Quantum emulation of classical dynamics

Quan tum em ulation of classical dynamics Norman Margolus ∗ Massachusetts Institute of T e chnolo gy (Dated: No vem ber 27, 2024) In statistic al me chanics, it is wel l known that finite-state classic al lattic e mo dels c an b e r ec ast as quantum mo dels, with distinct classic al c onfigur ations identifie d with ortho gonal b asis states. This mapping makes classic al statistic al me chanics on a lattice a sp e cial c ase of quantum statistic al me- chanics, and classic al combinatorial entr opy a sp e cial c ase of quantum entr opy. In a similar manner, finite-state classic al dynamics c an b e r e c ast as finite-energy quantum dynam- ics. This mapping tr anslates c ontinuous quantities, c onc epts and machinery of quantum me chanics into a simplifie d finite-state context in which they have a pur ely classic al and c ombinatorial inter- pr etation. F or example, in this mapping quantum aver age energy b e comes the classic al up date r ate. Interp olation the ory and c ommunication the ory help explain the truce achieve d her e b etwe en p erfe ct classic al determinism and quantum unc ertainty, and b etwe en discr ete and c ontinuous dynamics. INTR ODUCTION In this pap er w e discuss a mapping b et w een classical and quantum systems that lets us regard quan tum dy- namics as a generalization of finite state classical dynam- ics, and that allows us to identify equiv alen t quan tities and concepts in classical and quantum systems. A similar mapping has long b een known in statistical mec hanics [1] that establishes classical lattice models and their com binatorial entrop y as simple examples of quan- tum statistical mechanics. There is an obvious candidate for the comparable dy- namical mapping: classical computations are equiv alen t to a subset of quan tum computations [2]. Most w ork on quantum computation is, how ever, based on hybrid classical/quan tum models in whic h macroscopic classical op erations con trol the sequencing of quan tum operations. Suc h systems do not pro vide a purely quan tum target for a classical/quantum mapping. Instead, early w ork show- ing that autonomous quantum systems can p erform clas- sical computation [3] forms the basis for the dynamical mapping presen ted here. This mapping allo ws ph ysical quan tities such as energy and momentum to be iden tified with finite-state classical quan tities, with the aid of classical interpolation theory . Related issues are addressed in [4], but a general dynam- ical mapping is not provided there. As a preliminary to discussing dynamics w e first review a canonical metho d for mapping classical lattice mo dels on to quantum lattice mo dels in statistical mec hanics. ST A TISTICAL MECHANICS In statistical mechanics, it is well known that classical lattice mo dels can b e recast as quantum mo dels, with distinct classical configurations identified with orthogo- nal basis states [1]. Consider, for example, the well known ferromagnetic 2D Ising model. In this mo del eac h of M lattice sites in a square lattice is o ccupied by a classical tw o-state “spin,” and eac h state S n of the N = 2 M p ossible configurations of the lattice is assigned a classical configurational en- ergy E classical n that dep ends only on how many pairs of adjacen t lattice sites hav e the same spin v alue and how man y hav e opp osite v alues. A quantum lattice model corresp onding to suc h a clas- sical lattice model can be constructed b y iden tifying eac h of the N distinct classical states S n with a distinct basis v ector | n i in an N dimensional Hilb ert space. A hamilto- nian op erator H is defined by taking each configuration state | n i to b e an energy eigenstate of H with energy eigen v alue E classical n : H | n i = E classical n | n i . (1) In quantum statistical mechanics the energy eigen- states are also eigenstates of the density operator ρ , with eigen v alues that giv e the statistical weigh t to attach to eac h energy eigenstate. F or example, for a canonical en- sem ble of quan tum mec hanical systems, ρ is prop ortional to e − β H . F rom (1) this becomes the usual classical Boltz- mann factor when applied to a configuration state | n i , and quan tum statistical mechanics