psi ({𝒯}_n),\Phi ({𝒯}_n)\}$ to $\{1,0\}$, the better the test.

3. Question: (hypothesis testing)
Note: This question is a combination of question 1 and 2.

Given

1.	set of states $\Theta$ and
2.	single state $\Phi$.

Is there a sequence of sets $\{{𝒯}_n{\}}_{n\in \mathit{N}}$ such that

1.	universally typical for all states in $\Theta$ and
2.	${\Phi }^{(n)}({𝒯}_n)\to 0$?

Answer yes:
Provided: Convergence to $0$ not too fast
Optimal convergence: exponential law
Best exponent: $s(\Theta ,\Phi )$ (rel.entropy).

Remark on Classical - Quantum

	Classical 1-bit algebra: Diagonal 2x2 Matrices, functions $={\mathit{C}}^{\{0,1\}}$
	Classical state $\psi (d)=d(0)p+d(1)(1-p)$
	Quantum 1-q-bit algebra: 2x2 Matrices
	Quantum state $\psi (A)=\text{trace}(\psi A)$

Classical algebra of phase space functions (diagonal matrices) is abelian subalgebra of quantum algebra (full matrix algebra).

Remark on proof:

1.	One state: Basic starting point (Shannon, McMillon)
2.	Two states: Stein's lemma
3.	Many + 1 states: Sanov's theorem

2 to 3: Slice the set $\Theta$ along level sets.

2 Setup and Basic Concepts

Def: Discrete Information Source $({𝒜}^{\infty },\Psi ,T)$

	$𝒜={\mathit{M}}^{[2,2]}$ Quantum 1-Bit Algebra
	$𝒟=D^{[2,2]}={\mathit{C}}^{\{0,1\}}$ Classical 1-Bit Algebra (functions on $\{0,1\}$)
	${𝒜}^{[m,n]}:={\otimes }_{i=m}^n{𝒜}_i$ for $m,n\in \mathit{Z},{𝒜}_i\simeq 𝒜,\forall i\in \mathit{Z}$
	${𝒜}^{(n)}:={𝒜}^{[1,n]}$
	$$\begin{array}{rlll}\forall r<s\in \mathit{N}:& {𝒜}^{[-r,r]}& \hookrightarrow & {𝒜}^{[-s,s]}\\ & A& \mapsto & 1\otimes A\otimes 1\end{array}$$
	${𝒜}^{\infty }:={{\bigcup }_{n\in \mathit{N}}{𝒜}^{[-n,n]}}^{\left|\right|·\left|\right|}$ local quantum spin chain algebra
	${𝒟}^{\infty }:={{\bigcup }_{n\in \mathit{N}}{𝒟}^{[-n,n]}}^{\left|\right|·\left|\right|}$ local classical spin chain algebra
	$\forall m<n\in \mathit{N}$ ${𝒜}^{[m,n]}\hookrightarrow {𝒜}^{\infty }$
	$\Psi$ state on ${𝒜}^{\infty }$: $\iff \Psi :{𝒜}^{\infty }\to \mathit{C}\text{linear, positiv and}\Psi (1)=1$
	$n\in \mathit{N}:{\Psi }^{(n)}(A^{(n)}):=\Psi (A^{(n)}),\forall A^{(n)}\in {𝒜}^{(n)}$
	$T\text{right shift},*$-automorphism on ${𝒜}^{\infty }$: $T(a)=1_{{𝒜}_m}\otimes a\in {𝒜}^{[m,n+1]},\forall a\in {𝒜}^{[m,n]},$ extends to ${𝒜}^{\infty }$

Def: Stationary, Ergodic, IID States

	$\Psi \text{is stationary}:\iff \Psi (Ta)=\Psi (a),\forall a\in {𝒜}^{\infty }$
	$\Psi \text{ is ergodic}:\iff \Psi \text{extremal point in the set of stationary states}\text{(Choquet simplex)}$
	$\Psi$ is IID :$\iff \exists \text{a state}\psi \text{on}𝒜\text{such that}\Psi \text{is the product state of}\psi$

