Localized Excitations from Localized Unitary Operators
aa r X i v : . [ h e p - t h ] J un Localized Excitations from Localized UnitaryOperators
Allic Sivaramakrishnan ∗ Department of Physics and Astronomy,University of California, Los Angeles,Los Angeles, CA 90095-1547, USA
Abstract
Localized unitary operators are basic probes of locality and causality in quantumsystems: localized unitary operators create localized excitations in entangled states.Working with an explicit form, we explore properties of these operators in quantummechanics and quantum field theory. We show that, unlike unitary operators, localnon-unitary operators generically create non-local excitations. We present a local pic-ture for quantum systems in which localized experimentalists can only act throughlocalized Hamiltonian deformations, and therefore localized unitary operators. Wedemonstrate that localized unitary operators model certain quantum quenches exactly.We show how the Reeh-Schlieder theorem follows intuitively from basic propertiesof entanglement, non-unitary operators, and the local picture. We show that a recentquasi-particle picture for excited-state entanglement entropy in conformal field theoriesis not universal for all local operators. We prove a causality relation for entanglemententropy and connect our results to the AdS/CFT correspondence. ∗ [email protected] ontents In this work, we study correlation functions in time-dependent states and in theories withtime-dependent Hamiltonians. Our results apply to pure states in quantum mechanics andlocal quantum field theory. Our primary goal is to detail the universal role of localized unitaryoperators in creating localized excitations. A brief overview of our results is as follows. Justas every observable is represented by some Hermitian operator, every localized excitationis created by some localized unitary operator. If a localized operator is non-unitary, theexcitation it creates is not necessarily localized. For example, a localized unitary operator e iα O ( x ) creates a localized excitation while the localized non-unitary operator e α O ( x ) does1ot. Localized experimentalists can only act on states by deforming the Hamiltonian by alocalized quantity, H → H + H loc ( t ), and this is equivalent to acting with the Heisenberg-picture localized unitary operator T (cid:0) e i R dt H loc ( t ) (cid:1) on the state. As they create localizedexcitations, localized unitary operators are tied to basic questions of causality.This manuscript extends contemporary studies of excited-state entanglement entropyin conformal field theories (CFTs) [1–8]. Our work builds upon the large body of workon localized excitations and real-time perturbation theory (see, for example, [9–19]). Inaddition to presenting our own results, we reinterpret some well-known results from thisbody of literature and from quantum information theory that are relevant. We revisit theseresults to give a coherent picture for the connection between localized unitary operators andlocalized excitations. We present all results in elementary terms and eschew a complete oraxiomatic treatment of the topics discussed. Our purpose is to make contact with modernstudies of entanglement entropy in CFT, and a more formal treatment of locality is outsidethe scope of this work.While we explore how localized unitary operators create localized excitations, these op-erators play many other well-known roles in quantum systems. Specific forms of localizedunitary operators create squeezed states, coherent states, generalized coherent states, andimplement local gauge transformations [20–24]. Localized unitary operators are also impor-tant in large-N and large-dimension limits in quantum mechanics, gauge theories, and theAdS/CFT correspondence.We now summarize each section. In section 2, we define what we mean by localized andreview related concepts. Suppose a Hilbert space H can be written as a tensor productHilbert space H = H ⊗ H . An excitation of state | Ψ i ∈ H can be represented by actingwith some operator O e on | Ψ i , where O e is suitably normalized so that O e | Ψ i has unit norm.This excitation is localized in H if h Ψ |O † e OO e | Ψ i = h Ψ |O| Ψ i (1.1)for all operators O local in H . O is local in H when O can be written as O ≡ I × O . (1.2)Here I is the identity in H and O : H → H . In field theory, H , H can be chosenas the Hilbert spaces of the theory restricted to a subregion A of a Cauchy surface andits complement A c . An excitation localized to A does not affect correlation functions ofoperators inserted at points spacelike-separated from all points in A . The familiar localoperators O ( x ) in field theory are localized to the Hilbert space of every arbitrarily smallneighborhood of x . Localized operators can be built from operators that are local in different2oints or Hilbert spaces. For example, if f ( x ) has support in region R , smeared operator R dxf ( x ) O ( x ) is localized in R . We address subtleties involved in defining localization forgauge theories.In section 3, we review real-time perturbation theory [9, 10]. This perturbation theorygives corrections to correlation functions in time-dependent states perturbatively in a time-dependent interaction Hamiltonian. Perturbation theory for the S-matrix calculates in-out matrix elements h Ψ out | Ψ in i , while real-time perturbation theory calculates in-in matrixelements h Ψ in |O . . . O n | Ψ in i . Real-time perturbation theory makes manifest how localizedinteraction Hamiltonians create localized excitations.In section 4, we present a coherent picture for time-dependent operations in quantummechanics and field theory. We call this picture the local picture of quantum systems. In thelocal picture, an experimentalist can only alter states through deforming the Hamiltonian.A localized experimentalist can only make localized deformations. The excitations thatare natural in the local picture are created by acting with a time-ordered localized unitaryoperator on a state, for example the Heisenberg-picture operator T (cid:0) e i R dtJ ( t ) O ( t ) (cid:1) . As weshow later, the familiar local non-unitary operators of field theory generically create non-localized excitations, so the local picture reveals that local experimentalists cannot act withgeneric local non-unitary operators.In section 5, we present results in quantum mechanics. We show how local unitary oper-ators alter entangled states locally, while local non-unitary operators alter entangled statesnon-locally. Local non-unitary operators can be written as state-dependent non-local uni-tary operators. In the language of quantum information, non-unitary operators implementnon-local quantum gates. Our results explain the Reeh-Schlieder theorem intuitively. Thesuperposition of two local excitations may not be a local excitation itself in entangled states.For instance, the sum of two unitary operators U + U is not necessarily unitary. We showthat the natural way to combine localized excitations created by operators U , U to pro-duce another localized excitation is by acting with the operators in succession: U U . Thisprescription for combining localized excitations follows from the local picture.In section 6, we move on to quantum field theory, our main focus. We show that localizedunitary operators create localized excitations. The reason is as follows. If x, y are spacelike-separated, local operators O , O ′ inserted at x, y commute:[ O ( x ) , O ′ ( y )] = 0 . (1.3)It follows immediately that h Ψ | e i O ( x ) O ′ ( y ) e − i O ( x ) | Ψ i = h Ψ |O ′ ( y ) | Ψ i . (1.4)3f x, y are not spacelike-separated, then the above equality generically does not hold. Forexample, in the vacuum of a free real scalar field, the Baker-Campbell-Hausdorff lemma gives h | e iαφ ( x ) φ ( y ) e − iαφ ( x ) | i = iαG R ( x − y ) + . . . , (1.5)where α can be treated as an expansion parameter and x is restricted to the future of y . Theretarded Green’s function G R ( x − y ) vanishes when x − y is spacelike. More generally, thecommutator of operators h [ O ( x ) , O ′ ( y )] i diagnoses causality in field theory, and we see thatthis commutator is in fact the order α correction to hO ′ i in the excited state e − iα O ( x ) | Ψ i .We explore the properties of localized unitary operators, including what we call separableand non-separable localized unitary operators. Separable unitary operators like e i ( O ( x )+ O ( y )) create excitations at x, y that are not entangled with each other. Acting with non-separableunitary operators like e i O ( x ) O ( y ) create excitations at x, y that may be used to violate causality.As such, non-separable unitary operators cannot be applied to states under time evolutionin local quantum field theory. We give a criterion to test separability.We show that local non-unitary operators can create non-local excitations in field theory.We provide examples and show where the intuition that arbitrary local operators create localexcitations breaks down. We provide evidence that certain local non-unitary operators docreate local excitations and give examples of others that do not.In section 7, we apply lessons from the previous sections to give new results concerningthe entanglement entropy of excited states in field theory. Recently, a compelling quasi-particle picture has emerged from calculations of entanglement entropy in CFTs [1, 3–5, 25].It has been suggested that local operators create entangled pairs of quasi-particles at theirinsertion point [4]. We provide evidence that this picture applies to local operators withdefinite conformal dimension. It is known that the quasi-particle picture is invalid for certaintheories [5, 26], and using the example of the operator e α O ( x ) , we show how this picture failsto extend to all local operators even within theories for which the picture is expected tobe accurate. We extend a result in ref. [27] by proving a general causality relation forentanglement entropy. We will review locality and causality criteria in quantum mechanics and quantum field theory.Causality in field theory is a statement about the commutators of operators. If spacetimepoints x, y are spacelike-separated, then any two local operators O ( x ) , O ( y ) commute.[ O ( x ) , O ( y )] = 0 . (2.1)4n analogous statement holds if the operators are smeared out over some spacetime region.The vanishing of the commutator for spacelike separation is equivalent to a statement aboutthe branch cuts of all Euclidean correlators that contain O , O .We may state a locality condition based on whether local operations affect observablesnon-locally. We state a version of this condition first in quantum mechanics. Express theHilbert space H of some system as a tensor product Hilbert space H = H A ⊗H B of subsystems A, B . First we define local operators.
Definition 1.
The operator O ( B ) is local in H B when it can be written as O ( B ) ≡ I A ⊗ O B . (2.2) Here, O B : H B → H B and I A is the identity in H A . As we will show in section 5, acting with operator O ( B ) may change the expectationvalue of some operator O ( A ) local in H A , and therefore some measurement performed byan experimentalist with access to subsystem A but not B . We define the relevant notions oflocal and non-local changes in state. Definition 2.