reduces to classical. CLASSICAL DYNAMICS Since the definition of H used in the statistical mec han- ics mapping makes eac h classical configuration a time- in v arian t state under unitary time evolution, w e use a differen t definition of H to emulate classical dynamics. Finite-state dynamics An inv ertible classical finite-state dynamics is a dis- crete idealization of classical dynamics [5]. Perfect digi- tal degrees of freedom are up dated at discrete times ac- cording to a sequence of inv ertible transformations. The total amount of state in the system, including that used 2 to define the dynamics, is finite. Here w e take the time b et ween update ev en ts to alw ays be τ , so that the system is up dated at the constant rate ν = 1 /τ . The finite set of p ossible configurations of the system is partitioned b y the in v ertible dynamics in to a collec- tion of disjoint dynamic al orbits , with eac h dynamical orbit consisting of a set of configurations that turn into eac h other under the dynamics ( cf. [6, 7]). F or each dy- namical orbit d the num b er of configurations N d in the orbit determines the p eriod T d = τ N d of the orbit. One configuration of each orbit is lab eled with the integer 0. The configuration obtained from 0 by one up date step is lab eled 1, and so on. W e identify configuration n of dynamical orbit d with the basis state | n, d i . Because the orbit is p eriodic, | 0 , d i = | N d , d i . This mapping identifies each possible configuration of the classical dynamics with a basis state: w e call this the configuration basis. Hamiltonian dynamics Giv en an in vertible classical finite-state dynamics, we construct a contin uous quantum hamiltonian dynamics isomorphic to the classical dynamics at regularly-spaced times. W e b egin b y defining a discrete F ourier trans- formed set of basis states. Let | E : m, d i = 1 √ N d N d − 1 X n =0 e 2 π inm/ N d | n, d i (2) for integer m , where “ E ” is the name of the new basis. The inv erse transformation is | n, d i = 1 √ N d N d − 1 X m =0 e − 2 π inm/ N d | E : m, d i . (3) W e define a hamiltonian H by assigning the | E : m, d i states to b e its energy eigenstates and E m,d = m h/T d to b e the corresp onding energy eigenv alues [26]: H | E : m, d i = m h T d | E : m, d i . (4) If we let U = e − i H τ / ~ b e the time evolution op erator for the time interv al τ , then U | n, d i = 1 √ N d N d − 1 X m =0 e − 2 π i ( n +1) m/ N d | E : m, d i = | n + 1 , d i . (5) Av erage energy The configuration state (3) is a uniform sup erposition of all N d energy eigenstates | E : m, d i with eigenv alues mh/T d , and so the av erage energy is E = h ( N d − 1) 2 T d . (6) W e’ve taken E 0 ,d = 0 in the construction ab o ve, but the fact that the system has a harmonic-oscillator-lik e en- ergy sp ectrum suggests that we should really add h/ 2 T d to all the energy eigenv alues. This is in fact the small- est energy allow ed b y quantum distinguishability b ounds, assuming the ground state energy of a muc h larger sys- tem encompassing this one sets the zero of the energy scale [9]. Adding h/ 2 T d mak es the av erage energy (6) indep enden t of T d , E = hν 2 . (7) This is the least p ossible av erage energy compatible with a dynamics that trav erses distinct states at the a verage rate ν [9]. Th us our construction is energetically ideal, and the a verage energy is identified with the classical up date rate of the finite-state dynamics. If a lattice dynamics is up dated sequentially—one lo- cation at a time in a rep eating cycle—the frequency with whic h a given lo cation is up dated determines a lo cal en- ergy . T otal up date frequency (total energy) is the sum of the lo cal frequencies [27]. Differen t kinds of up dates ( e.g. , ones in v olving particle or b ond motion, and ones that don’t [12, 13]) define different kinds of energy [4]. F or a large system with a very long perio d, h/ 2 T d ≈ 0, and so for simplicity we will revert to taking E 0 ,d = 0 in the remainder of the discussion. BANDLIMITED ST A TES W e hav e pro vided a prescription for constructing a con tinuous-time quan tum hamiltonian description for an y in v ertible classical finite-state dynamics—turning discrete-time mo dels into contin uous-time mo