Meta tags:

Headings (most frequently used words):

entropy, subadditivity, von, neumann, contents, fundamentals, properties, change, under, time, evolution, thermodynamic, meaning, generalizations, and, derived, quantities, history, footnotes, references, strong, minimum, shannon, holevo, quantity, unitary, measurement, conditional, relative, entanglement, measures, quantum, rényi, entropies,

Text of the page (most frequently used words):
the (245), rho (95), #entropy (86), displaystyle (83), quantum (67), and (65), state (51), von (41), neumann (41), that (39), 2017 (30), for (30), system (26), states (24), can (24), doi (23), edit (22), theory (21), entanglement (21), density (20), pure (20), isbn (19), sum (19), 2010 (18), chuang (18), nielsen (18), are (18), this (17), with (17), physics (17), subadditivity (16), 978 (15), życzkowski (15), measurement (15), rangle (15), information (14), bengtsson (14), matrix (14), space (14), holevo (13), alpha (13), shannon (12), bibcode (12), sigma (12), left (12), mixed (11), leq (11), frac (11), lambda (11), statistical (10), press (10), rau (10), relative (10), not (10), hilbert (10), mathematical (9), mechanics (9), arxiv (9), wilde (9), peres (9), 2001 (9), all (9), given (9), operatorname (9), then (9), its (9), right (9), from (8), university (8), 1993 (8), change (8), operator (8), inequality (8), upon (8), pmatrix (8), log (8), which (8), povm (8), wikipedia (7), was (7), rényi (7), theorem (7), lieb (7), strong (7), 1007 (7), properties (7), maximally (7), quant (7), other (7), known (7), one (7), any (7), only (7), measure (7), operators (7), when (7), psi (7), langle (7), about (6), under (6), john (6), cambridge (6), 2011 (6), evolution (6), s2cid (6), history (6), subsystem (6), entropies (6), rank (6), reduced (6), also (6), over (6), main (6), dagger (6), positive (6), matrices (6), toggle (5), page (5), description (5), gibbs (5), polak (5), rieffel (5), 267 (5), trace (5), elliott (5), journal (5), 1103 (5), der (5), entangled (5), lüders (5), some (5), function (5), classical (5), version (5), example (5), possible (5), each (5), used (5), written (5), article (5), where (5), channel (5), unitary (5), contents (4), search (4), mechanical (4), boltzmann (4), field (4), correlation (4), process (4), particles (4), thermodynamic (4), maximum (4), 2022 (4), zwiebach (4), computation (4), 2021 (4), springer (4), 540 (4), convex (4), ruskai (4), mary (4), beth (4), revmodphys (4), reviews (4), modern (4), physik (4), measures (4), systems (4), proved (4), introduced (4), describing (4), measurements (4), combination (4), between (4), set (4), defined (4), bipartite (4), quantity (4), abe (4), probability (4), begin (4), end (4), two (4), zero (4), difference (4), conditional (4), quantities (4), eigenvalues (4), named (4), they (4), but (4), outcome (4), changes (4), time (4), numbers (4), bigg (4), finite (4), dimensional (4), abc (4), hide (4), move (4), sidebar (4), view (3), terms (3), may (3), using (3), august (3), date (3), different (3), retrieved (3), potential (3), functions (3), introduction (3), 1973 (3), january (3), 1063 (3), pdf (3), inequalities (3), max (3), delbrück (3), general (3), die (3), translated (3), albert (3), einstein (3), foundations (3), help (3), cite (3), landau (3), pair (3), 264 (3), resource (3), gerhart (3), triangle (3), composite (3), hand (3), product (3), result (3), distribution (3), maximal (3), results (3), strongly (3), same (3), described (3), thus (3), than (3), equal (3), constant (3), associated (3), generalizations (3), derived (3), eigenvectors (3), provide (3), self (3), adjoint (3), case (3), kraus (3), projectors (3), have (3), qubit (3), projection (3), prob (3), side (3), probabilities (3), minimum (3), three (3), subadditive (3), spin (3), concept (3), basis (3), eta (3), tools (3), subsection (3), topic (2), languages (2), table (2), contact (2), privacy (2), policy (2), text (2), apply (2), use (2), categories (2), short (2), wikidata (2), applications (2), critical (2), mean (2), equation (2), force (2), models (2), partition (2), thermodynamics (2), mit (2), 262 (2), 2nd (2), scientific (2), jochen (2), oxford (2), kluwer (2), 7923 (2), asher (2), 521 (2), references (2), mathematics (2), dyson (2), talk (2), conference (2), 2013 (2), proof (2), 1968 (2), robinson (2), derek (2), lanford (2), oscar (2), 1970 (2), communications (2), araki (2), huzihiro (2), oral (2), 361 (2), alfred (2), april (2), 221 (2), über (2), zeitschrift (2), für (2), szilard (2), leo (2), princeton (2), 1997 (2), 691 (2), eds (2), 291 (2), incompatibility (2), book (2), berlin (2), 1927 (2), 2016 (2), 012003 (2), 698 (2), 5022 (2), physical (2), review (2), letters (2), 2002 (2), 197 (2), 514 (2), 515 (2), rule (2), daniel (2), kirkpatrick (2), 2006 (2), 1002 (2), andp (2), annalen (2), 443 (2), 102 (2), 1975 (2), equality (2), 516 (2), 513 (2), subsystems (2), 4614 (2), footnotes (2), more (2), earlier (2), vector (2), just (2), family (2), additive (2), vanishes (2), continuous (2), here (2), logarithm (2), minimizing (2), writing (2), average (2), based (2), such (2), facilitates (2), performing (2), saying (2), knowledge (2), does (2), parts (2), always (2), products (2), cannot (2), degree (2), unique (2), let (2), monotonic (2), decreases (2), joint (2), expressed (2), bounded (2), above (2), increase (2), value (2), hamiltonian (2), fixed (2), exp (2), meaning (2), will (2), further (2), specified (2), element (2), into (2), specification (2), initial (2), important (2), special (2), elements (2), taken (2), define (2), linear (2), never (2), leave (2), unchanged (2), leaves (2), collection (2), unity (2), spaces (2), textstyle (2), find (2), outcomes (2), semi (2), definite (2), while (2), discussed (2), below (2), uparrow (2), downarrow (2), independent (2), dimension (2), point (2), within (2), unit (2), ball (2), quad (2), freedom (2), fundamentals (2), after (2), denotes (2), appearance (2), upload (2), file (2), links (2), read (2), create (2), account (2), donate (2), menu (2), add, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, available, additional, site, you, agree, registered, trademark, non, profit, organization, wikimedia, foundation, inc, creative, commons, attribution, sharealike, license, rendered, parsoid, last, edited, 2026, utc, hidden, cs1, errors, articles, https, org, index, php, title, von_neumann_entropy, oldid, 1369953359, machine, chaos, complex, condensed, matter, superfluidity, elementary, particle, tsallis, size, scaling, length, exponents, phase, transition, phenomena, conformal, stochastic, bbgky, hierarchy, vlasov, approaches, lennard, jones, depletion, percolation, heisenberg, potts, ising, ferromagnetism, maxwell, relations, equations, ensembles, ergodic, principle, 04613, mastering, essentials, barton, 316, 80997, 1017, 9781316809976, 1106, 1445, mark, wolfgang, engineering, mass, 01506, computing, gentle, eleanor, 289630, processing, approach, 959506, 2549, concepts, methods, 10th, anniversary, univ, 63503, isaac, michael, 42082, lecture, notes, monographs, structure, alexander, ingemar, 107, 02625, geometry, karol, 527, 288, 1016, 0001, 8708, 90011, advances, wigner, yanase, conjecture, invited, honour, 90th, birthday, freeman, institute, advanced, studies, nanyang, technological, singapore, 2014, world, original, 2020, archived, youtube, com, fundamental, 1941, 1666274, 1973jmp, 1938l, 1938, july, 1125, 1664685, 1120, 170, 189832417, bf01646092, 1970cmaph, 160a, 160, harding, carolyn, 2024, caltech, archives, project, interview, 365, wehrl, 1978, 260, december, 1925, ausdehnung, phänomenologischen, thermodynamik, auf, schwankungserscheinungen, 788, bf01331713, 753, engel, 01734, collected, papers, volume, 