Locality and entanglement in bandlimited quantum field theory
LLocality and entanglement in bandlimited quantum field theory
Jason Pye ∗ Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
William Donnelly † Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
Achim Kempf ‡ Departments of Applied Mathematics and Physics and Institute for Quantum Computing,University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada (Dated: November 7, 2018)We consider a model for a Planck scale ultraviolet cutoff which is based on Shannon sampling.Shannon sampling originated in information theory, where it expresses the equivalence of continuousand discrete representations of information. When applied to quantum field theory, Shannon sam-pling expresses a hard ultraviolet cutoff in the form of a bandlimitation. This introduces nonlocalityat the cutoff scale in a way that is more subtle than a simple discretization of space: quantum fieldscan then be represented as either living on continuous space or, entirely equivalently, as living on anyone lattice whose average spacing is sufficiently small. We explicitly calculate vacuum entanglemententropies in 1+1 dimension and we find a transition between logarithmic and linear scaling of theentropy, which is the expected 1+1 dimensional analog of the transition from an area to a volumelaw. We also use entanglement entropy and mutual information as measures to probe in detail thelocalizability of the field degrees of freedom. We find that, even though neither translation norrotation invariance are broken, each field degree of freedom occupies an incompressible volume ofspace, indicating a finite information density.
PACS numbers: 03.67.-a, 03.70.+k, 04.60.-m, 89.70.-a
I. INTRODUCTION
It is generally thought that quantum theory and gen-eral relativity, when combined, imply the existence ofa minimum length in nature, see e.g., [1–6]. One lineof argument is to consider that when attempting to re-solve distances with a smaller and smaller uncertainty∆ x , the thereby necessarily increasing momentum un-certainty will induce a correspondingly increasing curva-ture uncertainty. Eventually the growing curvature un-certainty should make it impossible to resolve distancesany further, leading to ∆ x (cid:38) (cid:96) P , where (cid:96) P is at thePlanck scale, or perhaps the string scale.Another line of argument comes from black holephysics. Black holes (and more general causal hori-zons [7]) are believed to carry a finite entropy givenby the Bekenstein-Hawking formula, S BH = A G (we set c = (cid:126) = 1). This entropy may have its origin in the en-tanglement entropy of quantum fields [8–10] (see [11] fora review): the entanglement entropy has the same scalingbehavior as S BH and the dependence on the number ofparticle species could also be matched when taking intoaccount how the renormalized gravitational coupling de-pends on the number of species [12, 13]. Crucially, theentanglement entropy also matches the order of magni- ∗ [email protected] † [email protected] ‡ [email protected] tude of S BH , if there exists a natural ultraviolet cutoffwhich is close to the Planck length, (cid:96) P = √ G .Given these and other arguments, (see e.g., [14]), forthe existence of a natural ultraviolet cutoff at the Plancklength, the question arises as to the nature of this ultra-violet cutoff. In which sense might the density of degreesof freedom in quantum fields be finite? How are thesedegrees of freedom distributed and how can they be spa-tially resolved?To completely answer these questions would requireknowledge of the still-to-be-developed theory of quantumgravity. More feasible at the present stage is to studyhow the natural ultraviolet cutoff may first manifest itselfas one approaches the Planck scale from low energies,i.e., when coming from the safe ground of low energyphysics where conventional quantum field theory (QFT)still holds. The question then is how can one model thefirst modifications to quantum field theory that expressthe impact of a Planck length cutoff?One simple model is that of QFT on discretized spaceor spacetime. A difficulty with this approach is the as-sociated complete breaking of local Poincar´e symmetry,though there are interesting methods to generate dis-cretizations whose statistics are Poincar´e invariant [15–17]. Also, when lattices evolve in time, there tend to beproblems with non-adiabaticity and an associated exces-sive particle production from the vacuum, in particular,in an expanding universe [18].Here, we will consider a different model for a natu-ral ultraviolet cutoff in QFT with which at least the lo- a r X i v : . [ qu a n t - ph ] A ug cal Euclidean symmetries are preserved and which doesnot necessarily induce non-adiabaticities in an expandinguniverse. The model is that of QFT with a hard cutoffin the form of a finite spatial bandwidth, i.e, a finitesmallest wavelength [19–22]. How does this compare tothe models of QFT on a lattice? It is well-known thatthe discretization of space gives rise to a minimum wave-length. It is less well-known that, vice versa, to impose alower bound on wavelengths does not automatically leadto a discretization of space. Instead, one can also ob-tain a so-called sampling theoretic cutoff which does notbreak symmetries such as translation invariance. Withthis cutoff, the theory does not live on a single dedicatedlattice. Instead, it lives on continuous space and, entirelyequivalently, it can be written as living on any one latticewhose average spacing is sufficiently small.The mathematical structure that underlies the sam-pling theoretic ultraviolet cutoff is Shannon samplingtheory. Shannon sampling plays a central rˆole in in-formation theory because it establishes the equivalenceof continuous and discrete representations of information[23].Consider, for example, a music signal, f ( t ), that is low-pass filtered to some finite bandwidth, Ω. Even thoughthe bandlimited music signal is a continuous function intime, it suffices to know the signal’s amplitudes f ( t n ) onany single lattice of points { t n } , if the lattice has an av-erage spacing at or below the critical spacing π/ Ω, whichis called the Nyquist spacing. From the amplitude sam-ples { f ( t n ) } , the signal f ( t ) can then be reconstructedfor all t , in principle, without error. This establishes theequivalence of continuous and discrete representations ofinformation. The fact that none of the set of sufficientlydensely-spaced lattices is preferred allows the preserva-tion of full translation invariance (and in higher dimen-sions also rotation invariance).Applied to physics, this means that spacetime couldbe simultaneously continuous and discrete in mathemat-ically the same way that information (such as a musicsignal) possesses simultaneously both a continuous repre-sentation and equivalent discrete representations. Phys-ical fields then possess a representation on continuousspace, while being fully equivalently represented also bytheir amplitudes on any one lattice of sufficiently densespacing. The democracy among these lattices allowstranslation and rotation invariance and more generallyalso Killing vector fields to be preserved with the cutoff.Our aim here is to study the implications of this sam-pling theoretic ultraviolet cutoff for localization and en-tanglement in quantum field theory.We find that the sampling-theoretic ultraviolet cutoffimplies a particular kind of nonlocality in QFT whichmanifests itself through small but non-vanishing equal-time commutators at spacelike separation. This typeof nonlocality appears naturally in perturbative quan-tum gravity [24, 25] and in string field theory [26]. Wealso show how correlation functions are affected by thebandlimit, both classically at finite temperature, and quantum-mechanically.Further, in order to probe the spatial localization of de-grees of freedom with this UV cutoff, we examine the be-haviour of the entanglement entropy in a quantum field.For comparison, recall that in a QFT regulated on a lat-tice, the local field oscillators are coupled via the dis-cretized spatial Laplace operator in the Lagrangian. Thismakes their joint ground state, i.e., the vacuum state, anentangled state. Tracing over a region therefore yieldsan entanglement entropy. Due to the generally shortrange of the vacuum entanglement, the contributions tothe entanglement entropy are dominated by correlationsbetween those local field oscillators that are close theboundary of the considered region. This usually leads toan area law for the entanglement entropy, see e.g., [27].In the 1+1 dimensional massless field theory that we willconsider, the correlation length is infinite and the en-tanglement entropy therefore grows logarithmically (see[28]).Unlike in such lattice theories, a bandlimited QFT doesnot have a preferred lattice representation. This makesthe splitting of space into distinct regions nontrivial and,as a consequence, the Hilbert space does not automati-cally factorize into subsystems. Instead, more subtly, anydiscrete subset of points in space now defines a subsys-tem, and one can calculate its entanglement entropy withthe rest of the system. In particular, we show how to cal-culate the entanglement entropy for a subset of degreesof freedom in a Gaussian state. Our result generalizesknown formulae for the entropy of Gaussian states to thesetting where the commutation relations are nontrivial.With samples placed at the Nyquist spacing, π/ Ω, werecover the usual logarithmic scaling behaviour of the en-tanglement entropy. When the spacing is larger than theNyquist spacing (undersampling), we find that the entan-glement entropy crosses over to a volume law. Further,unlike in a lattice theory, in the bandlimited theory it ispossible to probe the entanglement between degrees offreedom that are arbitrarily closely spaced. Surprisingly,when the spacing of samples is smaller than the Nyquistspacing (oversampling), we find no reduction in entan-glement entropy. Since the sampling points can occupyan arbitrarily small region, this, naively, appears to indi-cate that the field still carries an infinite density of localdegrees of freedom (local field oscillators). We show thatthe resolution of this apparent paradox is that the regionof space being probed by the sample points does not ac-tually decrease as the samples are taken closer together.In effect, each local field degree of freedom occupies anincompressible volume.Finally, we briefly examine the infrared behaviour ofthe entanglement entropy in the 1+1 dimensional ban-dlimited theory. In addition to the well-known logarith-mic growth of the entanglement entropy, there is a sub-leading double logarithmic infrared divergence. This di-vergence appears whenever there is a continuum of modesabove the infrared cutoff scale; thus it appears when reg-ulating the infrared with a mass or a hard momentumcutoff, but not if regulating by imposing periodic bound-ary conditions.We conclude with a discussion of implications and di-rections for future work.