dels. This construction can b e regarded as an application of band- limited in terp olation theory [14]. Bandlimited dynamics Let us choose our unit of time suc h that τ = 1, so that our configuration basis states are simply the states seen in the dynamics at integer v alues of time starting from | 0 , d i . A t a contin uous moment of time t the state is | t, d i = e − i H t/ ~ | 0 , d i = 1 √ N d N d − 1 X m =0 e − 2 π itm/ N d | E : m, d i , (8) 3 � 4 � 2 0 2 4 0.2 0.4 0.6 0.8 1.0 FIG. 1: | S ( N , u ) | 2 probabilit y distribution (solid) v ersus nor- malized gaussian of unit height (dashed). whic h is just (3) with t replacing n . W e can express the con tinuous-time state | t, d i as a function of the N d in teger-time states | n, d i by replacing | E : m, d i with its definition (2): | t, d i = N d − 1 X n =0 S ( N d , n − t ) | n, d i (9) where S ( N , u ) = 1 N N − 1 X m =0 e 2 π imu/ N . (10) The function S ( N d , n − t ) equals the Kronec ker delta δ n,t for integer v alues of t b et w een 0 and N d − 1 but is also defined for non-integer v alues. S ( N , u ) is a p eriodic ver- sion of the sinc function [15], which is the foundation of bandlimited in terp olation theory: S ( N , u ) = 1 for integer v alues of u that equal 0 mo dulo N and S ( N , u ) = 0 for other in teger v alues of u . In fact, if w e sum the geometric series we reco ver sinc times a phase for large N , lim N →∞ S ( N , u ) = e iπ u sin π u π u . (11) A p ortion of the probabilit y distribution | S ( N , u ) | 2 is sho wn in Figure 1 for N = 100 (solid). Near its cen- ter it is approximately gaussian (dashed). Reconstruction from samples Using S ( N , u ), any p eriodic function f ( t ) with p erio d T and a bandlimited F ourier sp ectrum with N frequen- cies can be reconstructed from N equally spaced samples. Because of the p eriodicity all frequencies must be integer m ultiples of 1 /T , and if the lo west frequency is 0, then f ( t ) = N − 1 X m =0 a m e 2 π imt/T (12) for some set of a m . Using τ = T / N , f ( t ) is also given by f ( t ) = N − 1 X n =0 f ( nτ ) S ( N , t τ − n ) . (13) This is obviously true at the N sample times t = nτ and so it m ust be true at all times, since S ( N , t τ − n ) is comp osed of the same frequency comp onen ts as f ( t ), and the N co efficien ts a m are completely determined by the v alues of f ( t ) at the N sample times t = nτ (in fact, the a m ’s are the F ourier transform of the f ( nτ )’s). If the lo west frequency is k /T rather than zero, use S k ( N , u ) = e 2 π iku/ N S ( N , u ) instead of S ( N , u ) ab o v e. Th us (9) can b e regarded as an exact reconstruction of a con tinuous but bandlimited dynamics in Hilb ert space from N d samples. The bandlimit on the energy sp ec- trum erases the distinction b et w een con tinuous-time and discrete-time dynamics (and field op erators [16]), since a bandlimited p erio dic function is completely determined b y a finite num b er of sample p oin ts. If g ( t ) has the same p eriod and bandwidth as f ( t ) (per- haps with a different lo west frequency) then (13) implies 1 T Z T 0 dt f ( t ) g ( t ) = 1 N N − 1 X n =0 f ( nτ ) g ( nτ ) , (14) and so a bandlimit also erases some of the distinction b e- t ween contin uous and discrete analysis of the dynamics. CONTINUOUS ISOMORPHISM Rather than just hav e integer time states of a classical finite-state dynamics corresp ond to in teger time states of a quantum finite-energy dynamics, we can also extend the classical finite-state dynamics to in termediate times and ha ve the tw o systems b e isomorphic at all times. Con tinuously extended dynamics In classical finite-state lattice dynamics it is often use- ful to imagine that, when a 1 represen ting a particle hops from one lattice site to another, it mo ves contin uously in b et w een. This extension of the dynamics allows us to extend classical-mechanical conserv ations asso ciated with con tin uous spatial symmetries to discrete particle motion in order to define, for example, momentum con- serving lattice gases [28]. Con tinuously extended lattice dynamics hav e a con- tin uous evolution in b oth time and space but, at every 4 momen t, only a finite amoun t of state: if there are n sp ots in space that can hav e a 1 or