Fundamental work cost of quantum processes

1607.03104 1709.00506 Fundamental work cost of quantum processes Philippe Faist 1,2, Renato Renner 1 1 Institute for Theoretical Physics, ETH Zurich 2 Institute for Quantum Information and Matter, Caltech QIP 2018, Delft, January 2018

electromagnetic radiation steam engines gases chemistry solid state physics black holes QIP 2018, Delft, January 2018 2

electromagnetic radiation steam engines gases chemistry solid state physics black holes Maxwell s demon anomalous heat flows Jennings & Rudolph, PRE, 2010 actually OK, just need to be careful side information del Rio et al., Nature, 2011 QIP 2018, Delft, January 2018 2

electromagnetic radiation steam engines gases chemistry solid state physics black holes Maxwell s demon anomalous heat flows Jennings & Rudolph, PRE, 2010 side information del Rio et al., Nature, 2011 actually OK, just need to be careful What is the most general formulation of thermodynamics? QIP 2018, Delft, January 2018 2

Information and Thermodynamics 1 bit of information can be traded for work Szilárd engine Szilárd, 1929 Landauer: Irreversible information processing incurs thermodynamic cost Landauer, 1961 Bennett, 1982, 2003 QIP 2018, Delft, January 2018 3

Maxwell s Demon Demon lets particles go from right to left only QIP 2018, Delft, January 2018 4

Maxwell s Demon Demon lets particles go from right to left only Net decrease in entropy of gas violation of the second law of thermodynamics? QIP 2018, Delft, January 2018 4

Maxwell s Demon Demon lets particles go from right to left only Net decrease in entropy of gas violation of the second law of thermodynamics? Idea: The demon has access to microscopic information QIP 2018, Delft, January 2018 4

Maxwell s Demon memory The demon stores the measurement results Resetting this memory costs work! Landauer, 1961 Bennett, 1982, 2003 QIP 2018, Delft, January 2018 4

Resource Theory of thermal operations Allowed any ancilla in a Gibbs state QIP 2018, Delft, January 2018 5

Resource Theory of thermal operations Allowed any ancilla in a Gibbs state Allowed any energy-conserving unitaries: QIP 2018, Delft, January 2018 5

Resource Theory of thermal operations Allowed any ancilla in a Gibbs state Allowed any energy-conserving unitaries: Allowed to discard any system QIP 2018, Delft, January 2018 5

Known Results Necessary and sufficient conditions for (thermo-majorization, block-diagonal states) Horodecki & Oppenheim, Nat. Comm. 2013 Conversion rates Brandão et al., PRL, 2013 Rényi- entropies monotones: second laws Brandão et al., PNAS, 2015 Generalized thermodynamic baths Yunger Halpern & Renes, PRE, 2016,... Catalytical transformations, correlations... Ng et al., NJP, 2015; Lostaglio et al., PRL, 2015...... QIP 2018, Delft, January 2018 6

Thermodynamic cost of any process? mapping of input states to output states AND, XOR, gate any classical or quantum computation any physical process (completely positive, tracepreserving map) QIP 2018, Delft, January 2018 7

Thermodynamic cost of any process? mapping of input states to output states AND, XOR, gate any classical or quantum computation Fundamental thermodynamic limit any physical process (completely positive, preserving erving trace- map) to the cost of implementing? QIP 2018, Delft, January 2018 7

Our approach A restriction on what we can do QIP 2018, Delft, January 2018 8

Our approach A restriction on what we can do Free operations must preserve the thermal state (most generous set of maps) QIP 2018, Delft, January 2018 8

Our approach A restriction on what we can do Free operations must preserve the thermal state (most generous set of maps) A resource which we can use to overcome the restriction QIP 2018, Delft, January 2018 8

Our approach A restriction on what we can do Free operations must preserve the thermal state (most generous set of maps) A resource which we can use to overcome the restriction Battery system QIP 2018, Delft, January 2018 8

Framework (2) To each system of interest associated an operator is QIP 2018, Delft, January 2018 9

Framework (2) To each system of interest associated an operator is Choose QIP 2018, Delft, January 2018 9

Framework (2) To each system of interest associated an operator is Allowed only trace-nonincreasing CPMs satisfying Choose QIP 2018, Delft, January 2018 9

Framework (2) To each system of interest associated an operator is Choose Allowed only trace-nonincreasing CPMs satisfying Can always be dilated to trace-preserving, QIP 2018, Delft, January 2018 19

Framework (2) To each system of interest associated an operator is Choose Allowed only trace-nonincreasing CPMs satisfying Can always be dilated to trace-preserving, Information battery QIP 2018, Delft, January 2018 9

Information Battery stored work Landauer/Szilárd: 1 pure qubit = work QIP 2018, Delft, January 2018 10

Step #1: Limit for an exact process QIP 2018, Delft, January 2018 11

Step #2: Optimize effective process QIP 2018, Delft, January 2018 12

Step #2: Optimize effective process process matrix of QIP 2018, Delft, January 2018 12

Step #2: Optimize effective process process matrix of (purified/fidelity distance) QIP 2018, Delft, January 2018 12

Step #2: Optimize effective process process matrix of (purified/fidelity distance) limit for specific = Step #1 QIP 2018, Delft, January 2018 12

Step #2: Optimize effective process process matrix of (purified/fidelity distance) fundamental limit = Step #2 Step #1 QIP 2018, Delft, January 2018 12

Results fundamental limit = Step #2 Step #1 QIP 2018, Delft, January 2018 13

Results fundamental limit = Steps #1 & #2 QIP 2018, Delft, January 2018 13

Results coherent relative entropy Ultimate maximum extractable work for implementing a map with process matrix close to = QIP 2018, Delft, January 2018 13

