Quantum Entangled-Probe Scattering Theory

We develop an entangled-probe scattering theory, including quantum detection, that extends the scope of standard scattering approaches. We argue that these probes may be revolutionary in studying entangled matter such as unconventional phases of strongly correlated systems. Our presentation focuses on a neutron beam probe that is mode-entangled in spin and path as is experimentally realized in [1], although similar ideas also apply to photon probes. We generalize the traditional van Hove theory [2] whereby the response is written as a properly-crafted combination of two-point correlation functions. Tuning the probe's entanglement length allows us to interrogate spatial scales of interest by analyzing interference patterns in the differential cross-section. Remarkably, for a spin dimer target we find that the typical Young-like interference pattern observed if the target state is un-entangled gets quantum erased when that state becomes maximally entangled.


I. INTRODUCTION
For more than a century, scattering techniques have been successfully employed to extract information about structural and dynamical properties of matter. Different types of probes (X-rays, electrons, neutrons, for example) reveal different (classical or quantum) characteristic properties of the target system depending on the nature of the probe-target interaction. So far, no quantum probe has exploited the characteristic trait of quantum mechanics: entanglement. Can one realize entangled-beams of particles? What information do entangled probes extract from the target? In this work we develop an entangledprobe scattering theory that addresses some of these issues at a fundamental level.
Recent work [1,3,4] has demonstrated two types of entanglers capable of preparing a beam of neutrons in a state exhibiting mode entanglement in two (spin-path) or three (spin-path-energy) distinguishable subsystems. These probes can (and will) be used in scattering experiments to examine condensed matter systems in a way similar to standard neutron scattering [5]. Ideally, one would like to develop quantum measurements that identify/detect the entanglement present in the target matter. Thus, extension of the standard textbook theory of scattering [6][7][8] to include entanglement of the probe (or projectile) is necessary. Typically, projectiles are counted by detectors arranged spatially (see Fig. 1). The nature of those detectors may vary depending on the property of the projectile one is trying to unveil, and the counting rate as a function of scattering angle from the direction of incidence defines the differential cross section (DCS).
Mode entanglement in the state of a single particle refers to the existence of non-local correlations between Figure 1. Scattering layout for the entangled probe of entanglement length ξ compared to an un-entangled wave packet, of transverse width ∆, and a plane wave, of momentum k0. Scattered waves from the target, with energies E λ and eigenstates |λ , are detected at distance r and solid angle dΩ.
its different distinguishable subsystems (path, spin, energy, etc.); alternatively, if non-local correlations between multiple particles are present one speaks of particle entanglement [9,10]. The latter is realized, for instance, in beams of entangled photon pairs [11]. As we will see, these two different forms of scattering probe entanglement differ in the type of information they extract from the target. While mode-entangled scattering uncovers distinctively crafted two-point correlations of the sample being probed, matrix elements in multiparticle scattering include two-body interaction operators for each particle, thus contributing to a multi-point correlation function. Importantly, by tuning the probe's entanglement one can in principle vary the correlation function and unveil entanglement in the target system.
A main result of this paper is the formulation of a scattering theory for a mode-entangled probe. When applied to a magnetic interaction potential, such formulation represents an extension of van Hove's theory. To gain understanding of the kind of information one can ex-tract using this kind of probe, we illustrate our findings in the case of a spin-dimer. Depending on the relation between the various length scales involved in the scattering process, one obtains radically different responses allowing detection of entanglement in the target material. For instance, the typical Young-like interference pattern observed when the target state is un-entangled gets quantum erased when the target state is maximally entangled. In addition, to obtain ancillary information on the nature of entanglement in the target we propose to alter the way one detects the outgoing scattered probe, as happens when one measures spin polarization using a spin-echo device in neutron scattering.
The paper is organized as follows. Section II starts with the formal introduction of a multimode-entangled probe and then sets the scattering framework by generalization of the standard T -matrix formalism (details of the derivation can be found in Appendix A). Section III discusses the particular case of magnetic interaction between the target and a neutron probe and derives the extension of van Hove's theory. To identify the kind of information the entangled probe can extract from a target state, this magnetic scattering is further specialized in Section IV using the example of a spin-dimer target. In this section, we illustrate how the DCS changes under different experimental conditions and discuss the prospects of detecting the varied outcomes. Finally, in Section V we summarize main results and implications of this work. An Appendix B briefly discusses the general scattering theory for the multiparticle-entangled fermionprobe case.