not at integer times, there are still only n sp ots that can at non-integer times. Since these n bits don’t change their v alues while they’re mo ving betw een in teger lo cations, the non-in teger-time states are really just a fixed sequence of rearrangements of the bits of the integer-time state. These extra inter- mediate states are distinct classically since the bits are in differen t places but they are redundant informationally . Note that a contin uously extended lattice dynamics can still be described as a repeated cycle of local updates, but in this case eac h up date mo ves a bit only infinitesi- mally . After any finite interv al of time all of the bits will ha ve mov ed by equiv alen t amounts. Con tinuously extended isomorphism In a contin uously extended classical lattice dynamics, an y unit-time separated sequence of states pro vides a complete description of the logical dynamics: since the bits of state don’t change b et w een integer times, exactly when we sample them do esn’t matter. Similarly , any unit-time separated sequence of states from the contin uous unitary evolution (8) constitute a complete orthonormal basis set, since (9) implies h t 0 , d | t, d i = S ( N d , t 0 − t ) . (15) Th us we are free to define a distinguished basis at any time t consisting of the unit-time separated set of N d states from the evolution (8) that includes the current state | t, d i . If w e identify these basis states with cor- resp onding unit-time separated classical configurations, then the classical and quantum dynamics are isomorphic at all times. In analyzing finite-state dynamics, the | t, d i ’s act m uc h lik e a complete contin uous basis since, again from (9), Z T d 0 dt | t, d ih t, d | = N d − 1 X n =0 | n, d ih n, d | = I . (16) Moreo ver, the inner pro duct (15) acts like a Dirac delta function in an integral with a bandlimited function f ( t ). F rom (14), Z T d 0 dt f ( t ) h t 0 , d | t, d i = f ( t 0 ) . (17) The contin uously extended isomorphism can be used to compute a v erage v alues for op erators, such as momen- tum, defined on contin uous sets of configurations. CONTINUOUS HAMIL TONIAN Rather than use N d orthonormal quan tum states to de- scrib e a classical orbit with N d informationally distinct configurations, it is sometimes conv enient to use more. In the contin uous-basis limit this yields a con tinuous- hamiltonian description. Ov ersampled dynamics Supp ose that, starting with a classical finite-state dy- namics, w e add M − 1 redundant intermediate-time states in the unit interv al betw een eac h pair of consecutive in teger-time states. Eac h orbit d of the corresponding quan tum dynamics (generated by the hamiltonian H M ) no w visits M N d basis states rather than just the N d of the original dynamics (generated b y H 1 ), and the state of the new H M dynamics at a contin uous mome n t of time b ecomes, from (9), | t, d, M i = M N d − 1 X k =0 S ( M N d , k − M t )   k M , d, M  , (18) where the basis state   k M , d, M  has b een lab eled b y the time k/ M when it is reached in an evolution starting from | 0 , d, M i . Since this extended dynamics trav erses distinct states at a rate ν M that is M times the original rate ν , it has M times the av erage energy . As the num- b er of intermediate states added in a fixed time p eriod go es to infinit y , the hamiltonian H M approac hes a con- tin uous hamiltonian H ∞ and the av erage energy of the state | t, d, M i go es to infinity . Bandlimited basis By putting a bandlimit on the energy sp ectrum of the configuration basis states we can mak e the H M dynamics isomorphic to the original H 1 dynamics, with the same a verage energy: a bandlimit on energy can correct for an o versampling of the underlying classical dynamics. The F ourier transform relationship (2) b et w een energy eigenstates and configurational basis states is left un- c hanged but we construct, in addition, a new set of band- limited configurations | n, d, M i N d whic h are the F ourier transforms of the low est N d energy eigenstates of H M , | n, d, M i N d = 1 √ N d N d − 1 X m =0 e − 2 π inm/ N d | E : m, d, M i , (19) with n an integer. These states constitute an orthonor- mal basis for bandlimited superp ositions of