Suppose operator O is normalized in a state | Ψ i so that h Ψ |O † O| Ψ i = 1 .Suppose also that there exists an operator O (B) local in H B such that h Ψ |O † O ( B ) O| Ψ i 6 = h Ψ |O ( B ) | Ψ i . (2.3) If for all O ( A ) local in H A , h Ψ |O † O ( A ) O| Ψ i = h Ψ |O ( A ) | Ψ i , (2.4) then O changes the state | Ψ i locally in H B . Otherwise, O changes the state non-locally. It should be understood that when we assess locality by expectation values of operators,we are considering only operators that correspond to observables. We can also assess localitywith the reduced density matrix ρ A . If acting with O does not change ρ A , in other words ρ A ( O | Ψ i ) = ρ A ( | Ψ i ) , (2.5)then O changes the state locally in H B . Local is a special case of localized. A localizedoperator is local in more than one Hilbert space.The definitions we have given for local operators and changes in state apply to fieldtheory. So-called local operators in field theory are local in the quantum-mechanical sensewe have defined. Operator O ( x ) is local in H ( x ), the Hilbert space of the theory restricted to5oint x . When referring to local operators O ( x ), we will refer to the point x as the “insertionpoint” of O . The insertion points of a Wilson loop are the points along its integration path.An operator inserted at multiple points is non-local. For example O ( x ) O ( y ) is non-local butlocalized to x, y .We define localized excitations in field theories. This definition is the same as definition1 but we state it using field theory terminology for clarity. First, we define what we meanby an excitation. Definition 3.
In field theory, we call
O | Ψ i an excitation of the state | Ψ i with operator O . The following definition of localized excitations is also known as “strict localization” [12].
Definition 4.
Consider O ( A ) inserted in subregion A of a Cauchy surface. The complementof A on the Cauchy surface is subregion B . An operator O creates an excitation that islocalized to B if h Ψ |O † O ( A ) O| Ψ i = h Ψ |O ( A ) | Ψ i ∀ O ( A ) . (2.6)The definitions we provided extend in an obvious way to describe operators and excita-tions localized to a region of spacetime, rather than just a region of a Cauchy surface. Alocal excitation is an excitation that is localized to a single point. We will sometimes referto local and non-local excitations of the state in quantum mechanics if we make statementsthat apply to both quantum mechanics and field theory. Various statements we will makealso apply in a natural way to non-localized operators, which are operators inserted in everypoint in a Cauchy surface.In gauge theories, we must fix a gauge before checking the above condition, or we maysimply work in terms of gauge-invariant operators. We must also fix a gauge in order touse the reduced density matrix ρ A to diagnose locality as the density matrix is not gauge-invariant. In this work, we will rarely mention these subtleties involved with gauge theories,as our statements can often be extended in an obvious way to these theories.The ability to change a state through a non-local excitation should not be confusedwith the inability for localized experimentalists to transmit information between spacelike-separated entangled systems by performing local measurements. We will explain how thesetwo features of locality are different and consistent in section 5.3.We state the Reeh-Schlieder theorem, a theorem in quantum field theory that is importantfor understanding locality considerations. Consider the set of all operators O ( B ) that arelocalized to an open subregion B of a Cauchy surface. These operators generate an algebra A ( B ) of the subregion B . The complement of B on the Cauchy surface is B c . Supposethe Hilbert space of the theory on the full Cauchy surface is H . Consider the vacuum stateof some quantum field theory | Ω i . The Reeh-Schlieder theorem is that states A ( B ) | Ω i aredense in H [11]. In other words, one can act with operators that are localized to B to change6he state in B c . Moreover, acting with operators localized in B can prepare a state in B c that is arbitrarily close to any state in H even if that state is an excitation localized entirelyin B c . The Reeh-Schlieder theorem is paradoxical if one assumes that any state O ( B ) | Ω i in principle represents the action of an experimentalist localized to region B on the state | Ω i . The Reeh-Schlieder theorem holds for states other than the vacuum as well. Standardreferences to the Reeh-Schlieder theorem, as well as other aspects of locality in algebraic andaxiomatic quantum field theory include refs. [28, 29]. We review real-time perturbation theory, otherwise known as the in-in formalism [9,10]. Theformalism is called in-in in contrast to perturbation theory for S-matrix elements, which canbe called in-out perturbation theory as it calculates transition amplitudes between initialand final states. Real-time perturbation theory calculates correlation functions h Ψ |O . . . O n | Ψ i (3.1)perturbatively in an interaction Hamiltonian with arbitrary time dependence and makesaspects of locality and causality manifest. Beginning with some initial time-independentHamiltonian H and initial state | Ψ( t ) i , time evolution commences, and an interactionHamiltonian may be turned on. The calculations proceed purely in Lorentzian signature,but the initial state | Ψ( t ) i may be prepared in various standard ways including by Euclideanpath integral. The real-time formalism is also known as the closed-time or Keldysh formalismbecause the same calculation can be performed using a path integral with a closed-time(Keldysh) contour. The real-time formalism is an inherent part of cosmology and AdS/CFT[30–32].Just as in-out perturbation theory may be obtained from a Euclidean path integralthrough Wick rotation, real-time perturbation theory may be obtained from the same Eu-clidean path integral by deforming the purely imaginary-time contour into a closed-timecontour. Calculations proceed similarly for in-out and real-time perturbation theory, bothin the use of Feynman diagrams and the treatment of divergences.To illustrate how real-time perturbation theory works, we calculate the expectation valueof Heisenberg-picture operator O ( t, x ) in a theory with a time-dependent interaction Hamil-tonian H int ( t ). We work in ( d + 1)-dimensional spacetime throughout this manuscript. Indefining the Heisenberg picture, we use reference time t . The associated Schrodinger-pictureoperator defined at time t is O ( t , x ). The full Hamiltonian is H ( t ) = H + H int ( t ) , (3.2)7here H is time-independent. We use a perturbation that is zero at time t : H int ( t ) = 0 . (3.3)We now work in the interaction picture, denoting interaction-picture operators with a sub-script I . The interaction picture is defined in terms of Schrodinger-picture states and oper-ators as | Ψ I ( t ) i = e iH ( t − t ) | Ψ( t ) i . (3.4) O I ( t, x ) = e iH ( t − t ) O ( t , x ) e − iH ( t − t ) . (3.5)The interaction Hamiltonian H int ( t ) in the interaction picture is H I ( t ). As we are workingin real time, time evolution is unitary and preserves the norm of the state, which we chooseto be h Ψ( t ) | Ψ( t ) i = 1. The time evolution operator U ( t, t ) is U ( t, t ) = T (cid:16) e − i R tt dt ′ H ( t ′ ) (cid:17) . (3.6)The interaction-picture evolution operator is U I ( t, t ) = T (cid:16) e − i R tt dt ′ H I ( t ′ ) (cid:17) = e iH ( t − t ) U ( t, t ) . (3.7)We may now calculate h Ψ( t ) |O ( t, x ) | Ψ( t ) i perturbatively in H I . Explicitly, h Ψ( t ) |O ( t, x ) | Ψ( t ) i = h Ψ( t ) |U † ( t, t ) O ( t , x ) U ( t, t ) | Ψ( t ) i = h Ψ( t ) | (cid:0) U † ( t, t ) e − iH ( t − t ) (cid:1) (cid:0) e iH ( t − t ) O ( t , x ) e − iH ( t − t ) (cid:1) × (cid:0) e iH ( t − t ) U ( t, t ) (cid:1) | Ψ( t ) i . (3.8)Passing into the interaction picture, h Ψ( t ) |O ( t, x ) | Ψ( t ) i = h Ψ( t ) |U † I ( t, t ) O I ( t, x ) U I ( t, t ) | Ψ( t ) i . (3.9)Expanding in H I , h Ψ( t ) |O ( t, x ) | Ψ( t ) i = h Ψ( t ) |O ( t , x ) | Ψ( t ) i + i Z t −∞ dt h Ψ( t ) | [ H I ( t ) , O I ( t, x )] | Ψ( t ) i− Z t −∞ dt Z t −∞ dt h Ψ( t ) | [ H I ( t ) , [ H I ( t ) , O I ( t, x )]] | Ψ( t ) i + . . . . (3.10)8he all-order expression for real-time perturbation theory is given by Weinberg [31]. h Ψ( t ) |O ( t, x ) | Ψ( t ) i = ∞ X N =0 i N Z t −∞ dt N Z t N −∞ dt N − . . . Z t −∞ dt × h Ψ( t ) | [ H I ( t ) , [ H I ( t ) , [ . . . [ H I ( t N ) , O I ( t, x )] . . . ]]] | Ψ( t ) i . (3.11)The interaction-picture operators are the Heisenberg-picture operators of the unperturbedtheory at time t .Real-time perturbation theory makes manifest how turning on a localized interactioncreates a localized excitation. Suppose the interaction Hamiltonian is given by some localoperator O ′ smeared over a spatial region: H I ( t ′ ) = Z d d yf ( t ′ , y ) O ′ I ( t ′ , y ) . (3.12)The interaction H I ( t ′ ) can only change hO ( t, x ) i if f ( t ′ , y ) has support on points that arenull or time-like separated from the point ( t, x ). Only the perturbations for which t ′ ≤ t contribute. If f ( t ′ , y ) has support only at points spacelike-separated from the point ( t, x ),then [ H I ( t ′ ) , O I ( t, x )] = 0 . (3.13)Each term in (3.11) will also vanish for the same reason.Suppose H is a free Hamiltonian and | Ψ( t ) i = | i , the vacuum of H . At each orderin perturbative expansion, the nested commutators will produce various contractions mul-tiplied by an overall retarded Green’s function G R ( x − y ), which has precisely the correctcausality properties. Diagrammatic rules that make retarded Green’s functions manifest arediscussed in ref. [33]. In practice, one can obtain the different terms in (3.11) from differentanalytic continuations of the appropriate Euclidean correlators. The structure of real-timeperturbation theory and the presence of retarded Green’s functions parallels a problem inclassical field theory, calculating corrections to the value of a free field perturbatively in asource. We present a coherent picture for time-dependent operations on pure states in quantumsystems. We will refer to this picture as the “local picture”. This picture is generated bythe assumptions that all physical interactions occur through terms in the Hamiltonian, and9ocalized experimentalists deform the Hamiltonian in a localized region. We define the localpicture because it unites several different manifestations of locality and causality into oneconcrete framework. The local picture will provide simple explanations for results in latersections.We first briefly review the two types of systems we may consider in quantum mechanics.Closed quantum systems are pure states that undergo unitary time evolution. For example,the Hilbert space of a closed quantum system H may be a tensor-product Hilbert space of asystem, experimentalist, and environment: H = H env ⊗ H exp ⊗ H sys . (4.1)The experimentalist can be described as an observer and an interaction apparatus: H exp = H obs ⊗ H app . (4.2)Open quantum systems are systems that can be acted upon by some external experimentalist.Pure states of an open quantum system are elements of the Hilbert space H sys . In closedquantum systems, measurement is described by an interaction term in the Hamiltonianthat entangles states between H exp , H sys . This is a unitary process and there is no statecollapse. Projecting onto one of the states