II. SAMPLING THEORY FOR QUANTUMFIELDSA. Overview of classical sampling theory
The central result of classical sampling theory is Shan-non’s sampling theorem [23]. This theorem establishes anequivalence between discrete and continuous representa-tions of information and it is, therefore, in ubiquitoususe in communication engineering and signal processing.Let us consider a bandlimited signal, φ ( x ), i.e., a sig-nal whose Fourier transform is supported on an interval( − Ω , Ω), whose width is the bandwidth φ ( x ) = (cid:90) Ω − Ω dk π ˜ φ ( k ) e ikx (1)Shannon’s theorem states that such a function is com-pletely determined by its values φ ( x ( α ) n ) on a discretelattice of points { x ( α ) n } , where x ( α ) n = (2 πn − α ) / (2Ω)and n ∈ Z . Here, α ∈ [0 , π ) is an arbitrary fixed con-stant that labels lattices. Given the values, { φ ( x ( α ) n ) } ,of the function on this lattice, φ ( x ) can be recovered forany x by the reconstruction formula: φ ( x ) = (cid:88) n ∈ Z sinc[( x − x ( α ) n )Ω] φ ( x ( α ) n ) (2)Thus, the space of bandlimited functions is completelyequivalent to the space of functions defined on this lat-tice. This has the practical implication of allowing usto work with concrete and therefore computationally-convenient representations of functions on a lattice (aswell as quantum fields, as we will see below) while pre-serving translation invariance (because the function re-tains its equivalent continuous representation).Another important finding in sampling theory is thatthe samples of the function do not have to be takenequidistantly: a bandlimited function can be recon-structed on any discrete set of points as long as thesepoints are chosen with a sufficiently dense average spac-ing (technically, the Beurling average spacing [29, 30]).Here, this maximum average spacing is π/ Ω. In gen-eral, such a set of points is called a sampling lattice , andthe case of equidistant samples with separation π/ Ω iscalled a
Nyquist lattice . In the case of sampling on a lat-tice other than a Nyquist lattice, perfect reconstruction isstill possible but the reconstruction formula is more com-plicated than (2) and the reconstruction becomes moresensitive to noise in the samples. Because the degrees offreedom of the function (values at the sample points) are not confined to any particular sampling lattice, the in-formation contained in bandlimited signal is subtly non-local.Sampling theory generalizes readily to bandlimitedfunctions in higher dimensions, whose Fourier transformsare supported in a compact region of momentum space,see [31–34] and it generalizes also to curved space [35].Now we will briefly outline some of the functional ana-lytic properties of the space of bandlimited functions. Werefer the reader to Appendix A for more details regardingthe functional analytic structure of sampling theory.The sampling theorem for a Nyquist lattice { x ( α ) n } n ∈ Z implies that the collection of sinc functions centred at thelattice points forms a basis for the space of bandlimitedfunctions. Moreover, the coefficients of a function in thisbasis are simply the values of the function at the corre-sponding lattice points. This basis is orthonormal in theinner product( φ, ψ ) := Ω π (cid:90) dx φ ∗ ( x ) ψ ( x ) . (3)The orthogonality follows from the identityΩ π (cid:90) dx sinc[( x − x ( α ) n )Ω] sinc[( x − x ( α ) m )Ω] = δ nm . (4)Different values of α corresponds to translated versionsof the lattice. The sinc functions centred at lattice pointswith a different value of α also form an orthogonal basis.Crucially, however, sinc functions from different latticesare not orthogonal:Ω π (cid:90) dx sinc[( x − x ( α ) n )Ω] sinc[( x − x ( α (cid:48) ) m )Ω]= sinc[( x ( α ) n − x ( α (cid:48) ) m )Ω]= sinc[ π ( n − m ) + ( α − α (cid:48) ) / (cid:54) = δ nm , for α (cid:54) = α (cid:48) (5)Here α, α (cid:48) ∈ [0 , π ). This fact will be important when westudy the localization of field degrees of freedom.Let us denote the vector in the function space corre-sponding to sinc[( x − x ( α ) n )Ω] by | x ( α ) n ). For fixed α , wethen obtain a resolution of the identity: (cid:88) n ∈ Z | x ( α ) n )( x ( α ) n | = (6)Taking the union of these bases over α , we get an over-complete basis for the function space with the corre-sponding x ( α ) n ’s covering R . Thus, we can write downan overcomplete resolution of identity:Ω π (cid:90) R dx | x )( x | = , (7)where the measure Ω dx/π gives the density of degreesof freedom in space, given by the Nyquist rate. Thisovercomplete resolution of the identity is analogous tothat for coherent states | α (cid:105) π (cid:90) C d α | α (cid:105) (cid:104) α | = , (8)where the measure d α/π = dxdp/ π (cid:126) gives the densityof independent states in phase space.The space of bandlimited functions is a reproducingkernel Hilbert space [31, 32, 34]. This means that anyfunction in the space can be recovered by means of thereproducing kernel K ( x, x (cid:48) ) as follows: ψ ( x ) = (cid:90) dx (cid:48) K ( x, x (cid:48) ) ψ ( x (cid:48) ) ,K ( x, x (cid:48) ) := Ω π ( x | x (cid:48) ) = Ω π sinc[( x − x (cid:48) )Ω] . (9) B. Reconstruction formula for quantum fields
We now apply the sampling theorem to a quantumfield ˆ φ in 1+1 dimensions, with a Hilbert space H . Let | φ (cid:105) ∈ H be an eigenstate of the field operator obeyingˆ φ ( x ) | φ (cid:105) = φ ( x ) | φ (cid:105) ∀ x ∈ R for some real-valued functionof eigenvalues, φ . Note that in a continuum field theorywithout a cutoff, ˆ φ is not an operator but an operator-valued distribution which must be smeared with a suit-able test function to obtain an operator. In the ban-dlimited theory, the cutoff acts as an effective smearingfunction, and ˆ φ is a genuine operator, though still un-bounded.Now let H ( − Ω , Ω) be the subspace spanned by the eigen-states of ˆ φ where the corresponding functions of eigen-values φ are functions bandlimited by Ω. Because φ isa bandlimited function, the eigenvalue φ ( x ) at any point x ∈ R can be determined by the knowledge of the eigen-values φ ( x ( α ) n ) at all of the points x ( α ) n on a samplinglattice { x ( α ) n } n ∈ Z . Thus, the action of the operator ˆ φ ( x )on an eigenstate of the field is determined from its actionon a sampling lattice. Explicitly,ˆ φ ( x ) | φ (cid:105) = φ ( x ) | φ (cid:105) = (cid:88) n ∈ Z sinc[( x − x ( α ) n )Ω] φ ( x ( α ) n ) | φ (cid:105) = (cid:88) n ∈ Z sinc[( x − x ( α ) n )Ω] ˆ φ ( x ( α ) n ) | φ (cid:105) . (10)Of course, this is true for all eigenstates of ˆ φ in H ( − Ω , Ω) .Since H ( − Ω , Ω) is the span of these eigenstates, the ac-tion of ˆ φ ( x ) is determined by that of { ˆ φ ( x ( α ) n ) } n ∈ Z for allstates in H ( − Ω , Ω) . Thus, we can write:ˆ φ ( x ) = (cid:88) n ∈ Z sinc[( x − x ( α ) n )Ω] ˆ φ ( x ( α ) n ) . (11)Therefore, the operators { ˆ φ ( x ( α ) n ) } n ∈ Z form a completeset of commuting observables for any α . The fact that there are many lattices on which the field can be rep-resented means that the localization of the degrees offreedom is non-trivial. In the next section we will brieflyexamine this, but it will be studied further in later sec-tions of the paper. C. A first look at localization
In a bandlimited field theory, the local harmonic os-cillators, i.e., the local degrees of freedom φ ( x ), are notall independent. Namely, a set of degrees of freedom { φ ( x n ) } is linearly independent only if the x n all belongto the same sampling lattice. The field φ ( x ) at any otherspatial point x can then be reconstructed from the am-plitudes { φ ( x n ) } . In this section we will construct a afunction that describes the spatial volume occupied by aset of degrees of freedom { φ ( x ) } .To this end, consider the subspace spanned by N po-sition eigenvectors {| x n ) } Nn =1 of first quantization (i.e.,they can be represented as number-valued function overspace). We will allow that they are not all from the sameNyquist lattice, and therefore they are generally not or-thogonal. Intuitively, the vector | x n ) characterizes thespatial profile of the field degree of freedom ˆ φ ( x n ) lo-cated at this point. Let us now construct the projectoronto this subspace, in the basis of these N position vec-tors. First, we map the position vectors to an orthogonalbasis | e i ) = (cid:88) j B ij | x j ) . (12)The projector onto this subspace is given by N := (cid:88) i | e i )( e i | = (cid:88) j,k (cid:32)(cid:88) i B ij ( B † ) ik (cid:33) | x j )( x k | . (13)We can express the elements of the projector in the non-orthogonal basis {| x n ) } Nn =1 using the reproducing kernel, K ( x j , x k ) := Ω π ( x j | x k )= Ω π (cid:32)(cid:88) i ( e i | ( B − † ) ji (cid:33) (cid:32)(cid:88) l ( B − ) kl | e l ) (cid:33) = Ω π (cid:88) i ( B − † ) ji ( B − ) ki . (14)Viewing K ( x j , x k ) as the ( j, k ) th element of an N × N matrix K N , we can write N = Ω π (cid:88) j,k ( K − N ) kj | x j )( x k | . (15)Inserting the resolution of identity (7), we can writethis projector in the continuum basis: N = (cid:90) R dxdx (cid:48) Ω π (cid:88) jk K ( x, x j )( K − N ) kj K ( x k , x (cid:48) ) | x )( x (cid:48) | (16)Note that the trace of this operator is N , since it is sim-ply the identity on an N -dimensional subspace. Thistrace is represented in the continuum basis as an integralover the diagonal elements of the integral kernel. Wecan interpret the diagonal elements of this projector inthe continuum basis (i.e., as a function of x ∈ R ) as thespatial profile of the N vectors {| x n ) } Nn =1 . We can thenvisualise the spatial profile of N degrees of freedom forvarious spacings between the degrees of freedom; this isillustrated in Figure 1, for 5 points. − − − − − x ∆ x . . . . FIG. 1. Spatial profile (in the horizontal axis) for 5 degreesof freedom as a function of their spacing (vertical axis). Bothaxes are scaled so that the cutoff length, or Nyquist spacing, π/ Ω is equal to 1. The sample points are centred at x =0. We see that for spacings above the Nyquist spacing, thedegrees of freedom occupy a region consisting of 5 disjointintervals of length ∼ ∼
