Abstract
We can see that multiplying the state vector by a complex phase factor yields the same density matrix, i.e., the same q-state. Hence the phase of the state vector can be deliberately altered, still the same pure q-state is obtained. In the conservative q-theory, contrary to the classical theory, not even the pure state is interpreted on a single system but on the statistical ensemble of identical systems
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Notes
- 1.
Our lectures use the Schrödinger-picture: the q-states \(\hat{\rho}\) evolve with t, the q-physical quantities \(\hat{A}\) do not.
- 2.
Equivalent terminologies, like spectral or diagonal decomposition, or just diagonalization, are in widespread use.
- 3.
We use the notion of irreversibility as an equivalent to non-invertibility. We discuss the entropic-informatic notion of q-irreversibility in Sect. 10.8.
- 4.
Cf., e.g., the monograph by Joos et al. [2].
- 5.
Note that, in the q-literature, the post-measurement states are usually specified in a more general form \((1/p_{n})\hat{U}_{n}\hat{\Pi}_{n}^{1/2}\hat{\rho} \hat{\Pi}_{n}^{1/2}\hat{U}_{n}^{\dag},\) to include the arbitrary selective post-measurement unitary transformations \(\hat{U}_n.\)
- 6.
This definition of q-separability was introduced by Werner [5].
References
von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)
Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, K., Stamatescu, I.O.: Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin (2003)
Aharonov, Y., Albert, D.Z., Vaidman, L.: Phys. Rev. Lett. 60, 1351 (2008)
Diósi, L.: Weak measurements in quantum mechanics. In: Françoise, J.P., Naber, G.L., Tso, S.T. (eds) Encyclopedia of Mathematical Physics, vol. 4. Elsevier, Oxford pp. 276–282 (2006)
Werner, R.F.: Phys. Rev. A 40, 4277 (1989)
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Diósi, L. (2011). Foundations of Q-Physics. In: A Short Course in Quantum Information Theory. Lecture Notes in Physics, vol 827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16117-9_4
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DOI: https://doi.org/10.1007/978-3-642-16117-9_4
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