in H exp ⊗ H sys shows the state that one particularexperimentalist has access to. In an open quantum system, this measurement process ismodelled by projection operators that implement the collapse of the state, which is theCopenhagen interpretation of quantum mechanics, together with a re-normalization of thestate. In principle, an open quantum system can be obtained from a closed quantum system,and the details of this process are the subject of current research. We will assume this well-known description is valid formally. In short, to describe the measurement process withoutcollapse and state re-normalization, we must use the closed quantum system. To describeoperations performed on the state by an external experimentalist and calculate expectationvalues, the open quantum system is the natural choice.We now state the local picture, which governs the evolution of the pure state | Ψ( t ) i ofsome system prepared at time t . Physical operations on the state are described by deforma-tions of the Hamiltonian. Any norm-preserving operation can be treated as a Hamiltoniandeformation, but deforming the Hamiltonian by functions of local operators are natural waysto implement physical operations. Localized experimentalists can only deform the Hamil-tonian by localized operators, and therefore only act with localized unitary operators onthe state. Operators that create non-localized excitations can only be implemented by non-localized experimentalists. Non-localized experimentalists are experimentalists that haveaccess to the entire Cauchy surface, and should not be confused with experimentalists who10ay depart from the principles of local quantum field theory. We have given the local picturefor open quantum systems, but these principles describe closed quantum systems as well.We give an example that makes the elements of the local picture concrete and showshow they arise. We work in quantum field theory for convenience. The expectation value ofoperator O ( x , t ) evolves as hO ( x , t ) i = h Ψ( t ) |U † ( t, t ) O ( x , t ) U ( t, t ) | Ψ( t ) i . (4.3)In order to describe the effect of some interaction Hamiltonian H int , we may pass into theinteraction picture. As we reviewed in section 3, h Ψ( t ) |O ( t, x ) | Ψ( t ) i = h Ψ( t ) |U † I ( t, t ) O I ( t, x ) U I ( t, t ) | Ψ( t ) i (4.4)The above expresion is equivalent to the following calculation, in the Heisenberg picturedefined by evolution from t with Hamiltonian H : hO ( t, x ) i = h Ψ e |O ( t, x ) | Ψ e i , | Ψ e i ≡ T (cid:16) e − i R tt dt ′ H int ( t ′ ) (cid:17) | Ψ( t ) i . (4.5)We use H int to denote H int in the Heisenberg picture. It is therefore natural that localizedunitary operators U of the form U = T (cid:16) e − i R tt dt ′ H int ( t ′ ) (cid:17) (4.6)create localized excitations, and this follows from real-time perturbation theory. This con-clusion is independent of perturbation theory, as we will show in Section 6.As we have shown, there is a correspondence between excitations of the state and Hamil-tonian deformations. We focus on localized unitary operators, but this correspondence holdsfor non-localized unitary operators and their associated non-localized Hamiltonian deforma-tions in the same way. Localized Hamiltonian deformations take the form of local operatorssmeared over some compact spacetime region, as in equation (3.12), while for non-localizedHamiltonian deformations the smearing function has support on all of spacetime. It is ofcourse not obvious how to find an explicit H int or unitary operator U to represent the ac-tion of an arbitrary operator N O on a state. Here N is the state-dependent normalizationconstant |N | = 1 / h Ψ( t ) |O † O| Ψ( t ) i .We comment on what an experimentalist cannot easily do according to the principlethat she may only interact with the system through H int . At any given time, she may onlyinteract with the state through operators evaluated at that time, and interactions that lastfor some finite time must be time-ordered. Other operations, while mathematically valid,11re not as natural. For example, it is natural to act with the operator U but not U : U = T (cid:16) e − i R dt O ( t ) (cid:17) U = e − i R dt O ( t ) Calculating correlation functions of operators O ( t ) at time t represents experiments con-ducted at t , and excitation of the state that occur after this time do not contribute. Thesenaturalness conditions for operator excitations follow automatically from the local picture.The local picture reveals that localized experimentalists cannot act with operators thatcreate non-localized excitations. If the experimentalist is localized to some spacelike region,she can only use H int also localized in this region, which means acting with unitary operatorslocalized to that same region. These operators create localized excitations. In sections 5 and6, we will find that local non-unitary operators N O can create non-localized excitations,and so a localized experimentalist cannot act with these operators. In fact, if O creates anon-local excitation, the experimentalist must know the state on the entire Cauchy surfacein order to calculate N . Operator N O can only be acted on the state by a non-localizedexperimentalist. An example of such an operator is a normalized projection operator thatimplements a measurement, but to discuss locality in the context of measurements, one isusing a closed quantum system either implicitly or explicitly. Projection operators in an openquantum system are simply models of the process. We will discuss measurements explicitlyin a later section.
We discuss unitary and non-unitary operators in quantum mechanics. Locality properties ofoperators depend on whether or not they are unitary. Our conclusions in quantum mechanicsapply to quantum field theory as well. For a Hilbert space H = H ⊗ H , we address whetheracting with an operator local in H may affect expectation values taken in H . In this section,we will use a two-particle system of spin 1 / |±i .Non-unitary operators generically do not preserve the normalization of states, so to rep-resent their action on the state, we must include a normalization factor along with eachoperator. This normalization factor must be state-dependent, and so in general non-unitaryoperators are state-dependent. We will refer to these norm-preserving non-unitary operatorsas non-unitary operators for short. 12onsider an operator local in H : O = I ⊗ O . (5.1)We will act with O on different states and calculate the reduced density matrix of particle1, ρ . If ρ changes, O has changed the state non-locally. Following the examples, we willprove various general results. The proofs are elementary, and we use elementary methods inorder to make certain properties explicit.We will refer to product states and entangled states of, for example, H . A state | Ψ i ∈ H is a product state if there exist states | Ψ i ∈ H , | Ψ i ∈ H such that | Ψ i = | Ψ i ⊗ | Ψ i .Product states are also known as separable states. A pure state that is not a product stateis entangled. In this section, we show how both local unitary and non-unitary operators change productstates locally. We show an example and then prove this statement. Choose O to be diagonalfor convenience: O = N a b ! . (5.2)Here, a, b ∈ C . If a, b are pure phases then O is unitary. Here N is a normalization factor.First, consider the product state | Ψ p i = | + i |−i . (5.3)The reduced density matrix ρ p is ρ p = | + i h + | . (5.4)Acting with O , O | Ψ p i = N | + i ( b |−i ) ≡ | Ψ p ′ i . (5.5)To normalize the state, |N | = 1 / | b | . The reduced density matrix is unchanged: ρ p ′ = | + i h + | . (5.6)Acting with O does not change measurements performed on particle 1 regardless of whatvalues a, b take.We now prove that all local operators, including non-unitary operators, change productstates locally. Consider two Hilbert spaces H , and n orthonormal basis elements | ψ n , i .Consider the arbitrary product state | Ψ p i ∈ H with H = H ⊗ H , and an arbitrary norm-13reserving operator O local in H . | Ψ p i = X i c i | ψ i i X j c j | ψ j i . (5.7)The state is normalized: X i | c i | ! × X j | c j | ! = 1 . (5.8)The reduced density matrix associated with H is ρ = X i c i | ψ i i ! X k c k ∗ h ψ k | ! X j | c j | . (5.9)Acting with O on the state, O | Ψ i = X i c i | ψ i i X j d j | ψ j i , (5.10)where the d j are defined by the action of O on states in H in the chosen basis: d j ≡ P i O ji c i .The normalization condition is X i | c i | ! × X j | d j | ! = 1 . (5.11)Acting with O is a unitary operation in H : X j | d j | = X j | c j | . (5.12)We may see that the new reduced density matrix ρ ′ is equal to ρ : ρ ′ = X i c i | ψ i i ! X k c k ∗ h ψ k | ! X j | d j | = X i c i | ψ i i ! X k c k ∗ h ψ k | ! X j | c j | = ρ . (5.13)This concludes the proof. In entangled states, local unitary operators affect the state locally but local non-unitaryoperators may affect the state non-locally. As an example, we will act with O on entangled14tate | Ψ e i and find that generically O will change the state non-locally unless O is unitary. | Ψ e i = 1 √ | + i |−i − |−i | + i ) . (5.14)The reduced density matrix ρ e is ρ e = 12 ( | + i h + | + |−i h−| ) . (5.15)Acting with O on the state | Ψ e i gives O | Ψ e i = N√ b | + i |−i − a |−i | + i ) ≡ | Ψ e ′ i . (5.16)The normalization factor satisfies |N | = | a | + | b | . The reduced density matrix is now ρ e ′ = 1 | a | + | b | ( | b | | + i h + | + | a | |−i h−| ) . (5.17)Acting with O changes ρ e unless | a | = | b | , which would make O unitary.Local operators mix states within H , but if these states are coupled with different weightsto states in a different Hilbert space H , mixing states within H will generically change therelative weights of the states in H . Only local unitary operators mix states in H in theway that leaves the partial trace unchanged.The action of any non-unitary operator N O on a state may by definition be written asa unitary operator U acting on the state. In the state | Ψ i , |N | = h Ψ |O † O| Ψ i − . For every O and | Ψ i there exists a unitary operator U such that N O | Ψ i = U | Ψ i . (5.18)This equality follows from the fact that acting with N O does not change the norm of thestate. It follows that this norm-preserving operation on a state can be implemented byacting with some unitary operator U , as the set of all unitary operators is the space of allpossible norm-preserving operations on the state. The non-unitary operator N O is of coursenot equal to the corresponding unitary operator U , but their actions on the state | Ψ i arethe same. The same O is represented by different U on different states | Ψ i . If the operator O changes the state non-locally, U is non-local. The equivalence between non-unitary andunitary operators acting on the state shows how non-unitary operators may be applied toa state through time evolution. According to the local picture, the unitary operator U isapplied by a non-local experimentalist with access to both systems, as U is not local in H or H alone. 15et us see an explicit example of the equivalence between non-unitary and unitary op-erators using the operator O and state | Ψ e i . We now work in the basis of the full Hilbertspace: ( | + i | + i , | + i |−i , |−i | + i , |−i |−i ) where we have dropped the subscripts 1 ,
