Deep inelastic scattering as a probe of entanglement
Abstract
Using nonlinear evolution equations of QCD, we compute the von Neumann entropy of the system of partons resolved by deep inelastic scattering at a given Bjorken and momentum transfer . We interpret the result as the entropy of entanglement between the spatial region probed by deep inelastic scattering and the rest of the proton. At small the relation between the entanglement entropy and the parton distribution becomes very simple: . In this small , large rapidity regime, all partonic microstates have equal probabilities – the proton is composed by an exponentially large number of microstates that occur with equal and exponentially small probabilities , where is defined by . For this equipartitioned state, the entanglement entropy is maximal – so at small , deep inelastic scattering probes a maximally entangled state. We propose the entanglement entropy as an observable that can be studied in deep inelastic scattering. This will require eventbyevent measurements of hadronic final states, and would allow to study the transformation of entanglement entropy into the Boltzmann one. We estimate that the proton is represented by the maximally entangled state at ; this kinematic region will be amenable to studies at the Electron Ion Collider.
pacs:
13.60.Hb, 12.38.CyI Introduction and summary
In almost fifty years that ensued after the birth of the parton model Bjorken:1968dy ; Feynman:1969ej ; Feynman:1969wa ; FEYN ; BJ ; Gribov , it has become an indispensable building block of high energy physics. The picture of “quasifree” partons “frozen” in the infinitemomentum frame due to the Lorentz dilation is clear and intuitively appealing. Even more importantly, the parton model combined with the QCD factorization Collins:1989gx allows to describe a vast variety of hard processes in terms of universal parton distributions. The renormalization group (RG) flow of QCD Gross:1973id ; Politzer:1973fx ; DeRujula:1974mnv in terms of the parton model can be recast in the form of parton splitting Gribov:1972ri ; Altarelli:1977zs ; Dokshitzer:1977sg , leading to the evolution of parton densities in Bjorken and virtuality . Nevertheless, in spite of a spectacular success of the parton model, it raises a number of conceptual questions:

The hadron in its rest frame is described by a pure quantum mechanical state with density matrix and zero von Neumann entropy . How does this pure state evolve to the set of “quasifree” partons in the infinitemomentum frame? If the partons were truly free and thus incoherent, they would be characterized by a nonzero entropy. Since the Lorentz boost cannot transform a pure state into a mixed one, what is the precise meaning of “quasifree”? What is the rigorous definition of the parton distribution when applied to a pure quantum state?

Deep inelastic scattering (DIS) at Bjorken and momentum transfer probes only a part of the proton’s wave function; let us denote it . In the proton’s rest frame, where it is definitely described by a pure quantum mechanical state, the DIS probes the spatial region localized within a tube of radius and length Gribov:1965hf ; Ioffe:1969kf , where is the proton’s mass. The inclusive DIS measurement thus sums over the unobserved part of the wave function localized in the region complementary to , so we have access only to the reduced density matrix , and not the entire density matrix . Is there an entanglement entropy associated with the DIS measurement? If there is, how does it relate to the conventional parton distribution?

What is the relation between the parton distribution and the multiplicity of final state hadrons in deep inelastic scattering? Is the “parton liberation” Mueller:1999fp picture universal, or does it apply only in the parton saturation domain? What is the interpretation of parton saturation Gribov:1984tu and color glass condensate McLerran:1993ni ; McLerran:1993ka ; Gelis:2010nm in terms of von Neumann entropy?
Answering these questions would also allow to interpret the inelastic electron scattering measurements in the domain of strong coupling (relevant at small and moderate ), where the concept of quasifree partons does not apply. Even at very large momentum transfer, this domain is relevant in the interpretation of deep inelastic scattering – this is because the parton evolution equations describing the renormalization group (RG) flow in QCD depend on the initial conditions at some moderate initial . There is another reason for relating parton distributions to the entanglement entropy – there exist quantum bounds on entropy (see for example Bekenstein:1980jp ; Bousso:1999xy ; Ryu:2006bv ; Hubeny:2007xt ), whereas a priori there is no bound on the growth of parton distributions^{1}^{1}1The growth of the total cross section is limited by the Froissart theorem; however the relation of parton distributions to the cross section gets modified in the domain of high parton densities due to shadowing corrections to the scattering amplitude. Let us emphasize from the beginning that here we will view the parton distribution as the multiplicity of partons at a given and . at small and large .