II. ENTANGLED-PROBE SCATTERING THEORY
For simplicity, we focus on mode entanglement and consider a coherent beam with two distinguishable subsystems: path and spin. In distinguishing these subsystems, the full Hilbert space of our probe state is constructed as H probe = H path ⊗ H spin . The two pathways of the probe are indistinguishable alternatives, and given a separation in paths ξ (Fig. 1), a wave packet description must be employed.
We define the simplest initial, t 0 < 0, single-particle entangled-probe state to be (see Fig. 2). Equivalently, in plane wave components [12] Φ in (r, t 0 ) = 1 where ω(k) = E k = 2 k 2 /2m, m is the mass of the probe, Θ k = k·ξ + 2φ, φ is a phase introduced by the Figure 2. Schematic of probe wave packets of mean momentum k0, illustrating that the spin and path subsystems are entangled. This figure also shows the scattering to momentum k by a particular orientation of the dimer vector, d = |d|ŷ, which is investigated in Section IV.
experimental apparatus (i.e., the entangler [1]), and the quantization box of the momentump states |k is taken to be of linear size L. We choose normalizations such that The entangled wave packet is characterized by the distributiong(k) with mean momentum k 0 , transverse spatial width ∆, and energy Φ in |Ĥ p |Φ in = E p , withĤ p =p 2 2m . The probe Φ in becomes unentangled with respect to the H probe decomposition if ξ = 0. The orthogonal spin-1/2 states |χ α ν , ν = 0, 1, are defined along a particular spinquantization axis α = x, y, z, with σ α representing Pauli matrices. Starting from states quantized along α, the effective spinor associated with the k component of Φ in will be fully aligned along a particular direction given by where the (k · ξ)-dependent axesχ α are found to bê Since we are interested in both elastic and inelastic scattering we must include dynamical properties of the target HamiltonianĤ t of spectral representationĤ t |λ = E λ |λ . Then, the total Hamiltonian of the probe-target system isĤ =Ĥ 0 +V , whereĤ 0 =Ĥ p +Ĥ t , withV representing their interaction potential. The Hilbert space of the probe-target system is H probe ⊗ H target , and has basis elements denoted by |kχλ ≡ |k ⊗ |χ ⊗ |λ . We assume that the probe-target initial state is the mixed stateρ in = |Φ in Φ in | ⊗ρ t , withρ t = λ p λ |λ λ|, where p λ is a Boltzmann weight if the target is in thermodynamic equilibrium at t = t 0 .
We next extend the T -matrix formalism to include entanglement in the probe. In the interaction picture the propagator obeysÛ I (t, t 0 ) = 1− i t t0 dt 1VI (t 1 )Û I (t 1 , t 0 ), withÛ I (t 0 , t 0 ) = 1 andV I (t) = e iĤ0t/ V e −iĤ0t/ . The T matrix, describing the transition from the state |ψ k = |k χ k·ξ λ to the basis state |ψ = |k χ λ , is defined by in which we notateT ψ ψ k = ψ |T |ψ k , ω(ψ , ψ k ) ≡ E ψ − E ψ k , E ψ = ψ|Ĥ 0 |ψ , and > 0 a regulator. In the equation above we have introduced the matrix element per quantization volume L 3 ; eventually one has to perform the L → ∞ limit. The un-scattered state is decomposed in terms of the |ψ k states as Note that this state does not contain t dependence explicitly, as we are working in the interaction picture. Then, Although the wave packet is not monoenergetic by design, the spread ofg(k) about its central value k 0 (= k 0ẑ in the geometry of Fig. 1) is supposed to be small. As such, when measuring scattering away from the propagation axis of the wave packet, k is far from k 0 and sog(k ) ≈ 0 [6,7]. The forward scattering term, δ λλ g(k ) χ |χ k ·ξ , is thus omitted. Using this approximation and the density of states ρ(E k ) = mk 2 L 2π 3 dΩ k with dΩ k the solid angle in the direction of k , the probability of transition per final-state energy is shown to be (see Appendix A) represents the transition rate [6,7]. The Born approximation is effected by replacingT bŷ V and these regular matrix elements are given byT ψ ψ k ≈ V ψ ψ k = V ψ ψ k /L 3 , with for a local interaction r|V |r = δ(r − r )V (r).