5. This interval does not decreasein size even as the sampling points are taken on top of oneanother.
We see that for sampling point separations larger thanthe Nyquist spacing (shown in the region > N pointspushed close together still describe an interval of length N . Intuitively, it is clear that, because of the cutoff,placing the sampling points closer than a Nyquist spac-ing should not uncover new degrees of freedom. What issurprising but in hindsight plausible is that even when N sample points are closer than the Nyquist spacing theystill access N degrees of freedom.We will return to this issue in the context of quantumfields in section V. D. Bandlimited correlation functions
We would now like to see the effect that bandlimiting aquantum field has on the commutation relations and cor-relation functions of the ground state of the field. Theparticular fields we will hereafter be considering are 1+1dimensional massless scalar quantum fields with an ultra-violet cutoff Ω and infrared cutoff ω imposed on the spa-tial momentum, so that ω < | k | < Ω. Note that bandlim-ited functions which have support on ( − Ω , − ω ) ∪ ( ω, Ω)in Fourier space form a subspace of the space of func-tions with support on ( − Ω , Ω). Thus the above samplingtheorem (2) also applies to these functions.We consider the system described by the free Klein-Gordon Hamiltonian, H [ φ, π ] := 12 (cid:90) R dx (cid:0) π ( x ) + φ ( x )( − ∆) φ ( x ) (cid:1) , (17)where ∆ = ∂ x is the scalar Laplacian operator. Here andbelow we will not write the ˆ on top of operators. Thefield φ and its conjugate momentum π can be expressedwith the usual mode expansion (see, e.g., [36]), exceptthat the field modes outside of the range ω < | k | < Ω areremoved: φ ( x ) = (cid:90) ω< | k | < Ω dk π (cid:112) | k | (cid:16) a k e ikx + a † k e − ikx (cid:17) , (18) π ( x ) = (cid:90) ω< | k | < Ω dk π i (cid:114) | k | (cid:16) a k e ikx − a † k e − ikx (cid:17) . (19)Here a k , a † k are the usual annihilation and creation op-erators obeying the commutation relations [ a k , a † k (cid:48) ] =(2 π ) δ ( k − k (cid:48) ) and all other commutators vanish. No-tice that the Hilbert space H ( − Ω , Ω) is unitarily preservedunder free time evolution because the Fourier modes ofthe field are uncoupled . From the mode expansions we In an interacting theory the modes would couple and, naively, onemay expect this to generate shorter-than-cutoff (i.e. transplanck-ian) wavelengths. Realistically, however, such high-energetic par-ticle collisions would necessarily excite the very gravitational de-grees of freedom that are thought to enforce the ultraviolet cutoffin the first place, which may well save unitarity. can now calculate commutation relations and two pointfunctions for the field at two arbitrary points x, x (cid:48) ∈ R .The equal-time commutation relations between φ and π become[ φ ( x ) , π ( x (cid:48) )] = i (cid:18) Ω π sinc(Ω∆ x ) − ωπ sinc( ω ∆ x ) (cid:19) , (20)where we have written ∆ x := x − x (cid:48) . Note that the com-mutator can be related to the reproducing kernel whichwe used in the first quantized picture (9) after projectingthe kernel onto the space of bandlimited functions withsupport on ( − Ω , − ω ) ∪ ( ω, Ω) in Fourier space:[ φ ( x ) , π ( x (cid:48) )] = iK ( x, x (cid:48) ) . (21)The remaining commutators all vanish: [ φ ( x ) , φ ( x (cid:48) )] =[ π ( x ) , π ( x (cid:48) )] = 0 for any two points.The correlation function for the ground state of thefield at two distinct points is: (cid:104) | φ ( x ) φ ( x (cid:48) ) | (cid:105) = 12 π [Ci(Ω∆ x ) − Ci( ω ∆ x )] . (22)In the coincidence limit this becomes: (cid:104) | φ ( x ) | (cid:105) = 12 π log (cid:18) Ω ω (cid:19) . (23)Similarly, the correlation function for the conjugate mo-mentum at two distinct points is (cid:104) | π ( x ) π ( x (cid:48) ) | (cid:105) = cos(Ω∆ x ) − cos( ω ∆ x )2 π ∆ x + Ω sin(Ω∆ x ) − ω sin( ω ∆ x )2 π ∆ x , (24)which becomes in the coincidence limit (cid:104) | π ( x ) | (cid:105) = Ω − ω π . (25)The correlation functions between φ and π at equal timesall vanish, (cid:104) | { φ ( x ) , π ( x (cid:48) ) } | (cid:105) = 0.From Eq. (23), we see that the infrared cutoff ω is nec-essary to regulate the infrared divergence of the (cid:104) φφ (cid:105) cor-relator. It will therefore be necessary to keep ω strictlypositive for our calculations, and we will examine infraredeffects in more detail in section VI. Since we are moreinterested in the effect of the ultraviolet cutoff Ω, we dis-play the correlation function in the limit ω → ω/ Ω (cid:28)
1, we haveCi (cid:16) ω Ω Ω∆ x (cid:17) = γ +log(Ω∆ x )+log( ω/ Ω)+ O ( ω/ Ω) . (26)Therefore if we add a term log( ω/ Ω) / (2 π ) to the (cid:104) φφ (cid:105) correlator, in the limit ω → (cid:104) | φ ( x ) φ ( x (cid:48) ) | (cid:105) (cid:55)→ π [Ci(Ω∆ x ) − γ − log(Ω∆ x )] . (27) We also want this correlator to give the usual nonban-dlimited correlator when Ω → ∞ . The nonbandlimitedcorrelator is (cid:104) | φ ( x ) φ ( x (cid:48) ) | (cid:105) = (cid:90) R dk π | k | e ik ∆ x = − π ( γ + log | ∆ x | ) . (28)Thus, since Ci(Ω | ∆ x | ) → → ∞ , we must addanother factor of log(Ω) / (2 π ) to (cid:104) φφ (cid:105) to get (cid:104) | φ ( x ) φ ( x (cid:48) ) | (cid:105) = 12 π [Ci(Ω∆ x ) − γ − log | ∆ x | ] , (29)and at a single point we have (cid:104) | φ ( x ) | (cid:105) = 12 π log(Ω) . (30)The correlator (cid:104) ππ (cid:105) is not divergent in the infrared, thuswe can directly take the limit ω/ Ω → (cid:104) | φ ( x ) φ ( x (cid:48) ) | (cid:105) and (cid:104) | π ( x ) π ( x (cid:48) ) | (cid:105) (re-spectively) as a function of ∆ x , with the infrared cutoffremoved. In both figures, we see that the correlationsdecay with distance, similarly to the correlations whenthere is no ultraviolet cutoff. We notice that the ban-dlimited correlation functions oscillate with wavelengthsof the order of the ultraviolet cutoff length. As we willsee later, similar oscillations also occur in the correlationsof thermally distributed bandlimited classical signals. ∆ x h φφ i UV cutoffUV cutoff, Nyquist latticeNo cutoff
FIG. 2. φ - φ correlations as a function of their separation.The horizontal axis is scaled by Ω /π so that integer valuescorrespond to Nyquist spacings. The bandlimited correlationsare blue with the Nyquist spacings indicated by red dots. Theblack dashed line shows the ultraviolet-divergent correlationswithout the ultraviolet bandlimit. We see that for points onthe Nyquist lattice, the bandlimited correlators are in closeragreement to the correlation functions without the bandlimit.A counterterm of log( ω ) / (2 π ) is added to (cid:104) φφ (cid:105) to cancel theinfrared divergence. ∆ x h ππ i UV cutoffUV cutoff, Nyquist latticeNo cutoff
FIG. 3. π - π correlations as a function of their separation.The horizontal axis is scaled by Ω /π so that integer valuescorrespond to Nyquist spacings. The vertical axis is scaled by1 / Ω . The bandlimited correlations are blue with the Nyquistspacings indicated by red dots. The black dashed line showsthe ultraviolet-divergent correlations without the ultravioletbandlimit. III. ENTROPY OF GAUSSIAN STATES
We would like to understand how the delocalizationof degrees of freedom of the bandlimited quantum fieldtheory affects the localization of information in space. Auseful tool to probe the distribution of degrees of freedomin space is the von Neumann entropy of localized subsys-tems, and derived quantities such as the mutual infor-mation. Suppose we have a state of a bandlimited field,whose wavenumbers are in the range ω < | k | < Ω, andwe are able to make measurements of the field amplitude φ and its conjugate momentum π at a finite number, N ,of points. Any set of points defines a subsystem in thisway, regardless of their relative positions in space, andwe can define a von Neumann entropy associated withthis subsystem. In this section we show how to calculatethe von Neumann entropy associated with an arbitrarysubset of points using the formalism of Gaussian states[37]. We leave concrete numerical calculations for sectionIV.First in section III A, we consider a classical field in athermal state. For this system there is no entanglementor quantum noise and all of the entropy comes from ther-mal fluctuations. Then in III B, we consider a thermalstate of a quantum system, which in the limit of zerotemperature becomes the ground state of the quantumfield. In this case the entropy includes both entangle-ment entropy and thermal fluctuations, and in the zerotemperature limit is entirely due to quantum entangle-ment. This will demonstrate how the classical resultsemerge from the quantum results at high temperatures. A. Classical entropy
Let us first consider the classical situation where thefield we are given is chosen from some given classicaldistribution. If, for example, the statistical distributionis that of Gaussian white noise, the values of the fieldat different sample points are uncorrelated. Since theentropy is additive for uncorrelated degrees of freedom,the entropy associated with N samples is linear in N .The classical case which we will be more interested in,because it is more comparable to the quantum case, isthe case where the statistical distribution of the signalsis a thermal distribution of a 1+1 dimensional classicalKlein-Gordon Hamiltonian, with fields that have bothan ultraviolet cutoff Ω and infrared cutoff ω . For thisdistribution, the degrees of freedom of the field at thesample points will be coupled, causing the values of thefield at these points to be correlated. We therefore ex-pect that the entropy of the interval of N samples has acontribution which is nonlinearly dependent on N . Wenow explicitly calculate the entropy created for these N samples.We will go through the entropy calculation in detailsince many of the steps are reproduced in the quantumcalculation. We begin with the Hamiltonian in momen-tum space, H [ ˜ φ, ˜ π ] = 12 (cid:90) ω< | k | < Ω dk π (cid:16) | ˜ π ( k ) | + k | ˜ φ ( k ) | (cid:17) . (31)Both the ultraviolet and infrared cutoffs are explicitlyenforced in this description by the restriction on the mo-mentum k . For a thermal state, the probability distri-bution of the fields in momentum space is given by theBoltzmann distribution p [ ˜ φ, ˜ π ] = 1 Z e − βH [ ˜ φ, ˜ π ] . (32)We seek the entropy of a reduced probability distribu-tion for a set of N sample points. To this end, we choosea lattice { x n } n ∈ Z and we write ∆ x mn := x m − x n . Wedenote the values of the field at these points φ n := φ ( x n ), π n := π ( x n ). To find the probability distribution for thesamples, we perform a change of phase space coordinatesfrom the fields in momentum space to a sampling lattice: φ ( x n ) = (cid:90) ω< | k | < Ω dk π e ikx n ˜ φ ( k ) ,π ( x n ) = (cid:90) ω< | k | < Ω dk π e ikx n ˜ π ( k ) . (33)Instead of explicitly performing this change of variables,it will be more convenient to make use of the fact that theprobability distribution in terms of φ n and π n is Gaus-sian. This follows from the fact that the probability dis-tribution in momentum space is Gaussian and the changeof phase space variables (33) is linear. The distributionis therefore characterized entirely by the two-point func-tions.First, note that one can easily calculate the power spec-tra of the fields from the probability distribution (32)where the fields are represented in momentum space: (cid:104)| ˜ φ ( k ) | (cid:105) = 1 βω k , (34) (cid:104)| ˜ π ( k ) | (cid:105) = 1 β . (35)Then we can obtain the correlators of the fields betweentwo arbitrary points by taking the Fourier transform ofthe power spectra: (cid:104) φ ( x ) φ ( x (cid:48) ) (cid:105) = 1 β (cid:20) cos( ω ∆ x ) πω − cos(Ω∆ x ) π Ω (cid:21) + ∆ xβ [Si( ω ∆ x ) − Si(Ω∆ x )] , (36) (cid:104) π ( x ) π ( x (cid:48) ) (cid:105) = 1 β (cid:20) Ω π sinc(Ω∆ x ) − ωπ sinc( ω ∆ x ) (cid:21) . (37)These correlators are plotted as a function of ∆ x inFigures 6 and 7 alongside the finite-temperature quan-tum correlators calculated below. In a similar manneras for the vacuum correlators in the previous section,these functions are displayed with added infrared coun-terterms. The (cid:104) ππ (cid:105) correlator is again finite as ω/ Ω → ω ∆ x ) → ω/ Ω →