configura- tions. They hav e the same av erage energy as the con- figuration basis states of the H 1 dynamics: the amount of time T d tak en for one p erio d of the orbit is b eing k ept constan t, and so from (4) the first N d energy eigenv alues mh/T d of H M are the same as for H 1 . The con tin uous time states | t, d, M i N d that ev olv e from | 0 , d, M i N d are given by (19) with n replaced b y t . As in 5 (9) they ob ey | t, d, M i N d = N d − 1 X n =0 S ( N d , n − t ) | n, d, M i N d , (20) so the evolution of bandlimited states is isomorphic with that of | t, d i . Moreov er, from (19) with n → t and ex- pressing | E : m, d, M i in terms of the M N d configura- tional basis states using (2), | t, d, M i N d = 1 √ M M N d − 1 X k =0 S ( N d , k M − t )   k M , d, M  (21) = 1 √ M M − 1 X m =0 N d − 1 X n =0 S ( N d , n + m M − t )   n + m M , d, M  . (22) The bandlimited state is, at all times, an equally w eighted sup erposition of M equiv alen t states, each of whic h corresponds to the extended classical configuration at time t represented in a different unit-time separated basis. Thus the corresp ondence of | t, d, M i N d to classical configurations is the same as for | t, d i . The state (21) is a sum o ver configurations separated in time b y du = 1 / M . If we normalize eac h configuration state to length √ M instead of to length 1, this b ecomes delta-function normalization in the limit M → ∞ and | t, d, ∞i N d = Z T d 0 du S ( N d , u − t ) | u, d, ∞i . (23) F rom this and (15), h t 0 , d, ∞| t, d, ∞i N d = h t 0 , d | t, d i , (24) and so we can use the isomorphic | t, d i states to deter- mine amplitudes in the contin uous configuration basis. P AR TICLE MOTION A classical finite-state lattice dynamics is naturally de- scrib ed as a rep eating sequence of inv ertible gate op er- ations [18]. In mapping this onto a quantum dynamics, the classical mo del can b e implemented isomorphically as a sequence of lo cal unitary op erations. F undamental physics is, ho wev er, normally describ ed as particle dynamics. T o make contact with this view- p oin t we can recast finite-state lattice dynamics as parti- cle mechanics, fol lowing the motions of individual 1’s as if they wer e distinguishable p articles . Single particle Consider a classical lattice dynamics in which a single particle, represented by a 1, hops in the + x direction from lattice site to adjacent lattice site at a constan t rate, with av erage sp eed v = 1. The motion is p eriodic in space, trav ersing N lattice sites in a distance L b efore rep eating. At t = 0 the particle is at x = 0. F or this classical evolution, we can take the state of the system to b e the integer p osition n of the 1 at inte- ger time n . In an isomorphic H 1 quan tum ev olution, the distinct classical configurations b ecome integer-position basis states | n i . F rom (9) w e get a description of inter- mediate configurations in terms of integer-time ones, | x i = N − 1 X n =0 S ( N , n − x ) | n i , (25) where | x i is the configuration obtained by evolving for a time t = x/v from the configuration | 0 i . W e identify the non-integer | x i with the non-in teger p ositions of the con tinuously extended dynamics. In the quantum description of a classical particle at a non-in teger p osition v t mo dulo L , there is some ampli- tude for the particle at more than one integer p osition. F rom (15) and using (24) we can interpret ψ ( x, t ) = h x | v t i = S ( N , x − v t ) (26) to be the amplitude to find the particle at an y contin uous p osition x at time t , and compute the a verage momentum directly from ψ ( x, t ). Alternativ ely , w e can instead start with an infinite- dimensional quantum hamiltonian that generates a con- tin uous shift in space in the + x direction at sp eed v : H ∞ = v p , with p = − i ~ ∂ ∂ x . (27) The direction of the shift is apparent from noting that H ∞ ψ = i ~ ∂ ψ /∂ t implies ψ ( x, t ) = ψ ( x − v t, 0) [29]. Now w e can make this dynamics isomorphic to the H 1 discrete shift by bandlimiting the initial state so that the evolu- tion only tra v erses N distinct states in the width L . Then from (23) the state corresponding to a classical particle at p osition v t in the p osition basis is ψ ( x, t ) = S ( N , x − v t ), with energy E = hN / 2 T just as in H 1 . F rom (27), p = E v = h 2 λ , (28) where λ = L/ N . The state S ( N , x − v t ) achiev es a general b ound λ ≥ h/ 2 p on the a v erage separation of distinct states of a moving particle [9]. Since this description applies to an y particle shifting uniformly in a lattice dynamics, (28) giv es the corre- sp onding momentum. Of course only lattice up date op- erations that ac tually mo ve a particle contribute to the shift-energy E = v p p ortion of its total energy [30]. Classical mechanics W e can often consider a classical lattice-gas dynamics to b e a discrete-time sampling of an idealized classical- 6 mec hanical particle dynamics [5, 21] that ob eys Hamil- ton’s equations, ∂ H ∂ q j = − dp j dt , ∂ H ∂ p j = dq j dt . (29) T o make the lattice dynamics run faster by a factor κ w e reduce the interv al betw een the discrete even ts, τ → τ /κ . F rom (29), this can b e accomplished b y letting H → κH , whic h is exactly the energy scaling required b y (7). W e can’t just rescale τ arbitrarily while k eeping the p j ’s and q j ’s unc hanged, how ever, because particle veloc- ities are limited b y the sp eed of light. W e can, instead, run the dynamics faster by putting the discrete even ts closer together in b oth time and space, leaving veloci- ties unc hanged. If the distance b et ween ev ents λ → λ/κ , then the scale of the p j ’s m ust b e multiplied by κ to get an ov erall scaling of H b y κ in (29). This is exactly the momen tum scaling required by (28). Indistinguishable particles T reating 1’s in a classical finite-state lattice dynamics as distinguishable particles—and keeping trac k of the dis- crete p osition and velocity of each 1—dramatically ov er- represen ts the num b er of distinct states: all states with the same spatial pattern of 1’s and velocities corresp ond to a single state of the original lattice mo del. W e can fix this ov er-represen tation in a quantum description of the distinghishable particle dynamics b y merging equiv a- len t states, adding them together to form new o c cup ation numb er basis states, and using only these to describ e the ev olution. If w e antisymmetrize each sum under parti- cle in terchange, the new basis states will each hav e at most one 1 with a given p osition and velocity—w e can symmetrize instead to allow more [19]. T o describ e a dynamics in which the num b er of ones c hanges with time, we can use creation and annihilation op erators to add and remo ve particles from the state, while maintaining symmetrization. These field op erators inherit fermionic or b osonic commutation rules from the symmetrization [31]. As we see from (25) (or from (20) for H ∞ ), a finite set of bandlimited basis states allows a particle to b e added centered at any con tinuous p osition in space. In one dimension with one velocity , for example, the creation op erator Ψ † ( x ) for any x is a sup erposition of the creation op erators Ψ † ( n ) for integer p ositions n , Ψ † ( x ) = N − 1 X n =0 S ( N , n − x ) Ψ † ( n ) . (30) Of course nothing essenti al is gained by using a con- tin uous space and time description, since a bandlimited con tinuous state is completely determined b y its v alues at discrete positions and times. Similarly , nothing es- sen tial is gained b y introducing fermionic field op erators: there w ould b e no need to main tain the antisymmetry of equiv alent states if the original dynamics were describ ed isomorphically in terms of lo cal unitary op erations [32]. UNCER T AINTY The particle described by (26) mo v es at a constant sp eed and is lo calized to a single p osition basis state of a finite-dimensional basis at all times ( cf. [22]). This in no w ay conflicts with the uncertaint y relations of quantum mec hanics, which can be regarded as b ounds on repre- sen ting information using limited bandwidth. Bandwidth b ounds Constrain ts on time or p osition determine the mini- m um width of the energy or momentum eigenfrequency distribution needed to describe a state that meets the constrain ts. In the usual uncertaint y b ounds we also as- so ciate a width with the time or p osition amplitude dis- tribution [23], but in general other constrain ts on time or p osition can b