To compute the total probability of scattering we sum over λ and χ , and average over λ, assuming the initial state of the target is the stateρ t . In taking the L → ∞ limit, (L/(2π)) 3/2g (k) → g(k) and L −3 k → (2π) −3 dk, and normalizing by the time-integrated average flux I = lim t→∞ t −∞ dt  z (t ), we obtain the DCS where the energy transfer from the constituent, incoming plane wave component to the target is ω ≡ E k1 − E k and the integration measure is dΛ ki = g(k i )dΩ ki . Note that the constraint δ ω(k 1 ) − ω(k 2 ) in Eq. (3) enforces k 1 = k 2 . The probe's entanglement is encoded in the matrix elements V ψ ψ k , which is enhanced in magnitude whenever the relevant length scales of entanglement in the target match ξ. As we will see in the application (Section IV), there are subtle interference effects hidden in those matrix elements which are linked to entanglement.
After scattering, the probe state becomes entangled with the target state, and the Lippmann-Schwinger equation describes the resulting outgoing state [6,7]. The outgoing scattered probe state in a given direction is given byρ where Tr λ is the partial trace over the target state space. A remark on the main result, Eq. (5), in the standard limit of plane wave scattering: when ξ = 0 and the incident state is a plane wave normalized in a box, the flux is uniform z (t) = k 0 /mL 3 . Examining the limit L → ∞, we impose the concurrent restriction that ∆ ∼ L. This ensures that the (now un-entangled) wave packet asymptotes to a plane wave while, for finite L, remaining square-integrable. Performing this calculation, the known form of the standard plane wave (pw) cross section as reported e.g. in [5] is recovered As said at the beginning of this section, we focused on the single-particle, mode-entangled probe. For a comparison of equivalent results when the probe is multiparticleentangled see Appendix B.

III. MAGNETIC SCATTERING
We next extend van Hove's theory [2,5] to the case of an entangled probe. Consider the particular case of a neutron probe, with spin 1 2σ and mass m = m n , interacting magnetically with electrons (of mass m e ) of the target. The interaction potential is given bŷ where {A, B} = A·B+B·A, γ = −1.913 is the neutron's gyromagnetic ratio, µ B = e 2mec and µ N = m e µ B /m n are the Bohr and nuclear magnetons, respectively. The spin and momentum of the j-th electron areŝ j (eigenvalues ± 1 2 ) andp j , and the position vectors R j are directed from the j-th electron to the neutron probe. Using the identity R = − R 3 2π 2 ∇ dq 1 q 2 e iq·R and completeness λ |λ λ | = 1, evaluation of the matrix elements in Eq. (5) yields the form with scattering vector κ 1,2 = k 1,2 − k , and r 0 = γe 2 mec 2 . Although the two momenta, k 1 , k 2 , involved in this integration have the same magnitude, the variation in their direction generates a dependence of the response function S(κ 1 , κ 2 , ω) on two momentum transfers, as opposed to the single transfer present in a plane wave treatment. The response function is given by andκ = κ/κ. The matrixρ α k1,k2 = |χ k2·ξ χ k1·ξ | encodes the spin states of the entangled-probe wave packets along with the ξ-dependent phase-shift. Written in terms of the individual wave packets basis, whose trace is given by and it is a pure-state density matrix when k 1 = k 2 :ρ α k1 ≡ ρ α k1,k1 = (1 +σ ·χ α )/2. Using this formalism, the trace in Eq. (6) gives a decomposition of S(κ 1 , κ 2 , ω) into the four combinations of spin components involved, so that spin-diagonal (ν = ν ) entries describe scattering due to each individual wave packet of the probe while the off-diagonal (ν = ν ) entries describe interference between them. Similarly one can derive the polarization vector P [5] of the scattered neutron, defined as , to be given by . Magnetic scattering of an entangled-probe with entanglement length ξ, by a dimer of size |d|, into scattering angles (θ, ϕ) for a general orientation of the dimer. The spinquantization axis for the probe is α = x, i.e., |↑ x,|↓ x. The initial state of the probe-target system isρ in = |Φ in Φ in | ⊗ρt.