2. Wewish to find a unitary operator U that satisfies the following: U | Ψ e i = O | Ψ e i . (5.19)Writing this condition in the chosen basis, O | Ψ i = N √ b − a = U √ − = U | Ψ i . (5.20)We can now write down a solution. U rotates components into one another, and may producean arbitrary phase. U ( θ, φ , φ ) = e iφ cos θ − e iφ sin θ e iφ sin θ e iφ cos θ
00 0 0 1 . (5.21)The above matrix is the product of the rotation matrix with diag(1 , e iφ , e iφ , a, b are: e iφ cos θ + e iφ sin θ = b N (5.22) e iφ sin θ − e iφ cos θ = − a N . (5.23)Elements a N , b N have three degrees of freedom: an overall phase, a relative phase, and arelative magnitude. Operator U has the same three degrees of freedom as well: two phases φ , φ , the angle θ that controls the components’ magnitudes. The normalization condition( | a N | + | b N | ) / U depends on O andthe state.We will now show that local unitary operators change entangled states locally and genericlocal non-unitary operators change entangled states non-locally. This is an elementary prop-erty of the partial trace, but we will find a proof in component notation useful. We beginwith an arbitrary entangled state | Ψ e i . We use repeated index summation notation in this16roof. Label states in H , by | ψ , i . | Ψ e i = C ia | ψ i i | ψ a i . (5.24)The normalization condition is C † ai C ai = 1. The reduced density matrix of H is ρ = C ia C † aj | ψ i i h ψ j | . (5.25)Now act with an operator O that is local in H on the state. | Ψ e i = O ba C ia | ψ i i | ψ b i . (5.26)Assume that O is normalized to satisfy the normalization condition C † di O † db O ba C ia = 1. If O † = O − then O † db O ba = I da and the state’s norm is automatically preserved. The newreduced density matrix ρ ′ is ρ ′ = C † dj O † db O ba C ai | ψ i i h ψ j | . (5.27)If operator O is unitary, then ρ ′ = ρ . If not, the state may change non-locally. In thelanguage of quantum information, non-unitary operators O implement non-local quantumgates, as O ’s action can be represented by a non-local unitary operator.There are states for which non-unitary operators acting on a subspace do leave ρ un-changed. Suppose that for a > k , C ai = 0 for every i . To leave ρ unchanged, ( O † O ) db = I db for d, b ≤ k suffices, but this condition not necessary for d, b > k . A simple example is if O is local and unitary in subspace P ∈ H but non-unitary in the rest of H : O will create alocal excitation of states in P despite being non-unitary. When the two Hilbert spaces havethe same dimensionality, such states require the density matrix to have at least one zeroeigenvalue.We have not addressed the most general condition for operators to leave entanglemententropy unchanged, which is a weaker condition than leaving ρ unchanged, and an inter-esting direction for future work. Entanglement entropy is the von Neumann entropy of areduced density matrix. The entanglement entropy S of the subsystem with Hilbert space H is S = − tr( ρ ln ρ ) (5.28)In a pure state, entanglement entropies for the two subsystems must be equal: S = S . Itfollows that acting with a local unitary operator on one subsystem or the other does notchange S , S because the density matrix of the subsystem not acted upon is unchanged.There can be local non-unitary operators O that alter a state | Ψ i non-locally, but leave17xpectation values of operators O ′ unchanged. In field theory this condition is sometimessatisfied by the modular Hamiltonian, O ′ = − ln( ρ ), whose expectation value is entangle-ment entropy [4]. It would be interesting to investigate this question further in quantummechanics and field theory to understand the basis of the apparent quasi-particle picture forentanglement entropy. So far, we have shown how local non-unitary operators act as unitary operators on entangledstates, and that these unitary operators must be non-local. According to the local picture,operators that change states non-locally can only be implemented by non-local experimen-talists. Experimentalists act with the non-local unitary operators through time evolution.Non-unitary operators are intrinsically non-local in entangled states.Our results can be viewed another way, motivated by the Reeh-Schlieder theorem in fieldtheory. In a state in H = H ⊗ H that is entangled between H , H , non-unitary operators O local in H can prepare states in H . We may see how this works for | Ψ e i . | Ψ e i = 1 √ | + i |−i − |−i | + i ) . (5.29)We may use suitably normalized operators L ± , L z acting on particle 2 that are the angularmomenta operators for spin 1 / L + | Ψ e i = | + i | + i .L − | Ψ e i = |−i |−i . (1 + L z ) | Ψ e i = |−i | + i . (1 − L z ) | Ψ e i = | + i |−i . (5.30)Our results show that seemingly counter-intuitive features of the Reeh-Schlieder theo-rem are perfectly straightforward in quantum mechanics. The fact that local non-unitaryoperators create non-local excitations in entangled states is the origin of the non-local statepreparation in the Reeh-Schlieder theorem. The theorem is consistent with causality becauselocal experimentalists act only with local unitary operators, which do not permit non-localstate preparation. The example we gave of preparing a state non-locally was used as aquantum-mechanical model for the Reeh-Schlieder theorem in the leading interpretation [15].Discussions of the Reeh-Schlieder theorem have previously been centered on non-local statepreparation through measurements in open quantum systems.Our conclusions about the non-locality of non-unitary operators may appear to contra-dict a standard statement of causality for entangled states that allows non-local changes in18tate, but in fact the two are compatible. Measurements in an open quantum system can bedescribed in the associated closed quantum system by projecting onto an experimentalist-system state with a particular measurement outcome. It is obvious that replacing a superpo-sition of states with one of its constituent states is a non-local change of state, but questionsof locality are more clearly formulated in the closed quantum system, in which it is manifesthow measurement does not change states non-locally. These statements are standard, butfor completeness we now make them concrete with an explicit example.Consider an initial state of a closed quantum system with Hilbert space H = H exp ⊗ H sys , | Ψ i = | Ψ i | Ψ i ( a | + i |−i − b |−i | + i ) . (5.31)We label the experimentalists by the outcome they observe as | Ψ exp ( ± ) i . After experimen-talist 2 measures the spin of particle 2, the state is | Ψ i ′ = | Ψ i ( a | Ψ ( − ) i | + i |−i − b | Ψ (+) i |−i | + i ) . (5.32)Experimentalist 1 may now measure the spin of particle 1. Once again, this is an interactionthat couples experimentalist states to system states. The new state is | Ψ i ′′ = a | Ψ (+) i | Ψ ( − ) i | + i |−i − b | Ψ ( − ) i | Ψ (+) i |−i | + i ) . (5.33)We now could project onto various states to determine what each experimentalist measures.However, tracing out experimentalist 2 and particle 2, the reduced density matrix ρ exp ⊗ sys for experimentalist 1 and particle 1 is ρ exp ⊗ sys = | a | (cid:0) | Ψ (+) i | + i h + | h Ψ (+) | (cid:1) + | b | (cid:0) | Ψ ( − ) i |−i h−| h Ψ ( − ) | (cid:1) . (5.34)The reduced density matrix is unchanged by the measurement that experimentalist 2 per-formed. The unitary operator that implements experimentalist 2’s measurement is localizedto H ⊗ H , and so it does not change ρ exp ⊗ sys . In entangled states, the superposition of two localized excitations is generically not itselfa localized excitation. One can represent each localized excitation as a localized unitaryoperator acting on some reference state. The sum of two unitary operators need not beunitary, and so their superposition need not create a localized excitation. For example,by adding two unitary matrices of the form diag( e iφ , e iφ ), we may change the magnitudeof the sum’s diagonal entries. The non-locality of the superposition measures a kind of19nterference between the two unitary operators. We can state the general condition for whichthe superposition of localized excitations implemented by U , U must itself be a localizedexcitation in entangled states. U U † + U U † = 0 (5.35)We will not explore this condition. It follows that a local experimentalist cannot superimposetwo local excitations of an entangled state. The condition for superpositions of local unitaryoperators to be local has been addressed on a more formal level [13].There is a natural way to combine local excitations without superposition. Acting withtwo localized unitary operators on the same state creates a localized excitation. For example,the operator U U creates an excitation localized to the same Hilbert spaces in which U , U are localized. In the local picture, these two operators should be time-ordered: T ( U U ).This method of combining local excitations is natural in the local picture, as it correspondsto an interaction that occurs in two subsystems: T (cid:0) e i ( O + O ) (cid:1) and is implemented by turningon sources for both O and O . We will elaborate on this point in a later section when wediscuss separable and non-separable localized unitary operators.Superpositions of local unitary operators create non-local excitations, which are asso-ciated more naturally with non-unitary operators. This relationship between unitary andnon-unitary operators can be viewed from another direction. Non-unitary operators canbe written as superpositions of unitary operators. Acting with a non-unitary operator isequivalent to superimposing states formed by acting with unitary operators. As is standard,any operator O can be written as a linear combination of an Hermitian operator H + andanti-Hermitian operator H − . O = H + + H − . (5.36) H ± = 12 ( O ± O † ) . (5.37)In fact, H − can be written in terms of a Hermitian operator H ′ + simply: H ′ + = iH − . Wenow show that any Hermitian operator can be written in terms of a unitary operator and itsadjoint, subject to a certain condition. Suppose the spectrum of some Hermitian operator H is bounded from above and below. This is not always the case for the Hamiltonian, but thisis true for many other operators, especially those relevant in systems with a finite numberof spins. There exists λ which is at least as large as H ’s largest-magnitude eigenvalue, butfinite. Operator H is given by H = 12 | λ | ( U + U † ) . (5.38)To prove this, first suppose H is diagonal. Its entries are its eigenvalues, which are real. We20ay choose U = diag( e iφ , e iφ , . . . ). U + U † = diag(2 cos( φ ) , φ ) , . . . ) . (5.39)One may then choose each φ i to match each eigenvalue in H . Hermitian matrices arediagonalized by unitary matrices, so we may use unitary V to produce any other Hermitianoperator from H that has the same eigenvalues: V HV † = 12 | λ | ( V U V † + V U † V † ) (5.40) ≡ | λ | ( U ′ + U ′† ) . (5.41)Note that U ′ is also unitary. This concludes the proof. We now turn to field theory and our main result: localized unitary operators create local-ized excitations, while more familiar local non-unitary operators generically create non-localexcitations. We explore the properties of these operators. As generic states in field theoryare entangled over spatial regions, field theory is often the study of operators in entangledstates. The properties we found in section 5 will apply in field theory as well. An initialinvestigation into the locality of certain local unitary operators was conducted in the contextof free field theory, and our work extends this investigation [12].