0, the first term in Eq. (36) isthe only infrared divergent term, whose divergence maybe cancelled by subtracting 1 / ( πβω ) since as ω/ Ω → ω ∆ x ) πβω ∼ πβω . (38)The function plotted in figure 6 is then (cid:104) φ ( x ) φ ( x (cid:48) ) (cid:105) = − β (cid:20) cos(Ω∆ x ) π Ω + ∆ x Si(Ω∆ x ) (cid:21) . (39)For the entropy calculation, we require the reducedprobability distribution for N sample points obtained bymarginalizing over a complementary set of phase spacecoordinates. In a lattice theory, this would simply meanintegrating over the field values at all lattice sites notunder consideration. In a bandlimited theory, the com-plementary subsystem to a set of points is not generallyassociated to any set of points; it is nonlocal. As pre-viously noted, we do not actually need to carry out themarginalization, as the reduced probability distributionon the N samples is Gaussian, and hence entirely deter-mined by the two-point functions of the N points. Nev-ertheless it is instructive to carry out the decompositionof the total system into the N sample points and theircomplement.We will perform this calculation for any Gaussian stateof the form p ( { φ n , π n } n ) = 1 Z e − (cid:80) m,n π m A mn π n × e − (cid:80) m,n φ m B mn φ n (40) with positive-definite, symmetric matrices A and B . Forthe above example, the φ n ’s and π n ’s would representthe field amplitudes on a sampling lattice. We can splitthe phase space into a subsystem describing the N samplepoints and its complementary subsystem by performing achange of variables which splits the matrices encoding thePoisson bracket and symplectic form into a direct sum ofthe matrices which act on the individual subspaces sep-arately. It will be convenient to write the phase spacevariables in a vector (cid:126)r = ( φ , π , φ , π , . . . ). Then thePoisson bracket for the total phase space with the ul-traviolet and infrared cutoff constraints enforced can beencoded in an anti-symmetric matrix Λ ij := { r i , r j } P B ,where { φ i , π j } P B = Ω π sinc(Ω∆ x ij ) − ωπ sinc( ω ∆ x ij ) , { φ i , φ j } P B = { π i , π j } P B = 0 . (41)Note that this consistent with the quantum mechanicalcommutation relations (20).First we write the Poisson bracket as a block matrix,Λ = (cid:20) α η − η T γ (cid:21) , (42)where α T = − α and γ T = − γ , ensuring that Λ T = − Λ.The blocks are arranged such that the upper left blockcontains the first 2 N indices which correspond to the N sample points which will remain after marginalizing.Now consider a change of phase space variables (cid:126)r (cid:48) = Q(cid:126)r ,where Q := (cid:20) I η T α − I (cid:21) . (43)The transformed Poisson bracket isΛ (cid:48) := Q Λ Q T = (cid:20) α η T α − η + γ (cid:21) . (44)Now the phase space splits as a direct sum of the N degrees of freedom remaining after marginalizing and itscomplement, with symplectic form F := N (cid:88) n,m =1 ( α − ) nm dφ (cid:48) n ∧ dπ (cid:48) n + (cid:88) n,m (cid:54)∈{ ,...,N } [( η T α − η + γ ) − ] nm dφ (cid:48) n ∧ dπ (cid:48) n . (45)This construction requires that the matrix α be invert-ible, but we will now show that in the bandlimited theorythis is always the case. We can express α as the matrix α ij = K ( x i , x j ) = Ω π ( x i | x j ), where | x i ) , | x j ) are the ban-dlimited position eigenfunctions defined in II C. Supposethat α is not invertible. Then we can find a vector a i such that0 = a i α ij a j = (cid:88) i,j ( x i | a i a j | x j ) = (cid:107) (cid:88) i a i | x i ) (cid:107) , (46)and hence (cid:80) i a i | x i ) = 0. This implies that for any ban-dlimited function f , we have0 = ( f | (cid:88) i a i | x i ) = (cid:88) i a i ( f | x i ) = (cid:88) i a i f ∗ ( x i ) . (47)However given any finite set of points x i and target val-ues y i , we can find a bandlimited function f such that f ( x i ) = y i [38]. This contradicts (47), hence α must beinvertible.Now, the probability distribution after performing themarginalization is of the form: p red( { φ n , π n } Nn =1 ) = 1 Z red e − (cid:80) Nm,n =1 π m ( A | N ) mn π n × e − (cid:80) Nm,n =1 φ m ( B | N ) mn φ n , (48)where A | N and B | N are positive-definite symmetric ma-trices given by ( B | − N ) mn = (cid:104) φ m φ n (cid:105) , (49)( A | − N ) mn = (cid:104) π m π n (cid:105) . (50)These correlators are just (36) and (37) evaluated at thepoints x = x m and x (cid:48) = x n . Note that the matrices A | N and B | N are simply the original matrices A and B withindices restricted to the subsystem m, n ∈ { , . . . , N } .Before we can calculate the partition function by in-tegrating over the phase space variables { φ n , π n } Nn =1 , wemust first choose a measure for this phase space. Whencalculating the entropy for a classical probability distri-bution with continuous random variables, there is an am-biguity in the entropy caused by an ambiguity in thechoice of measure. Here, we will fix the measure tobe proportional to the symplectic form in this subspace(Eq. (45)) dφ n ∧ dπ n / (2 π ). This is sufficient to resolvethe ambiguity in the entropy, and ensures that the en-tropy invariant under symplectic transformations of thephase space. As we shall see, the factor of 2 π gives anentropy which matches the high temperature limit of thevon Neumann entropy in the quantum setting.Now we find that the reduced partition function is Z red = (cid:90) det( α ) − N (cid:89) n =1 dφ n dπ n π e − (cid:80) Nm,n =1 π m A mn π n × e − (cid:80) Nm,n =1 φ m B mn φ n = det( α ) − (cid:112) det ( A − ) det ( B − ) . (51)Then the entropy of the distribution of the remainingsamples is: S = (cid:90) det( α ) − N (cid:89) n =1 dφ n dπ n π p red( { φ n , π n } Nn =1 ) × log (cid:2) p red( { φ n , π n } Nn =1 ) (cid:3) = N + log (cid:104) det( α ) − (cid:112) det ( A − ) det ( B − ) (cid:105) . (52)Note that the first term is linear in the number of sam-ples N , but the second term, which encodes the coupling between the sample points, is not exactly linear. In Fig-ure 4, we plot the entropy for a massless bandlimitedfield in a thermal state. We find that the entropy is veryclose to linear in the number of lattice points, regardlessof the temperature. This simply reflects the extensivityof thermal entropy. NS T/ Ω = 10 T/ Ω = 10 T/ Ω = 10 T/ Ω = 10 T/ Ω = 10 − T/ Ω = 10 − FIG. 4. Entropy of a set of sample points for a thermally-distributed classical bandlimited Klein-Gordon field for sev-eral temperatures. We see that the magnitude of the entropygrows linearly with the number of points, illustrating a vol-ume law. Notice also that this figure shows negative entropyat low temperatures, which simply reflects the fact that con-tinuous probability distributions can have negative entropy.We fix the inherent ambiguity in these entropies by requiringthat it matches the quantum entropy at high temperature.
It turns out to be more convenient to rewrite the for-mula for the entropy in terms of the following matrices:Λ | N := (cid:20) λ − λ (cid:21) , (53)Σ | N := (cid:20) B | − N A | − N (cid:21) , (54)where | N denotes restriction of the matrix indices to1 , . . . , N (i.e. the indices corresponding to the remain-ing sample points after marginalizing). The matrix Λ | N encodes the Poisson bracket with the indices of the phasespace vector (cid:126)r rearranged so that all of the φ n ’s occur be-fore the π n ’s, and where λ mn := { φ m , π n } P B . Note that λ is just the matrix α from Eq. (42) after a permutationof the indices. Similarly, the matrix Σ | N is simply thematrix of correlators between the phase space variables.It is then straightforward to verify that the entropycan be written as S = N + log (cid:20)(cid:113) det(Λ | − N Σ | N ) (cid:21) = N (cid:88) i =1 (1 + log( d i )) , (55)0where { d i } Ni =1 are the positive imaginary parts of theeigenvalues of Λ | − N Σ | N which come in pairs ± id i . There-fore, the entropy is determined entirely in terms of thePoisson bracket (41) and two-point correlators (36) and(37). Notice that since we can write down the bracketand correlators between any two points of the field (notnecessarily on a Nyquist lattice), we can calculate theentropy associated with N arbitrary points of the field. B. Quantum entropy
Let us now proceed to the case of calculating the corre-sponding entropy for N samples of a quantum field. Herewe will review a method for calculating the entanglemententropy for Gaussian states of systems of quantum har-monic oscillators. The Gaussian formalism for calculat-ing the entropy for Gaussian states goes back to [8–10]with significant simplifications developed in [37, 39]. Cru-cial for our purposes is that this formalism also general-izes naturally to the case where the commutation rela-tions between φ and π are nonlocal. As in the classicalcase, the state of a region, and hence its entropy, is de-termined entirely by the matrix of Poisson brackets andthe covariance matrix.We begin with the Hamiltonian for a set of M coupledoscillators (where M may be finite or infinite): H := 12 M (cid:88) m,n =1 π m A mn π n + 12 M (cid:88) m,n =1 φ m B mn φ n . (56)Here A and B are positive-definite, symmetric matrices.As in section III A it will be convenient to combine φ n and π n into a single vector (cid:126)r = ( φ , π , φ , π , . . . ). Thenwe define the correlation matrixΣ ij := (cid:104){ r i , r j }(cid:105) ≡ tr ( { r i , r j } ρ ) , (57)where ρ is the density matrix for the state. Also, sincethe commutators are c-numbers, we can define the matrixΛ that encodes the commutation relations: i Λ ij := [ r i , r j ] . (58)Now we would like to find the state of a subset ofoscillators, i = 1 , . . . , N (with N < M ) obtained bytracing over the complementary set of degrees of free-dom. For the bandlimited theory, the complementaryset of degrees of freedom are not simply the complemen-tary set of lattce points N + 1 , . . . , M . Notice that ifwe remove the infrared cutoff, i.e., set ω = 0, then ona Nyquist lattice we have the canonical commutationrelations [ φ ( x m ) , π ( x n )] = i Ω π δ mn . Therefore, on thisNyquist lattice the Hilbert space H ( − Ω , Ω) factors into thetensor product ⊗ n H x n of the Hilbert spaces H x n gener-ated by the field operators { φ ( x n ) , π ( x n ) } . However, fora general sampling lattice the commutation relations arenonlocal at the cutoff scale. Thus, in the case of non-Nyquist sampling, the Hilbert space H ( − Ω , Ω) does not simply factor into a tensor product of Hilbert spaces gen-erated by the field operators at the sample points; i.e., H ( − Ω , Ω) (cid:54) = ⊗ n H x n . This is because for points, say x n and x m , which do not lie on the same Nyquist lattice, the op-erator φ ( x n ) acts non-trivially on H x m since this sectorof the Hilbert space is generated by φ ( x m ) and π ( x m ),but [ φ ( x n ) , π ( x m )] (cid:54) = 0. However, if we identify a finitesubset of lattice points it is possible to factor the Hilbertspace into a tensor product of a subspace describing thesubsystem and its complement.Factoring the Hilbert space is similar to the phasespace splitting performed in the classical case. We pro-ceed using the change of variables defined by the matrix(43), so that Λ takes the form:Λ (cid:48) := Q Λ Q T = (cid:20) α η T α − η + γ (cid:21) . (59)Note