The first law of general quantum resource theories

We extend the tools of quantum resource theories to scenarios in which multiple quantities (or resources) are present, and their interplay governs the evolution of physical systems. We derive conditions for the interconversion of these resources, which generalise the first law of thermodynamics. We study reversibility conditions for multi-resource theories, and find that the relative entropy distances from the invariant sets of the theory play a fundamental role in the quantification of the resources. The first law for general multi-resource theories is a single relation which links the change in the properties of the system during a state transformation and the weighted sum of the resources exchanged. In fact, this law can be seen as relating the change in the relative entropy from different sets of states. In contrast to typical single-resource theories, the notion of free states and invariant sets of states become distinct in light of multiple constraints. Additionally, generalisations of the Helmholtz free energy, and of adiabatic and isothermal transformations, emerge. We thus have a set of laws for general quantum resource theories, which generalise the laws of thermodynamics. We first test this approach on thermodynamics with multiple conservation laws, and then apply it to the theory of local operations under energetic restrictions.

[1]  Renato Renner.  Symmetry implies independence , 2007 .

[2]  D. Janzing,et al.  Thermodynamic Cost of Reliability and Low Temperatures: Tightening Landauer's Principle and the Second Law , 2000, quant-ph/0002048.

[3]  Mark Fannes,et al.  Entanglement boost for extractable work from ensembles of quantum batteries. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  M. N. Bera,et al.  Entanglement and Coherence in Quantum State Merging. , 2016, Physical review letters.

[5]  E. Lieb,et al.  The entropy concept for non-equilibrium states , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  R. Renner,et al.  Axiomatic Relation between Thermodynamic and Information-Theoretic Entropies. , 2015, Physical review letters.

[7]  A. J. Short,et al.  Thermodynamics of quantum systems with multiple conserved quantities , 2015, Nature Communications.

[8]  Rolf Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[9]  Michal Horodecki,et al.  LETTER TO THE EDITOR: On asymptotic continuity of functions of quantum states , 2005 .

[10]  'Alvaro M. Alhambra,et al.  Entanglement fluctuation theorems , 2017, Physical Review A.

[11]  Andreas J. Winter,et al.  Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints , 2015, ArXiv.

[12]  'Alvaro M. Alhambra,et al.  Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality , 2016, 1601.05799.

[13]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[14]  M. Christandl.  The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography , 2006, quant-ph/0604183.

[15]  G. Gour,et al.  Quantum resource theories , 2018, Reviews of Modern Physics.

[16]  P. Hayden,et al.  Universal entanglement transformations without communication , 2003 .

[17]  Paul Erker,et al.  Autonomous quantum clocks: how thermodynamics limits our ability to measure time , 2016, 1609.06704.

[18]  J Eisert,et al.  Positive Wigner functions render classical simulation of quantum computation efficient. , 2012, Physical review letters.

[19]  T. Rudolph,et al.  The Wigner–Araki–Yanase theorem and the quantum resource theory of asymmetry , 2012, 1209.0921.

[20]  R. Spekkens,et al.  The resource theory of quantum reference frames: manipulations and monotones , 2007, 0711.0043.

[21]  K. Audenaert,et al.  Asymptotic relative entropy of entanglement. , 2001, Physical review letters.

[22]  V. Vedral,et al.  Entanglement measures and purification procedures , 1997, quant-ph/9707035.

[23]  David Jennings,et al.  Description of quantum coherence in thermodynamic processes requires constraints beyond free energy , 2014, Nature Communications.

[24]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .

[25]  Zahra Baghali Khanian,et al.  Thermodynamics as a Consequence of Information Conservation , 2017, Quantum.

[26]  M. Horodecki,et al.  Reversible transformations from pure to mixed states and the unique measure of information , 2002, quant-ph/0212019.

[27]  Eric Chitambar,et al.  Relating the Resource Theories of Entanglement and Quantum Coherence. , 2015, Physical review letters.

[28]  Fernando G. S. L. Brandão,et al.  A Reversible Theory of Entanglement and its Relation to the Second Law , 2007, 0710.5827.

[29]  Jack K. Wolf,et al.  Noiseless coding of correlated information sources , 1973, IEEE Trans. Inf. Theory.

[30]  Andreas Winter,et al.  Quantum reference frames and their applications to thermodynamics , 2018, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[31]  J. Eisert,et al.  Limits to catalysis in quantum thermodynamics , 2014, 1405.3039.

[32]  C. H. Bennett,et al.  Unextendible product bases and bound entanglement , 1998, quant-ph/9808030.

[33]  Uttam Singh,et al.  Maximally coherent mixed states: Complementarity between maximal coherence and mixedness , 2015, 1503.06303.

[34]  Joseph M. Renes,et al.  Work cost of thermal operations in quantum thermodynamics , 2014, 1402.3496.

[35]  Lidia del Rio,et al.  Resource theories of knowledge , 2015, 1511.08818.

[36]  F. Brandão,et al.  A Generalization of Quantum Stein’s Lemma , 2009, 0904.0281.

[37]  Gerardo Adesso,et al.  Quantum coherence fluctuation relations , 2018, Journal of Physics A: Mathematical and Theoretical.

[38]  F. Brandão,et al.  Reversible Framework for Quantum Resource Theories. , 2015, Physical review letters.

[39]  M. Horodecki,et al.  Quantum information can be negative , 2005, quant-ph/0505062.

[40]  E. Rains.  Bound on distillable entanglement , 1998, quant-ph/9809082.

[41]  M. Horodecki,et al.  Fundamental limitations for quantum and nanoscale thermodynamics , 2011, Nature Communications.