In a ( d + 1)-dimensional theory, we give a general form for time-ordered localized unitaryoperators U , or localized unitary operators for short. This form arises naturally in the localpicture. Foliate the spacetime by Cauchy surfaces and define a timelike coordinate thatparameterizes motion across these surfaces. For any given foliation, all operators of thefollowing form are localized unitary operators: U = T (cid:18) e − i P n (cid:16)Q { in } R Rin d d +1 x in (cid:17) J n ( x ,x ,...,x in ) ( Q { in } O in ( x in ) ) (cid:19) . (6.1)For each n , there is a set denoted by { i n } which specifies the source function J n andoperators O i n ( x i n ) appearing in the product. Functions J n can have dimensions and includea small expansion parameter. The term in the exponent multiplying i is Hermitian. Unitary21perators can always be placed in exponential form but the expression we present is sim-ply the general time-ordered exponential of products and sums of localized operators withsmearing functions. Operators O ( x i n ) need not be local in space but must be local in time,and we have labelled operators O i n by their insertion points schematically. R i n is defined as the spacetime region in which the smearing function J n ( x , . . . , x i n ) isnon-zero. The localized unitary operator U creates an excitation localized to spacetime region R = ∪ i n R i n . Correlators of operators inserted at spacelike separation from all points in R do not change. Consider hO ( y ) i in an excited state formed by U . If y is spacelike-separatedfrom all points in R , h Ψ |U † O ( y ) U | Ψ i = h Ψ |U † U O ( y ) | Ψ i = h Ψ |O ( y ) | Ψ i . (6.2)If y is not spacelike from all of R , the above equality may not hold. While we may find itconvenient to use perturbation theory to calculate correlators in this state, this result is truenon-perturbatively, and follows from (2.1).There is an operator-excited state correspondence for subregions. For every Hermitianoperator O inserted in a subregion A of a Cauchy surface, there is an excited state whoseexcitation is localized to A and is given by acting e i O on the original state.Just as H int is not normal ordered, the operators in the exponent of U are not normal-ordered. Correlators in this state will generically diverge. Treating H int as a perturbativecorrection, calculating correlators in a state created by U amounts to a calculation in real-time perturbation theory, and the divergences are treated using standard methods.A simple example of a local unitary operator is U ( x ) = e − iα O ( x ) , O † ( x ) = O ( x ) . (6.3)The parameter α can be chosen to be α ≡ α ′ ǫ , where α ′ may have dimensions and the dimen-sionless parameter ǫ may be taken small. Exciting a state with this operator is equivalentto introducing the interaction Z H int = α Z d d +1 x ′ δ d +1 ( x ′ − x ) O ( x ′ ) . (6.4)The first-order correction to hO ( y ) i is a familiar quantity in time-dependent systems, thecommutator. h Ψ |U † ( x ) O ( y ) U ( x ) | Ψ i = h Ψ |O ( y ) | Ψ i − iα h Ψ | [ O ( y ) , O ( x )] | Ψ i + . . . . (6.5)Conversely, calculations of the commutator of two operators are also the first-order correc-tion to the one-point function in an excited state. In general, the correspondence between22ocalized unitary operators and Hamiltonian deformations is U = T (cid:18) e − i P n (cid:16)Q { in } R Rin d d +1 x in (cid:17) J n ( x ,x ,...,x in ) ( Q { in } O in ( x in ) ) (cid:19) l Z H int = X n Y { i n } Z R in d d +1 x i n J n ( x , x , . . . , x i n ) Y { i n } O i n ( x i n ) . (6.6)This correspondence is clear from the interaction picture and section 4.Deformations of the Hamiltonian cannot always be represented by localized unitary op-erators acting on states. For example, if H int has not turned off at the time operators areinserted, the calculation of an unequal-time correlator will involve insertions of operators e − i R t H int between operators inserted at different times. Also, the time t should not be takenlater than the latest time at which the operators in the correlation function are inserted.These rules follow from the interaction picture.In gauge theories, it is natural to restrict the operators in the exponent of a localizedunitary operator to be gauge invariant. For example, the operator U ( x ) = e iF ( x ) with F µν ( x ) being the field strength tensor of a gauge theory creates a local excitation. Notall quantities diagnose locality well in gauge theories. For example, entanglement entropyis not a gauge-invariant measure, but relative entropy and mutual information are free ofambiguities associated with the gauge theories, and so may prove useful [34–36].Localized excitations can change conserved quantities. For example, a localized excitationcreated by U ( x ) will generically change the total energy h R d d yT ( y ) i . The energy addedby U ( x ) is injected at x and can spread within the forward lightcone of x . The amount bywhich a localized unitary operator changes a conserved quantity is a property of both theoperator and the state. It has been argued that localized excitations of fixed particle numberare of limited applicability [12, 17, 18, 37].We may also prepare a localized excitation by sending in an “ingoing excitation” ratherthan by deforming the Hamiltonian. So far, we have described how to create a localizedexcitation by applying a localized unitary operator U to some state | Ψ i . Applying thisoperator can change conserved quantities. The same localized excitation can be prepared bybeginning with some initial state and evolving time. Conserved quantities will not change inthis case. The initial state | Ψ( t ) i that encodes the ingoing excitation is found by evolvingthe state U ( x ) | Ψ i backwards in time. Here, U ( t, t ) is the time evolution operator and U ( x )is a local unitary operator inserted at spacetime point x = ( t, x ). | Ψ( t ) i = U ( t , t ) U ( x ) | Ψ i . (6.7)23ven in an interacting field theory in an arbitrary number of dimensions, the ingoing ex-citation must be a pulse that leaves no imprint on the state as it passes through a space-time region, because the regions it passes through are causally connected to points thatare spacelike-separated from x . While such excitations are familiar in classical theories, inquantum field theories they can require extensive fine-tuning, and may not be possible inpractice.The dynamics of entanglement at different scales within a local excitation can be investi-gated through the time-dependence of entanglement density [38]. This investigation may beuseful in the AdS/CFT correspondence through the Hubeny-Rangamani-Takayanagi conjec-ture, and as part of the entanglement tsunami picture [26, 39–41]. There are two qualitatively different types of excitations created by localized unitary opera-tors, separable and non-separable. We explore their properties. We use the labels separableand non-separable for reasons which will become clear.We have stated a general form for useful localized unitary operators is U = T (cid:18) e − i P n (cid:16)Q { in } R Rin d d +1 x in (cid:17) J n ( x ,x ,...,x in ) ( Q { in } O in ( x in ) ) (cid:19) . Separable unitary operators U create separable excitations, and take the form U = T (cid:16) e − i P n R Rn d d +1 x n J n ( x n ) O n ( x n ) (cid:17) . (6.8)Non-separable unitary operators contain products of operators in the exponent that areinserted at different points. For example, consider separable and non-separable local unitaryoperators U s , U ns , with U s = e − i ( O ( x )+ O ( y )) . (6.9) U ns = e − i O ( x ) O ( y ) . (6.10)Points x, y are spacelike-separated and so time-ordering has no effect for these two operators.We require that O † = O . Operator U s can be separated into the product of two local unitaryoperators while U ns cannot. This will become obvious shortly.We can understand separable and non-separable unitary operators through their quantum-mechanical analogs. Consider an entangled state of two spin 1/2 particles. A separableoperator is U s = e − i ( S z + S z ) , which amounts to acting with a local unitary operator on eachparticle. Separable operators are the natural way to concatenate local excitations of a sys-tem. A non-separable operator is U ns = e − iS z ⊗ S z . This is equivalent to turning on a spin-spin24oupling between the two systems. Separable operators represent interaction of an externalsystem with the state and non-separable operators represent the coupling of two subsystems.In the local picture, separable operators are implemented by a non-local experimentalist.Separable unitary operators represent uncorrelated localized excitations while non-separableunitary operators represent correlated excitations. We will explore this statement in fieldtheory. If two excitations are correlated, correlators affected by one excitation will alsodepend on the value of the field at the location of the second excitation. Consider the expec-tation value of operator O ( z ) which is altered by the excitation at x but not y . Suppose z and x are timelike-separated and z is in the future of x . Point z is spacelike-separated from y . For the separable operator, h Ψ |U † s ( x, y ) O ( z ) U s ( x, y ) | Ψ i = h Ψ | e i O ( x ) O ( z ) e − i O ( x ) | Ψ i . (6.11)For the non-separable operator, h Ψ |U † ns ( x, y ) O ( z ) U ns ( x, y ) | Ψ i = h Ψ |O ( z ) + i [ O ( x ) O ( y ) , O ( z )] + . . . | Ψ i = h Ψ |O ( z ) + i O ( y )[ O ( x ) , O ( z )] + . . . | Ψ i . (6.12)Both separable and non-separable unitary operators create localized excitations, just as U s , U ns change the state only in the forward lightcones of their insertion points x, y . With U ns , the correction to hO ( z ) i depends on an operator O ( y ) inserted at spacelike separation.Just as in quantum mechanics, the non-separable operator has coupled the state at x and y .We can understand what this coupling entails. If we first alter the state at y with anotherlocal excitation, we will affect hO ( z ) i only when the next excitation is non-separable. Createthis first excitation with a local unitary operator U ( y ′ ) where y ′ and y are timelike-separatedbut y ′ is spacelike-separated from x and z . Point y ′ is earlier in time than y . The operator U ( y ′ ) when acted alone changes the state at y , but not x or z . The expectation value hO ( z ) i does not change for the separable excitation: h Ψ |U † ( y ′ ) U † s ( x, y ) O ( z ) U s ( x, y ) U ( y ′ ) | Ψ i = h Ψ | e i O ( x ) O ( z ) e − i O ( x ) e i O ( y ′ ) e i O ( y ) e − i O ( y ) e − i O ( y ′ ) | Ψ i = h Ψ | e i O ( x ) O ( z ) e − i O ( x ) | Ψ i . (6.13)The expectation value hO ( z ) i does change for the non-separable excitation: h Ψ |U † ( y ′ ) U † ns ( x, y ) O ( z ) U ns ( x, y ) U ( y ′ ) | Ψ i = h Ψ |U † ( y ′ )( O ( z ) + i O ( y )[ O ( x ) , O ( z )] + . . . ) U ( y ′ ) | Ψ i = h Ψ |O ( z ) + i U † ( y ′ ) O ( y ) U ( y ′ )[ O ( x ) , O ( z )] + . . . | Ψ i As y and y ′ are timelike-separated, the above expectation value of an operator at z has25hanged in response to an excitation at y ′ which is spacelike-separated from z .The behavior we have identified for non-separable excitations violates the causality prop-erties of local quantum field theory at the time the operator acts. Measurements at z areaffected by a local excitation at y ′ , which was spacelike-separated from z . This violation ofcausality is no surprise, as acting with U ns corresponds to turning on an interaction O ( x ) O ( y )in the Hamiltonian, which couples the field at spacelike-separated points. This term is notallowed in the Hamiltonian of a local quantum field theory. Even a non-local experimentalistin the closed system cannot act with this operator as long as the theory that describes theexperimentalist and system are both local quantum field theories. We conclude that thereis a restriction on localized operators and excitations in local quantum field theories: theoperators and excitations must be separable. Separability can be tested using the criteriawe have used in this section.Separable unitary operators localized to two different regions of a Cauchy surface do notchange the entanglement entropy of those regions, as the operators can be expressed as theproduct of unitary operators, each local in a different subregion. For example, U s ( x, y ) doesnot change the entanglement entropy of region A or B for x ∈ A, y ∈ B . We give examples of familiar quantities that are localized unitary operators. Squeezed states,coherent states, generalized coherent states are examples of excitations that can be createdby localized unitary operators, and their interpretations are well-understood [20–23]. In afree ( d + 1)-dimensional field theory, a coherent state is (cf. [42]) | Ψ c ( π c ( x ) , φ c ( x )) i = e i R d d xπ c ( x ) ˆ φ ( x ) − φ c ( x )ˆ π ( x ) | i . (6.14)The state is labelled by its expectation values h Ψ c ( π c ( x ) , φ c ( x )) | φ ( y ) | Ψ c ( π c ( x ) , φ c ( x )) i = φ c ( y ) . (6.15) h Ψ c ( π c ( x ) , φ c ( x )) | π ( y ) | Ψ c ( π c ( x ) , φ c ( x )) i = π c ( y ) . (6.16)A coherent state is a localized excitation when both φ c ( x ) , π c ( x ) have compact support.Generalized coherent states are analogous constructions for arbitrary Lie groups [43] andalso create localized excitations.Path-ordered exponentials can be localized unitary operators for certain choices of path.If the path is nowhere spacelike, then the path ordering is a time ordering. If the path iseverywhere spacelike, then the operator creates an excitation at a single time. Wilson loopswith spacelike integration paths are examples of these operators, and they create flux tubesalong their path. 26ocalized unitary operators model certain quantum quenches exactly. Quantum quenchesare abrupt changes in the Hamiltonian. For instance, the coefficient λ ( x ) of some operator O ( x ) in the Hamiltonian may change suddenly. If the state was the ground state of theHamiltonian, the state after the quench is an excited state of the new Hamiltonian. Inglobal quenches, λ ( x ) is constant in space. In inhomogeneous quenches, λ ( x ) varies in space.Two different types of quenches go by the name “local quenches”. One type of local quenchinvolves preparing two different states in half the space, joining them, and evolving withtime [44, 45]. Another type of local quench is given by changing λ ( x ) at one spacetimepoint [1, 2, 4]. We give the localized unitary operators that describe this second type of localquench. Global and inhomogeneous quenches are described in a similar way, although thecorresponding unitary operators are fully non-localized. In a ( d + 1)-dimensional field theory,Global quench : U = e − i R d d x O ( x ) . (6.17)Inhomogeneous quench : U = e − i R d d xf ( x ) O ( x ) . (6.18)Local quench : U = e − i O ( x ) . (6.19)Motivated by the properties of non-separable unitary operators, we see that non-separableoperators model a “non-separable quantum quench”Non-separable quench : U = e − i O ( x ) O ( y ) . (6.20)Introducing a small parameter α in the exponent to control the strength of the quench, thefirst, second, and third-order corrections to operator expectation values are straightforwardto calculate in CFTs as they often involve two, three, and four-point functions. In this section, we show that the familiar local operators in field theory do not alwayscreate localized excitations. While this may seem counter-intuitive, locality in field theoryis enforced through commutators, and expectation values in states
O | Ψ i do not involve anycommutators with O . Moreover, the statement that two operators commute at spacelikeseparation is not a statement about expectation values in the state created by acting with oneof those operators. In section 5, we showed that localized finite-norm operators genericallycreate non-localized excitations. An operator O has a finite norm if O | Ψ i has a finite normfor all normalized states | Ψ i . A related conclusion is that local non-unitary operators do notalways model a local quench exactly. Infinite-norm operators will be treated more carefully,and we give a specific infinite-norm operator that creates a non-local excitation.27e first address an intuition that is sometimes held about local operators creating localexcitations. Consider a real scalar field in (3 + 1) dimensions. We may ask about theinterpretation of the state φ ( x ) | i = Z d p (2 π ) E p e − i p · x | p i . (6.21)We will paraphrase the interpretation of this state given in a well-known field theory textbook[46]. For small (non-relativistic) p , E p is approximately constant, and in this case φ ( x ) | i approaches the non-relativistic expression for a position eigenstate | x i in basis | p i . Toquote the authors, “we will therefore put forward the same interpretation, and claim thatthe operator φ ( x ), acting on the vacuum, creates a particle at position x .” Moreover, thisinterpretation is corroborated by calculating h | φ ( x ) | p i = h | Z d p ′ (2 π ) p E p ′ (cid:16) a p ′ e i p ′ · x + a † p ′ e − i p ′ · x (cid:17) p E p a † p | i (6.22)= e i p · x . (6.23)This is the same as the inner product h x | p i in non-relativistic quantum mechanics. We mayperform another check to learn that the analogy with quantum mechanics is only valid inthe non-relativistic limit.QM : h x | y i = δ (3) ( x − y ) (6.24)QFT : h | φ ( x ) φ ( y ) | i = Z d p (2 π ) e i p · ( x − y ) E p = D ( x − y ) . (6.25)In the non-relativistic approximation, E p is approximately constant, and both expressionsare delta functions. The authors of course never make an erroneous claim, for examplethat φ ( x ) creates a particle only at x but nowhere else. In fact, particles themselves areapproximate notions, and it has been shown that localizing a finite number of particles ina single region is in tension with causality [12, 17, 37]. We have reproduced a textbookargument here to make explicit what considerations and terminology may lead one to theincorrect intuition that if a field theory operator is local it creates a local excitation.In field theory, the Dirac orthogonality condition does not diagnose locality as we havedefined it. The condition that the inner product between two states h | φ ( x ) φ ( y ) | i = D ( x − y ) grows small as the separation between x, y grows large is known as asymptotic locality [47].We have seen how localized unitary operators create localized excitations, and even localizedunitary operators are not Dirac orthogonal. Consider operators of the form U = e − i O : h Ψ |U † ( y ) U ( x ) | Ψ i = h Ψ | Ψ i + i h Ψ |O ( y ) − O ( x ) | Ψ i + . . . (6.26)28he failure of local excitations in field theory to obey the Dirac orthogonality conditionillustrates that not all quantum-mechanical measures of locality are useful measures in fieldtheory.Our discussion of the Reeh-Schlieder theorem in section 5.3 applies to field theory aswell. The Reeh-Schlieder theorem is a consequence of local non-unitary operators actingin entangled states. It is widely accepted that the source of Reeh-Schlieder theorem infield theory is the entanglement between spatial regions [15]. To make the connection withour quantum-mechanical explanation of the Reeh-Schlieder theorem, we should considerfinite-norm operators in field theory localized to some region. Local operators genericallyhave infinite norm, and must be smeared over some region to have finite norm. Just asin quantum mechanics, these finite-norm non-unitary operators may be localized, but theycreate the non-localized excitations described by the Reeh-Schlieder theorem.Some local infinite-norm operators create non-localized excitations. For example, con-sider the infinite-norm operator O e = e α O ( x ) . (6.27)Here, α is real and O is Hermitian. Consider how the expectation value of some operator inthe vacuum h |O ′ | i changes in the excited state O e | i . To first order in α this excitationdoes not change the state’s norm as h |O ( x ) | i = 0. So, for the first-order calculation, wedo not have to regulate the operator. The expectation value to first order is therefore h |O e ( x ) O ′ ( y ) O e ( x ) | i ≈ h |O ′ | i + α h | {O ( x ) , O ′ ( y ) } | i + . . . (6.28)The anticommutator of two operators does not vanish for spacelike separations and so thislocal infinite-norm operator creates a non-localized excitation. While not all infinite-norm local operators create local excitations, we show some which do.The locality of these operators comes from the singularity structure of the infinite-normstates they create. This is in contrast to unitary operators, which obtain their localitythrough operator commutators. To calculate correlators in excitations created by infinite-norm operators, we must first regulate the norm. One way to regulate is to dampen thehigh-energy modes, which is equivalent to inserting the operator at complex time [4, 48]: e − δH O ( x ) | i = O ( x − iδ ) | i . (6.29)We have used the shorthand x ± iδ ≡ ( t ± iδ, x ). Expectation values are taken with the limit δ →
