introduction, position, verication, consider, situation, verify, entity, solelyby, lo, cation, provide, inaspecic, area, example, stadium, us, imagine, haveasports, replays, spectators, smartphones, organisers, concerned, notpossible, passwords, printed, distributed, crucial, access, information, decided, devices, bytheirlocation, simplest, achieving, device, verier, victor, role, unit, reachesthe, border, provers, region, alice, shestartsthe, pro, cedure, tokenxto, token, immediately, received, timet, thenvictorwould, assume, called, distance, ounding, desirable, spherical, approach, vericationistousemultiple, veriersvi, thatareplacedaroundtheproversregion, convex, hull, veriers, locations, protocol, lethbe, function, takesrinputs, cho, ose, random, stringsxi, proverp, allxiand, calculatesy, xr, sendsyto, stringyi, theycheckif, yi, xi, alli, ifthisistrueandthebit, stringsindeed, property, soundness, regionpif, malicious, impersonate, prover, unfortunately, besecure, classical, impossibility, shown, ability, copiesofthe, multiple, isnotpossible, arbitrary, quantum, thatposition, ossible, wasshown, theoretically, secure, protocols, exist, restrictions, allowed, entanglement, storage, capabilities, oracle, model, ve, introduced, functionh, veriersvipick, stringsxiof, stringyof, encodeyin, basis, calculates, measuresin, basisand, resulty, hethensendsy, ifyes, location, bounding, vs, using, regionp, whereallxiare, theregionp, illustrated, dotted, wherexiare, solid, reachvi, contribution, acrucialpointintheproofofthepositionvericationtheorem, advantage, monogamy, phenomenon, described, describing, entangled, inuence, measurements, outcomes, therefore, achieve, measurement, outcome, systems, additional, communication, thenalice, charlie, result, stated, strategy, charliealways, referee, alices, revealed, common, theorem, adversaries, tworeceiving, listed, op, en, problem, couldbe, generalised, receiving, precision, smaller, thesis, added, dene, needed, assumption, probability, winning, intended, indeedahigher, nve, ierent, provethatfor, negligible, dierent, discuss, assumptionsneededforthe, proofs, denition, proven, formal, proof, thatis, dependent, nt, rmonogamy, intuition, missing, theproof, eded, current, progress, qubit, wo, ke, ast, qubits, ifthe, increase, theoremstraight, notdecrease, growing, roundcomplexity, error, sending, monogamygameform, ving, optimal, highest, adding, possiblein, similar, manner, thescope, gametheorem, describ, ed, generalisable, structure, rst, chapter, mathematical, preliminaries, denitions, imp, ortant, results, mainly, lack, detail, wewill, dierences, briey, serves, tolookupmathematical, chapters, anddiscusses, possibilities, generalise, versions, generalisedmonogamytheorems, sketch, sketches, underline, changed, accommodate, obtain, conclusion, introducing, mechanical, concepts, inthis, introduce, frequently, resultswithoutproof, vector, anonemptysetvwith, relations, complex, ifx, zvanda, yz, xy, xyyx, bx, ab, axay, xaxbx, byaxwe, denote, scalar, ductax, abasisofa, spacevis, elementsvi, vivsuch, elementxof, vcan, represented, combination, elements, thereare, elementsaicsuch, aivix, sndimensional, hasnelements, multiplebases, contain, nite, dimensional, letxbe, element, ann, spacevand, vnbe, ofv, thismeans, thatthereareelementsai, iaivix, sinceaiuniquely, denexwe, represent, xas, conjugate, transpose, vectorxis, wherexdenotes, ofx, ifxabi, wherea, thenxabi, wecannowdenetheinnerproductx, yof, vectors, xandyby, yxy, inner, product, numb, er, properties, letx, zbe, anda, bcomplex, yx, axbz, yax, ybz, anormis, spacev, everyx, yvanda, axax, xyxy, thatx, xis, valid, norm, measured, asd, wecallasetmametricspaceifwecancalculatethedistance, elementx, wesaythatametricspaceiscompleteif, cauchy, sequence, limit, inm, thisbasicallymeans, converges, complete, hilbert, avectorspacehwith, induced, metric, letaandbbe, matrices, tensor, producta, bofaandbis, vectorsxandyin, spaceh, suchthat, xn, yn, productx, yis, xnyn, thathis, vnis, umis, themdimensional, thenthebasisofhis, vnum, thathisnm, operators, transformationo, linear, preserves, addition, multiplication, yhanda, ao, nh, itisimportanttonoticethat, thenthereis, aone, onecorrespondencebetweenoperatorsonhandn, nmatrixes, matrix, notation, xed, recall, special, types, leth, byn, nmatrices, identity, operator, onh, thisoperatorissuchthatforeveryvectorx, operatorp, thatsatisespppis, jector, operatorpis, positive, semi, denite, vectorx, lemma, fo, xcthe, matrixccis, ifcaij, aji, rhis, hermitian, ifh, ruis, unitary, analyse, aect, eigenvectors, letobe, oxax, vectorxhanda, thenxiscalledaneigenvectorofoanda, eigenvalue, ofo, letooij, tr, oii, cyclic, operatorsa, andc, thentr, traceisalsoadditivei, aa, atr, whrere, hadamard, operatorhis, section, encountered, tabe, spacehthen, usual, allv, schatten, tabifabis, anbe, sem, de, ite, projectors, letkk[n]be, ofnmutually, orthogonal, permutations, of[n], i[n], ai, k[n], max, aiak, computing, alternatives, relation, cryptography, physics, experiment, consists, photon, source, splitters, detectors, photons, onto, splitter, located, angle, reected, bothdirections, opposite, register, graphical, description, abeamsplittercanbeconsideredasippingacoinandthenreecting, randomly, statistics, detectorb, experienceand, models, asan, ips, thestartingstate, asseeninfigure, modelled, bitcan, computers, detector, setup, reproduced, repro, duced, starting, scription, probabilistic, probabilities, theseprobabilisticbitslieonthelinebetween, zero, applyingthesecondbeamsplitterwillresultin, coincide, oftheexperiment, itlies, sphere, lies, inthequantum, whereandare, sincethese, multiplying, passing, superposition, ifweweretomeasure, spitter, thiscorrespondswiththeexperimentalresultthat, thebphoton, encoded, ofthephotons, encode, informationas, encodes, anelectron, polarisation, concernuswiththe, physical, representation, concentrate, themathematical, abstraction, mathematics, discussed, previous, dirac, orbra, ket, bra, notationis, andis, bracket, heresdenotes, hencestis, statessandt, ss, aquantumstatesis, ofh, inthefollowingwewillalways, meansthatthegiven, hilbertspaceisn, suitable, expressed, twe, thenabasis, adensitymatrixis, nmatrix, density, statesisss, itisimportantto, formss, quantumstates, formssare, outer, mixed, statical, ensemble, wecan, purify, statein, spacehby, extension, hof, thenthereisapurestate, trr, wheretrris, partial, overhr, thisisalsocalledtracingoutthe, systemhr, ments, aprojectivemeasurementisasetofprojectorspi, piare, thequantumstatescan, wewanttomeasureifthesatesis, analogously, measurementpi, thentheprobability, measuringion, statesis, pr[, isi, ]spis, mere, observations, system, disturb, thenswill, pis, spis, perform, collapses, letbe, amixedquantumstate, inthiscasetheprobabilityofmeasuringiis, pi, jective, repeatable, sincepipjijpi, moregeneralmeasurementsthatarenotrepeatable, erator, valued, anndimensional, ofkoperatorseksuch, specic, stateinacertain, computational, wesay, themeasurement, basisthebasis, aspect, bases, thismeansthatifwedonotknow, hauabax, bcx, circuit, certainty, nxn, circuits, illustrate, images, thatshowquantum