Def: entropy, relative entropy

The entropy of ${\Psi }^{(n)}$ is defined by $$S({\Psi }^{(n)}):=-\text{trace}({\Psi }^{(n)}log({\Psi }^{(n)})$$ For stationary states entropy is subadditiv. Hence its rate converges: $$s(\Psi ):=\underset{n\to \infty }{\text{lim}}\frac{1}{n}S({\Psi }^{(n)})$$

Relative entropy of ${\Psi }^{(n)},{\Phi }^{(n)}$ is defined by $$S({\Psi }^{(n)},{\Phi }^{(n)}):=\left\{\begin{array}{cc}\text{trace}({\Psi }^{(n)}(log({\Psi }^{(n)})-log({\Phi }^{(n)})),\hfill & \text{if trace exists}\hfill \\ \infty ,\hfill & \text{otherwise}\hfill \end{array}\right.$$ and its rate by $$s(\Psi ,\Phi ):=\underset{n\to \infty }{\text{lim}}\frac{1}{n}S({\Psi }^{(n)},{\Phi }^{(n)})$$ provided the limit exists.

Def: The Hiai-Petz condition

	$\Psi ,\Phi$ states on ${𝒜}^{\infty }$
	Sequence of optimally seperating exponent: $${\beta }_{\epsilon ,n}(\Psi ,\Phi ):=min\{log\Phi (q)|q\in {𝒜}^{(n)}\text{ proj., }\Psi (q)\ge 1-\epsilon \},\epsilon \in (0,1)$$
	${\overline{\beta }}_{\epsilon }(\Psi ,\Phi ):=\underset{n\to \infty }{limsup}\frac{1}{n}{\beta }_{\epsilon ,n}(\Psi ,\Phi )\in -[0,\infty ]$
	$(\Psi ,\Phi )$ is HP: $\iff {\overline{\beta }}_{\epsilon }(\Psi ,\Phi )\le -s(\Psi ,\Phi ),\forall \epsilon \in (0,1)$ Hiai and Petz: $\Psi$ completely ergodic, $\Phi \ast$-mixing
	$\Phi \text{HP-state}:\iff \forall \Psi \in {𝒮}_{\text{erg}}({𝒜}^{\infty })\text{the pair}(\Psi ,\Phi )\text{is HP}$

3 Sanov's theorem, the ergodic case

Theorem (Sanov, ergodic, for classical case see (8):

Let $\Phi$ be a state on ${𝒜}^{\infty }$ and $\Theta \subseteq {𝒮}_{\text{erg}}({𝒜}^{\infty })$.

Statements $1$ and $2$ are equivalent:

1.

For each $\Psi \in \Theta$ the pair $(\Psi ,\Phi )$ is HP.

2.

$\forall \Psi \in \Theta s(\Psi ,\Phi )\le +\infty$ exists. for each $\Omega \subseteq \Theta$ and $\eta >0$ there exists a test sequence of projectors $\{p_n{\}}_{n\in \mathit{N}},p_n\in {𝒜}^{(n)}$ such that $$\begin{array}{rll}\underset{n\to \infty }{lim}{\Psi }^{(n)}(p_n)& =& 1,\forall \Psi \in \Omega (\ast )\\ \underset{n\to \infty }{\text{limsup}}\frac{1}{n}log{\Phi }^{(n)}(p_n)& \le & -s(\Omega ,\Phi )+\eta (\ast \ast )\end{array}$$ for each sequence of projections $\left\{{\tilde{p}}_n\right\}$ with $(\ast )$: $$\underset{n\to \infty }{\text{liminf}}\frac{1}{n}log{\Phi }^{(n)}({\tilde{p}}_n)\ge -s(\Omega ,\Phi )\text{.}$$

Remarks:

	if $s(\Omega ,\Phi )=\infty$ inequality $(\ast \ast )$ is replaced by $$\underset{n\to \infty }{\text{limsup}}\frac{1}{n}log{\Phi }^{(n)}(p_n)\le -\frac{1}{\eta }.$$
	$-s(\Omega ,\Phi )$ is $liminf$ of all sequences of separation exponents.

