The bosonic birthday paradox

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1 msp Geometry & Topology Moographs 18 (2012) 1 7 The bosoic birthday paradox ALEX ARKHIPOV GREG KUPERBERG We motivate ad prove a versio of the birthday paradox for k idetical bosos i possible modes. If the bosos are i the uiform mixed state, also called the maximally mixed quatum state, the we eed k p bosos to expect two i the same state, which is smaller by a factor of p 2 tha i the case of distiguishable objects (boltzmaos). While the core result is elemetary, we geeralize the hypothesis ad stregthe the coclusio i several ways. Oe side result is that boltzmaos with a radomly chose multiomial distributio have the same birthday statistics as bosos. This last result is iterestig as a quatum proof of a classical probability theorem; we also give a classical proof. 60C05; 05A10, 81P99 The traditioal birthday paradox says that give a caledar with days, there is a sigificat chace (bouded away from 0) that a room with. p / people with uiformly radom birthdays has two with the same birthday. Aaroso ad Arkhipov [1] discuss the same paradox for radomly chose bosos. Here we preset a differet treatmet of the same problem. I fact we will preset two paradoxes". The first result (which Aaroso ad Arkhipov derived, i a less geeral form) is that although bosos prefer to have the same birthday, they have the same asymptotic behavior i the birthday problem, up to costat factors, as distiguishable particles (boltzmaos). The secod result is that they have exactly the same behavior, o-asymptotically, as i.i.d. boltzmaos whose commo distributio is a radomly chose poit i the simplex of all distributios o cofiguratios. This leads to a iterestig result i classical probability with a quatum probability proof. We assume that the Hilbert space for oe particle is H D C. We assume a self-adjoit birthday operator BW H! H with eigevalues 1; 2; : : : ; i some basis. The Hilbert space of k bosos is the the symmetric power S k.h/ Š C.. k// ; Published: 14 October 2012 DOI: /gtm

2 2 Alex Arkhipov ad Greg Kuperberg usig the multiset coefficiet otatio (1) k def D C k 1 I the termiology used for idetical particles, the states of a basis of H are called modes. I the traditioal versio of the classical birthday problem, we assume the uiform distributio o all k choices of the birthdays of the k people. The uiform distributio uif.x / o ay fiite set X ca be characterized i either of two ways: It is the uique distributio with the most etropy, log jx j; ad the uique distributio with the most symmetry, Sym.X /. We will cosider a aalogue of the uiform distributio for a quatum system with a Hilbert space H: the mixed state uif.h/ whose desity matrix is the scaled idetity o H. Like the classical state uif, the quatum state uif.h/ is the uique state o H with the most etropy, log dim H; ad the uique state with the most symmetry, U.H/. Moreover, uif.h/ is the uique state that yields the distributio uif.x / for ay complete measuremet that takes values i a set X. We will use the uiform state uif D uif.s k.h// o the joit Hilbert space of k bosos. The, the measuremet S k.b/ of all birthdays of uif yields the uiform distributio uif o cofiguratios of k ulabelled people with possible birthdays. (It is also stadard to refer to ulabelled balls i labelled boxes, but we will stick to the birthday metaphor.) Moreover, this particular uiform state ca be justified usig less symmetry tha the largest available uitary group U.S k.h//: Propositio 1 The state uif o S k.h/ is the uique state which is ivariat uder the uitary group U.H/. Proof Suppose that is a U.H/ ivariat state o S k.h/, ie, a U.H/ ivariat desity operator. Schur s Lemma says that if V is a irreducible complex represetatio of a group G, the every G ivariat operator o V is proportioal to the idetity. Thus it is sufficiet (ad also ecessary, if either V is uitary or G is compact) for V to be irreducible. It is a stadard fact of represetatio theory, Fulto ad Harris [3, Sectio 6.1], that S k.c / is a irreducible represetatio of GL.; C/. It is aother stadard fact [3, Sectio 26.1] that GL.; C/ ad U./ have the same irreducible represetatios, sice the former is the complexificatio of the latter. This symmetry implies that uif is the U.H/ average of ay state, sice such a average must be ivariat with respect to the actio of U.H/. k : Geometry & Topology Moographs, Volume 18 (2012)