0. 29s a simple example of an expectation value in an infinite-norm state, we consider thetwo-point function h φ ( x ) φ ( y ) i of a free scalar field in state φ ( z ) | i , and we work in d + 1 > φ ( z ) creates a local excitation at z . Suppose x, y are spacelike-separated from z . This way the δ → h φ ( x ) φ ( y ) i ≡ h | φ ( z + iδ ) φ ( x ) φ ( y ) φ ( z − iδ ) | ih | φ ( z + iδ ) φ ( z − iδ ) | i = D ( z + iδ, x ) D ( y, z − iδ ) + D ( z + iδ, y ) D ( x, z − iδ ) D ( z + iδ, z − iδ ) + D ( x, y ) . (6.30)Here, D ( x − y ) = h | φ ( x ) φ ( y ) | i . The two-point function is unchanged by the excitation incomparison to its vacuum expectation value, as only the φ ( z + iδ ) φ ( z − iδ ) contraction inthe numerator has the same divergence as the denominator in the δ → x, y are spacelike-separated from z :lim δ → h φ ( x ) φ ( y ) i = D ( x, y ) . (6.31)The same conclusion holds for an n-point function in this state. The infinite-norm localoperator φ ( z ) creates a local excitation at z even though it is not unitary.In general CFTs, we can prove the same behavior we saw in the free scalar case, thata non-unitary infinite-norm local operator with definite conformal dimension creates a localexcitation. When calculating a correlation function in the state O ( x − iδ ) | Ψ i , the OPEbetween O and O † may be used when x is spacelike separated from the locations of theother operators in the correlation function. When | Ψ i is a conformally-invariant state, forexample the vacuum, the identity dominates the OPE in the δ → O . We will see an explicit example of this processin section 7.2. This argument applies when O has definite non-zero conformal dimension.For example, this excludes the operator φ of the free scalar in (1 + 1) dimensions, whichhas conformal dimension zero and creates an excitation that is not asymptotically local [47].Explicitly, for x spacelike-separated from all y i , hO ( y ) O ( y ) . . . O ( y n ) i ≡ h Ψ |O † ( x + iδ ) O ( y ) O ( y ) . . . O ( y n ) O ( x − iδ ) | Ψ ih Ψ |O † ( x + iδ ) O ( x − iδ ) | Ψ i = X ∆ k ,s k h Ψ | h C O † OO k O k ( x + iδ )(2 iδ ) − ∆ k i O ( y ) O ( y ) . . . O ( y n ) | Ψ ih Ψ |O † ( x + iδ ) O ( x − iδ ) | Ψ i . (6.32)The sum is over all operators O k , which we have indexed by dimension ∆ k and spin s k , andthe C O † OO k are theory-dependent coefficients. If O k is a primary operator, then C O † OO k isthe three-point coefficient. 30s the state | Ψ i is conformally invariant, h Ψ |O p | Ψ i = 0 for all local primary operators O p .Therefore, O † O must contain the identity in its OPE in order for the two-point function inthis state, h Ψ |O † ( x + iδ ) O ( x − iδ ) | Ψ i , to be non-zero. We will assume the identity is presentin this OPE. It follows that only the identity’s contribution to the OPE in the numerator of(6.32) survives the δ → hO ( y ) O ( y ) . . . O ( y n ) i = X ∆ k ,s k h Ψ | h C O † OO k O k ( x + iδ )(2 iδ ) − ∆ k i O ( y ) O ( y ) . . . O ( y n ) | Ψ ih Ψ |O † ( x + iδ ) O ( x − iδ ) | Ψ i = X ∆ k ,s k h Ψ | h C O † OO k O k ( x + iδ )(2 iδ ) − ∆ k i O ( y ) O ( y ) . . . O ( y n ) | Ψ i (2 iδ ) − δ → −→ h Ψ |O ( y ) O ( y ) . . . O ( y n ) | Ψ i . (6.33)Even if the field theory is not a CFT, an argument similar to (6.33) shows that O ( x )creates local excitations if we assume a certain short-distance factorization. The only non-zero contribution to a correlator evaluated a state created by O ( x − iδ ) comes from the δ → x is spacelike from the insertion points of all the other operators, then the δ → δ → O ( x ) createsa local excitation, and the argument proceeds similar to the CFT case: hO † ( x + iδ ) O ( y ) O ( y ) . . . O ( y n ) O ( x − iδ ) ihO † ( x + iδ ) O ( x − iδ ) i δ → −→ hO † ( x + iδ ) O ( x − iδ ) i hO ( y ) O ( y ) . . . O ( y n ) ihO † ( x + iδ ) O ( x − iδ ) i = hO ( y ) O ( y ) . . . O ( y n ) i , y i − x spacelike . (6.34)Strictly speaking, local operators themselves are only operator-valued distributions. Whilecertain infinite-norm operators may create local excitations, these operators must be smearedwith some test function to create a physical excitation with finite norm. But once the non-unitary operator has finite norm, the finite-norm excitation will generically not be localizedto the region of smearing. As such, the conclusions drawn from the locality properties ofinfinite-norm operators must be treated with care, as they may not extend to the operators’smeared counterparts. Entanglement entropy has recently emerged as a useful probe of excited-state dynamics in(1 + 1)-dimensional conformal field theories. A compelling quasi-particle picture has been31roposed for local operators, wherein generic local operators create local excitations that canbe interpreted as entangled pairs of quasi-particles [4]. In this section, we revisit the results inthe literature and show how, while some infinite-norm operators create local excitations thatmay admit a quasi-particle description, entanglement and Renyi entropies change non-locallyfor other infinite-norm operators, and so the quasi-particle picture is not universal for alllocal operators. It is known that the quasi-particle picture fails for some theories [5, 26], andwe show its failure in theories in which the picture is expected to be accurate. We show theresults in the literature are consistent with evidence we have presented that local operatorswith definite conformal dimensions create local excitations. We also prove a causality relationfor entanglement entropy.
The details of how localized excitations are implemented by localized unitary operatorsmotivate a general causality condition for entanglement entropy in quantum field theories.The condition applies to pure states. Our result extends the result proved in ref. [27].This earlier result makes use of the fact that, for a localized excitation within domain ofdependence D ( A ) of subregion A of a Cauchy surface, one can always find a Cauchy surface A ′ of D ( A ) such that the state on A ′ is unaffected by the excitation. The excitation’s support R is in the future of A ′ and to the past of A . As a reminder, the domain of dependence is definedas the region D ( A ) that, if an inextendible curve that is nowhere spacelike passes throughthe region, then this curve must intersect A . The domain of dependence D ( A ′ ) = D ( A ).The reduced density matrices on A, A ′ are unitarily related and so the entanglement entropydoes not change.However, having a Cauchy surface A ′ that is unaffected by the excitation is not a nec-essary condition. For example, no such A ′ exists for local excitations prepared by ingoingexcitations, yet these excitations still do not change entanglement entropy. We provide aproof that does not rely on the existence of A ′ , but on the properties of the excitationregardless of how it was prepared.Consider a quantum field theory in some pure state | Ψ i . We choose a purely spatialsurface at time t as a Cauchy surface for simplicity. Divide the spatial surface into tworegions A, B with reduced density matrices ρ A , ρ B . Consider an excitation localized within A at time t . This excitation can be created by acting with a unitary operator U ( A ) localizedin A. This localized excitation does not change ρ B . The entanglement entropy S B of region B therefore does not change either. As the state | Ψ i is pure, S A = S B , and so S A does notchange. If the perturbation is localized within B , the excitation does not change ρ A or S A .Only when the excitation is localized to a region that is causally connected to both A and B will these arguments fail. In this case, the entanglement entropy may change.32he complement of D ( A ) ∪ D ( B ) to the past of t is precisely the correct region. Thisregion includes its boundary, which consists of null rays. We recover the causality conditionof ref. [27]. Some excitations with support in both A and B can leave the entanglemententropy unchanged. As we showed in section 6.2, the excitations created by separable unitaryoperators accomplish this. In light of our conclusions that some infinite-norm local operators can create non-localizedexcitations, the results of recent entanglement entropy calculations may seem surprising. Weshow how these calculations are consistent with our results.Calculations of entanglement entropy in excited states created by infinite-norm operatorshave shown that Renyi and entanglement entropies change only when the operator insertionis null or timelike to the subregion [1–8]. States of the form O ( x ) | i were considered in(1 + 1)-dimensional CFTs. It was suggested that the jumps in entanglement entropy revealsa local quasi-particle picture. In this picture, a local operator creates quasi-particle pairs thatpropagate at the speed of light from the operator’s insertion point. Entanglement entropychanges only when one member of the pair is inside the subregion, but not both members.The (1 + 1)-dimensional calculations we address use the replica trick to calculate en-tanglement entropy. In the replica trick, entanglement entropy of interval A is calculatedfrom the replicated density matrix tr ρ nA , and conveniently given by correlators with twistoperators Φ n [49, 50]. The path integral for a field φ on an n -sheeted Riemann surface isgiven by a path integral for fields φ i living on C with certain boundary conditions relatingthe φ i . These boundary conditions can be represented by inserting twist operators at theendpoints of the interval. Twist operators are primary. Correlators are taken in the theorywith the n fields φ i . For details, see ref. [51].We consider a single interval A with endpoints u, v . The replica trick for excited stateshas been established [1–4]. Take O to be an operator creating an excited state. For example,one such operator could be O ( x ) = Q ni φ i ( x ). Renyi entropies are calculated fromTr( ρ nA ) = h |O † ( x + iδ )Φ n ( u ) ¯Φ n ( v ) O ( x − iδ ) | ih |O † ( x + iδ ) O ( x − iδ ) | i . (7.1)The normalization is such that Tr( ρ nA ) = 1 for n = 1. Entanglement entropy is calculatedfrom the Renyi entropy. In generic excited states, when the Renyi entropy changes, theentanglement entropy will change as well.Suppose O is an operator with definite conformal dimension. If x is spacelike-separatedfrom u, v , we can use the O ( x + iδ ) O † ( x − iδ ) OPE to understand what happens in the limit33 →