that the block α := Λ | J corresponding to thedegrees of freedom { φ i , π i } Ni =1 is unchanged under thistransformation. The covariance matrix also transformsas Σ (cid:48) := Q Σ Q T . (60)It is also easy to check that the block Σ | J is unchangedunder this transformation.Now, by Darboux’s theorem, we can find a change ofvariables via a transformation T which brings the matrixΛ (cid:48) into its canonical form. That is, we can find a set ofcanonical coordinates satisfying canonical commutationrelations [ q i , p j ] = iδ ij . Since we have split the phasespace into two pieces, we can find such a matrix T in theform T = (cid:20) T T (cid:21) . (61)Now, Λ (cid:48)(cid:48) := T Λ (cid:48) T T = M (cid:77) i =1 (cid:20) − (cid:21) (62)Σ (cid:48)(cid:48) := T Σ (cid:48) T T . (63)An advantage of working with covariance matrices is thatthe tracing operation can be implemented by simply re-stricting the indices of Λ (cid:48)(cid:48) and Σ (cid:48)(cid:48) to the set { , . . . , N } .Also, by construction of the matrices Q and T , the re-duced commutator and covariance matrices can be deter-mined solely from the reduced commutator and covari-ance matrices of the original variables, i.e.,Λ (cid:48)(cid:48) | N = T Λ | N T T (64)Σ (cid:48)(cid:48) | N = T Σ | N T T . (65)Therefore, we see that since we are only mixing the coor-dinates for the degrees of freedom 1 , . . . , N among them-selves before the tracing operation, the Hilbert space af-ter the tracing operation is the same as the Hilbert space1generated by the original operators { φ n , π n } Nn =1 . Theimportance of this fact is that it does not matter wherethe remainder of the samples are taken, as we can simplyidentify the entire Hilbert space H ( − Ω , Ω) as the tensorproduct of the Hilbert space generated by the operatorscorresponding to { φ n , π n } Nn =1 and the Hilbert space ofthe complementary set of degrees of freedom. This alsoallows us to choose a set of samples which do not all lie ona Nyquist lattice, as we can simply identify the reducedHilbert space as the space generated by the operators atthese points.Now that we have the reduced commutator and co-variance matrices in a Darboux basis, as shown byWilliamson [40, 41], we can make a further (symplec-tic) transformation S that preserves the form of Λ (cid:48)(cid:48) | N ,and puts Σ (cid:48)(cid:48) | N into diagonal form: S Λ (cid:48)(cid:48) | N S T = Λ (cid:48)(cid:48) | N , (66) S Σ (cid:48)(cid:48) | N S T = N (cid:77) i =1 (cid:20) d i d i (cid:21) . (67)Here the diagonal entries d i come in pairs, and are calledthe symplectic eigenvalues of Σ (cid:48)(cid:48) | N .In these coordinates, the density matrix is a prod-uct of uncorrelated thermal density matrices of N har-monic oscillators with canonical coordinates q i , p i satis-fying canonical commutation relations, in a state where (cid:104) q i (cid:105) = (cid:104) p i (cid:105) = d i . In particular, the i th oscillator is inthe state ρ i = (cid:88) n ≥ d i + (cid:18) d i − d i + (cid:19) n | n (cid:105)(cid:104) n | . (68)Note that the uncertainty relation implies d i = ∆ q i ∆ p i ≥ | [ q i , p i ] | = . (69)Thus the symplectic eigenvalues are all bounded belowby d i ≥ .The entropy is then the sum of entropies of each indi-vidual oscillator: S = (cid:88) i S ( d i ) ,S ( d ) := ( d + ) log( d + ) − ( d − ) log( d − ) . (70)Thus we can determine the entropy entirely from thesymplectic eigenvalues d i .Note that one does not need to carry out this symplec-tic diagonalization in order to find the symplectic eigen-values. Under the sequence of similarity transformations Q , T , S , the eigenvalues of Λ | − N Σ | N are invariant, i.e.,spec (cid:32) Λ (cid:48)(cid:48) | − N N (cid:77) i =1 (cid:20) d i d i (cid:21)(cid:33) = spec (cid:0) [( ST )Λ | N ( ST ) T ] − ( ST )Σ | N ( ST ) T (cid:1) = spec (cid:0) Λ | − N Σ | N (cid:1) , (71) where spec( A ) denotes the spectrum of A . We thereforesee that the eigenvalues of Λ | − N Σ | N coincide with theeigenvalues of the matrixΛ (cid:48)(cid:48) | − N N (cid:77) i =1 (cid:20) d i d i (cid:21) = N (cid:77) i =1 (cid:20) d i − d i (cid:21) , (72)which are the values ± id i . Thus we can find the sym-plectic eigenvalues simply by finding the eigenvalues ofΛ | − N Σ | N . See [37], as well as the related result [42].When the symplectic eigenvalues { d i } are large (whichin a thermal state can be shown to correspond to ahigh temperature limit for thermal states), the expres-sion S ( d ) becomes S ( d ) ≈ d ) , (73)so the quantum entropy formula approaches the classicalformula (55) obtained above. This fact is illustrated infigure 5, which shows the classical and quantum formulasfor S ( d ). dS ( d ) ClassicalQuantum
FIG. 5. The entropy of a harmonic oscillator as a functionof the symplectic eigenvalue, both classically and quantum-mechanically. In quantum mechanics, the uncertainty prin-ciple requires d ≥ /
2, which is saturated by the vacuumstate for which the entropy is zero. Classically, the symplec-tic eigenvalue can be any positive number, but the entropybecomes negative for small d . This again is a consequence ofthe fact that the entropy of a continuous probability distri-bution is not bounded from below. IV. VON NEUMANN ENTROPY
We now calculate the entropy associated to a uniformlyspaced lattice of field samples. There are three distinctregimes, depending on whether the spacing is equal to theNyquist spacing, larger than Nyquist (undersampling), orsmaller than Nyquist (oversampling). We will considereach of these possibilities in turn, both in vacuum and atfinite temperature.2
A. Nyquist sampling
As shown in section III, the entropy of a set of samplepoints in a Gaussian state is determined by the commu-tators and two-point functions at the sample points. Wenow calculate these for a quantum field at finite temper-ature.Before we perform the tracing operation, the densitymatrix associated with a single mode of the field is ρ k = 1 Z k ∞ (cid:88) n k =0 e − βω k ( n k + ) | n k (cid:105) (cid:104) n k | . (74)Formally, the total density matrix for the field is ρ = (cid:79) ω< | k | < Ω ρ k . (75)This leads to the power spectra (cid:104)| φ k | (cid:105) = tr( | φ k | ρ ) = 1 ω k (cid:18) e βω k − (cid:19) , (76) (cid:104)| π k | (cid:105) = tr( | π k | ρ ) = ω k (cid:18) e βω k − (cid:19) . (77)From the power spectra we find the two-point functionsfor a massless bandlimited field are (cid:104) φ ( x ) φ ( x (cid:48) ) (cid:105) = (cid:90) Ω ω dkπ cos( k ∆ x ) 1 k (cid:18) e βk − (cid:19) , (78) (cid:104) π ( x ) π ( x (cid:48) ) (cid:105) = (cid:90) Ω ω dkπ cos( k ∆ x ) k (cid:18) e βk − (cid:19) . (79)These correlators are plotted in Figures 6 and 7 alonsidethe classical correlators. We notice that, as expected,the classical and quantum correlators agree at high tem-peratures. Both classical and quantum correlations ex-hibit oscillations at the ultraviolet scale. At temperatures T / Ω (cid:46)
1, the quantum field exhibits stronger correlationsthan its classical counterpart due to vacuum fluctuations.Using these thermal correlation functions togetherwith the commutation relations (eq. (20)), we numeri-cally calculate the entanglement entropy of a finite latticeof points separated by the Nyquist spacing π/ Ω. Thenwe consider the change in the entropy as the number ofpoints increases. The entropy as a function of the num-ber of sample points is illustrated in Figure 8 for a rangeof temperatures. The plot shows clearly the transitionbetween linear and logarithmic growth of the entropy.Comparing the entropy in the quantum theory (Fig. 8)with the corresponding classical result (Fig. 4), we seethe expected agreement of the scaling behaviour at hightemperatures, with the entropy growing linearly with thenumber of lattice points. This simply reflects the exten-sivity of the thermal entropy. However, at low tempera-tures the entropy of the quantum field increases only log-arithmically with the number of points removed (e.g. thezero temperature state in Figure 8). This is a well-known ∆ x T h φφ i ClassicalQuantum, T/ Ω = 0.01Quantum, T/ Ω = 0.1Quantum, T/ Ω = 1
FIG. 6. Classical and quantum φ - φ correlations as a functionof their separation at various temperatures. The horizontalaxis is scaled by Ω /π so that integer values correspond toNyquist spacings. The vertical axis is scaled by Ω /T . Theclassical correlator is plotted as black dots. The scaling ofthe vertical axis absorbs all of the temperature dependenceof the classical correlator (since it grows proportionally to T /
Ω), thus the single graph of the classical correlator com-pletely characterises its behaviour. The quantum correlatorsare shown as lines for temperatures up to
T /
Ω = 1, at whichpoint the quantum correlator converges to the classical corre-lator. result for conformal field theories, such as the masslessscalar field, for which the entropy scales as S ∼ log(Ω L )[28]. Here, for the thermal state at zero temperature (i.e.,the ground state) the fitted curve to the data is S ( N ) = 0 .
334 log N + 3 .
25 (80)which is in agreement with the expected leading orderterm log( N ). This result is for an infrared scale of ω/ Ω = 10 − .We therefore see that the total entropy of the quantumfield is a combination of the thermal and entanglemententropy. At high temperatures the entropy is primarilythermal, while at low temperatures the thermal entropyvanishes, leaving only the entanglement entropy. B. Undersampling
Now we examine how the entanglement entropy of theground state of the quantum field scales with the numberof samples N when the distance between adjacent sam-ples is independent of Ω. In sampling theory terminology,this corresponds to undersampling when ∆ x > π/ Ω andoversampling when ∆ x < π/
Ω.First, we will examine the case of undersampling. Re-call that the strength of correlations between the field attwo separate points decays with the distance between the3 ∆ x T h ππ i ClassicalQuantum, T/ Ω = 0.01Quantum, T/ Ω = 0.1Quantum, T/ Ω = 1
FIG. 7. Classical and quantum π - π correlations as a functionof their separation at various temperatures. The horizontalaxis is scaled by Ω /π so that integer values correspond toNyquist spacings. The vertical axis is scaled by Ω /T . Theclassical correlator is plotted as black dots. The scaling ofthe vertical axis absorbs all of the temperature dependenceof the classical correlator (since it grows proportionally to T /
Ω), thus the single graph of the classical correlator com-pletely characterises its behaviour. The quantum correlatorsare shown as lines for temperatures approaching
T /
Ω = 1, atwhich point the quantum correlator converges to the classicalcorrelator. points and the entropy is dominated by local correlations.Thus, for points separated by more than the Nyquistspacing any given point is most strongly correlated withdegrees of freedom in the complementary subsystem. Asa result, the entanglement entropy in this regime is pro-portional to the number of points traced out. Hence weshould recover a volume law for the entropy similar tothe high temperature thermal state.The procedure to calculate the entanglement entropy isthe same as before, except now the correlators and com-mutators are taken between points which do not all lie ona Nyquist lattice. Figure 9 shows the dependence of theentanglement entropy on the number of points traced outfor spacings that vary between 1 and 2 Nyquist spacings.We see that for Nyquist and near-Nyquist sample spac-ings the entanglement entropy grows logarithmically withthe number of points. As the sample separation increasesthere is a transition to a linear scaling with the numberof points. This corroborates our intuition for the scalingbehaviour of the entanglement entropy when undersam-pling.