, gatesandmeasurements, reversible, correspond, reversibleandtheyhave, outputs, output, outcomeand, presentationof, operations, hcwith, operatorsuab, uabcacting, onha, hcandhrespectively, measurementsaxandbcxacting, andhb, hcrespectively, lethaandhb, es, hbis, ssa, sb, wheresahaandsb, erst, measuring, measurment, basisthe, wegetthesameresultifwemeasureinthe, ofthesystem, determines, correlated, anticorrelated, monogamous, statesandare, statecannot, wediscussmonogamyofentanglementinthenext, monogamyofentanglement, thatha, hcandhfare, moreover, thathaveaccessto, spacesha, hbandhcrespectively, spacehf, consist, andadversaries, howto, stateof, thereferee, auniformlyrandombasisandrevealsittotheadversaries, andoutput, ya, yb, adversariesa, andyf, ifyaybyf, amonogamyofentanglementgameg, xconsists, spacehfand, projective, measurementsxon, thatxis, distinguish, bythestrategies, dened, analysed, original, strategies, sis, andxandxare, haandhbrespectively, wherexis, ameasurementonannqubit, corresponding, shownin, bypwin, strategyswin, gameg, maximal, sups, pwin, wheresupremumis, strategiess, although, formore, namely, berepresented, dimensions, thatpwin, heretheoptimal, sn, itiseasytosee, cos, sections, analysing, transformations, shared, registers, performed, measurementsandinsome, registersatthesame, wherex, andxare, evident, thereforepwin, adversary, guess, ethat, overlapping, xandxare, hcandha, guarantees, theresultsofthe, identical, communicate, preparing, containing, andoutputting, thisguarantees, directly, commute, measurementsx, xandx, mu, surementsxandxif, xandy, xqyypx, xandxif, commutative, xandqyrycommute, thatpxq, ypx, implies, prop, erty, impliesthe, based, type, implications, propertiesand, xbx, analogous, appendix, corollary, concerns, assumptions, reasonable, ofrestrictionon, ensures, adversariesare, transfer, assuming, modied, seegure, itis, basically, requirement, thirdadversary, outputting, alternative, looking, fairly, winthe, transferring, thewires, simply, create, paris, ofrstqubits, unfortunatelythissimplestrategyisnot, suggest, assumes, commutativeproperty, slightly, weaker, thanthe, stronger, whichrisesthe, concern, regarding, theforms, wherexandxare, spacehxand, xandywe, thataxandb, ycommute, illustration, fullling, equivalent, carried, outonseparate, andacare, hcand, andx, independent, andwe, xcommute, andacandb, xbxcx, takeab, andbcto, identities, allowedstrategyisof, letss, xbe, sinceacdoes, supplying, discard, wehave, identify, gaps, lled, explainourintuition, diculties, lling, refereefwith, measurementsxxxassume, thismeansthatthereexistsastateand, bandcwith, xandxsuch, contradiction, required, adversariesaandband, oseasjunk, wewouldthenuseabto, measurexand, basistoaandb, thecomparison, whenaandbguess, input, stateshould, related, tojunk, steering, sates, observed, correlations, deduce, existence, local, observables, functions, ofa, whichactonthebipartite, statesuch, exists, isometry, aand, statejunkfasuch, eminrelation, observable, xand, xare, paulizandxgates, thatzfandxfanticommute, lp, observablesxpandypis, forxpas, application, tbe, adversaryaguessing, correctly, adversariesbandcthe, ofandis, ierence, betweenthe, inequality, thatxaandxfcommute, operate, sincexfandzfanticommute, calculate, equation, observablesxf, xaandzawith, eigenvalues, statejunkabsuch, usingtheequalitiesabovewehave, statejunkfais, tdoes, thebcwire, preparation, ofjunkand, sbcbandbcccannot, thefwire, adversarybguesses, isometrybthat, theaca, andaccwires, showing, thef, abaandabbwires, diculty, thebcbandbccwires, withaba, acaandaccwires, situations, separated, trivially, specicplace, apositionvericationprotocolisaprotocolthatveriesprover, secureinthe, demonstrate, exactly, connecting, channelsothatthey, asapreparation, performs, calculation, hethen, channel, proto, col, insecure, toshow, setofadversaries, convince, cated, thehonestprovers, dimension, theoriginal, theyarenowbothabletocalculatef, asexpected, respectively, isat, isalsothecase, main, easily, cloning, disturbing, ifadversaries, hashfunctions, oracles, hash, theadversary, limited, query, guessingthehashvalue, ofxis, guessingxitself, thecorrect, andoverlapping, tighten, honestproversregion, precise, actions, asspacetime, spacetime, circuitsthatarelocated, species, executed, isawirefrom, gateato, gatebif, travel, fromatobwithout, defying, gionpin, inpis, vrbe, letnbe, challenges, fraction, errors, execution, xrof, theyalso, lengthn, stateby, enco, dingyin, veriervisends, bi, tstringxito, hecalculatesthebasis, heobtainstheoutcomey, event, ofpat, messagesxifrom, allvi, andsend, ifthemaliciousproveris, mostqqueries, herepis, allxican, inanylocationinspacetimethatisoutsideofpeither, ofxiis, theveriers, ofpmay, ok, wherewelooked, regionpwould, een, itwas, construct, correctyto, copying, forwarding, classicalchallenges, instantaneously, regionpbecomes, thismeansthattherewillbenogatesinpand, sub, xp, prep, wecanthereforedivideall, subcircuitintuition, prepre, computation, pgates, postafter, circuitp, preinvolves, circuitsp, postis, gure, intop, preas, prehappens, therecanbeno, leavingp, postasp, preallxiare, betweenp, ifthiswasnotthe, inp, ers, accepting, bitstringsxiof, lenght, ra, tri, gyof, lenghtn, pickarandomfunctionh, execute, ofp, asithasnoeect, ournextgoalistodelaythechoiceofthebasis, currentlyitisused, executing, delay, statey, ordiagonalbasis, uniformly, furthermorewhenperformingthe, epr, pre, picking, randomyand, stateyis, basisto, obtainy, picka, un, ti, prepareeprpairs, executecircuit, sbut, reprogram, oraclehto, completely, returnon, theproverwillnotnoticethisunlesshequeries, reprogrammed, ofthis, happening, guessing, knowing, allxi, reprogrammingh, bitstringxiof, arandomfunctionh, wiggly, reprogramh, sothat, executep, thej, th, toh, measuretheargumentxof, lettheregistersbecalleda, letarefer, enteringp, letbrefer, ofon, entering, letfbe, lowest, abe, consisting, ingeneralthesemeasurementsarepovm, loss, generality, increasing, underlying, pickrandombasis, measurementm, bandmfre, ul, weseethat, parallelly, repeated, combining, regionpis, allxiare, larger, givetheprover, calculations, timeis, regionbecomes, complicated, intersection, originating, fromviat, future, becomesmaller, reached, veriervi, wherexiis, protocolthat, bitstringyof, bitstringxito, ifthemalicious, ofxi, oneofthe, higher, emphasise, subcircuits, wewouldlike, gatep, division, preprecomputation, forp, include, twop, iand, jeach, interpret, ofyiandyj, thiswaywecanpartitionthequantumcircuit, asp, ifthiswasnot, formp, thenp, pickingarandomyand, halfof, delayed, basisbut, stringxiof, preuntil, letm, bandmcrefer, operatorsab, andac, thiswaywegetthreemeasurements, raon, spaceha, writeaif, becomesha, althoughitwould, writea, performing, wont, ensure, ifmiis, measurementsm, bandmcare, andmfre, acceptif, wesee, formalise, statement, npin, ifor, thefollowingholds, inan, interaction, andp, theprobabilitythattheveriers, aprotocol, regionpof, letpbe, rbe, events, causally, precedep, ispecies, challenge, thatp, whenviexpects, response, letn, integers, modeled, communicating, channels, precedesp, visendsxitopat, eventvi, eventp, pwill