In this paper we attempt to address these questions in the framework of high energy QCD, see KOLEB for an introduction. Before we proceed to presenting the derivation, let us state our main results:

We have found the relation between the von Neumann entropy and the gluon distribution^{2}^{2}2Here the parton distribution is defined as the number of gluons at a given . accessed in deep inelastic scattering. At small this relation becomes very simple:
(1) Equation (1) implies that all microstates of the system are equally probable, and the von Neumann entropy is maximal. We argue that this equipartitioning of microscopic states that maximizes the von Neumann entropy corresponds to the parton saturation.

At small , we find that the von Neumann entropy diverges logarithmically at small :
(2) where is the longitudinal distance probed in DIS ( is the proton mass) and is the proton’s Compton wavelength, see Fig.1; is defined by . This expression reminds the well known result for the entanglement entropy in conformal field theory (CFT) HLW ; CARDY
(3) where is the length of the studied region, is the regularization scale describing the resolution of the measurement, and is the central charge of CFT that counts the number of degrees of freedom. We argue that this agreement is not coincidental, and propose that the parton distributions, and the entropy associated with them, arise from the entanglement between the spatial domain probed by DIS and the rest of the target. Therefore the maximal value of the entanglement entropy attained at small implies that the corresponding partonic state is maximally entangled. Unlike the parton distribution, the entanglement entropy is an appropriate observable even at strong coupling when the description in terms of quasifree partons fails.

Assuming that the second law of thermodynamics applies to entanglement entropy (see e.g. Hubeny:2007xt for a discussion), we get an inequality for the entropy of final state hadrons and the entropy of the initial state in DIS: . There are indications from holography that the entropy may not increase in the realtime evolution of strongly coupled systems (for a recent result, see Megias:2016vae ); this would imply the proportionality in accord with the “parton liberation” picture Mueller:1999fp and “local partonhadron duality” Dokshitzer:1987nm . This relation can be tested in deep inelastic scattering experiments by using eventbyevent measurements of hadronic final state.
The entanglement entropy between the large and small components of the wave function due to QCD evolution has recently been addressed in Ref. Kovner:2015hga , see also Peschanski:2012cw . This “momentum space entanglement” Balasubramanian:2011wt characterizing the renormalization group flow is different from the entanglement between the spatial region probed by deep inelastic scattering and the rest of the proton that is the object of our present study.
Ii Entanglement entropy and parton distributions
ii.1 Quantum mechanics of parton entanglement
The entropy of a macroscopic state is given by the logarithm of the number of distinct microscopic states that compose it – the Boltzmann formula forms the basis of statistical physics (and is appropriately inscribed on Boltzmann’s tombstone). Since partons are introduced as the microscopic states that compose the macroscopic state of the proton, it seems natural to evaluate the corresponding entropy. However, the proton as a whole is a pure quantum state with a zero von Neumann entropy – so we come to the apparent contradiction described in the Introduction.
To resolve it, let us consider a region of space (for simplicity of notation, onedimensional) probed by deep inelastic scattering (DIS). In the proton’s rest frame, it is a segment of length , where is the proton mass and is the Bjorken Gribov:1965hf ; Ioffe:1969kf . Let us introduce the coordinates and , where is the region of space complementary to . The wave function^{3}^{3}3In the rest frame of the proton, it can be interpreted as the wave function of the incident photon. can thus be written down as , and the corresponding density matrix is
(4) 
The state described by (4) is a pure quantum state with zero entropy.
Let us now introduce the density matrix describing the state probed in DIS. Since the region outside of is inaccessible to the measurement, we have to integrate over the coordinates Landau :
(5) 
or in operator form, . The density matrix (5) describes a mixed state with a nonzero von Neumann entropy.
