Experimental Rectification of Entropy Production by Maxwell s Demon in a Quantum System

Transcription

1 Experimental Rectification of Entropy Production by Maxwell s Demon in a Quantum System Tiago Barbin Batalhão SUTD, Singapore Work done while at UFABC, Santo André, Brazil Singapore, January 11th, tiago_batalhao@sutd.edu.sg APS/Alan Stonebraker Web: Brazilian National Institute for Science and Technology on Quantum Information

2 This talk is based on

3 Collaborators Financial support CBPF John Peterson Alexandre Souza Roberto Sarthour Ivan Oliveira São Paulo Rio de Janeiro UFABC Patrice Camati Tiago Batalhão Kaonan Micadei Roberto Serra

4 Basic theories Nonequilibrium Thermodynamics Information Theory Fluctuation theorems for work and heat, treated as stochastic quantities. Entropy production as a central quantity. Big advances since Created in 1948 by Shannon. Connections with thermodynamics established by Jaynes in the 1950's.

5 Maxwell demon (1860 s) First connection between information and thermodynamics. Intelligent being that can measure microscopic variables and act on a system depending on the result of said measurements.

6 Szilard engine (1929) Measure Erase memory W = kb T log 2 Extract wrok Act depending on result

7 Landauer principle (1962) Measure Erase memory W = kb T log 2 Extract wrok Act depending on result

8 Nonequilibrium thermodynamics Thermodynamic variables are stochastic and obey fluctuation theorems. Crooks theorem PRE 60, 2721 (1999) P F (+W )=e W F P B ( W ) Jarzynski relation e W = e PRL 78, 2690 (1997) F Second Law from Jensen inequality hw i F Also valid in quantum domain. arxiv:cond-mat/ (2000) arxiv:cond-mat/ (2000)

9 Feedback process General schematics of the experiment.

10 Feedback process Unitary Unital, conditional

11 Fluctuation theorems Modified fluctuation theorems to handle information about the system. Sagawa-Ueda theorem D e W + F (k) I (k,l)e =1 PRL 104, (2010) PRE 88, (2013) Free energy difference depends F (k) F (k) = 1 log Z(k) 2 on measurement result Z 0 I (k,l) Unaveraged mutual information I (k,l) = log P p (k l) l 0 p (k l 0 ) p (l 0 )

12 Mean entropy production Using Jensen inequality h i D F (k)e D I (k,l)e D I (k,l)e Mutual information between system and memory (measurement result). Right-hand side is independent of the specifics of the feedback protocol. While this equation does not rule out the possibility of negative mean entropy production, it does not provide an explicit way to achieve it.

13 Mean entropy production Equality using information-theoretic quantities h i = I gain + DS KL (k,l) 2 k (k,eq) 2 E + D S (k,l)e F Information gain: average information the demon obtains reading the outcomes of the measurement. I gain = S ( 1 ) X l p (l) S (l) 1 Always non-negative for projective measurements. Kullback-Leibler divergence between the resulting state of the feedback process and an equilibrium state. S KL (k,l) 2 k (k,eq) 2 =tr (k,l) 2 log (k,l) 2 log (k,eq) 2

14 Feedback trade-off relation A feedback process is effective (leads to negative mean entropy production) if I gain D S KL (k,l) 2 k (k,eq) 2 E + D S (k,l)e F Unitary feedback processes cancel the second term, but leads to big values on the first term. A non-unitary process can cancel the first term, but may lead to big values on the second term. There s a trade-off to minimize the right-hand side.

15 Nuclear Magnetic CBPF, Rio de Janeiro 13 1 C Deviation matrix H C'1 C' H Frequency (Hz) H Frequency (Hz)

16 Nuclear Magnetic Resonance Hamiltonian description as two 2-level systems. Ĥ = (! H! H,rf ) ÎH z (! C! C,rf ) ÎC z +2 JÎH z ÎC z Offset Offset Scalar coupling Initial state = 1 Z e Ĥ/k BT = 1 4Î Compensated by having many molecules in sample NMR frequencies! H MHz! C MHz J 215 Hz Spin-lattice relaxation (Energy damping) T1 H T1 C 7.36 s s Transverse relaxation (Dephasing) T2 H T2 C 4.76 s 0.33 s Typical one-qubit gate: 10 µs Typical two-qubit gate: 10 ms Excellent for small-scale quantum computation or simulation.

17 Implementation

18 Implementation Unitary quench (change of Hamiltonian) PRL 113, (2014) PRL 115, (2015)

19 Implementation Projective measurement with a mismatch between measurement basis and feedback basis. Introduces a controlled error in the feedback protocol.

20 Implementation Non-unitary (but unital) feedback process. Unital processes preserve the maximally-mixed state and never decrease von Neumann entropy F (k) (I) =I

21 Experimental results Readout of entropy production was done using interferometric strategy. PRL 110, (2013) PRL 110, (2013) Negative mean entropy production was observed under different initial temperatures and different feedback implementation errors.

22 Experimental results Readout of information-theoretic quantities was done using full Quantum State Tomography. Changing the spin temperature degrades information gain, but does not affect mutual information.

23 Experimental results Readout of information-theoretic quantities was done using full Quantum State Tomography. Changing the feedback error (basis mismatch) degrades mutual information, but does not affect information gain.

24 Trade-off relations Changing spin temperature Changing feedback error The trade-off relation is more clearly seen when changing the feedback error.

25 Discussion Employing an information-to-energy trade-off relation, we designed an entropy rectification protocol based on Maxwell s demon. This protocol was experimentally carried out using Nuclear Magnetic Resonance, with a quantum 2-level system performing the role of the demon s memory. Understanding the trade-off between information and entropy production at the quantum scale is important to develop applications of quantum technologies with high efficiency and enhance the performance of the thermal machines of the future. If you want a more extended discussion, you can me at tiago_batalhao@sutd.edu.sg