[42]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[43]  Victor Veitch,et al.  The resource theory of stabilizer quantum computation , 2013, 1307.7171.

[44]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[45]  F. Brandão,et al.  Entanglement theory and the second law of thermodynamics , 2008, 0810.2319.

[46]  M. Plenio,et al.  Quantifying Entanglement , 1997, quant-ph/9702027.

[47]  Robert W. Spekkens,et al.  A mathematical theory of resources , 2014, Inf. Comput..

[48]  A. Winter,et al.  Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges , 2015, Nature Communications.

[49]  C. H. Bennett,et al.  Quantum nonlocality without entanglement , 1998, quant-ph/9804053.

[50]  R. Renner,et al.  Inadequacy of von Neumann entropy for characterizing extractable work , 2009, 0908.0424.

[51]  Nicole Yunger Halpern,et al.  The resource theory of informational nonequilibrium in thermodynamics , 2013, 1309.6586.

[52]  A. Harrow.  Entanglement spread and clean resource inequalities , 2009, 0909.1557.

[53]  Antonio Acín,et al.  Entanglement generation is not necessary for optimal work extraction. , 2013, Physical review letters.

[54]  R. Spekkens,et al.  Measuring the quality of a quantum reference frame: The relative entropy of frameness , 2009, 0901.0943.

[55]  Joseph M. Renes,et al.  Relative submajorization and its use in quantum resource theories , 2015, 1510.03695.

[56]  Rahul Jain,et al.  Quantifying Resources in General Resource Theory with Catalysts. , 2018, Physical review letters.

[57]  M. Mohseni,et al.  Thermodynamic resource theories, non-commutativity and maximum entropy principles , 2017 .

[58]  E. Rains.  Entanglement purification via separable superoperators , 1997, quant-ph/9707002.

[59]  Jonathan Oppenheim,et al.  A Resource Theory for Work and Heat , 2016, ArXiv.

[60]  David Jennings,et al.  Thermodynamic resource theories, non-commutativity and maximum entropy principles , 2015, 1511.04420.

[61]  Quantum thermodynamics with local control. , 2016, Physical review. E.

[62]  Jonathan Oppenheim,et al.  Are the laws of entanglement theory thermodynamical? , 2002, Physical review letters.

[63]  M. Horodecki,et al.  The Uniqueness Theorem for Entanglement Measures , 2001, quant-ph/0105017.

[64]  J. Emerson,et al.  Corrigendum: Negative quasi-probability as a resource for quantum computation , 2012, 1201.1256.

[65]  G. Lindblad.  Completely positive maps and entropy inequalities , 1975 .

[66]  M. Kim,et al.  Clock-Work Trade-Off Relation for Coherence in Quantum Thermodynamics. , 2017, Physical review letters.

[67]  Christopher T. Chubb,et al.  Energy cost of entanglement extraction in complex quantum systems , 2017, Nature Communications.

[68]  R. Spekkens,et al.  The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations , 2011, 1104.0018.

[69]  F. Brandão,et al.  Resource theory of quantum states out of thermal equilibrium. , 2011, Physical review letters.

[70]  R. Werner,et al.  Entanglement measures under symmetry , 2000, quant-ph/0010095.

[71]  J. Renes,et al.  Beyond heat baths: Generalized resource theories for small-scale thermodynamics. , 2014, Physical review. E.

[72]  Jonathan Oppenheim,et al.  A general derivation and quantification of the third law of thermodynamics , 2014, Nature Communications.

[73]  Lídia del Rio,et al.  Currencies in Resource Theories , 2016, Entropy.

[74]  S. Popescu,et al.  Thermodynamics and the measure of entanglement , 1996, quant-ph/9610044.

[75]  A. Miranowicz,et al.  Closed formula for the relative entropy of entanglement , 2008, 0805.3134.

[76]  E. Lieb,et al.  The physics and mathematics of the second law of thermodynamics (Physics Reports 310 (1999) 1–96)☆ , 1997, cond-mat/9708200.

[77]  J. Eisert,et al.  Thermodynamic work from operational principles , 2015, 1504.05056.

[78]  M. Plenio,et al.  Quantifying coherence. , 2013, Physical review letters.

[79]  J. Åberg.  Catalytic coherence. , 2013, Physical review letters.

[80]  TOBIAS FRITZ,et al.  Resource convertibility and ordered commutative monoids , 2015, Mathematical Structures in Computer Science.

[81]  A. Winter,et al.  Operational Resource Theory of Coherence. , 2015, Physical review letters.

[82]  M. Horodecki,et al.  QUANTUMNESS IN THE CONTEXT OF) RESOURCE THEORIES , 2012, 1209.2162.

[83]  R. Renner,et al.  Inadequacy of von Neumann entropy for characterizing extractable work , 2011 .

[84]  Michal Horodecki,et al.  The second laws of quantum thermodynamics , 2013, Proceedings of the National Academy of Sciences.

[85]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[86]  Nicole Yunger Halpern.  Beyond heat baths II: framework for generalized thermodynamic resource theories , 2014, 1409.7845.

[87]  J. Åberg.  Quantifying Superposition , 2006, quant-ph/0612146.

[88]  A. J. Short,et al.  Work extraction and thermodynamics for individual quantum systems , 2013, Nature Communications.

[89]  Paul Skrzypczyk,et al.  Thermodynamic cost of creating correlations , 2014, 1404.2169.

[90]  Johan Aberg,et al.  The thermodynamic meaning of negative entropy , 2010, Nature.

[91]  J Eisert,et al.  Second law of thermodynamics under control restrictions. , 2016, Physical review. E.