Entropy from Entanglement. Sid Parameswaran

Transcription

1 Entropy from Entanglement Sid Parameswaran Saturday Mornings of Theoretical Physics Oxford, November 17, 2018

2 Usually: system exchanges heat with environment

3 Intuitively: coupling to environment can limit the menu of possible phenomena Forbidden City, Beijing Images: businessinsider.com, captainsvoyage-forum.com

4 Isolated systems may offer more exotic possibilities. Images: abovetopsecret.com/, tahitibycarl.com/

5 Isolated Quantum Systems T ~ 10-8 K Atoms trapped in ultrahigh vacuum: almost perfect isolation Macroscopic number of weakly interacting atoms Image: David Weld, UCSB

6 Can isolated systems self-generate environment? Can a system be its own heat bath (need to go to the roots of statistical physics)

7 Statistical Physics of Classical Systems Try to directly simulate a monoatomic ideal gas Statistical Mechanics is how we solve this! ~10 24 atoms 6 coordinates/atom (x, y, z, px, py, pz) 6 bits/molecule Tb!

8 Microstates vs. Macrostates P V T E {x i (t), p i (t)}

9 Fundamental Postulate of Statistical Mechanics An isolated system in equilibrium is equally likely to be in any of its accessible microstates (given a macrostate) Josiah Willard Gibbs ( ) Ludwig Boltzmann ( ) James Clerk Maxwell ( ) Images: Wikipedia

10 Ergodicity How do we go from microstates to macrostates? t =0 t = t " t = t + " {x i (0), p i (0)} System forgets which microstate it started in

11 Ergodicity and Entropy Given enough time, systems explore all accessible microstates consistent w/ macrostate Entropy: how many microstates correspond to given macrostate? S = k B log W

12 Thermalization Given enough time, systems explore all accessible microstates consistent w/ macrostate Time evolution is towards higher entropy.

13 Image: Molecular Biology of the Cell, Alberts B, Johnson A, Lewis J, et al. New York: Garland Science; 2002.

14 Image: David Weld, UCSB What about isolated quantum systems?

15 The math is right. It s just in poor taste. * *Translated from the American by S. Parameswaran

16 ~ =1 rest of this talk Quantum Mechanics Reminder State of system described by state vector Properties of system captured by Hamiltonian i Ĥ Dirac notation Time evolution: Schrödinger equation: Ĥ i i Solution simple in terms of special eigenstates : Ĥ i = E i i 7! e ie t/~ i (t = 0)i = X c i! (t)i = X c e ie t/~ i

17 Quantum Superposition Simplest quantum system: single spin, up or down "i #i analogous to horizontal/vertical polarization: superposition means some polarization in between i = cos "i +sin #i #i Align detector along up /horizontal i "i See signal w/ probability cos 2 θ no signal w/ probability sin 2 θ Image: Wikipedia/Bob Mellish

18 Quantum Superposition Simplest quantum system: single spin, up or down "i #i analogous to horizontal/vertical polarization: superposition means some polarization in between i = cos "i +sin #i #i Align detector along down /vertical i "i See signal w/ probability sin 2 θ no signal w/ probability cos 2 θ Image: Wikipedia/Bob Mellish

19 Isolated quantum systems Microstates: single eigenstate of the Hamiltonian Ĥ i = E i Macrostate : set approximate energy ~ E i E Must also fix an initial state: (0)i = X c i X c 2 =1 (probabilities add to 1)

20 Isolated quantum systems Time evolution: (t)i = X c e ie t i Probability to be in microstate (eigenstate) α: P ( ) = c 2 doesn t change with time! As long as it had different probabilities of being in different microstates initially, system never forgets! We need to think differently about statistical physics in isolated quantum systems

21 Observables Recall that quantum mechanics is a theory of measurement Consider measuring an observable hô(t)i h (t) Ô (t)i = X, c c e i(e E )t h O i Off-diagonal terms oscillate, diagonal terms constant: hô(t)i t!1 X c 2 h Ô i Even observables seem to remember the microstate! How can the system thermalize, i.e. forget its initial state?

22 Thermalization Only possible if expectation values just depend on macrostate properties h Ô i f(e ) hô(t)i t!1 X c 2 f(e ) f(e) X c 2 f(e) (as long as the initial state is not too spread out in energy) The logical corollary is that we could just work with a single eigenstate, dispensing with the c α How can we define entropy in this setting?

23 Entanglement Simplest example: 2 quantum spins, A, B, each up or down Two distinct states: 1i AB = 1 p 2 ( "i A "i B + #i A "i B ) 2i AB = 1 p 2 ( "i A #i B + #i A "i B ) One is entangled, the other is not. What s the difference? How can we tell?

24 Entanglement Constructive ignorance : give up some information. Agree never to measure the spin B; I ll keep handing you copies of the state, and you can measure spin A all you like. i AB θ Measure only A (in direction θ) How does signal depend on θ? Image: Wikipedia/Bob Mellish

25 Entanglement 1i AB = 1 p 2 ( "i A "i B + #i A "i B ) i AB θ Measure only A (in direction θ) signal ~ cos 2 θ Measurement result ~ polarized light: quantum superposition 1i AB = ui A vi B unentangled product state Image: Wikipedia/Bob Mellish

26 Entanglement 2i AB = 1 p 2 ( "i A #i B + #i A "i B ) θ Measure only A i AB (in direction θ) signal ~ 1/2 (independent of θ) Measurement result ~ unpolarised light: classical randomness 2i AB 6= ui A vi B entangled, can t write as product state Image: Wikipedia/Bob Mellish

27 Entanglement For an unentangled state, giving up information on spin B doesn t change the fact that spin A is in a quantum superposition state For an entangled state, giving up information on spin B makes spin A have classical uncertainty. The classical uncertainty yields an entropy of ln 2: This is the entropy of entanglement, denoted SE

28 Entanglement Local observables: ignorant about most of the system: see a lot of classical uncertainty - entropy even in a single state! most of system unmeasured local observable In a typical quantum state of a many-spin system, SE is extensive ( volume), just like the usual thermal entropy

29 Entanglement Growth The analogue of the special low-entropy state is a product state of spins: e.g. "#"""##"...i (zero entanglement between any of the spins) What is the analogue of entropy growth?

30 Entanglement Growth Consider two spins with eigenstates and energies as follows: +i = "i A #i B + #i A "i B p 2 E + =+E i = "i A #i B #i A "i p B 2 E = E Initial state: (t = 0)i = "i A #i B = +i + i p 2 (t)i = e iet +i + e iet i p 2 = cos(et) "i A #i B i sin(et) #i A "i B entangled! Many spins: each spin highly entangled with several others

31 To sum up Isolated quantum systems can thermalize by self-generating classical uncertainty and thus entropy via entanglement. Image: David Weld, UCSB

32 Started this talk by suggesting isolated systems may allow us to study exotic things Physics by the Lake 2018 Moorea, French Polynesia The news is not good (?) Images: abovetopsecret.com/, tahitibycarl.com/

33 Started this talk by suggesting isolated systems may allow us to study exotic things Physics by the Lake 2018 Moorea, French Polynesia The news is not good (?) Images: abovetopsecret.com/, tahitibycarl.com/

34 Amazingly, some quantum systems can and do evade the tyranny of entropy. Counter-intuitively, these are often imperfect, so that spins are no longer able to cooperatively generate entanglement. This is one of the frontiers of research today.

35 It is by avoiding the rapid decay into the inert state of equilibrium that an organism appears so enigmatic. - E. Schrödinger, What is Life? Image: Wikipedia