0. For finite δ the excitation created by O has a finite-norm and can be non-local.Indeed, entanglement entropy changes at spacelike separations for finite δ [1–4]. The leadingcontribution to the OPE for small δ is from the identity operator, and we showed in (6.33)how this implies the locality of certain operator excitations. We will revisit and providecontext for this statement shortly, comparing it to the result in ref. [6] to understand whenthe leading contribution to the full correlator comes from the identity and when it can comefrom the full identity block. For small δ ,Tr( ρ nA ) = h |O † ( x + iδ ) O ( x − iδ ) | i h | Φ n ( u ) ¯Φ n ( v ) | i + subleading h |O † ( x + iδ ) O ( x − iδ ) | i . (7.2)For δ →
0, Tr( ρ nA ) = h | Φ n ( u ) ¯Φ n ( v ) | i . (7.3)The excitation created by O does not affect the Renyi entropy. This argument was also givenin section 6.5.As x becomes null-separated from u or v , the OPE of O † ( x + iδ ) O ( x − iδ ) is not convergentbecause the twist operators are within what would be the neighborhood of convergence. Inref. [6], the authors instead consider the vacuum block approximation to the four-point func-tion, which is valid under certain assumptions and in a particular limit. They observe thatthis function has a certain branch cut, that when performing the continuation to real time,causes the entanglement entropy to pick up an additional contribution when the excitationis not spacelike-separated from the subregion. This is an example of how the entanglemententropy changes when the subregion becomes null and timelike to x .Our statement that for spacelike-separations the identity operator and not also its descen-dants dominates the correlation function as δ → δ → ρ nA ) reduces to the two-pointfunction of twist operators in the vacuum as long as the excitation is spacelike-separatedfrom the interval. The Renyi entropy is therefore unchanged by the excitation, just as wefound in (7.3). In the expression for the vacuum conformal block used in ref. [6], δ → z → z − δ →
0, only the leading divergence to the vacuum block gives a non-zerocontribution to the correlator. Only when the excitation is not spacelike-separated from theinterval does ¯ z pass to its second sheet, and the branch cut in the conformal block causesthe Renyi entropy to change.The argument we have given that infinite-norm local operators create local excitationsfails when O does not have a definite scaling dimension. As an example, instead excite the34acuum with the operator O e = n Y i e α O i ( z ) (7.4)Here α is real and contains a small dimensionless parameter. For simplicity, take O i = O † i .To first order in α , O e = 1 + α n X i O n ( z ) ≡ α O ( z ) . (7.5)We denote P ni O n ( z ) = O ( z ) for short. Notice that this operator does not change thestate’s norm to first order in α . The correction to the Renyi entropy is proportional to h | (cid:8) O ( z ) , Φ n ( u ) ¯Φ n ( v ) (cid:9) | i , and unless the three-point function vanishes, the anti-commutatorgenerically is not zero for z spacelike-separated from u, v . For an explicit example, choose O = P ni T n ( z ), the stress tensor. The anticommutator is known [49]. This calculation maybe performed with the entanglement first law. Alternatively, replace O i ( z ) with a non-localoperator O i ( z ) O i ( z ) to see a case in which the Renyi entropy will be non-zero.The argument we gave that uses the OPE to show that Renyi and entanglement entropieschange in response to a local excitation does not apply to O e . For example, to first order in α , the four-point function is a three-point function involving one O , and so there is no OO OPE to take. Said another way, as the OPE O e ( z + iδ ) O e ( z − iδ ) contains no divergence tofirst order in α , the contribution of the identity operator to the OPE does not determine thecorrelator’s behavior. While we must introduce a regulator δ for the state’s infinite norm,we need not introduce δ if we are working to first order in α .The calculations we have shown are consistent with our arguments in section 6.5, asoperators O which have definite conformal dimension change entanglement entropy onlywhen O is in causal contact with the interval.We have shown the quasi-particle picture does not describe excitations created by alllocal operators, but we have provided evidence that operators with definite conformal di-mension have a quasi-particle interpretation for certain conformal field theories. Others havedemonstrated that the quasi-particle picture is invalid for some field theories [5, 26]. Thequasi-particle picture remains a compelling description of certain excitations in certain theo-ries, and understanding its origin may reveal important properties of entanglement entropy. In this work, we have shown how localized unitary operators are fundamental building blocksof time-dependent quantum systems in entangled states. Localized unitary operators cre-ate localized excitations. We have detailed various features of localized unitary operators,including their locality properties, their behavior under superposition, and the difference be-35ween separable and non-separable unitary operators. We found that non-separable unitaryoperators, and their associated non-separable localized excitations, are in conflict with theprinciples of local quantum field theory. We gave a criterion to test for separability.We have shown how, unlike local unitary operators, local non-unitary operators can cre-ate non-local excitations in entangled states. As a reminder, generic states in field theoryare entangled over spatial regions. Local non-unitary operators are state-dependent and canhave infinite norm. We provided an example of an infinite-norm local non-unitary operatorthat creates a non-local excitation. We gave arguments that suggest that certain infinite-norm local non-unitary operators do create local excitations. However, these operators mustbe smeared to have finite norm, and the resulting finite-norm operators can create fully non-localized excitations. Consequently, one must be careful when drawing conclusions aboutlocality properties based on those of infinite-norm local operators. In practice, however, cor-relators in excited states created by a non-unitary operator O ( x ) can be simpler to calculatethan correlators in excited states created by a unitary operator e iα O ( x ) , which can involveperturbation theory in α and a treatment of divergences.We defined a local picture for quantum systems that unifies several different manifesta-tions of locality and causality into a simple description. The local picture follows naturallyfrom real-time perturbation theory and the definitions of open and closed quantum systems.According to the local picture, experimentalists can only act through deforming the Hamil-tonian, and localized experimentalists can only deform the Hamiltonian locally. Localizedunitary operators are central features of the local picture. Deforming the Hamiltonian in alocalized region is equivalent to acting with a localized unitary operator on the state, andthis operator will create a localized excitation. Generic non-unitary operators create non-localized excitations, so in order to act with these operators on the state, the experimentalistmust be fully non-localized herself.Using the local picture and our analysis of unitary and non-unitary operators, we dis-tilled more formal results in algebraic quantum field theory into elementary statements inquantum mechanics, and demonstrated their underlying mechanisms. We showed how thenon-local state preparation described by the Reeh-Schlieder theorem comes from the factthat local non-unitary operators create non-localized excitations in entangled states. Thelocal picture makes clear how the Reeh-Schlieder theorem is intuitive and consistent withcausality. Localized experimentalists can only create localized excitations, and so cannot actwith the local non-unitary operators that create non-localized excitations.We applied our results to entanglement entropy in field theory. We used properties oflocalized excitations to prove a causality condition for entanglement entropy that extends anearlier result [27]. Our proof applies to separable excitations and states prepared with ingoingexcitations. We addressed recent calculations of entanglement entropy in (1 + 1)-dimensionalconformal field theories [1–4, 6], and provided evidence that the locality properties demon-36trated by these calculations are only properties of operators with definite conformal di-mension. We showed consistency between these calculations and our conclusions about thelocality of operator excitations. We provided an example of a local non-unitary operatorthat changes entanglement non-locally. While the quasi-particle picture is known to fail incertain theories [5, 26], we concluded that the quasi-particle picture does not describe exci-tations created by all local operators in theories in which the picture is expected to hold.Understanding whether the picture applies to all localized excitations may provide insightsinto entanglement entropy.We connect our results to the AdS/CFT correspondence in the limit in which the bulk issemiclassical. Non-normalizable modes of bulk fields φ with dual CFT operators O are turnedon at the boundary by acting with the localized unitary operators T (cid:16) e − i R d d +1 xφ ( x ) O ( x ) (cid:17) in the CFT. An excitation of the CFT on a Cauchy surface S is associated with a bulkexcitation in Q S ∪ S , where the causal shadow Q S is the set of points spacelike-separatedfrom all points in S . This is because the region Q S ∪ S is the union of all possible bulkCauchy surfaces which intersect the boundary at S , and there is generically no preferred wayto choose one of these Cauchy surfaces for the bulk theory. Work on operator reconstructionis fully compatible with the fact that local non-unitary operators generically create non-localized excited states. For instance, for every local Hermitian operator O ( x ) there isa unitary operator e iα O ( x ) which creates a local excitation at x . Recent work sheds lighton these considerations through a bulk exploration of the Reeh-Schlieder theorem [52, 53].Recall that unlike unitary operators, non-unitary operators are state-dependent operators.State-dependent operators in AdS/CFT have been explored in detail [54–57].We expect that our results, along with our elementary treatment of related discussionsin diverse branches of the literature will help clarify investigations into locality, causality,entanglement entropy, and the AdS/CFT duality in the future. We would like to thank our advisor Per Kraus for invaluable guidance, support, and discus-sions. We also thank River Snively and Eliot Hijano for enjoyable, helpful conversations andcomments on the draft. We thank Detlev Buchholz for stimulating correspondence.
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