1914, beiträge, zur, quantentheorie, 820, deutsche, physikalische, gesellschaft, verhandlungen, stöltzner, 6812, 017, 2012, 0_7, math, 0102013, rédei, petz, henderson, leah, 2003, reply, shenker, 296, 3541968, jstor, 1093, bjps, british, philosophy, science, 271, 1955, 02893, 1932, 59207, mathematische, grundlagen, quantenmechanik, das, daempfungsproblem, wellenmechanik, 464, 125732617, bf01343064, 1927zphy, 430l, 430, 369, 370, 477, 475, 474, 471, enríquez, wintrowicz, march, 1088, 1742, 6596, 2016jphcs, 698a2003e, series, multipartite, brief, survey, 505, 447, hill, bits, 5025, physrevlett, 1997phrvl, 5022h, 9703041, wootters, role, 234, 6370982, 2002rvmp, 197v, 0102094, vedral, vlatko, plenio, martin, virmani, shashank, 2007, 2005quant, 4163p, 0504163, fuchs, christopher, 130, 19926, coming, age, 131, 106, 218, 435, 524, 511, yunger, halpern, nicole, 12051, 1038, ncomms12051, nature, microcanonical, theoretic, derivations, thermal, noncommuting, charges, 266, 380, 381, lahti, pekka, 2009, greenberger, hentschel, klaus, weinert, friedel, heidelberg, 358, 70622, 70626, 7_110, 356, compendium, busch, paul, concerning, due, 670, 119103479, 200610207, 2006anp, 518, 663l, 0403007, 663, 1950, zustandsänderung, durch, den, messprozeß, 322, 19504430510, 1950anp, 322l, 531, 534, terno, 2004, relativity, 123, 7481797, 2004rvmp, 93p, 0212023, 0356797, 1090, s0002, 9904, 13621, bulletin, american, society, convexity, conditions, 4375, 1497701, 0205064, 4358, 364, 220, 236, 519, 216, 217, 126, february, schrödinger, hjw, 15995449, 0894, 9875, issn, s10702, 006, 1852, 2006fophl, 95k, 0305068, hall, brian, multiple, vol, 440, 7115, 7116, 5_19, 419, graduate, texts, mathematicians, 1957, techniques, 1957rvmp, 74f, fano, 700, 360, 355, concavity, 1936, hisaharu, umegaki, 1962, were, difficult, conjectured, gert, molière, motivations, motivation, inspired, impossibility, order, develop, both, expression, now, arguing, probabilistic, analogous, mixture, ideal, gases, first, published, his, argument, built, work, lev, member, broader, too, generalized, limit, recovers, maximized, nonincreasing, parameter, weak, proven, hartley, reasonable, entropic, character, nonentangled, ways, those, idea, extending, larger, partial, yields, finds, choices, infimum, squashed, formation, separable, achieves, uniform, therefore, said, diagonal, ddots, operations, tasks, involve, communication, definition, paraphrased, whole, imply, individual, describes, half, correlated, however, understood, daily, life, instead, thought, generate, actual, appropriate, experiment, expressible, local, constituents, single, term, provides, tool, quantify, overall, sense, satisfies, certain, axioms, required, superposition, energy, greater, equals, unlike, itself, remains, part, traced, total, marginals, mutual, marginal, alone, words, conditioning, found, maximizing, constraint, expected, temperature, factor, canonical, generally, bring, complete, necessary, describe, remedy, decomposing, necessarily, obtained, updated, themselves, onto, vectors, respectively, formula, simplifies, preserving, summing, post, without, normalisation, interpreted, expressing, how, performed, lost, channels, projective, decrease, every, input, sends, towards, amplitude, damping, unital, completely, map, mathrm, karl, isolated, takes, follows, fact, valid, weighted, attained, spanned, their, corresponding, nonzero, orthogonal, appears, supports, see, calculate, mathematically, simplest, number, obtaining, making, proportional, achievable, minimization, povms, identity, valued, technique, establishes, show, equivalent, following, etc, ordinary, symmetry, tripartite, less, sometimes, lower, take, has, considered, individually, because, indicates, bell, followed, generalization, automatically, means, otimes, geq, concave, invariant, transformation, uρu, being, represents, prototypical, whose, arbitrary, real, coordinates, lies, surface, attains, tfrac, times, pauli, experiments, made, calculated, specifically, infinite, line, alternatively, occurs, degrees, representation, acting, designated, assigning, implies, certainty, assigned, though, quantifies, extent, way, wavefunctions, square, integrable, widely, study, uncertainty, extends, counterpart, merely, form, seen, reinterpreted, dots, natural, whom, type, free, encyclopedia, item, projects, printable, download, print, export, switch, legacy, parser, get, shortened, url, permanent, link, related, what, actions, english, українська, slovenščina, português, polski, 日本語, עברית, فارسی, español, català, العربية, top, personal, pages, recent, community, portal, learn, contribute, random, current, events, navigation, jump, content,

Text of the page (random words):
entropy 38 generalizations and derived quantities edit conditional entropy edit let ρ a b displaystyle rho _ ab be a joint state for the bipartite quantum system ab then the conditional von neumann entropy s a b displaystyle s a b is the difference between the entropy of ρ a b displaystyle rho _ ab and the entropy of the marginal state for subsystem b alone s a b s ρ a b s ρ b displaystyle s a b s rho _ ab s rho _ b this is bounded above by s ρ a displaystyle s rho _ a in other words conditioning the description of subsystem a upon subsystem b cannot increase the entropy associated with a 39 quantum mutual information can be defined as the difference between the entropy of the joint state and the total entropy of the marginals s a b s ρ a s ρ b s ρ a b displaystyle s a b s rho _ a s rho _ b s rho _ ab which can also be expressed in terms of conditional entropy 40 s a b s a s a b s b s b a displaystyle s a b s a s a b s b s b a relative entropy edit main article quantum relative entropy let ρ displaystyle rho and σ displaystyle sigma be two density operators in the same state space the relative entropy is defined to be s ρ σ tr ρ log ρ log σ displaystyle s rho sigma operatorname tr rho log rho log sigma the relative entropy is always greater than or equal to zero it equals zero if and only if ρ σ displaystyle rho sigma 41 unlike the von neumann entropy itself the relative entropy is monotonic in that it decreases or remains constant when part of a system is traced over 42 s ρ a σ a s ρ a b σ a b displaystyle s rho _ a sigma _ a leq s rho _ ab sigma _ ab entanglement measures edit just as energy is a resource that facilitates mechanical operations entanglement is a resource that facilitates performing tasks that involve communication and computation 43 the mathematical definition of entanglement can be paraphrased as saying that maximal knowledge about the whole of a system does not imply maximal knowledge about the individual parts of that system 44 if the quantum state that describes a pair of particles is entangled then the results of measurements upon one half of the pair can be strongly correlated with the results of measurements upon the other however entanglement is not the same as correlation as understood in classical probability theory and in daily life instead entanglement can be thought of as potential correlation that can be used to generate actual correlation in an appropriate experiment 45 the state of a composite system is always expressible as a sum or superposition of products of states of local constituents it is entangled if this sum cannot be written as a single product term 46 entropy provides one tool that can be used to quantify entanglement 47 48 if the overall system is described by a pure state the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems for bipartite pure states the von neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure 49 50 it is thus known as the entanglement entropy 51 it is a classical result that the shannon entropy achieves its maximum at and only