4 Sanov's theorem, the stationary case

Can be reduced to the ergodic case.

Recall:

1.	(${𝒮}_{\text{stat}}({𝒜}^{\infty })$ is Choquet simplex,
2.	${𝒮}_{\text{erg}}({𝒜}^{\infty })$ is the set of extremal points.
3.	Any stationary $\Psi \in {𝒮}_{\text{stat}}({𝒜}^{\infty })$ has ergodic decomposition $$\Psi =\underset{{𝒮}_{\text{erg}}({𝒜}^{\infty })}{\int }\Xi {\gamma }_{\Psi }(d\Xi )$$ ${\gamma }_{\Psi }\text{: prob. meas. on}[{𝒮}_{\text{erg}}({𝒜}^{\infty }),ℬ({{\rm Y}}_{{𝒜}^{\infty }})]$ ${{\rm Y}}_{{𝒜}^{\infty }}$ weak-$\ast$-topology $ℬ({{\rm Y}}_{{𝒜}^{\infty }})$ Borel $\sigma$-field.

Notation

	Let $\Phi \in 𝒮({𝒜}^{\infty }),\Theta \subseteq {𝒮}_{\text{stat}}({𝒜}^{\infty })$ $$\begin{array}{ccc}\hfill \underset{¯}{s}(\Psi ,\Phi ):& \hfill =\hfill & {\text{essinf}}_{{\gamma }_{\Psi }(d\Xi )}s(\Xi ,\Phi ),\hfill \end{array}$$
	Let $\Omega \subseteq \Theta$ $$\begin{array}{ccc}\hfill \underset{¯}{s}(\Omega ,\Phi ):& \hfill =\hfill & {inf}_{\Psi \in \Omega }\underset{¯}{s}(\Psi ,\Phi ).\hfill \end{array}$$

Theorem (Sanov, stationary):

Let

	$\Phi$ state on ${𝒜}^{\infty }$
	$\Theta \subseteq {𝒮}_{\text{stat}}({𝒜}^{\infty })$
	$\forall \Psi \in \Theta$ and ${\gamma }_{\Psi }$-almost all $\Xi$ the pair $(\Xi ,\Phi )$ is HP.

Then

	$\underset{¯}{s}(\Psi ,\Phi )\le +\infty$ exists for each $\Psi \in \Theta$
	$\forall \Omega \subseteq \Theta ,\eta >0$ there exists a test sequence of projectors $\{p_n{\}}_{n\in N},p_n\in {𝒜}^{(n)}$ with $$\begin{array}{rll}\underset{n\to \infty }{lim}{\Psi }^{(n)}(p_n)& =& 1,\forall \Psi \in \Omega (\ast )\\ \underset{n\to \infty }{\text{limsup}}\frac{1}{n}log{\Phi }^{(n)}(p_n)& \le & -s(\Omega ,\Phi )+\eta (\ast \ast )\end{array}$$
	for each sequence of projections $\left\{{\tilde{p}}_n\right\}$ with $(\ast )$: $$\underset{n\to \infty }{\text{liminf}}\frac{1}{n}log{\Phi }^{(n)}({\tilde{p}}_n)\ge -s(\Omega ,\Phi )\text{.}$$

Remarks:

	if $s(\Omega ,\Phi )=\infty$ inequality $(\ast \ast )$ is replaced by $$\underset{n\to \infty }{\text{limsup}}\frac{1}{n}log{\Phi }^{(n)}(p_n)\le -\frac{1}{\eta }.$$
	$-s(\Omega ,\Phi )$ is $liminf$ of all sequences of separation exponents.

