Rises in closed quantum systems. ⢠to the correct thermodynamic value. ⢠Future: Gibbs paradox, information engines, work extraction, fluctuation theoremsâ¦
Quantum coarse-grained entropy and thermodynamics Dominik Šafránek in collaboration with
Joshua M. Deutsch and Anthony Aguirre
University of California Santa Cruz
Overview • Why thermodynamics of closed quantum systems? • Classical systems and Boltzmann entropy • Quantum generalization = Observational entropy • Does it work? …connection to real thermodynamics
2
Why? • Imagine a bunch of particles in a box.
3
Why? • Imagine a bunch of particles in a box. • In a bigger box
4
Why? • Imagine a bunch of particles in a box. • In a bigger box • Let’s put a hole in there
5
Why? • Imagine a bunch of particles in a box. • In a bigger box • Let’s put a hole in there • After a while…
6
Why? • Imagine a bunch of particles in a box. • In a bigger box • Let’s put a hole in there • After a while… • Described by Boltzmann entropy
7
Why? • Imagine a bunch of particles in a box. • In a bigger box • Let’s put a hole in there • After a while… • Described by Boltzmann entropy • No analog quantum description!
8
Why? • Imagine a bunch of particles in a box. • In a bigger box • Let’s put a hole in there • After a while… • Described by Boltzmann entropy • No analog quantum description! • Some approaches • Entanglement entropy • Diagonal entropy …do not quite work
9
Boltzmann entropy • Microstate (x,p) • In a macrostate of volume Vi • Attach Boltzmann entropy
10
Quantum physics recap • Hilbert space “system” • States of the system
• Vectors in Hilbert space “wave-function” • More generally, linear operators “density matrix”
x
• Direct sum of Hilbert spaces “partitioning” • Example: 3-dim space R3 =H1 + H2 • x-y plane, H1=span{(1,0,0),(0,1,0)} • z axis, H2=span{(0,0,1)}
y z
• Projectors “measurement basis”
• Example: projectors onto x-y plane (rank-2) or z axis (rank-1) 11
Observational entropy • Take projectors onto the subspaces from direct sum • Volume of the subspace
• Define Observational entropy with coarse-graining Probability of being in a macrostate
• In other words
Mean measurement outcome uncertainty
Volume of a macrostate
Mean uncertainty in a macrostate 12
Properties • For a state in a subspace (equivalent of Boltzmann entropy) • Bounded • Extensive on separable states for
,
• Always rises for a state starting in a macrostate (for a short time) • Tends to rise otherwise • Measures quantum equivalent of “spilling coffee on the table” 13
Visualization • Evolution starting in Macrostate 1 Point where probabilities are proportional to volumes of macrostates
Macrostate 2 (dim=3) Macrostate 3 (dim=8) Starting point
Macrostate 1 (dim=1) 14
Multiple coarse-grainings • Multiple sets of projectors (even non-commuting)
(Ordered) probability of being in a macrostate
• Again bounded!
(Ordered) volume of a macrostate
• Alternative expression: 15
Does it work? What about thermodynamics? • Two types of coarse-grainings work equally well • Factorized Observational entropy (FOE): • Measure local energies
H^(1): energy of the left box H^(2): energy of the right box
• S_xE
• Measure coarse-grained position, then energy.
X: E:
4 Particles on the left
0 Particles on the right
total energy
16
Simulations: 1-dim fermionic chain • Hamiltonian between sites k and l Nearest-neighbor interaction Next-nearest-neighbor interaction (this part adds non-integrability)
• Starting in a state confined in the left box 17
Time evolution • non-integrable system • (FOE and S_xE)
• integrable system • (FOE)
Canonical entropy
18
Thermality, and long-time convergence • non-integrable system • FOE (S_xE almost identical) • Red curve: long-time limit states Canonical entropy FOE (for various states) Micro-canonical entropy
19
Thermality, and long-time convergence • Same curves: S_xE and FOE comparison • Red S_xE • Blue FOE • Green microcanonical
20
Thermality, and long-time convergence • integrable system
• blue dots=S_xE • Green circles=FOE
• As expected, doesn’t look thermal
21
Conclusions Observational entropy = quantum generalization of Boltzmann entropy cool properties Inherently quantum (superposition, non-commutativity) Depends on observer’s abilities/ignorance Rises in closed quantum systems to the correct thermodynamic value Future: Gibbs paradox, information engines, work extraction, fluctuation theorems… • arXiv:1707.09722 [quant-ph] • • • • • • •