, thenpcomputes, ifso, byc, casual, eventai, eventa, inwords, thereisnoeventin, concretely, mostqoracle, queries, mostpwin, eciforc, iforc, subcircuitregion, spacetimeintuition, \iprecomputation, post\i\, preafter, specetime, circuitato, ifaandbare, eventseaandebrespectively, andeacasually, precedes, denotedea, transitive, leavingciand, enteringci, preconditions, aredisjointandtheir, union, wewillshowthatthefollowingequalitieshold, equality, thenx, whichmeansx, iandxi, reasoning, whichisacontradiction, thereforeequality, equalities, disjoint, theunionof, subset, thereforewehavetoshowthatis, thatxsuch, meansx, ppre, whichmeans, sincex, ppost, wehavexi, whichmeansthatjsuch, thatxj, rename, whichisagainacontradiction, thereforex, thatno, everyi, ij, kjthe, pij, pkj, unlesski, pjk, ifjik, herea, bdenotes, fromatob, aandbbe, bmeans, thata, bbsuch, abi, eventsaandbsuch, thatbfollowsa, assumep, thismeansthatx, jiandy\i, thatyfollowsx, thisisacontradiction, sincexi, xyimplies, withthesameargumentthereisalsonowirefrompijtop, wirep, thismeansxppost\i, andyppreisuch, thatxy, xyimpliesxi, thisargumentforalltheotherwiresin, permitted, theresultisagain, aquantumcircuitbutmaynotbeaspacetimecircuitanymore, reprogramming, protocolasasequence, funwhere, letaccept, xby, thepairs, thep, postcircuit, aecting, theupperwire, fromeprcarries, rstnqubits, lower, lastnqubits, gatemmeasuresnqubits, preparingxyfor, perfectly, indistinguishable, pairxyand, measuringyin, basiswill, calculated, returnh, theideabehindthis, queryh, afterp, thetimeof, preare, ofi, dom, algorithm, sharin, betw, vocation, thatperformatmost, qqueries, til, thenmeasuretheargumentof, orif, noj, occurs, pr[xx, executes, letbbe, executethecircuitafterthewigglyline, withoracleaccess, modiedhwithh, pcfrom, prepareeprand, wichmeans, andmcrefer, hawe, althoughit, transp, eratorsafterthe, thatm, antimes, pr[accept, guessingx, sxin, beingh, leteibe, thei, sincep, pre\, everyeithere, rsuch, thatei, thatxkis, toei, subsetsjk, qsuch, everyijkwe, ei, whichtheonlydierence, thatjis, qbut, fromji, ji, qwe, dividep, preinto, subcircuitspilowand, pihigh, letpilowbe, subcircuit, toxiandpihighthe, toxi, executingcircuitp, preis, circuitpilowand, circuitpihigh, noticethatforevery, jjimeasuring, ofpilow, sinceejis, ofci, executingpilowonly, circuitpilowuntil, argumentxof, sincexicannot, accessed, inpilowwe, intention, positionverication, showed, probabilitypwin, intuitive, attempt, proving, compared, wouldbe, ifthiswouldnotgiveaprobability, declines, growingn, thiswaywe, bound, includes, havent, considered, presented, bedoneusing, usable, currently, looked, scope, theses, vericationtheorem, personal, dominique, unruh, serge, fehr, appendices, letyxbe, onhx, thenpwin, pr, yfyabybcyac, wecancalculate, yfxyabxybcxyacx, yabxybcxyacx, ybcxyacx, yacx, dxandlet, yfyabxybcxyacx, letsabbe, measurementab, sbcthe, measurementbcandsacthe, measurementac, thenwehave, sac, sbc, sab, fromthedenitionofthenormwehavetr, maxk, permutationskare, determined, later, wherek, forxedandk, wedenotebythe, indices, whereanddier, bycwe, complement, bytthe, hamming, betweenand, whichmeanst, nowwedenethefollowing, jectors, xx, thatpand, recallthatabifb, matrixc, cis, cx, cy, andais, thereforeis, sincepandqwe, xyz, xxy, zz, xaczxzacxandxy, choosing, ow, transformationxonhxwe, namexif, transformation, hyinstead, iftheaected, context, fr, avetr, usinglemma, forthisweshowthatpis, calculatep, topthen, thatpis, yy, xxyy, yyy, xxx, operatorsaandb, whereaandbare, andab, analogouslywehaveqandr, estimate, sinceacwe, ycz, sincecxczxzcwe, xa