5 Sanov's theorem: A cradle of fundamental results in inf-theory

5.1 Shannon-McMillan, the ergodic case

Theorem (Shannon-McMillan, the ergodic case)

Consider an ergodic state $\Psi$.

Then

1.	exists a test sequence of projectores $\left\{p_n\right\}\in {𝒜}^{(n)}$, such that $\forall \epsilon >0\exists n_0$ with ${\Psi }^{(n)}(p_n)\ge 1-\epsilon ,$ for $n\ge n_0,$(typicality) $\text{trace}(p_n)\le 2^{n(s(\Psi )+\epsilon )},$ (achievability)
2.	For any sequence ${\tilde{p}}_n$ with $\underset{\to \infty }{\text{lim}}{\Psi }^{(n)}({\tilde{p}}_n)=1$ the lower bound holds: $$\begin{array}{ccc}\hfill \text{trace}{\tilde{p}}_n& \hfill \ge \hfill & 2^{n(s(\Psi )-\epsilon )}n\text{sufficiently large},\text{(optimality)}\hfill \end{array}$$

Proof of the statement:
Sanov, ergodic case $\Rightarrow$ Shannon McMillan, ergodic case:

1. Choice $\Omega :=\{\Psi \},\Phi$ equipartition (tracial state):

	choice of null hypothesis $\Omega :=\{\Psi \}$
	choice of alternative hypothesis $\Phi$ (equidistribution): $$\begin{array}{rll}\phi :& =& \frac{1}{2}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ {\Phi }^{(n)}:& =& \phi \otimes \phi \cdots \phi \otimes \phi \end{array}$$

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2. $(\Psi ,\Phi )$ is HP:

$\Phi$ is $\ast$-mixing by construction (product state)
$\Psi$ is ergodic by assumtion.
Due to a result by Hiai and Petz (1991) the statement holds. For details see section 6.3

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3. $s(\Psi ,\Phi )=-s(\Psi )+1<\infty$:

$$\begin{array}{rll}S({\Psi }^{(n),{\Phi }^{(n)}}& =& \text{tr}{\psi }^{(n)}(log(\Psi (n)-log{\Phi }^{(n)})\\ log({\Phi }^{(n)}& =& (log((\frac{1}{2}{)}^n)1=-n1\\ S({\Psi }^{(n),{\Phi }^{(n)}}& =& -S({\Psi }^{(n)})+n\\ s(\Psi ,\Phi )& =& lim\frac{1}{n}S({\Psi }^{(n)},{\Phi }^{(n)})\\ & =& -s(\Psi )+1\end{array}$$

Entropy $s(\Psi )$ exists and is finit (follows from subadditivity).

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4. $\exists \left\{p_n\right\},\text{s.t.}\Psi (p_n)\ge 1-\epsilon ,\text{tr}(p_n)\le 2^{n(s(\Psi )+\epsilon )},$ for $n$ large:

Applying Sanov's theorem (3 one gets:

$s(\Psi ,\Phi )=-s(\Psi )+1<+\infty$

$\exists \text{a test sequence of projectors}\{p_n{\}}_{n\in \mathit{N}},p_n\in {𝒜}^{(n)},$ such that $$\begin{array}{rll}\Psi (p_n)& \ge & 1-\epsilon ,n\text{large}\\ \text{and}& & \\ \frac{1}{n}log({\Phi }^{(n)}(p_n))& \le & -s(\Psi ,\Phi )+\epsilon \\ \frac{1}{n}log(\text{tr}(\frac{1}{2}{)}^n(p_n))& \le & -s(\Psi ,\Phi )+\epsilon \\ \frac{1}{n}(n+log(\text{tr}(p_n))& \le & -s(\Psi ,\Phi )+\epsilon \\ -1+\frac{1}{n}log(\text{tr}(p_n))& \le & s(\Psi )-1+\epsilon \\ \text{tr}(p_n)& \le & 2^{n(s(\Psi )+\epsilon )}\end{array}$$

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5. $\underset{n\to \infty }{\text{lim}}{\Psi }^{(n)}({\tilde{p}}_n)=1\Rightarrow \text{trace}{\tilde{p}}_n\ge 2^{n(s(\Psi )-\epsilon )},$ for $n$ large:

Due to a Sanov's theorem (3)

$$\frac{1}{n}log{\Phi }^{(n)}({\tilde{p}}_n)\ge -s(\Psi ,\Phi ),\text{(n large)}.$$

A computation analogous to the one in the last step proofs the claim.

