Abstract
The purpose of this chapter is to provide an overview of Quantum Information theory starting from Classical Information Theory, with the aim to: (1) define information mathematically and quantitatively, (2) represent the information in an efficient way (through data compression) for storage and transmission, and (3) ensure the protection of information (through encoding) in the presence of noise and other impairments. In Classical Information theory, the above goals are accomplished in accordance to the laws of Classical Physics. In Quantum Information theory, they are based on quantum mechanical principles and are intrinsically richer than in their Classical counterpart, because of intriguing resources, as entanglement; also, they are more interesting and challenging.
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Notes
- 1.
Considering that \(\lim _{p \rightarrow 0^+} p \log p=0\), the convention is assuming \(0 \log 0 = 0 \).
- 2.
An alternative prove is based on the concept of relative entropy (see [4]).
- 3.
The change of notation \(S(\rho _{AB})\rightarrow S(A,B)\), \(S(\rho _A)\rightarrow S(A)\), etc., are frequently used in the literature, where the density operator are replaced by the label of the system.
- 4.
Note the 2D matrix representing \(\rho _{AB}\) through the lexicographical order.
- 5.
Equivalently, an IID source may be viewed as a stationary random process \(\{A_\infty \}=(A_1,A_2,\ldots )\) with independent symbols and therefore completely specified by an ensemble \((\mathcal {A},p_A)\), where \(p_A(a), a\in \mathcal {A}\) is the common probability distribution, giving \(p_A(a)=\mathrm{{P}}[A_n=a]\) for any \(n\). From the random process one can extract words of any length, \((A_1,\ldots ,A_L)\), which, by the stationarity of the random process and the independence of its symbols, turn out to be \(L\)-tuples of IID random variables.
- 6.
We continue with the convention of denoting random quantities by upper case, as \(A\) and \(X^L\), and their realizations by the corresponding lower case letters, as \(a\) and \(x^L\).
- 7.
Intuitively, the tensor product of two channel maps \(\varPhi _1\) and \(\varPhi _2\) acts as the parallel of the two channels in a composite Hilbert space \({\mathcal {H}}_1\otimes {\mathcal {H}}_2\). Specifically, one has [2]
$$ \varPhi _1\otimes \varPhi _2=(\varPhi _1\otimes Id_2)\circ (Id_1\otimes \varPhi _2) $$where \(\circ \) is the concatenation (\(\varPhi _1\circ \varPhi _2\) is obtained by application of \(\varPhi _2\) at the output of \(\varPhi _1\)), \(Id_1\) and \(Id_2\) are the identity channels in \({\mathcal {H}}_1\) and \({\mathcal {H}}_2\), respectively. The interpretation becomes clear when \( \rho _{12}=\rho _{1}\otimes \rho _{2}\), where
$$ (\varPhi _1\otimes \varPhi _2)[\rho _{1}\otimes \rho _{2}]= \varPhi _1[ \rho _{1}]\otimes \varPhi _2[ \rho _{2}]. $$Roughly speaking, we can say that in the channel \(\varPhi ^{\otimes L}\) each “component” of the input state sees the channel \(\varPhi \).
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Cariolaro, G. (2015). Classical and Quantum Information Theory. In: Quantum Communications. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-15600-2_12
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DOI: https://doi.org/10.1007/978-3-319-15600-2_12
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