C. Oversampling
Instead of separating the sample points, we now take N samples which are spaced closer than a Nyquist spacingapart. Starting from Nyquist spacing and pushing the NS T/ Ω = 10 − T/ Ω = 10 − T/ Ω = 10 − T/ Ω = 10 − T/ Ω = 10 − T/ Ω = 0
FIG. 8. Entropy for N Nyquist spaced sample points fora thermal state of a quantum Klein-Gordon field. The plotillustrates the transition between the logarithmic behaviour atlow temperature to linear behaviour at higher temperatureswhere the entropy becomes an extensive variable. NS ∆ x = 2∆ x = 1 . x = 1 . x = 1 . x = 1 . x = 1 . x = 1 . x = 1 . x = 1 . x = 1 . x = 1 FIG. 9. The dependence of the entanglement entropy on thenumber of sample points, for sample spacings between 1 and2 Nyquist spacings. The ∆ x labels in the legend are scaled byΩ /π so that the Nyquist spacing is 1. We see that once theadjacent point spacing has reached twice the Nyquist spacing,the entropy has transitioned from the logarithmic scaling lawto a volume law. The infrared scale is ω/ Ω = 10 − . points closer together, we find that the entropy does notdepend very sensitively on the spacing between points.The logarithmic scaling of the entanglement entropy re-mains the same, and only the subleading constant ismodified. This is in contrast with the case of under-sampling, where we saw a transition from logarithmic tolinear growth.Unfortunately, with an oversampled set of latticepoints the above numerical calculation becomes unsta-4ble for a large number of sample points. This is becausethe matrices Λ and Σ defined in section III B becomeill-conditioned. However, it is possible to perform thesenumerical calculations in the regimes of a small amountof oversampling, or for only a small number of points.For a small amount of oversampling, that is, with lat-tice spacing between 97%-100% of the Nyquist spacing,we continue to find logarithmic scaling of the entangle-ment entropy with the number of points traced out: S ( N ) = c log N + c (81)where in each case c ∈ [0 . , . c = 3 .
25. Thisresult is for an infrared scale of ω/ Ω = 10 − .Although we were not able to extend far into the over-sampling regime, if we fix a small number of points it ispossible to calculate the entanglement entropy of thesepoints for an arbitrary amount of oversampling. Fig-ure 10 shows the entanglement entropy for 5 samplepoints as a function of the separation between the points.One sees from the figure that the entanglement entropyreaches a plateau for spacings below the Nyquist spacing,with the entropy depending only weakly on the spacingin this regime. This shows that the logarithmic scalinglaw for the entanglement entropy is also independent ofthe spacing between the points, if the spacing is belowthe Nyquist spacing.In figure 10 we also see that the entropy continues toincrease as the spacing is increased above the Nyquistspacing. This is because the entropy depends on localcorrelations, and as the degrees of freedom become moreseparated, more of their correlations are with the com-plementary degrees of freedom. In the 1+1 field theorywe consider, these correlations decay very slowly, so thatthe entropy continues to increase all the way up to theinfrared cutoff scale. The entropy is also slightly peakedat integer multiples of the Nyquist spacing: this is re-lated to the fact that the commutators vanish betweendegrees of freedom separated by multiples of the Nyquistspacing.We can also show that the plateau in the entropy atsmall spacings is not only a feature of the ground state,but also occurs in the thermal state considered in sec-tion IV A. Figure 11 shows the entanglement entropy offive points as a function of their spacing for various tem-peratures. We see that at any temperature, when thepoints are closer than a Nyquist spacing, the resulting en-tanglement entropy depends only weakly on the spacing.This plateau is therefore not a fundamentally quantum-mechanical effect: if we perform the same calculationwith five points for the classical thermal state, we findthe same plateau.The restriction to a small number of sample points is aconsequence of numerical instability that occurs for smallspacing and large numbers of points. In the followingsubsection we develop an analytic method that will allowus to consider a larger number of points, in the limit ofsmall spacing. ∆ xS FIG. 10. Entanglement entropy of five points as a functionof the spacing between them. The horizontal axis is scaledby Ω /π so that the Nyquist spacing is 1. For spacings be-low the Nyquist spacing, we see a plateauing effect indicatingthat the entanglement entropy is not sensitive to the spacingbetween the points for spacings below the Nyquist spacing.Above the Nyquist spacing the entropy tends to increase asthe spacing increases. In this plot the infrared to ultravioletratio is ω/ Ω = 10 − . D. Oversampling using derivatives
The above results suggest that even if the N samplepoints are arbitrarily close together, the correspondingentanglement entropy will scale as log( N ). However,these results are restricted either to a small number oflattice points, or to very mild oversampling. To show thisscaling behaviour we will develop a method that allows usto consider a larger number of sample points in the limitof small spacing. In the limit where the N points aretaken coincident, sampling at N points is equivalent tosampling the first N spatial derivatives at a single point.This is related to the result that in classical samplingtheory that instead of sampling the field at the Nyquistrate, one can instead sample the field and its derivativesat a fraction of the Nyquist rate [33]. For example, if ata sample point one measures both the amplitude of thefield and its first derivative, then the samples only needto be taken at half the Nyquist rate.First, consider a situation where we sample at just twonearby points. We can perform the following symplectictransformation on the phase space variables: φ ( x ) φ ( x + ∆ x ) π ( x ) π ( x + ∆ x ) → √ ( φ ( x + ∆ x ) + φ ( x )) √ ( φ ( x + ∆ x ) − φ ( x )) √ ( π ( x + ∆ x ) + π ( x )) √ ( π ( x + ∆ x ) − π ( x )) . (82)Thus, we see that sampling two points in the limit wheretheir separation vanishes, ∆ x →
0, can equivalently be5 ∆ xS Classical, T/ Ω = 10 Quantum, T/ Ω = 10 Quantum, T/ Ω = 10 − Quantum, T/ Ω = 10 − Quantum, T/ Ω = 10 − Quantum, T/ Ω = 10 − Quantum, T/ Ω = 0
FIG. 11. Entanglement entropy of five points as a functionof the spacing between them at various temperatures. Thehorizontal axis is scaled by Ω /π so that the Nyquist spacingis 1. We see here a plateau in the entropy at small spac-ings, similar to the plateau in the ground state entropy. Theanalogous classical calculation is also shown as black dots fortemperature T /
Ω = 1. At temperatures
T / Ω >
1, the entropybehaves the same as for
T /
Ω = 1 but shifted vertically by aconstant ∼ log( T /
Ω). In this plot the infrared to ultravioletratio is ω/ Ω = 10 − . viewed as sampling the field and its derivative at a singlepoint (as well as the conjugate momentum and its deriva-tive at that point). This motivates performing the cal-culation of the entanglement entropy after sampling thefield and its higher derivatives { φ, φ (cid:48) , φ (cid:48)(cid:48) , . . . , φ ( N − } ata single point, which is equivalent to sampling the fieldat N points in the limit of small spacing. This can bethought of as an extreme case of oversampling the field.The required matrix elements for the calculation of theentropy are:[( ∂ x ) n φ ( x ) , ( ∂ x ) m π ( x )] = (cid:40) iπ ( − n +3 m n + m +1 (cid:0) Ω n + m +1 − ω n + m +1 (cid:1) if ( n + m ) = 0 mod 20 if ( n + m ) = 1 mod 2 , (83) (cid:104) ( ∂ x ) n φ ( x ) · ( ∂ x ) m φ ( x ) (cid:105) = ( − n +3 m π log (cid:0) Ω ω (cid:1) if n + m = 0( − n +3 m π n + m (Ω n + m − ω n + m ) if n + m (cid:54) = 0 , ( n + m ) = 0 mod 20 if ( n + m ) = 1 mod 2 , (84)and (cid:104) ( ∂ x ) n π ( x ) · ( ∂ x ) m π ( x ) (cid:105) = (cid:40) ( − n +3 m π n + m +2 (cid:0) Ω n + m +2 − ω n + m +2 (cid:1) if ( n + m ) = 0 mod 20 if ( n + m ) = 1 mod 2 . (85)Figure 12 shows the entropy as a function of the num-ber of points in the limit of small spacing. For referencewe also show the entropy for Nyquist spacing. Both re-sults are in agreement with the curve 1 / N ) + c , butwith different constants.This is consistent with the interpretation that bothNyquist spacing and the small spacing limit calculate theentropy of an interval of the same length, but regulatedin a slightly different way. This different regularization inthe small spacing limit slightly disentangles the degreesof freedom very close to the entangling surface. This re-duces the constant coefficient in the entropy, while keep- ing the leading logarithmic behaviour. We will give someevidence for this interpretation in section V. V. LOCALIZATION OF DEGREES OFFREEDOM
We have shown how the entanglement entropy of a setof samples of a bandlimited quantum field depends on thenumber of samples taken and on the spacing betweensamples. For spacings larger than the Nyquist spacingthe entanglement grows linearly with the number of sam-6 NS Nyquist spaced pointsNyquist fitDerivativesDerivatives fit
FIG. 12. Entanglement entropy dependence of number ofderivatives traced out at a single point. Fitted curve toNyquist spaced points is S ( N ) = 0 .
334 log( N ) + 3 .
25, where N is the number of sampling points. Fitted curve to deriva-tive sampling points is S ( N ) = log( N ) + 3 .
14, where N − N sampled points). We see that both curves differ by a constant ≈ .