e used to determine a minimum width of energy or momentum eigenfrequencies. F or example, supp ose we ha v e an exactly p eriodic evo- lution with p erio d T . The state at time t can b e written | t i = X a n e − 2 π iν n t | E n i . (31) Exact p eriodicity requires that eac h ν n = E n /h b e an in- teger multiple of 1 /T . If this evolution passes through N m utually orthogonal states, then the sup erposition must in volv e at least N differen t | E n i ’s (since you can’t con- struct N distinct states out of fewer than N distinct states). Moreov er, there m ust also be at least N dis- tinct frequencies (since groups of | E n i ’s with the same frequency act like a single eigenstate in the construc- tion). T o hav e N distinct frequencies that are integer m ultiples of 1 /T , the bandwidth B (highest frequency in the sup erposition minus low est) must ob ey B ≥ N − 1 T . (32) This is a version of the bandwidth-time theorem of com- m unication theory [24]. If we let τ = T / N be the av er- age time b et ween distinct states, we see that this is also a version of the time-energy uncertaint y relation, using B directly rather than some other measure of the width of the energy eigenfrequency distribution. The definition (3) achi eves this b ound. Second-momen t b ounds In constructing uncertaint y b ounds, the standard de- viation of the eigenfrequency distribution is traditionally 7 c hosen to measure its width. This c hoice reflects b oth familiarit y from statistics and (for p osition and momen- tum) a simple connection betw een the comm utation rela- tion and the standard-deviation b ound [25]. This c hoice is often divergen t, how ever, and so fails to provide a use- ful b ound [23]. This is true in our case. Consider the bandlimited state h x | ¯ x i = S ( N , x − ¯ x ) cen tered at ¯ x . Limiting ourselves to spatial frequencies m/L with m ranging from 0 to N − 1, this state has the least p ossible information ab out what the momentum is, since all momentum eigenstates in the allo wed range ha v e equal amplitude. Correspondingly we might e xpect the p osition to b e as well-defined as p ossible, given the lim- ited bandwidth. It is clear from Figure 1 that the position lo calization of the probabilit y distribution | S ( N , x − ¯ x ) | 2 is similar to that of a gaussian (dotted line). This is not apparen t in the mean square p osition deviation, ho w ever, whic h can b e estimated for large N using (11) as  ( x − ¯ x ) 2  ≈ Z N 0 ( x − ¯ x ) 2 sin 2 π ( x − ¯ x ) π 2 ( x − ¯ x ) 2 dx = N 2 π 2 , (33) whic h diverges as N → ∞ ( i.e., on an infinitely wide space) [33]. Thus S ( N , x − ¯ x ), which is p erfectly dis- tinct from a unit shift of itself, is not lo calized at all on the infinite line if we use the traditional second-moment measure of the width of the distribution. The unit-height gaussian, which looks so similar in the figure, has a mean square deviation of 1 / 2 π . Other measures of the width ha ve b een prop osed that av oid this disparity [23]. First-momen t b ounds F or our purp oses, a m uch b etter measure of the width of the eigenfrequency distribution is twice the a v erage half-width: 2( ¯ ν − ν 0 ). Here ¯ ν is the av erage frequency ( e.g., E /h ) and ν 0 the lo west frequency used ( e.g., E 0 /h ). In general [9], 2( ¯ ν − ν 0 ) ≥ B min , (34) where B min is the minim um bandwidth compatible with the temp oral or spatial constraints on the system. F or example, if τ min is the minimum separation in time b et ween t wo m utually orthogonal states in the evolution, then the minimum bandwidth needed is B min = 1 / 2 τ min : there must b e at least tw o distinct frequencies and they m ust b e separated by at least half of 1 /τ min . The B min = 1 / 2 τ min b ound (34) is only achiev ed by the energy (6) for N = 2. F or N  2, the energy (6) is ab out twice as great as allow ed by this b ound. There is, ho wev er, the additional bandwidth constrain t (32) re- quired to hav e N distinct states in p erio d T . The energy (6) achiev es (34) with this constrain t. Uncertain states W e hav e seen examples where a quan tum hamiltonian describ es a classical finite-state dynamics, but also mak es extra distinctions not presen t in the original dynamics: A many particle