3 The bosoic birthday paradox 3 Corollary 2 Puttig k bosos i ay state o S k.h/, ad the applyig a Haarradom uitary matrix i U.H/ yields the state uif. Aaroso ad Arkhipov cosider such a average for a particular choice of, where is the pure state j i D j1; 2; 3; : : : ; ki i which the k bosos are i distict modes (which requires k ). Aother choice cosidered below is j i D j1; 1; 1; : : : ; 1i i which the bosos are all i the same mode. There are may choices for, but Corollary 2 says that they all become the same whe they are averaged. We will ow look at the asymptotics of j fold birthdays i uif. We will use the otatio f./ g./ to mea that f./=g./! 1, or equivaletly that f./ D g./.1 C o.1//. Theorem 3 Suppose that there are k bosos with modes, suppose that they are i the uiform state uif, ad suppose that k c.j 1/=j as! 1, for some iteger j 2 ad some costat c > 0. The the umber of j fold birthdays coverges i distributio to a Poisso radom variable with mea c j, while the umber of.j C 1/ fold-or-more birthdays coverges to 0. This is the same asymptotic aswer as i the case of boltzmaos, except that the mea i that case is c j =j!. I fact, our argumet i the case of bosos is very similar to a stadard argumet i the case of boltzmaos. Proof Recall that the joit measuremet S k.b/ of all of the birthdays yields the uiform distributio o k ulabelled people amog caledar days. The probability that the first birthday has at least j C 1 people is k jc1 k j 1 k. C k/ jc1 for fixed j ad ; k 1. Takig k D O..j 1/=j / ad summig over all days, the expected umber of.j C 1/ fold-or-more birthdays is O. 1=j /, which vaishes as! 1. Meawhile the probability that the first ` days each have at least j people is k j ` k D j` Y1 ad0 k a C k a k j`. C k/ ; j` Geometry & Topology Moographs, Volume 18 (2012)

4 4 Alex Arkhipov ad Greg Kuperberg where the approximatio holds for fixed j ad ` ad ; k 1. Summig over all ` ` choices of the ` days, we obtai that if X is a radom variable represetig `! the umber of j fold birthdays, the X cj` E ` `! : So i the limit, the `th factorial momet is c j`, which the same aswer i the limit as a Poisso radom variable with mea c j. To coclude the argumet, the Poisso distributio is determied by its momets. The calculatio for the arrow questio of the probability of at least oe repeated birthday is simpler. The probability that all of the birthdays are distict is k k D ky 1 ad0 1 a 1 C a e k2 = as log as k D o. 3=4 /. The approximatio is established by takig the logarithm of both sides ad the applyig the Taylor series estimate l 1 x 1 C x D 2x C O.x3 /: Corollary 4 For modes, we eed k p l 2 bosos to expect a repeated birthday with majority probability. This differs by oly a costat factor from the k p 2 l 2 people eeded to expect a repeated birthday i the classical birthday problem with distiguishable people. Remark We should say somethig about idepedet but o-uiform bosos. The otio of idepedece for bosos is subtle. Oe reasoable ad widely used otio is to first choose a distributio for the birthdays of oe boso, ad to model it by a diagoal desity matrix i the birthday basis. The there is a uique distributio o k bosos such that if k 1 of the bosos are fixed, the coditioal distributio of the last oe is give by. This distributio is also a thermal state, also kow as a Maxwell Gibbs state, for o-iteractig bosos. It was discovered by Bose ad Eistei that uder fairly mild assumptios o, almost all of the bosos have the most likely birthday. This paradox is commoly kow as Bose Eistei codesatio. Corollary 2 implies a iterestig secod model for the joit distributio of birthdays of k bosos. Geometry & Topology Moographs, Volume 18 (2012)