11. The infrared to ultraviolet ratio is ω/ Ω = 10 − . ples. At the Nyquist spacing there is a sharp transition.For samples spaced at a Nyquist spacing or closer, the en-tanglement entropy grows logarithmically with the num-ber of samples, with a coefficient that is independent ofthe spacing. We will now show how this is a consequenceof each mode of the field occupying an incompressiblevolume of space, which we will call a Planck volume.Since the degrees of freedom on all sample points of aNyquist lattice is equivalent to the entire field, one caninterpret each degree of freedom on the Nyquist lattice asrepresenting information roughly contained in a Planckvolume centered at the sample point. Thus, we can in-terpret sampling a set of points on a Nyquist lattice assampling the corresponding interval of the field. If wesample finitely many contiguous degrees of freedom ona Nyquist lattice, the resulting entanglement entropy isgenerated by the correlations cut across the boundary ofthe interval. Thus, the entanglement entropy varies loga-rithmically with the number of points sampled, N , sincethey describe an interval of length L = N π/
Ω.If the sample points we take are spaced farther than aNyquist spacing, the corresponding Planck volumes cen-tered at these sample points will be disjoint. Therefore,the local correlations around each point which contributeto the entanglement entropy will mostly be independent.Therefore, the entanglement entropy for points which arewell-separated will scale linearly with the number of sam-pled degrees of freedom, as we demonstrated above.The most interesting case is the case of oversampling.When the sampled degrees of freedom are pushed veryclose together, we find that the entropy varies logarith- mically with the number of degrees of freedom sampled.As the degrees of freedom are taken together one mightexpect to probe only a single Planck volume, and hencefind a smaller entropy. If the N points are almost ontop of one another, the correlations at the boundary ofthe overlapping Planck volumes would generate entan-glement entropy roughly equivalent to the entanglemententropy of a single sampled degree of freedom, ratherthan N degrees of freedom. However, from the results ofthe entropy calculation, it seems that the N degrees offreedom correspond to a larger, effective volume the sizeof the original volume described by the N contiguousNyquist-spaced points before they were pushed together.We can estimate the size of this effective volume asfollows. First, we take a set of N sample points withsome equidistant spacing, which we shall denote subsys-tem A N . Now we include another sample point whichwe initially place very far from these N sample points,which we shall denote subsystem P . This point will beused as a probe to localize the points in subsystem A N ,using the mutual information I ( A N : P ) = S ( A N ) + S ( P ) − S ( A N , P ) , (86)where S ( A N ), S ( P ) are the entanglement entropies of thesubsystems A N and P (respectively), and S ( A N , P ) isthe entanglement entropy of the combined system. Whilethe probe point P remains far away, the mutual informa-tion will be small because the correlations between A N and P are small. When P approaches the system of N points, the mutual information will increase as the twosubsystems become more correlated. We can use the in-crease of the mutual information I ( A N : P ) as an indi-cator of the extent of the subsystem A N . In particular,we can map the boundary of A N for various values of thespacing between the N points. If the points spaced farbelow the Nyquist spacing indeed describe an effectivevolume of ∼ N Planck volumes in size, then we shouldsee the boundary of the subsystem at the edge of thisregion.In figure 13 we see this is the case for a subsystemof N = 5 points. When the points in the interval arespaced farther than a Nyquist spacing (in the upper re-gion of the graph), we can clearly identify the region ofspace that they occupy, which is centered at each sam-ple point and on the order of a Planck volume in size.When the spacing approaches the Nyquist spacing, thepoints begin to occupy a single interval of roughly N = 5Planck volumes in size. As the spacing decreases belowthe Nyquist spacing, the size of this interval ceases todecrease, indicating that the sample points describe thesame volume of space regardless of their positions belowthe Nyquist spacing. Therefore, the sample points eachdescribe an independent volume of space of Planckiansize. In this sense, the Planck volumes described by theindividual degrees of freedom are incompressible.This behavior shows how the field is able to have afinite information density, while still allowing arbitrarilyclosely spaced probes. If we attempt to sample the field7 − − − − − x ∆ x . . . . . -20 -10 0 10 200.20.40.60.811.21.4 x I ( A N : P ) -20 -10 0 10 200.20.40.60.811.21.4 x I ( A N : P ) -20 -10 0 10 200.20.40.60.811.21.4 x I ( A N : P ) FIG. 13. Mutual information between subsystem A N and probe point P . The spatial axes are scaled so that the Nyquistspacing is 1. The variable x denotes the position of the probe point P , with x = 0 at the centre of the subsystem A N . Forspacings much larger than the Nyquist spacing, the degrees of freedom occupy independent intervals of the order of the Nyquistspacing in size. For spacings at or below the Nyquist spacing, the points in the interval describe a fixed volume of size N . at two points closer than a Nyquist spacing, we are reallyonly probing the degrees of freedom in a larger regionof space centered around these samples, whose volumeis determined by the number of samples rather than bytheir spacing. VI. INFRARED BEHAVIOUR
Above we have examined the effect of the ultravioletcutoff on the calculation of the entanglement entropy of abandlimited quantum field. Here we also briefly examinethe infrared behaviour of the entanglement entropy. Refs.[43, 44] calculated the entanglement entropy of a freescalar field in 1+1 dimensions with an infrared cutoff.They find that the entanglement entropy of a subset of8 N oscillators takes the form S = c log( N ) + 12 log (cid:18) log (cid:18) ω (cid:19)(cid:19) − log π, (87)where c is a constant and ω is the infrared cutoff. Herethe entropy diverges as a double logarithm with the in-frared cutoff. This infrared divergence is a special fea-ture of 1+1 dimensions and arises due to a build-up oflong range correlations in the field. We saw in sectionII that the φ - φ correlations decay very slowly with thepoint separation, and are only suppressed on the order ofthe longest wavelength 2 π/ω . If we remove the infraredcutoff, we allow for infinitely long wavelengths, and thusdivergent correlations.In our above numerical results for calculating the en-tanglement entropy of a quantum field on a Nyquist lat-tice, we perform a numerical fit of the data to determinethe coefficients of the equation S ( N ) = c log N + c . (88)This numerical fit was also performed for various valuesof the infrared cutoff, ω/ Ω ∈ [10 − , − ]. We findthat the leading order behaviour of c in this range is c = 12 log (cid:18) log (cid:18) Ω ω (cid:19)(cid:19) . (89)The coefficient c agrees with the result of Ref. [44] (upto a constant). Furthermore, in the limit ω (cid:28) Ω, theinfrared cutoff ω has the same effect on the entropy as asmall mass [43].It is possible to anticipate this result analytically byconsidering tracing out a system of one oscillator. Fromthe expressions found above for the two-point functions,we find the diagonal terms are[ φ ( x ) , π ( x )] = i Ω − ωπ , (90) (cid:104) φ ( x ) (cid:105) = 12 π log (cid:18) Ω ω (cid:19) , and (91) (cid:104) π ( x ) (cid:105) = 14 π (cid:0) Ω − ω (cid:1) . (92)The symplectic eigenvalues of the covariance matrix are λ ± = ± π Ω − ω (cid:115) π (Ω − ω ) log (cid:18) Ω ω (cid:19) . (93)Thus, the entanglement entropy is S = ( λ + + 12 ) log( λ + + 12 ) − ( λ + −
12 ) log( λ + −
12 ) , (94)which to leading order in the limit ω/ Ω → S ∼
12 log (cid:18) log (cid:18) Ω ω (cid:19)(cid:19) . (95)We see directly that the outer logarithm is due to theentanglement entropy formula, and the inner logarithm arises because the diagonal elements of the φ - φ correla-tion matrix diverge logarithmically in the infrared as (cid:104) φ ( x ) (cid:105) = (cid:90) Ω ω dk π k = 12 π log (cid:18) Ω ω (cid:19) . One may wonder how the infrared behaviour of the en-tanglement entropy changes with a different implementa-tion of the infrared cutoff. In particular, we will examinehow the entropy behaves when the field modes are dis-crete. Consider a scalar field on an interval [0 , L ] withperiodic boundary conditions (up to a phase), φ (0) = e iα φ ( L ) , (96)where α ∈ [0 , π ). Expanding the field in a spatialFourier series, φ ( x ) = (cid:88) k √ ω k L e ikx φ k (97)where ω k := | k | , one obtains a discrete set of momentummodes { k n = πn − αL } n ∈ Z . The ultraviolet cutoff will beimposed by restricting the set of allowable n to { n ∈ Z : ω k n = | k n | ≤ Ω } . We see that for α > φ - φ correlations are finite,but there will be an infrared divergence as α →
0. Itis straightforward to find that the Hamiltonian for thisKlein-Gordon field can be expressed as H = 12 (cid:88) n ω k n (cid:0) π k n + φ k n (cid:1) . (98)Expanding in terms of the usual creation and annihilationoperators, a k := 1 √ φ k + iπ k ) , (99)one finds[ φ ( x ) , π ( x (cid:48) )] = iL (cid:88) n e ik n ( x − x (cid:48) ) , (100) (cid:104) φ ( x ) φ ( x (cid:48) ) (cid:105) = 12 L (cid:88) n ω k n e ik n ( x − x (cid:48) ) , and (101) (cid:104) π ( x ) π ( x (cid:48) ) (cid:105) = 12 L (cid:88) n ω k n e ik n ( x − x (cid:48) ) . (102)These matrix elements are easy to calculate numericallysince the ultraviolet cutoff imposes a finite number of n to sum over.Similar to the calculation performed above, we shallcalculate the entanglement entropy associated with trac-ing out the field oscillator at a single point. The relevant9diagonal elements of the correlation matrices are:[ φ ( x ) , π ( x )] = i KL , (103) (cid:104) φ ( x ) (cid:105) = 12 α + 14 π (cid:104) ψ (cid:16) K − α π + 1 (cid:17) − ψ (cid:16) − α π (cid:17) + ψ (cid:16) K + α π (cid:17) − ψ (cid:16) α π (cid:17)(cid:105) , (104) (cid:104) π ( x ) (cid:105) = πK L , (105)where K := Ω L/ π , and ψ ( x ) is the digamma function.We see that now instead of a logarithmic infrared diver-gence of (cid:104) φ ( x ) (cid:105) , with a discrete set of modes the correla-tion function diverges as 1 /α as α →
0. In this limit, theleading order behaviour of the symplectic eigenvalues is λ ± ∼ (cid:114) π α . (106)The corresponding entanglement entropy is S ∼
12 log (cid:16) π α (cid:17) − log 2 + 1 . (107)Numerical calculations were performed with multiplepoints traced out, and the observed behaviour in the limit α → λ max = 2 π/ω . For periodic boundary conditions, themaximum wavelength is λ max = L/ πα . Thus, we cancompare and summarise the infrared behaviour of theentanglement entropy as • For a continuous set of modes: S ∼
12 log (log (Ω λ max )) as λ max → ∞ . (108) • For a discrete set of modes: S ∼
12 log (cid:18) λ max L (cid:19) as λ max → ∞ . (109)In the case of boundary conditions that are periodic upto a phase, the entropy diverges more quickly. This isplausible since this case is a description of a field on acircle, and any two points on this circle coupled to oneanother an arbitrary number of times in the limit λ max →∞ . VII. OUTLOOK