hamiltonian that keeps track of which iden tical 1-bit is where. A contin uous-shift hamiltonian that adds distinct states b et ween the discrete time steps. W e can eliminate ov er-represen tation and make the dynamics isomorphic to the original b y adding together equiv alent configurations with equal weigh t to construct truly distinct basis states. Starting from these, equiv- alen t configurations will alw a ys ha v e equal probability: e quivalenc e is r epr esente d as unc ertainty [34]. In the construction of the o ccupation num ber ba- sis states for identical 1-bits, a symmetrized or anti- symmetrized state represen ts equiv alent states as b eing equally probable. In the case of ov er-representation of in- termediate states, constructing a basis without the high frequency information needed to represent intermediate details also merges equiv alen t states (21), making them equally probable. The contin uous-hamiltonian representation of a dis- crete shift is an interesting limiting case of representing equiv alence as uncertaint y . A bandlimit with N distinct states yields (22) for finite M . F or a state centered at t = x/v and M → ∞ this b ecomes | x, ∞i N = Z 1 0 du N − 1 X n =0 S ( N , n + u − x ) | n + u, ∞i ! , (35) whic h is a uniform sup erposition of all the equiv alent w ays to represent a classical particle at p osition x if only N equally-spaced p ositions are distinct. The tradeoff b et w een bandwidth and minim um sep- aration in space determines the minimum uncertaint y v olume of phase space needed to represent each distinct state [35], and this is achiev ed by | x, ∞i N . DISCUSSION Classical finite-state dynamics that are inv ertible can b e mapp ed isomorphically onto the discrete time b eha v- ior of finite-energy quantum dynamics. A quantum evo- lution mapping an infinite num b er of distinct states into a finite time p eriod would hav e an infinite a verage energy . Quan tum-classical isomorphism challenges con ven- tional wisdom ab out essential differences b et ween quan- tum and classical systems: iden tical particles, ampli- tudes, frequencies, complementarit y and uncertain ty all pla y essen tial roles in describing and analyzing classical finite-state dynamics using contin uous language. Quan tum-classical mo dels also shed ligh t on the foun- dations of classical mechanics. They provide a quantum 8 substrate where in teresting classical b eha vior arises with- out approximation or decoherence. Physically meaning- ful energy and momentum scales are defined directly by the separation of classical even ts in time and space. 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[28] This idea is the basis of practical lattice gas mo dels of classical hydrodynamics [12] and of quantum fluids [17] (b oth with contin uous isotropy in the macroscopic limit). [29] F or motion in the − x direction we would use H ∞ = − v p instead. T o represen t the direction explicitly in the state w e would multiply it by | + i or |−i and let H ∞ = σ z v p . [30] T o make up dates local, we can use a part itioning dynam- ics, in which eac h particle motion inv olves a single up date that changes b oth old and new p ositions at once [4, 18]. [31] F or example, if particle lab els are generated sequen tially as particles are created, then interc hanging the order in which tw o particles are added to an antisymmetrized state is equiv alen t to interc hanging their particle lab els, and so creation op erators must an ticommute [20]. [32] This should apply equally to quantum lattice gas simu- lations of non-classical systems. [33] If ¯ x is near the middle of the p erio dic space, then the w av efunction go es to 0 at the b oundary as N → ∞ and so the usual uncertaint y relations apply [8, Eqn. 22]. [34] Ignorance of differences b et w een equiv alent states do esn’t coun t to w ard en trop y , whic h is one reason quantum prob- abilities must b e k ept separate from ordinary ones [4]. [35] In a p eriodic space of length L , momentum eigenfrequen- cies must be integer m ultiples of 1 /L . Thus to represent N distinct states a bandwidth B ≥ ( N − 1) /L is needed, and so the frequency-space volume p er distinct state is B L/ N ≥ ( N − 1) / N (uncertaint y tradeoff for B vs. L/ N ).