The Schmidt decomposition theorem Schmidt (see Peres for a discussion) states that the pure wave function of our bipartite system can be expanded as a single sum
(6) 
for a suitably chosen orthonormal sets of states and localized in the domains and , respectively, where are positive and real numbers. In parton model, we assume that this full orthonormal set of states is given by the Fock states with different numbers of partons.
The density matrix (5) of the mixed state probed in region can now be written down as
(7) 
where is the probability of a state with partons. The von Neumann entropy of this state is given by
(8) 
From our derivation it is clear that this entropy results from the entanglement between the regions and , and can thus be interpreted as the entanglement entropy. In terms of information theory, Eq. (8) represents the Shannon entropy for the probability distribution .
We will now evaluate the probabilities and the corresponding entropy in two cases: i) a toy dimensional model of nonlinear QCD evolution; and ii) in full dimensional case where the nonlinear evolution is described by the BalitskyKovchegov (BK) equation KOLEB .
ii.2 1 + 1 toy model of nonlinear QCD evolution
It will be convenient for us to describe the parton evolution using the dipole representation – in this representation, a set of partons is represented by a set of color dipoles. In this section we consider a dimensional toy model that emerges from the BK equation if one fixes the sizes of the interacting dipoles MUDI ; LELU . In this model the BFKL equation for the dipole scattering cross section at a rapidity is reduced to
(9) 
where is the BFKL intercept. The Eq. (9) reproduces the powerlike increase of the cross section with energy, .
Let us now introduce , which is the probability to find dipoles (of a fixed size in our model) at rapidity . For this probability we can write the following recurrent equation (see Fig. 2):
(10) 
This is a typical cascade equation in which the first term describes the depletion of the probability to find dipoles due to the splitting into dipoles, while the second one – the growth due to the splitting of dipoles into dipoles.
The Eq. (10) can be rewritten in a more convenient form by introducing the generating function
(11) 
At the initial rapidity we have only one dipole so and (so the state is pure); at ,
(12) 
We can now rewrite the Eq. (10) as the following equation for the generating function:
(13) 
It is instructive to observe that (13) implies a nonlinear equation for MUDI ; LELU . Indeed, the general solution to (13) is of the form ; if we substitute this function into (13), the derivatives on the l.h.s. and r.h.s. of (13) cancel, and we get a differential equation for the function . By using the first of the initial conditions (12), we can then rewrite (13) for rapidity near the one at which the initial condition is provided as
(14) 
Therefore, our parton cascade includes the interactions between the partons that lead to nonlinear evolution in QCD^{4}^{4}4The dipole scattering amplitude in our model is given by , where is the dipole scattering amplitude at . It also obeys the following nonlinear equation : LELU ; K ..
The solution to (13) with the initial and boundary condition of Eq. (12) takes the form LELU
(15) 
Comparing Eq. (15) with Eq. (11) one can see that
(16) 
We are now in a position to calculate the von Neumann entropy of the system given by the Gibbs formula (8) by identifying the probabilities of microstates with the probabilities to find dipoles inside the hadron given by (16), . The resulting entropy is given by
(17) 
By using the generating function, Eq. (17) can be rewritten in the following way:
(18) 
which leads to
(19) 
One can see that at large
(20) 
In Fig. 3 we show the dependence of the entropy on rapidity, as given by (19) – one can see that the asymptotic behaviour of Eq. (20) starts rather early, at .
To establish the relation between the entropy and the parton distribution, let us now evaluate the latter within the same framework. We will define the parton distribution as the average number of partons at a given Bjorken . Using Eq. (11) and Eq. (15) we can calculate this number:
(21) 
Note that in perturbative QCD obeys the BFKL evolution equation and grows at small as with , where . It should be stressed that the multiplicity of gluons () evolves in accord with the linear evolution equation in spite of the nonlinear equation Eq. (13) for the generating function .
By comparing (21) and (20) we can see that at small the relation between the entropy and the structure function becomes very simple:
(22) 
On the other hand for corresponding to a small increase in the number of partons, the entropy is given by
(23) 
where is the number of partons at an initial value of . In Eq. (23) we assumed that .