5 The bosoic birthday paradox 5 Theorem 5 The joit birthday distributio of k bosos i the uiform state uif is idetical to the average of k i.i.d. boltzmaos, if their commo distributio is give by a uiformly radom poit i the simplex of distributios o the birthdays. By combiig with the iduced uiform distributio o the birthday measuremet, we obtai a corollary of Theorem 5 that equates two distributios i classical probability. Corollary 6 Cosider a tow i which all families first agree to have childre accordig to a commo distributio o the days of the year, which itself is chose uiformly from the simplex of all distributios. The the childre s birthdays behave as if the childre were ulabelled, ie, if we make a table that oly gives the umber of childre bor o each day, the all such tables are equally likely. I other words, the uiform average of all multiomial distributios o multisubsets of size k i a set of size, is the uiform distributio o multisubsets. Proof of Theorem 5 Recall that the Hilbert space of k boltzmaos is H k. Cosider the state D.j ih j/ k, first for some fixed choice of j i 2 H. This yields idepedetly distributed birthdays for the k boltzmaos, ad the distributio of each oe is give by the measuremet of oe copy of j i. Meawhile, is evidetly a pure symmetric state, which meas that these boltzmaos are also bosos. By Corollary 2, the average of all choices of, with respect to Haar measure o U.H/, is the bosoic state uif. The Haar distributio of j i, or equivaletly oe colum of a matrix i U./, is give by Haar measure o the maifold of pure states CP 1. The iduced distributio of the birthday measuremet is give by the momet map mw CP 1! 1 to the simplex of distributios o cofiguratios, Caas de Silva [4, Sectio 6.4]. This momet map preserves ormalized measure [4, Sectio 6.6]. Thus a radom choice of amouts to a radom distributio o each birthday, draw uiformly from the simplex of distributios. This establishes the claim of the theorem. Theorem 5 yields a quatum proof of a classical probability result, Corollary 6. We also obtaied a classical proof of the same result. Classical proof of Corollary 6 The argumet uses a variatio of the stars-ad-bars otatio for multisets, Feller [2], that is also used to prove the idetity (1). Namely, we write a star for each of the k childre, with 1 separatig bars betwee the Geometry & Topology Moographs, Volume 18 (2012)

6 6 Alex Arkhipov ad Greg Kuperberg caledar days. For example, if there are k D 4 childre ad D 6 birthdays, the oe possible choice for all of the birthdays is?? j? jj? jj; i which two childre are bor o the first day, oe o the secod day, oe o the fourth day, ad oe o the other days. We first choose locatios of 1 bars idepedetly ad uiformly o the uit iterval I D Œ0; 1. This separates the iterval ito subitervals of legth p 1 C p 2 C C p D 1; ad we claim that the legths of these subitervals are give by a uiformly radom poit i the simplex of distributios. (Because, if we first take the bars to be umbered, they are distributed accordig to uiform measure o Œ0; 1 1. The, erasig the umbers yields the quotiet Œ0; 1 1 =S 1, which is a simplex ad also has uiform measure. The, takig the differeces of successive poits to obtai the probabilities p j is a liear isomorphism, which also preserves uiform measure.) The, if each child s birth is represeted by a star which is also at a uiformly radom positio i Œ0; 1, the probability of the j th birthday is exactly p j, the legth of the j th iterval. We ote that the orderig of the stars ad bars determies the umber of childre with each birthday. We claim that these multiset choices are all equally likely, as if the childre had bee bosos (with o distiguishig state other tha the date of birth). This is made clear if we equivaletly choose 1Ck poits idepedetly from I all at oce, ad the choose a radom subset of 1 poits to be the bars ad the other k poits to be the stars. These k D 1Ck k equally likely choices exactly correspod to a multiset choice of k ulabelled childre distributed amog days, as claimed. We coclude with a versio of the birthday paradox for fermios. Theorem 7 (Pauli) Give k fermios i ay state o the exterior power ƒ k.h/, there is o chace that ay two have the same birthday. We leave the questio of a ayoic birthday paradox, icludig o-abelia ayos, as a topic for future work. Ackowledgmets The authors would like to thak Scott Aaroso for suggestig the problem. Alex Arkhipov is supported by a Akamai Foudatio Fellowship. Greg Kuperberg is partly supported by NSF grat CCF Geometry & Topology Moographs, Volume 18 (2012)

7 The bosoic birthday paradox 7 Refereces [1] S Aaroso, A Arkhipov, The computatioal complexity of liear optics, from: Proceedigs of the 43rd aual ACM symposium o Theory of computig, STOC 11, ACM (2011) [2] W Feller, A itroductio to probability theory ad its applicatios. Vol. I, third editio editio, Joh Wiley & Sos, New York (1968) MR [3] W Fulto, J Harris, Represetatio theory, Graduate Texts i Mathematics 129, Spriger, New York (1991) MR [4] A Caas da Silva, Symplectic geometry, from: Hadbook of differetial geometry. Vol. II, Elsevier/North-Hollad, Amsterdam (2006) MR Departmet of Computer Studies, MIT, Cambridge MA 02139, USA Departmet of Mathematics, UC Davis, Davis CA 95616, USA arkhipov@mit.edu, greg@math.ucdavis.edu Received: 8 December 2011 Revised: 14 March 2012 Geometry & Topology Publicatios, a imprit of mathematical scieces publishers msp

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