Coherent relatve entropy: Special cases relative Datta, IEEE TIT (2009) Åberg, Nat Comm (2013) Horodecki & Oppenheim, Nat Comm (2013) QIP 2018, Delft, January 2018 14

Coherent relatve entropy: Special cases relative Datta, IEEE TIT (2009) Åberg, Nat Comm (2013) Horodecki & Oppenheim, Nat Comm (2013) coherent ( conditional ) for pure QIP 2018, Delft, January 2018 14

Examples (with trivial Hamiltonian) Pure information processing no internal energy cost discarded information output PhF, Dupuis, Oppenheim, Renner, Nat Comm, 2015 QIP 2018, Delft, January 2018 15

Examples (with trivial Hamiltonian) Pure information processing no internal energy cost cost: 1.6 bits cost: 1 bit extract: 2 bits cost: 1 bit PhF, Dupuis, Oppenheim, Renner, Nat Comm, 2015 QIP 2018, Delft, January 2018 15

The Coherent Relative Entropy Data processing inequality Chain rule Asymptotic equipartition property: For many independent copies QIP 2018, Delft, January 2018 16

The Coherent Relative Entropy Data processing inequality Chain rule Asymptotic equipartition property: For many independent copies The coherent relative entropy: measure of information has desirable properties reduces to known special cases QIP 2018, Delft, January 2018 16

Battery models Define for For : Reversibly interconvertable Some common battery models equivalent (information battery, wit, weight) Battery states are robust to smoothing (no need to smooth battery states) Brandão et al., PNAS (2015) QIP 2018, Delft, January 2018 17

Emergence of Macro Thermodynamics For a certain class of states (e.g. microcanonical): QIP 2018, Delft, January 2018 18

Emergence of Macro Thermodynamics For a certain class of states (e.g. microcanonical): derives from a potential! QIP 2018, Delft, January 2018 18

Emergence of Macro Thermodynamics For a certain class of states (e.g. microcanonical): derives from a potential! free isolated system: within class contact with heat bath: no i.i.d. assumption QIP 2018, Delft, January 2018 18

Observers in Thermodynamics QIP 2018, Delft, January 2018 19

Observers in Thermodynamics Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January 2018 19

Observers in Thermodynamics implies Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January 2018 19

Observers in Thermodynamics implies What if? possible apparent violation of second law Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January 2018 19

Observers in Thermodynamics implies What if Criterion? for when coarse-grained laws of thermodynamics hold: possible apparent violation of second law Petz, CMP (1986); Fawzi & Renner CMP (2015); Wilde PRSA (2015);... QIP 2018, Delft, January 2018 19

A picture of thermodynamics restriction: preserve information battery coherent relative entropy macroscopic states emergent thermodynamics QIP 2018, Delft, January 2018 20

Physics Information Theory Hamiltonian time evolution quantum state unitary operation QIP 2018, Delft, January 2018 21

Physics Information Theory Hamiltonian time evolution quantum state unitary operation energy, number of particles QIP 2018, Delft, January 2018 21

Physics Information Theory Hamiltonian time evolution quantum state unitary operation energy, number of particles thermodynamics (at least 2nd law) QIP 2018, Delft, January 2018 21

A simple yet general, single-instance, observer-dependent view of thermodynamics Outlook better understanding of universality of thermodynamics New measure of information non-i.i.d. version of relative entropy difference Achievability with thermal operations (+...)? Applications to information theory, coding? Applications to physical systems? QIP 2018, Delft, January 2018 22

Thank you for your attention!

Example: Maxwell s Demon microscopic picture macroscopic picture implicit demon explicit demon QIP 2018, Delft, January 2018 23

Example: Maxwell s Demon microscopic picture macroscopic picture implicit demon thermodynamic entropy is observer-dependent! explicit demon QIP 2018, Delft, January 2018 23

General case: non-trivial Hamiltonian Fundamental work cost for any process coherent relative entropy process matrix, characterizes and input state measures information relative to Gibbs weights QIP 2018, Delft, January 2018 24

Approaches to information thermodynamics statistical mechanics Piechocinska, PRA, 2000 resource theory approach Brandão et al., PRL, 2013 axiomatic approach Lieb & Yngvason, PR, 1999 Weilenmann et al., PRL, 2016 QIP 2018, Delft, January 2018 25

Approaches to information thermodynamics statistical mechanics Piechocinska, PRA, 2000 work probability distributions, time evolution, fluctuation relations axiomatic approach Lieb & Yngvason, PR, 1999 Weilenmann et al., PRL, 2016 resource theory approach Brandão et al., PRL, 2013 inherently one-shot, epsilonwork, general processes abstract, first-principles approach, structure of thermodynamics QIP 2018, Delft, January 2018 25

Smoothing QIP 2018, Delft, January 2018 26

Smoothing Ignore unlikely events up to total probability QIP 2018, Delft, January 2018 26

What About Statistical Mechanics? Model system s time evolution Average energy, von Neumann entropy; oneshot statements more tricky Count work? p(w) not well defined quantum Closer to applications than resource-theory approaches QIP 2018, Delft, January 2018 27

Work? Count work using a battery system Horodecki & Oppenheim, Nat. Comm. 2013 QIP 2018, Delft, January 2018 28

Work? Count work using a battery system Work extraction : Horodecki & Oppenheim, Nat. Comm. 2013 QIP 2018, Delft, January 2018 28

Work? Count work using a battery system Work extraction : Work cost of formation : Horodecki & Oppenheim, Nat. Comm. 2013 QIP 2018, Delft, January 2018 28

Work? Count work using a battery system Work extraction : Work cost of formation : valid for single instance of the process macroscopic limit Horodecki & Oppenheim, Nat. Comm. 2013 QIP 2018, Delft, January 2018 28