It is important to note that the small , large rapidity relation (22) emerges in the limit where all probabilities become equal, see (16) – in this regime, an exponentially large number of partonic microstates occur with equal and exponentially small probabilities . It is well known that this equipartitioning of microstates maximizes the von Neumann entropy and describes the maximally entangled state. We thus conclude that at small the proton represents a maximally entangled quantum state of partons.
In terms of information theory, the maximal value of the Shannon entropy (8) achieved at small means that all “signals” with different numbers of partons are equally likely, and it is impossible to predict how many partons will be detected. In other words, the information about the structure of the proton (provided through an initial condition at some large ) becomes completely scrambled at small – so nonlinear QCD evolution is a very noisy “communication channel” between large and small . A particular realization of this “information scrambling” is a chaotic behavior at small observed earlier Kharzeev:2005gn in a discrete model of nonlinear QCD evolution; however as we have shown the entropy attains it maximal value even if the chaos is not present. This implies that at sufficiently small the structure functions of all hadrons should become universal, i.e. independent of the initial conditions.
ii.3 Relation to the entanglement entropy in conformal field theory
At small , the formulae (22) and (21) yield the following result for the von Neumann entropy:
(24) 
where is the longitudinal distance probed in DIS ( is the proton mass) and is the proton’s Compton wavelength, see Fig.1. This expression looks very similar to the well known result for the entanglement entropy in conformal field theory (CFT) HLW ; CARDY :
(25) 
where is the length of the probed region, is the regularization scale describing the resolution of the measurement, and is the central charge of CFT that counts the number of degrees of freedom. The divergence in (25) reflects the growth of the number of states near the boundary of the probed region when the resolution of the probe increases.
The divergence in (2) is precisely of the same origin – the coherent quantum state of partons in DIS extends in the target rest frame over the distance , and is probed with the resolution given by the proton’s Compton wavelength . The limit in (25) is obviously equivalent to the small limit in (2), as in both cases .
First, (2) refers to the quantum state, since the nonlinear evolution equations that we used to derive (2) is a faithful representation of RG flow in quantum field theory. The divergence of (2) at small thus reflects the presence of the infinite number of states present in the theory. For the entire coherent quantum state, the entropy (2) should vanish – and it does: when the resolution of the measurement coincides with at , . This property is obviously shared between (2) and (25).
Second, in the limit when partons become incoherent the entropy (2) should become extensive in – for example, at high temperature the entropy of onedimensional gas is . The entanglement entropy (25) in CFT has been evaluated also at finite temperature CARDY , with the following result:
(26) 
At low temperatures (26) reduces to (25), whereas in the high temperature limit we indeed obtain as expected for the extensive Boltzmann entropy of a onedimensional gas.
The similarity of (2) and (25) makes it plausible that at small the field theory describing parton evolution approaches a fixed point corresponding to a CFT with the central charge . Let us discuss this in more detail. The length of the region probed in DIS grows at small (high energies), whereas stays fixed. To compare with the behavior in 2D field theory, we can however keep fixed and decrease the value of – the highenergy behavior of our model is thus mapped onto the ultraviolet behavior of the 2D field theory. In other words, as the energy increases, we resolve shorter distances. The number of degrees of freedom increases from infrared to ultraviolet, and in field theory this intuitive expectation is confirmed by the rigorous theorem ZAMO stating that the behavior of is monotonic under renormalization group flow.
