Chapter 2
Elements of Quantum Information
Theory
Fundamental measures of information arise as the answers to
fundamental questions about the physical resources required to
solve some information processing problem. Nielsen & Chuang
Abstract In this Chapter we review some fundamental concepts of quantum
mechanics and linear algebra using the Dirac notation and the density operator formalism, including the notions of qubits, entanglement and general quantum operations (Sects. 2.1–2.5). We then introduce the entropies characterizing informationprocessing tasks which commonly occur in quantum cryptography, in particular the
Shannon and von Neumann entropy (Sect. 2.6), the min- and max-entropy (Sect. 2.7)
and their smooth versions (Sect. 2.8). We emphasize the operational meaning of the
smooth min- and max-entropy for the cryptographic tasks of error correction and privacy amplification in Sects. 2.9 and 2.10, respectively. We additionally elaborate on
the notion of distance between quantum states and its relation to their distinguishability in Sect. 2.11 in the Appendix of this Chapter. The content of this Chapter is
mostly inspired by the following literature: [1–4].
2.1 Dirac Notation and Linear Algebra
The state of a quantum mechanical system, with d degrees of freedom, is represented
by a normalized vector |ψ in a d-dimensional Hilbert space H over the complex
numbers C, called the state space of the system. A Hilbert space is an inner product
space, which is also complete with respect to the norm induced by the inner product
if the space is infinite-dimensional.
The vector symbol |ψ is called a ket. To every vector |ψ in H corresponds a
unique dual vector ψ| in the dual Hilbert space H
∗ , i.e. the space of linear maps
from H to C. The symbol φ| of a dual vector is called a bra. Note that the dual of a
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
F. Grasselli, Quantum Cryptography, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-64360-7_2
7
Elements of Quantum Information
Theory
Fundamental measures of information arise as the answers to
fundamental questions about the physical resources required to
solve some information processing problem. Nielsen & Chuang
Abstract In this Chapter we review some fundamental concepts of quantum
mechanics and linear algebra using the Dirac notation and the density operator formalism, including the notions of qubits, entanglement and general quantum operations (Sects. 2.1–2.5). We then introduce the entropies characterizing informationprocessing tasks which commonly occur in quantum cryptography, in particular the
Shannon and von Neumann entropy (Sect. 2.6), the min- and max-entropy (Sect. 2.7)
and their smooth versions (Sect. 2.8). We emphasize the operational meaning of the
smooth min- and max-entropy for the cryptographic tasks of error correction and privacy amplification in Sects. 2.9 and 2.10, respectively. We additionally elaborate on
the notion of distance between quantum states and its relation to their distinguishability in Sect. 2.11 in the Appendix of this Chapter. The content of this Chapter is
mostly inspired by the following literature: [1–4].
2.1 Dirac Notation and Linear Algebra
The state of a quantum mechanical system, with d degrees of freedom, is represented
by a normalized vector |ψ in a d-dimensional Hilbert space H over the complex
numbers C, called the state space of the system. A Hilbert space is an inner product
space, which is also complete with respect to the norm induced by the inner product
if the space is infinite-dimensional.
The vector symbol |ψ is called a ket. To every vector |ψ in H corresponds a
unique dual vector ψ| in the dual Hilbert space H
∗ , i.e. the space of linear maps
from H to C. The symbol φ| of a dual vector is called a bra. Note that the dual of a
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
F. Grasselli, Quantum Cryptography, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-64360-7_2
7