Abstract
In this chapter, we cover the fundamentals of linear algebra and provide a mathematical formulation of quantum mechanics for use in later chapters. It is necessary to understand these topics since they form the foundation of quantum information processing discussed later. In the first section, we cover the fundamentals of linear algebra and introduce some notation. The next section describes the formulation of quantum mechanics. Further, we examine a quantum two-level system, which is the simplest example of a quantum-mechanical system. Finally, we discuss the tensor product and matrix inequalities. More advanced discussions on linear algebra are available in Appendix.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A vector space closed under commutation [X, Y] is called a Lie algebra. A vector space closed under the symmetrized product \(X\circ Y\) is called a Jordan algebra.
- 2.
In quantum mechanics, one often treats the state after the measurement rather than before it. The state change due to a measurement is called state reduction , and it requires more advanced topics than those described here. Therefore, we postpone its discussion until Sect. 7.1.
- 3.
- 4.
- 5.
Adaptive measurements are often called one-way LOCC measurements in entanglement theory. See Sect. 8.1.
- 6.
The uniqueness of this definition can be shown as follows. Consider the linear map \(X \mapsto \mathop {\mathrm{{Tr}}}\nolimits (X \otimes I_{\mathcal{H}_B}) \rho \) on the set of Hermitian matrices on \(\mathcal{H}_A\). Since the inner product \((X,Y)\mapsto \mathop {\mathrm{{Tr}}}\nolimits XY\) is non-degenerate on the set of Hermitian matrices on \(\mathcal{H}_A\), there uniquely exists a Hermitian matrix Y satisfying (1.26).
References
G.M. D’Ariano, P. Lo, Presti, M.F. Sacchi, Bell measurements and observables. Phys. Lett. A 272, 32 (2000)
M. Horodecki, P. Horodecki, R. Horodecki, General teleportation channel, singlet fraction and quasi-distillation. Phys. Rev. A 60, 1888 (1999)
H. Nagaoka, Information spectrum theory in quantum hypothesis testing, in Proceedings 22th Symposium on Information Theory and Its Applications (SITA), (1999), pp. 245–247 (in Japanese)
R. Bhatia, Matrix Analysis (Springer, Berlin, 1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hayashi, M. (2017). Mathematical Formulation of Quantum Systems. In: Quantum Information Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49725-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-662-49725-8_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-49723-4
Online ISBN: 978-3-662-49725-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)