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5.2 Shannon-McMillan for stationary states

Theorem (Shannon-McMillan, for stationary states)

Consider a stationary state

$$\Psi =\underset{{𝒮}_{\text{erg}}({𝒜}^{\infty })}{\int }\Xi {\gamma }_{\Psi }(d\Xi )$$

Then

1.	there exists a sequence of projectores $\left\{p_n\right\}$ in ${𝒜}^{(n)}$, such that $\underset{n\to \infty }{\text{lim}}{\Psi }^{(n)}(p_n)=1$ (typicality) $\underset{n\to \infty }{\text{lim}}\frac{1}{n}log(\text{trace}{p}_n)=\overline{s}(\Psi ):={\text{ess sup}}_{{\gamma }_{\Psi }(d\Xi )}s(\Xi )$
2.	For any sequence ${\tilde{p}}_n$ with $\underset{\to \infty }{\text{lim}}{\Psi }^{(n)}({\tilde{p}}_n)=1$ we have $$\begin{array}{ccc}\hfill \underset{n\to \infty }{liminf}\frac{1}{n}log(\text{trace}{\tilde{p}}_n)& \hfill \ge \hfill & \overline{s}(\Psi )\text{ (optimality).}\hfill \end{array}$$

Remarks:
1. in the stationary case equipartition does not hold!
2. The relevant entropy in Sanov's theorem for stationary states is not the mean entropy but the essential infimum of the ergodic constituents of all states in $\Theta$.

Proof of the statemtent:
Sanov, stationary case $\Rightarrow$ Shannon McMillan, the stationary case:

The proof follows very closely the one just given for the ergodic case. Again one chooses $\Omega =\{\Psi \}$ and $\Phi$ as the equidistribution. $s(\Psi )$ is replaced by $\overline{s}(\Psi )$ respectively $\underset{¯}{s}(\Psi )$.

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5.3 Kaltchenko-Yangs universality theorem for stationary states

Theorem (Kaltchenko-Yang, for stationary states)

Let

$${\Omega }_s:=\{\Psi \in {𝒮}_{\text{stat}}({𝒜}^{\infty }):\overline{s}(\Psi )<s\}$$

Then

1.	there exists a sequence of projectors $\{p_n{\}}_{n\in \mathit{N}},p_n\in {𝒜}^{(n)}$, such that $\underset{n\to \infty }{\text{lim}}{\Psi }^{(n)}(p_n)=1,\forall \Psi \in {\Omega }_s$ (typicality) $\underset{n\to \infty }{\text{lim}}\frac{1}{n}log(\text{trace}{p}_n)=s$ (max. ergodic entropy rate)
2.	For each sequence of projectors ${\tilde{p}}_n$ such that for all $\Psi \in {\Omega }_s$ $$\underset{n\to \infty }{\text{lim}}{\Psi }^{(n)}({\tilde{p}}_n)=1,$$ $s$ is the optimal lower bound: $$\underset{n\to \infty }{\text{liminf}}\frac{1}{n}log(\text{trace}{\tilde{p}}_n)\ge s$$

Proof of the statemtent:
Sanov, stationary case $\Rightarrow$ Kaltchenko-Yang theorem:

Set

	${\Omega }_s:=\{\Psi \in {𝒮}_{\text{stat}}({𝒜}^{\infty }):\overline{s}(\Psi )<s\}$
	$\Phi =$ equidistribution (as in the case of the proof of the Shannon McMillan theorem (5.1)).

Now, the statement is an immediate consequence of Sanov's theorem.