Shannon sampling, which in information theory estab-lishes the equivalence of continuous and discrete informa-tion, also yields a simple model for how a natural ultra-violet cutoff at the Planck scale might manifest itself in QFT. Namely, when approaching the Planck scale fromlower energies, where QFT is still valid, quantum fieldsare modeled as being bandlimited.Within this model, we probed the localization of thequantum field’s degrees of freedom by tracking the be-haviour of the vacuum entanglement entropy. We foundthat the degrees of freedom of the quantum field, i.e.,the local field oscillators, are non-local but can be local-ized down to the Planck-scale. In fact, we found that thelocal degrees of freedom occupy incompressible Planck-scale volumes, in the sense that N degrees of freedomalways describe N Planck volumes regardless of the dis-tance between them.The tools that we outlined in this paper are applicablealso to the study of bandlimited quantum fields in higherdimensions, and to fermionic fields.For the case of higher dimensions, we conjecture thatwe will continue to see incompressibility of the field de-grees of freedom, i.e., a plateau in the entanglement en-tropy for spacings below the Nyquist spacing. A systemof N lattice points should continue to probe a region of N Planck volumes even if they are sub-Nyquist spaced.In higher dimensions it will be interesting to see how theshape of this region changes with the relative locationof the sample points. Moreover, at larger than Nyquistspacings, we expect to find an infrared plateau. This isbecause the vacuum correlations decay over shorter dis-tances in higher dimensions and do, therefore, not exhibitthe infrared divergences that soften the area scaling to alogarithmic scaling in 1+1 dimensions.Our approach to studying localization and vacuum en-tanglement should also work for quantum fields on curvedspacetimes. To this end, the generalization of Shannonsampling theory to curved spaces developed in [35] canbe used. It will be interesting, in particular, to determinehow the localizability of the fundamental field oscillatordegrees of freedom is affected by curvature and horizons,and in particular, how this may affect Hawking radiation.A similar hard momentum cutoff was considered in [45],and it should be interesting to apply our new methodsto that case.A further direction of great interest would be to studythe entanglement entropy and the localization of degreesof freedom of fields with a fully covariant ultraviolet cut-off. In this case, the bandlimitation would be imposedby cutting off the spectrum of a covariant Laplacian(on Riemannian manifolds) or a d’Alembert operator (onLorentzian manifolds) [46]. To this end, it may be use-ful to express the entropy purely in terms of spacetimecorrelation functions [42], i.e., without reference to thecanonical formalism. We remark that it has been arguedthat if the entanglement entropy is finite and obeys aform of the Clausius relation, then Einstein’s equationemerges as a thermodynamic equation of state [47, 48].Thus having a regulator that is both covariant and cutsoff the entanglement entropy could be a key step towardquantum gravity.It should also be very interesting to study the inter-0action of Unruh-DeWitt detectors with the bandlimitedquantum fields. Unruh deWitt detectors would describeexplicit means to sample the quantum fields, and thiscould, therefore, constitute a further step towards thequantization of sampling theory. An interesting questionthat then arises is how the spatial profile of an Unruh-DeWitt detector interacts with the spatial profile of thedegrees of freedom. Another key question that ariseswith the quantization of sampling theory follows from thefact that, in classical sampling theory, the reconstructionof a function from a lattice with non-equidistant samplepoints is more sensitive to noise in the sample measure-ments than reconstruction from a lattice with equidis-tant sample points. This, therefore, raises the interestingquestion if or to what extent quantum noise may inter-fere with the ability to use significantly irregularly-spacedlattices.One of our central results is that the entanglement en-tropy does not decrease significantly as the spacing isdecreased to below the Nyquist spacing. In classical sam-pling theory too a curious phenomenon is known to ariseas the spacing is decreased to below the Nyquist spacing,namely the phenomenon of superoscillations.Superoscillations in a bandlimited function are a finiteset of oscillations that oscillate faster than the highestFourier component in the signal [38, 49–52]. Superoscil-lations are difficult to generate and rarely occur naturallybut they have become an active field of investigation, alsoin engineering, because of their potential for superreso-lution. An open problem in this field is to quantify theprevalence of superoscillations in random signals. In ouranalogous classical calculation, we obtained probabilitydistributions for observing particular field values on anarbitrary sampling lattice. These distributions could beused to calculate the probability for finding the field inany particular superoscillatory configuration. It might bepossible to find in this way a practical measure of classicalor quantum field fluctuations that is sensitive specificallyto the occurrence of superoscillatory behaviour.A key question that we addressed in this paper hasbeen the question of how a bounded continuous regioncan support only a finite number of degrees of freedom.In our model for a natural ultraviolet cutoff, this is ac-complished by the quantum field degrees of freedom atdistinct spacetime points not being independent. As aconsequence, probing with too many operators in a smallregion simply causes one to probe a larger region of space.This means that the degrees of freedom of the theory areencoded redundantly on a single spatial slice.In fact it has recently been argued that, similarly, sig-nificant redundancy is present in the bulk field theory inAdS/CFT [53]. The redundancy implied by holographyappears to be more extreme, however, since the numberof degrees of freedom is bounded by area rather than vol-ume. To obtain this more drastic reduction, the nontriv-ial commutation relations coming from gravitation inter-action may provide some clues [25]. One may also takea cue from holography by trying to redundantly encode the degrees of freedom of a quantum field in a space ofhigher dimension.Finally, we note that the hyperbolic geometry of anti-de Sitter space suggests that an encoding based onwavelets may have some connection to holography [54–57]. Wavelet theory and Shannon sampling theory areclosely related and the quantization of wavelet theoryshould be possible along the lines that we developed here.
ACKNOWLEDGEMENTS
AK and JP acknowledge support through the Discov-ery and NSERC-CGS-M (Canada Graduate Scholarship)programs of the Natural Sciences and Engineering Re-search Council (NSERC) of Canada. JP also acknowl-edges support through the Ontario Graduate Scholarhip(OGS) program.
Appendix A: Functional analytic structure ofsampling theory
Here we will briefly review the functional analyticstructure of sampling theory, as first shown in [19]. Thefunctional analytic view is powerful because it ultimatelyreduces sampling theory to the simple fact that whena Hilbert space vector is known in one basis then it isknown in all bases. A key point here is the distinctionbetween symmetric and self-adjoint operators. By thespectral theorem, each self-adjoint operator possesses aunique diagonalization, whereas a simple symmetric op-erator does not, see, e.g., [58]. A certain type of sim-ple symmetric operator, however, possesses a family ofself-adjoint extensions, and their diagonalizations andspectra provide the sampling lattices of sampling the-ory. (The functional analytic view therefore also allowsone to work with varying Nyquist rates, which can occurin signal processing as well as in curved spacetimes.)Indeed, when acting upon the space of functions ban-dlimited by some maximum frequency Ω, the usual po-sition operator from first quantization, ˆ X , is symmetricbut not self-adjoint. To see this, recall that for an oper-ator ˆ X to be symmetric, it must obey( ˆ Xφ | ψ ) = ( φ | ˆ Xψ ) (A1)for all | φ ) , | ψ ) ∈ D ( ˆ X ), where D ( ˆ X ) is the domain ofˆ X . Equivalently, an operator is symmetric if and onlyif its expectation value is real for all vectors in its do-main: ( φ | Xφ ) ∈ R ∀ φ ∈ D ( ˆ X ). For an operatorto be self-adjoint, it must be symmetric and its do-main must coincide with the domain of its adjoint, i.e.,( ˆ Xφ | ψ ) = ( φ | ˆ Xψ ) ∀| φ ) , | ψ ) ∈ D ( ˆ X ) and D ( ˆ X ) = D ( ˆ X † ).Indeed, the position operator ˆ X with domain restrictedto the space of bandlimited functions is symmetric, but1the domain of its adjoint is a larger space of functions.This is most easily seen in the momentum eigenbasis.In this basis, the space of physical wavefunctions, andtherefore the domain of ˆ X , is the space of functions D ( ˆ X ) = (cid:8) φ ∈ L [ − Ω , Ω] | φ ∈ AC [ − Ω , Ω] ,φ (cid:48) ∈ L [ − Ω , Ω] , φ ( − Ω) = φ (Ω) = 0 (cid:9) (A2)where AC [ − Ω , Ω] denotes the space of absolutely contin-uous functions. Notice that D ( ˆ X ) contains only func-tions which obey Dirichlet boundary conditions. In con-trast, the domain of ˆ X † is the larger function space ob-tained by not imposing any boundary conditions. Thiscan be seen from the definition of ˆ X † :( ˆ X † φ | ψ ) =( φ | ˆ Xψ )= (cid:90) dk ˜ φ ∗ ( k ) (cid:18) i ddk ˜ ψ ( k ) (cid:19) = i (cid:16) ˜ φ ∗ (Ω) ˜ ψ (Ω) − ˜ φ ∗ ( − Ω) ˜ ψ ( − Ω) (cid:17) + (cid:90) dk (cid:18) i ddk ˜ φ (cid:19) ∗ ( k ) ˜ ψ ( k ) . (A3)We see from the last line that as long as | ψ ) is in the do-main of ˆ X (thus obeys Dirichlet boundary conditions onthe interval [ − Ω , Ω] in Fourier space), then we can definethe adjoint of ˆ X † as i ( d/dk ) in the momentum represen-tation. However, since we did not need to impose bound-ary conditions on | φ ), we have that D ( ˆ X ) (cid:40) D ( ˆ X † ).Crucially now, it is possible to extend the domain ofˆ X so that the extension is a self-adjoint operator. Thisoperator is called a self-adjoint extension of ˆ X (e.g., see[58, 59]). In our situation, we can perform this extensionby enlarging the domain of ˆ X to include functions withperiodic boundary conditions up to a phase on the in-terval [ − Ω , Ω] in Fourier space, i.e., ˜ ψ (Ω) = e − iα ˜ ψ ( − Ω).We shall denote the self-adjoint extension correspondingto the particular phase e − iα as ˆ X ( α ) , thus the familyof self-adjoint extensions is parametrized by α ∈ [0 , π ).Now we will show that the operators ˆ X ( α ) are indeedself-adjoint. The definition of the adjoint of ˆ X ( α ) † gives( ˆ X ( α ) † φ | ψ ) = i (cid:16) ˜ φ ∗ (Ω) ˜ ψ (Ω) − ˜ φ ∗ ( − Ω) ˜ ψ ( − Ω) (cid:17) + (cid:90) dk (cid:18) i ddk ˜ φ (cid:19) ∗ ( k ) ˜ ψ ( k )= i (cid:16) ˜ φ ∗ (Ω) − e iα ˜ φ ∗ ( − Ω) (cid:17) ˜ ψ (Ω)+ (cid:90) dk (cid:18) i ddk ˜ φ (cid:19) ∗ ( k ) ˜ ψ ( k ) . (A4)We see that the adjoint of ˆ X ( α ) is defined as i ( d/dk ) inthe momentum basis provided that | φ ) also obeys theboundary condition ˜ φ (Ω) = e − iα ˜ φ ( − Ω). Therefore wesee D ( ˆ X ( α ) ) = D ( ˆ X ( α ) † ), and so ˆ X ( α ) is self-adjoint. Now, since the operators ˆ X ( α ) are self-adjoint, theyhave spectral decompositions. The spectrum of the oper-ator ˆ X ( α ) (for fixed α ) is discrete, spec( ˆ X ( α ) ) = { x ( α ) n := πn + α } n ∈ Z , and describes a one-dimensional lattice. Thecorresponding eigenvectors, {| x ( α ) n ) } n ∈ Z , are representedin the momentum eigenbasis as( k | x ( α ) n ) = 1 √ e − ikx ( α ) n . (A5)These eigenvectors are orthogonal and admit a resolutionof the identity: (cid:88) n ∈ Z | x ( α ) n )( x ( α ) n | = (A6). Crucially, the position eigenvectors from different self-adjoint extensions are not orthogonal:( x ( α ) n | x ( α (cid:48) ) n (cid:48) ) = sinc (cid:104) ( x ( α ) n − x ( α (cid:48) ) n (cid:48) )Ω (cid:105) . (A7)Note that the union of the spectra of the entire family ofself-adjoint extensions provides a covering of R . There-fore, it is possible to construct an overcomplete contin-uum basis by taking the union of eigenbases of the familyof self-adjoint extensions, i.e., | x ) := | x ( α ) n ) ⇐⇒ x = x ( α ) n := πn + α .It is then simple to write down the Shannon samplingtheorem for a bandlimited function ψ : ψ ( x ) = ( x | ψ ) (A8)= (cid:88) n ∈ Z ( x | x ( α ) n )( x ( α ) n | ψ ) (A9)= (cid:88) n ∈ Z sinc (cid:104) ( x − x ( α ) n )Ω (cid:105) ψ ( x ( α ) n ) . (A10)Therefore, we see that the function ψ is determined atany point x ∈ R from its values on one of the lattices { x ( α ) n } n .Also, we obtain an overcomplete resolution of identity,Ω π (cid:90) R dx | x )( x | = (A11)where π/ Ω is the density of degrees of freedom. 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