By Dénes Petz

Based on lectures given via the writer, this booklet specializes in supplying trustworthy introductory factors of key options of quantum details thought and quantum facts - instead of on effects. The mathematically rigorous presentation is supported by way of quite a few examples and routines and by means of an appendix summarizing the suitable facets of linear research. Assuming that the reader is aware the content material of normal undergraduate classes in quantum mechanics, likelihood thought, linear algebra and useful research, the publication addresses graduate scholars of arithmetic and physics in addition to theoretical and mathematical physicists. Conceived as a primer to bridge the distance among statistical physics and quantum details, a box to which the writer has contributed considerably himself, it emphasizes ideas and thorough discussions of the basic notions to organize the reader for deeper experiences, now not least throughout the collection of good selected exercises.

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**Quantum Information Theory and Quantum Statistics**

In accordance with lectures given by way of the writer, this booklet specializes in delivering trustworthy introductory motives of key recommendations of quantum details thought and quantum facts - instead of on effects. The mathematically rigorous presentation is supported by way of various examples and routines and by means of an appendix summarizing the correct facets of linear research.

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**Additional resources for Quantum Information Theory and Quantum Statistics**

**Example text**

N The type of a sequence is another name for the empirical distribution. Let Pn denote the set of all types Px when x ∈ X n . The elements of Pn are called n-types. The number of possible n-types is #(Pn ) = n + #(X ) − 1 ≤ (n + 1)#(X ) . #(X ) − 1 The upper estimate is useful in estimations. For P ∈ Pn the type class of P is defined as the set of all sequences of type P: Type(P) := {x ∈ X n : Px = P}. The cardinality of a type class Type(P) is a multinomial coefficient n! ∏x (nPx (x))! (Px ∈ P(T )) but the following exponential bounds are good enough: 1 2nH(P) ≤ #(Type(P)) ≤ 2nH(P) .

Then the inequality holds generally in the quantum case. For example. 27) 2D(μ1 ||μ2 ) ≥ ( μ1 − μ2 1 )2 is the Pinsker–Csisz´ar inequality, which extends to the quantum case. Consider the commutative subalgebra generated by ρ1 and ρ2 . The reduction of ρ1 and ρ2 can be viewed as probability distributions μ1 and μ2 and ρ1 − ρ2 1 = μ1 − μ2 1 . From the monotonicity of the quasi-entropy S f , we have S f (ρ1 ||ρ2 ) ≥ S f (μ1 ||μ2 ) ≥ G( μ1 − μ2 1 ) = G( ρ1 − ρ2 1 ). In particular, 2S(ρ1||ρ2 ) ≥ ( ρ1 − ρ2 1 )2 .

32). 13. Let β > 0 and H be a self-adjoint matrix. Use the relative entropy to show that the minimum of the functional F(ρ ) = Tr ρ H − 1 S(ρ ) β defined on density matrices is reached at 1 exp(−β H) , Z where the constant Z is for normalization. ) Chapter 4 Entanglement It is a widely accepted statement that entanglement is one of the most striking features of quantum mechanics. If two quantum systems interacted sometime in the past, then it is not possible to assign a single state vector to the subsystems in the future.