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5.4 Stein's lemma for stationary states

Theorem (Stein's lemma for stationary null hypothesis)

Let $\Phi \in 𝒮({𝒜}^{\infty })$ and $\Psi \in {𝒮}_{\text{stat}}({𝒜}^{\infty })$ such that for ${\gamma }_{\Psi }$ -almost all $\Xi$ the pair $(\Xi ,\Phi )$ is HP.

Then

1.	there exists a sequence $\left\{p_n\right\}$ of projections with $\underset{n\to \infty }{lim}{\Psi }^{(n)}(p_n)=1$ (typicality) $\underset{n\to \infty }{lim}\frac{1}{n}log{\Phi }^{(n)}(p_n)=-\underset{¯}{s}(\Psi ,\Phi )$ (achievability of the separation exponent $-\underset{¯}{s}(\Psi ,\Phi )$).
2.	For any sequence ${\tilde{p}}_n$ with $\underset{n\to \infty }{lim}{\Psi }^{(n)}({\tilde{p}}_n)=1$ we have $$\begin{array}{ccc}\hfill \underset{n\to \infty }{\text{liminf}}\frac{1}{n}log{\Phi }^{(n)}({\tilde{p}}_n)& \hfill \ge \hfill & -\underset{¯}{s}(\Psi ,\Phi )\text{ (optimality) }.\hfill \end{array}$$

Proof of the statement:
Sanov, stationary case $\Rightarrow$ Stein's lemma, stationary case:

Choose $\Theta =\{\Psi \}$ and apply Sanov's theorem.

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6 Proof of Sanov's theorem, the ergodic case

6.2 from Stein's lemma to Sanov

Construction of $\Theta$-universal sequencies of typical projections:
The main objective is the proof of formula (3). It asserts the existence of $\Psi$-typical test sequence of projectors for any $\Psi \in \Omega$ which is maximally non typical with respect to $\Phi$, in other words maximally separating.

Here is a sketch of the strategy of proof:

	partition $\Omega$ by level sets of the function $s(\Psi ,\Phi ),\Psi \in \Omega$ (see 1).
	use the Kaltchenko-Yang theorem asserting a uniform test sequence of projectores for all states with entropy below a given value $s$.
	apply Stein's lemma to each strip. Finally one makes partioning by level sets finer and finer.

(For details see (BDKSSS 20078).

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7 proof of Sanov's theorem, the stationary case

The proof of Sanov's theorem for stationary states proceeds essentially in 2 steps. First each $\Psi \in \Omega$ is decomposed into ergodic components $\Xi$:

Second, Sanov's theorem for ergodic states is appled to each ergodic component.

8 references

K.M.R.Audenaert, M.Nussbaum, A. Szkola, F.Verstraete: Asymptotic Error Rates in Quantum Hypothesis Testing, Commun.Math.Phys. 279, 251-283 (2008)

I. Bjelakovic, T. Krüger, Ra. Siegmund-Schultze, A. Szkola, The Shannon-McMillan theorem for ergodic quantum lattice systems, Invent. Math. 155 (1), 203-222 (2004)

I. Bjelakovic, J.-D. Deuschel, T. Krüger, R. Seiler, Ra. Siegmund-Schultze, A. Szkola: Typical support and Sanov large deviations fo correlated states. Commun. Math. Phys. 279, 559-584 (2008)

F. Hiai, D. Petz, The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability, Commun. Math. Phys. 143, 99-114 (1991)

A. Kaltchenko, E. H. Yang, Universal Compression of Ergodic Quantum Sources, Quant. Inf. and Comput. 3, 359-375 (2003)

M.Nussbaum, A. Szkola; Exponential error rates in multiple state discrimination on a quantum spin chain, J.Math.Phys. 51, 072203 (2010)

D. Ruelle: Statistical Mechanics, Benjamin Inc., 1969

I.N. Sanov, On the probability of large deviations of random variables, Mat. Sbornik 42, 11-44, 1957