at the uniform probability distribution 1 n 1 n 52 therefore a bipartite pure state ρ h a h b is said to be a maximally entangled state if the reduced state of each subsystem of ρ is the diagonal matrix 53 1 n 1 n displaystyle begin pmatrix frac 1 n ddots frac 1 n end pmatrix for mixed states the reduced von neumann entropy is not the only reasonable entanglement measure 54 some of the other measures are also entropic in character for example the relative entropy of entanglement is given by minimizing the relative entropy between a given state ρ displaystyle rho and the set of nonentangled or separable states 55 the entanglement of formation is defined by minimizing over all possible ways of writing of ρ displaystyle rho as a convex combination of pure states the average entanglement entropy of those pure states 56 the squashed entanglement is based on the idea of extending a bipartite state ρ a b displaystyle rho _ ab to a state describing a larger system ρ a b e displaystyle rho _ abe such that the partial trace of ρ a b e displaystyle rho _ abe over e yields ρ a b displaystyle rho _ ab one then finds the infimum of the quantity 1 2 s ρ a e s ρ b e s ρ e s ρ a b e displaystyle frac 1 2 s rho _ ae s rho _ be s rho _ e s rho _ abe over all possible choices of ρ a b e displaystyle rho _ abe 57 quantum rényi entropies edit just as the shannon entropy function is one member of the broader family of classical rényi entropies so too can the von neumann entropy be generalized to the quantum rényi entropies s α ρ 1 1 α ln tr ρ α 1 1 α ln i 1 n λ i α displaystyle s_ alpha rho frac 1 1 alpha ln operatorname tr rho alpha frac 1 1 alpha ln sum _ i 1 n lambda _ i alpha in the limit that α 1 displaystyle alpha to 1 this recovers the von neumann entropy the quantum rényi entropies are all additive for product states and for any α displaystyle alpha the rényi entropy s α displaystyle s_ alpha vanishes for pure states and is maximized by the maximally mixed state for any given state ρ displaystyle rho s α ρ displaystyle s_ alpha rho is a continuous nonincreasing function of the parameter α displaystyle alpha a weak version of subadditivity can be proven s α ρ a s 0 ρ b s α ρ a b s α ρ a s 0 ρ b displaystyle s_ alpha rho _ a s_ 0 rho _ b leq s_ alpha rho _ ab leq s_ alpha rho _ a s_ 0 rho _ b here s 0 displaystyle s_ 0 is the quantum version of the hartley entropy i e the logarithm of the rank of the density matrix 58 history edit the density matrix was introduced with different motivations by von neumann and by lev landau the motivation that inspired landau was the impossibility of describing a subsystem of a composite quantum system by a state vector 59 on the other hand von neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements 60 he introduced the expression now known as von neumann entropy by arguing that a probabilistic combination of pure states is analogous to a mixture of ideal gases 61 62 von neumann first published on the topic in 1927 63 his argument was built upon earlier work by albert einstein and leo szilard 64 65 66 max delbrück and gert molière proved the concavity and subadditivity properties of the von neumann entropy in 1936 quantum relative entropy was introduced by hisaharu umegaki in 1962 67 68 the subadditivity and triangle inequalities were proved in 1970 by huzihiro araki and elliott h lieb 69 strong subadditivity is a more difficult theorem it was conjectured by oscar lanford and derek robinson in 1968 70 lieb and mary beth ruskai proved the theorem in 1973 71 72 using a matrix inequality proved earlier by lieb 73 74 footnotes edit bengtsson życzkowski 2017 p 355 peres 1993 p 264 bengtsson życzkowski 2017 p 360 peres 1993 p 264 nielsen chuang 2010 p 700 fano u 1957 description of states in quantum mechanics by density matrix and operator techniques reviews of modern physics 29 1 74 93 bibcode 1957rvmp 29 