This allows us to conjecture a bound on the small behavior of the parton distributions. Indeed, in two dimensional CFTs the central charge assumes discrete values given by (see e.g. CARDY ):
(27) 
The largest value of corresponds to the free bosonic field theory – this is a likely fixed point in our theory as it corresponds to the asymptotically free behavior at short distances. Assuming that at small the partonic system indeed is described by a CFT, i.e. looks the same at all scales, we can thus put an upper bound on the value of :
(28) 
The growth of parton multiplicities at high energies should thus be limited by
(29) 
In fact, the value describes quite well the small behavior observed in DIS experiments PHEN1 ; PHEN2 ; PHEN3 ^{5}^{5}5The experimental data on the deep inelastic structure function show that , with that increases as a function of from 0.2 to 0.35 for (see Ref.PHEN2 ). The modern fits of experimental data are based on three ingredients: nonlinear evolution, running QCD coupling and nexttoleading order corrections to the BFKL kernel. Therefore, in general it is rather difficult to extract the value of effective value of for the linear BFKL evolution. However, for large both nonlinear and NLO corrections are rather small and the effective running QCD coupling turns out to be on the order of 0.1 (see Ref.PHEN1 ) leading to PHEN1 ; PHEN2 ; PHEN3 .. If we interpret this value in terms of the leading order BFKL result , it implies the strong coupling value .
The “asymptotic” small regime in which the formulae (22), (2) apply begins at (see Fig. 3), or at . It is accessible to the current and planned experiments, and can be investigated at the future ElectronIon Collider (EIC).
The small regime described by (22) and (2) implies the equipartitioning between the partonic microstates that maximizes the entropy. It can thus be viewed as an analog of thermal equilibrium for the parton system at small – just like statistical systems approach the thermally equilibrated macrostate with the largest entropy, small evolution leads to the universal state in which the entropy assumes the maximal value for a given . It is thus natural to associate this regime with parton saturation corresponding to the equilibrium between the parton splitting and recombination processes.
ii.4 Entanglement entropy from the dimensional BalitskyKovchegov equation
In this section we consider the evolution in dimensional QCD. We will see that the result for the entropy in this case is very similar to the one obtained above in the toy model. As discussed in Refs.MUDI ; LELU the parton cascade equation Eq. (10) in the case can be written down in the following form:
(30)  
where is the probability to have dipoles with size at rapidity . This QCD cascade leads to BalitskyKovchegov equation KOLEB for the amplitude and gives the theoretical description of the DIS.
Comparing Eq. (30) with Eq. (10) we see that the probability for one dipole to survive is not a constant as in Eq. (10) but depends on the dipole size:
(31) 
where is an infrared cutoff and . The probability for a dipole of the size to decay into two with the sizes and is equal to MUDI
(32) 
For Eq. (30) has the solution
(33) 
which reflects the fact that at we have only one dipole of size . It means also that .
Since is the probability to find dipoles , we have the following sum rule
(34) 
i.e. the sum of all probabilities is equal to 1.
Replacing by its Mellin image
(35) 
we reduce Eq. (30) to the form
(36) 
One can see that can be rewritten as
(37) 
with the following equation for :
(38) 
The solution is given by the recurrent equation
(39) 
in writing Eq. (39) we used the symmetry between and introduced the short notations: and .
We cannot solve Eq. (39) in an explicit way due to the term . However, we are able to do this for two instructive cases. The first case corresponds to the double log approximation of perturbative QCD in which and are both larger than . In this case Eq. (39) can be rewritten as
(40) 
The general solution to Eq. (40) takes the form
(41) 
The second case describes the decay of the large dipole into an asymmetric pair of dipoles, one large and one small. In this case, while . As noted in Ref.LETU this case corresponds to summation of terms for ; in other words, it describes the behavior of the parton cascade deep inside of the saturation region. In this kinematic region . The solution of Eq. (39) in this case takes the following form:
(42) 
Using Eq. (35) and Eq. (37) we find
(43) 
For we can rewrite Eq. (43) in the form
(44) 
Using Feynman parameters we can simplify Eq. (43) as follows:
(45) 
Eq. (44) now takes the form
(46) 
where and
(47) 
here is the Euler constant and is the incomplete gamma function. One can see that
(48) 
In Eq. (II.4) we used the usual ordering condition:
(49) 
From Eq. (II.4) one can see that satisfies the initial condition since . at
To find the entropy we need to evaluate the Gibbs formula
(50) 
It is hard to calculate the integrals in Eq. (50) in general case, but fortunately we can find the entropy at large . Indeed, at large values of ( reduces to the following form (see Eq. (II.4)) )
(51) 
Plugging Eq. (51) into Eq. (50) we obtain
(52)  
Using Eq. (34) and neglecting in Eq. (52) we reduce this equation to the form
(53)  
with . We emphasize that the normalization of the first term is fixed by the condition (34) and does not depend on the approximations we make in evaluating the integrals over .
