Abstract
The general idea that inspires all approaches to quantum computation is that information can be stored and transmitted by quantum physical systems. Thus, the quantum-theoretic formalism represents the natural mathematical environment for quantum computation theory. While classical information theory (as well classical mechanics) are naturally based on a twovalued semantics, the characteristic uncertainties of the quantum world have brought about some deep logical innovations, due to the divergence between the concepts of maximal information and logically complete information. Quantum pure states represent pieces of information that are at the same time maximal (since they cannot be consistently extended to a richer knowledge) and logically incomplete (since they cannot decide all the relevant properties of the objects under investigation). This chapter illustrates the basic quantum-theoretic concepts that play an important role in quantum information and in quantum computation.
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Notes
- 1.
\(\mathscr {M}({\mathscr {P}h}_\mathbf S)\) is the smallest subset of the power set of \(\mathscr {P}h_\mathbf S\) that contains all singletons, the total set, the empty set and is closed under the set-theoretic complement, countable intersections, countable unions. For the concepts of Boolean algebra, complete Boolean algebra and \(\sigma \)-complete Boolean algebra see Definitions 10.8 and 10.4 (in the Mathematical Survey of Chap. 10).
- 2.
\(\mathscr {B}({\mathbb R})\) is the set of all measurable subsets of \(\mathbb R\).
- 3.
For a detailed definition of Hilbert space see Definition 10.20 (in the Mathematical Survey of Chap. 10).
- 4.
- 5.
- 6.
- 7.
The concept of trace-functional is defined in Definition 10.34 (in the Mathematical Survey of Chap. 10).
- 8.
For the concepts of eigenvector and eigenvalue see Definition 10.24 (in the Mathematical Survey of Chap. 10). Note that all eigenvalues of a self-adjoint operator are real numbers.
- 9.
For the concept of trace-class operator see Definition 10.33 (in the Mathematical Survey of Chap. 10).
- 10.
For a more detailed definition of tensor product see Definition 10.37 (in the Mathematical Survey of Chap. 10).
- 11.
A more general concept of reduced state will be considered in Sect. 2.1
- 12.
For a long time non-destructive measurements have been considered a highly idealized concept. Interestingly enough, nowadays such “ideal” measurements can be experimentally realized by means of different technologies. For instance, one can manipulate some atoms by lasers and one can investigate their spectral features with high precision by means of optical clocks. In these experiments state-detection plays a crucial role: the fluorescence of an atom under laser-illumination reveals its internal quantum state.
- 13.
This definition is based on the so-called Kraus’ first representation theorem (See [20]). It is worth-while recalling that in the literature one can also find a different definition of quantum operation, where condition (1) (\(\sum _j E_j^\dagger E_j = \mathtt{I}\)) is replaced by the weaker condition: \(\mathtt{tr}(\rho \sum _j E_j^\dagger E_j) \le 1\), for every \(\rho \in \mathfrak D(\mathscr {H})\). In such a case, quantum operations are not trace-preserving. At the same time, quantum channels are defined as quantum operations that satisfy the stronger condition \(\sum _j E_j^\dagger E_j = \mathtt{I}\).
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- 18.
See, for instance, [10].
- 19.
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Dalla Chiara, M.L., Giuntini, R., Leporini, R., Sergioli, G. (2018). The Mathematical Environment of Quantum Information. In: Quantum Computation and Logic. Trends in Logic, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-030-04471-8_1
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