74f doi 10 1103 revmodphys 29 74 holevo 2001 pp 1 15 hall brian c 2013 systems and subsystems multiple particles quantum theory for mathematicians graduate texts in mathematics vol 267 springer pp 419 440 doi 10 1007 978 1 4614 7116 5_19 isbn 978 1 4614 7115 8 kirkpatrick k a february 2006 the schrödinger hjw theorem foundations of physics letters 19 1 95 102 arxiv quant ph 0305068 bibcode 2006fophl 19 95k doi 10 1007 s10702 006 1852 1 issn 0894 9875 s2cid 15995449 holevo 2001 p 15 wilde 2017 p 126 zwiebach 2022 22 2 rieffel polak 2011 pp 216 217 zwiebach 2022 22 2 holevo 2001 p 15 nielsen chuang 2010 p 513 rau 2017 p 32 holevo 2001 p 15 nielsen chuang 2010 p 513 rau 2017 p 32 wilde 2017 11 1 1 holevo 2001 p 15 nielsen chuang 2010 p 516 rau 2017 p 32 nielsen chuang 2010 p 514 rau 2017 pp 32 33 nielsen chuang 2010 p 519 nielsen chuang 2010 pp 515 516 bengtsson życzkowski 2017 p 361 rieffel polak 2011 p 220 rau 2021 p 236 bengtsson życzkowski 2017 p 364 ruskai mary beth 2002 inequalities for quantum entropy a review with conditions for equality journal of mathematical physics 43 4358 4375 arxiv quant ph 0205064 doi 10 1063 1 1497701 lieb elliott h january 1975 some convexity and subadditivity properties of entropy bulletin of the american mathematical society 81 1 doi 10 1090 s0002 9904 1975 13621 4 mr 0356797 peres asher terno daniel r 2004 quantum information and relativity theory reviews of modern physics 76 1 93 123 arxiv quant ph 0212023 bibcode 2004rvmp 76 93p doi 10 1103 revmodphys 76 93 s2cid 7481797 nielsen chuang 2010 p 90 wilde 2017 11 1 2 nielsen chuang 2010 pp 531 534 nielsen chuang 2010 p 102 zwiebach 2022 22 3 peres 1993 p 267 wilde 2017 11 1 1 wilde 2017 4 2 1 lüders gerhart 1950 über die zustandsänderung durch den messprozeß annalen der physik 443 5 8 322 bibcode 1950anp 443 322l doi 10 1002 andp 19504430510 translated by k a kirkpatrick as lüders gerhart 3 april 2006 concerning the state change due to the measurement process annalen der physik 15 9 663 670 arxiv quant ph 0403007 bibcode 2006anp 518 663l doi 10 1002 andp 200610207 s2cid 119103479 busch paul lahti pekka 2009 lüders rule in greenberger daniel hentschel klaus weinert friedel eds compendium of quantum physics springer berlin heidelberg pp 356 358 doi 10 1007 978 3 540 70626 7_110 isbn 978 3 540 70622 9 wilde 2017 4 4 1 wilde 2017 4 5 1 nielsen chuang 2010 p 515 bengtsson życzkowski 2017 pp 380 381 peres 1993 pp 266 267 rau 2017 p 37 yunger halpern nicole et al 2016 07 07 microcanonical and resource theoretic derivations of the thermal state of a quantum system with noncommuting charges pdf nature communications 7 12051 doi 10 1038 ncomms12051 peres 1993 p 267 rau 2017 pp 51 wilde 2017 11 4 nielsen chuang 2010 p 514 peres 1993 p 264 nielsen chuang 2010 p 511 nielsen chuang 2010 p 524 nielsen chuang 2010 p 106 rieffel polak 2011 p 218 bengtsson życzkowski 2017 p 435 rau 2021 p 131 fuchs christopher a 6 january 2011 coming of age with quantum information cambridge university press p 130 isbn 978 0 521 19926 1 rieffel polak 2011 p 39 plenio martin b virmani shashank 2007 an introduction to entanglement measures quantum information and computation 1 1 51 arxiv quant ph 0504163 bibcode 2005quant ph 4163p vedral vlatko 2002 the role of relative entropy in quantum information theory reviews of modern physics 74 1 197 234 arxiv quant ph 0102094 bibcode 2002rvmp 74 197v doi 10 1103 revmodphys 74 197 s2cid 6370982 holevo 2001 p 31 hill s wootters w k 1997 entanglement of a pair of quantum bits physical review letters 78 26 5022 5025 arxiv quant ph 9703041 bibcode 1997phrvl 78 5022h doi 10 1103 physrevlett 78 5022 bengtsson życzkowski 2017 p 447 nielsen chuang 2010 p 505 enríquez m wintrowicz i życzkowski k march 2016 maximally entangled multipartite states a brief survey journal of physics conference series 698 1 012003 bibcode 2016jphcs 698a2003e doi 10 1088 