Iii Discussion
Using nonlinear evolution equations of QCD, we have evaluated the von Neumann entropy of the system of partons that is resolved in a DIS measurement at a given Bjorken and momentum transfer . We have found that at small the relation between the entropy and the parton distribution becomes very simple and is given by (22). In this small regime (corresponding to and rapidity , as we estimated above), all partonic microstates have equal probabilities – the proton is composed by an exponentially large number of partonic microstates that occur with equal and exponentially small probabilities . In this equipartitioned state, the entropy (8) (that we have interpreted as resulting from the entanglement) is maximal – so the partonic state at small is maximally entangled.
If we interpret (8) in terms of information theory as the Shannon entropy, then the equipartitioning in the Maximally Entangled State (MES) means that all “signals” with different number of partons are equally likely, and it is impossible to predict how many partons will be detected in a given event. In other words, the information about the structure of the proton encoded in an initial condition becomes completely scrambled in the MES at small . Therefore the structure functions at sufficiently small should become universal for all hadrons.
Since the parton distribution and the entanglement entropy at small are related by (22), one may question the utility of entropy in characterizing the process of DIS. However there are several reasons to believe that the entanglement entropy is a useful DIS observable:

Identifying the entropy of partonic system as the entanglement entropy explains the apparent loss of quantum coherence in the parton model, solving an old conceptual problem described in the Introduction. The entropy that we have found originates from the entanglement between the spatial domain probed by DIS and the rest of the target, whereas the entire proton is in a pure quantum state with zero entropy.

Parton distributions have a welldefined meaning only for weakly coupled partons at large momentum transfer – but the entanglement entropy is a universal concept that applies to states at any value of the coupling constant.

Unlike the parton distributions, the entanglement entropy is subject to strict bounds – for example, if the small regime is described by a CFT, the growth of parton distributions should be bounded by , see (29).

If the second law of thermodynamics applies to entanglement entropy (and there are indications Hubeny:2007xt that it does), then the entropy of a final hadronic state cannot be smaller than the entropy accessed at a given Bjorken , and we expect the proportionality . The correspondence between the number of partons in the initial state and the number of hadrons in the final state is in accord with the “parton liberation” Mueller:1999fp and “local partonhadron duality” Dokshitzer:1987nm pictures. The link between the entropy in the initial state at small and the final state entropy has also been discussed in refs Baier:2000sb ; Kharzeev:2000ph ; Kharzeev:2001gp ; Kharzeev:2005iz ; Kharzeev:2006zm ; Fries:2008vp .
The entropy is a useful measure of information that can be obtained in an experiment – therefore it is an appropriate general characterization of the outcome of a DIS measurement. The entropic approach proposed here underlines the importance of measuring the hadronic final state of DIS. We thus encourage experimentalists to combine the measurements of the DIS cross sections with the determination of hadronic final state. The determination of the Shannon entropy of hadrons in the final state of DIS can be done using the eventbyevent multiplicity measurements, see e.g. Bialas:1999wi ; Atayan:2005cv . As we estimated above, the “asymptotic” small regime in which the formulae (22), (2) begins at . It is accessible to the current and planned experiments, and can be investigated at the future ElectronIon Collider (EIC) Accardi:2012qut .
We thank our colleagues at BNL, Stony Brook University, Tel Aviv University and UTFSM for stimulating discussions. We are grateful to A.H. Mueller and Yuri Kovchegov for positive and useful conversations. This work was supported in part by the U.S. Department of Energy under Contracts No. DEFG88ER40388 and DEAC0298CH10886, BSF grant 2012124, Proyecto Basal FB 0821(Chile), Fondecyt (Chile) grant 1140842, and by CONICYT grant PIA ACT1406.
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