Quantum thermodynamics and canonical typicality

Abstract

We present here a set of lecture notes on quantum thermodynamics and canonical typicality. Entanglement can be constructively used in the foundations of statistical mechanics. An alternative version of the postulate of equal a priori probability is derived making use of some techniques of convex geometry.

Typicality of Heisenberg scaling precision in multi-mode quantum metrology

“…We will show now that this is not the case for the pre-factor f (U, G ϕ ), thanks to the fact that it is a sufficiently well behaved function with respect to the random unitary U . In fact, by using results on concentration of measure in high-dimensional probability spaces, we prove in Appendix E that for an network with a large number M of ports the pre-factor f (U, G ϕ ) becomes typical, meaning that it becomes almost constant with respect to random choices of U ∈ U(M ) (according to the unitarily invariant measure), hence concentrating around its average value (34), bounded below by (36).…”

Peaceful coexistence of thermal equilibrium and the emergence of time

“…where β = dS(E)/dE is the inverse temperature with S(E) the environment entropy, Z = T r e −β ĤS and ΩU = d −1 U PU is the equiprobable mixed state in H U with d U the dimension of the space H U and PU the projection on H U . Notice that equal probabilities (and random phases) are assigned in this case to all the states within ΩU which is thus maximally mixed in H U [27]. Equation (3) implies that the thermal state of the small subsystem S can be derived from a (randomly chosen) pure state |Ψ ∈ H U or from the maximally mixed state ΩU .…”

“…Appendix F: Proof of equation (28) We prove here that equation (27) reduces to (28) in the case of t ≪ 1/|∆ i0 − ∆ k1 |. We start considering the second part of equation (27), that is: (F2)…”

Quantum thermodynamics and quantum coherence engines

“…Non-Markovian effects on the definition of temperature in open quantum systems have also been considered [67]. Thermalization and equilibration have also been discussed in closed quantum systems by introducing a generalized notion of Gibbs (maximum entropy) ensemble, quantum typicality [68], nonintegrability [69], and so-called eigenvalue thermalization hypothesis [70].…”