1742 6596 698 1 012003 holevo 2001 p 31 bengtsson życzkowski 2017 p 471 bengtsson życzkowski 2017 p 474 bengtsson życzkowski 2017 p 475 bengtsson życzkowski 2017 p 477 bengtsson życzkowski 2017 pp 369 370 landau l 1927 das daempfungsproblem in der wellenmechanik zeitschrift für physik 45 5 6 430 464 bibcode 1927zphy 45 430l doi 10 1007 bf01343064 s2cid 125732617 von neumann john 1932 mathematische grundlagen der quantenmechanik berlin springer isbn 3 540 59207 5 cite book isbn date incompatibility help von neumann john 1955 mathematical foundations of quantum mechanics princeton university press isbn 978 0 691 02893 4 cite book isbn date incompatibility help peres 1993 p 271 henderson leah 2003 the von neumann entropy a reply to shenker british journal for the philosophy of science 54 291 296 doi 10 1093 bjps 54 2 291 jstor 3541968 petz d 2001 entropy von neumann and the von neumann entropy in rédei m stöltzner m eds john von neumann and the foundations of quantum physics kluwer pp 83 96 arxiv math ph 0102013 doi 10 1007 978 94 017 2012 0_7 isbn 978 0 7923 6812 0 einstein albert 1914 beiträge zur quantentheorie deutsche physikalische gesellschaft verhandlungen 16 820 translated in the collected papers of albert einstein volume 6 translated by engel alfred princeton university press 1997 pp 20 26 isbn 0 691 01734 4 szilard leo december 1925 über die ausdehnung der phänomenologischen thermodynamik auf die schwankungserscheinungen zeitschrift für physik 32 753 788 doi 10 1007 bf01331713 wehrl alfred april 1978 general properties of entropy reviews of modern physics 50 2 221 260 doi 10 1103 revmodphys 50 221 bengtsson życzkowski 2017 pp 361 365 delbrück max harding carolyn max delbrück oral history interview caltech archives oral history project retrieved 2024 12 30 araki huzihiro lieb elliott h 1970 entropy inequalities communications in mathematical physics 18 2 160 170 bibcode 1970cmaph 18 160a doi 10 1007 bf01646092 s2cid 189832417 lanford oscar e robinson derek w july 1968 mean entropy of states in quantum statistical mechanics pdf journal of mathematical physics 9 7 1120 1125 doi 10 1063 1 1664685 lieb elliott h ruskai mary beth 1973 proof of the strong subadditivity of quantum mechanical entropy journal of mathematical physics 14 12 1938 1941 bibcode 1973jmp 14 1938l doi 10 1063 1 1666274 ruskai mary beth 10 january 2014 evolution of a fundamental theorem on quantum entropy youtube com world scientific archived from the original on 2021 12 21 retrieved 20 august 2020 invited talk at the conference in honour of the 90th birthday of freeman dyson institute of advanced studies nanyang technological university singapore 26 29 august 2013 lieb elliott h 1973 convex trace functions and the wigner yanase dyson conjecture advances in mathematics 11 3 267 288 doi 10 1016 0001 8708 73 90011 x nielsen chuang 2010 p 527 references edit bengtsson ingemar życzkowski karol 2017 geometry of quantum states an introduction to quantum entanglement 2nd ed cambridge university press isbn 978 1 107 02625 4 holevo alexander s 2001 statistical structure of quantum theory lecture notes in physics monographs springer isbn 3 540 42082 7 nielsen michael a chuang isaac l 2010 quantum computation and quantum information 10th anniversary ed cambridge cambridge univ press isbn 978 0 521 63503 5 peres asher 1993 quantum theory concepts and methods kluwer isbn 0 7923 2549 4 rau jochen 2017 statistical physics and thermodynamics oxford university press isbn 978 0 19 959506 8 rau jochen 2021 quantum theory an information processing approach oxford university press isbn 978 0 19 289630 8 rieffel eleanor polak wolfgang 2011 quantum computing a gentle introduction scientific and engineering computation cambridge mass mit press isbn 978 0 262 01506 6 wilde mark m 2017 quantum information theory 2nd ed cambridge university press arxiv 1106 1445 doi 10 1017...