IV. UNCOVERING ENTANGLEMENT FROM ENTANGLEMENT
To highlight the information entangled-probe scattering can provide, we apply our framework to the case of a target with two motionless interacting electrons, i.e., a dimer. Electrons are positioned at r j = (−1) j d/2, j = 0, 1 (Fig. 3), and their interaction is governed by the Heisenberg HamiltonianĤ t = −4Jŝ 0 ·ŝ 1 with exchange coupling J. Its Hilbert space is the direct sum of singlet and triplet subspaces: Table I. Variables entering Eqs. (11) and (12) for zero and finite temperature, T, which plays the role of adjusting the singlet/triplet admixture of the thermal state as described in the text. The normalized scattering vector isκ = κ/κ, s and t stand for singlet and triplet, respectively, andÃ τ τ andB τ τ represent the plane-wave limit of A τ τ and B τ τ .
A standard physical measure of multipartite entanglement is the purity [9, 10, 13], a pedagogical explanation of which is found in [14]. Given a normalized state |λ t = α c α |λ α ∈ H t , its purity is given by with c = (c x , c y , c z ) encoding the coefficients of the linear combination. A pure triplet state is maximally entangled (un-entangled) if and only if c * × c = 0, i.e. c is a realvalued vector (|c * × c| = 1). An example of a state exhibiting partial entanglement is 2 , 0 and purity of 1/4. This quantification allows for a simple identification of entanglement-induced features of the calculated cross section, as there will be terms proportional to the purity which then vary depending on the degree of entanglement.
The initial state of the entangled probe is defined by a Gaussian distribution with average momentum k 0 = k 0ẑ (k 0 ≈ 1.5 × 10 4 µm −1 in [1,4]), spatial width ∆, spin-quantization axis α = x, and tunable entanglement vector ξ in the y-direction (25nm < ξ < 25µm in [1,4], see Fig. 3 for the general setup and Fig. 2 for the specific setup used in the the plots of Figs. 6 and 7). We consider two types of initial target statesρ t : are Boltzmann factors which incorporate the effect of temperature T in the scattering cross section. Investigation of its pure state components will permit analysis of the effect of the target's entanglement on the DCS.
These terms can be factored in such a way as to isolate the information pertaining to the dimer from that of the entanglement of the probe (k 1 = k 2 ): (10) with ζ st = +1, ζ ts = −1, and ζ tt = 0. Conservation of energy implies In the above decomposition we have introduced real functions describing the dimer structure, and functions encoding the entanglement length of the probe, The expressions for A τ τ and B τ τ are summarized in Table I. The entanglement length enters into h τ τ via the first term as well as the ξ-dependence of the matrix element  (12), i.e., plane-wave limit, for the triplet |λt = |↑↑ z to singlet |λs transition, S t→s (κ, ω), when κx = 0 assuming k = πÅ −1 , d = 9Å, J = 1/4 meV, and k 2 = k 2 − 8mJ 2 . Top panels depict the case ξ = 0, corresponding to an incoming neutron of spin polarization in the +z direction, with forward (backward) scattering 0 ≤ θ < π/2 (π/2 ≤ θ ≤ π) on the right (left). Bottom panels display the same information when, by tuning ξ (k·ξ = 3π/2, φ = 0), the effective polarization is set along the +y direction.
There are three competing length scales in the problem: |d ⊥ |, that is the projection of d onto the x-y plane, ξ, and ∆ (axes and dimer orientation coincide with Fig. 2). To gain intuition into the kind of additional information entangled-beam scattering provides, we start by considering the magnetic response function of the dimer in the limit ∆ → ∞, k 1 = k 2 = k = kẑ, and ξ ∆ can be different from 0. As ∆ eclipses ξ, the wave packets overlap creating a wave function which is almost indistinguishable from the unentangled plane wave, but which still has a technical ξ dependence in the form of a phase. For this reason, we refer to this limit as the plane-wave (pw) limit. As for the entangled wave packet probe, the total response function can be decomposed into a sum of three components, where κ = k − k is the momentum transfer, and S (pw) τ →τ (κ, ω) = δ( ω + 4Jζ τ τ ) sin 2 κ·d + πδ τ τ 2 where expressions forÃ τ τ andB τ τ take the simpler form shown in Table I (B τ τ is purely imaginary). The term sin 2 ( κ·d+πδ τ τ 2 ) is typical of a two slit-type interference pattern, with the dimer playing the role of the slits.
Although the initial probe is in the plane-wave limit, there is still in principle a dependence of the scattered state on the path entanglement vector ξ. Notice, however, that when the target state is maximally entangled its purity |c * × c| 2 vanishes and, sinceB τ τ is directly proportional to the purity, the magnetic response S (pw) τ →τ is insensitive to ξ. Thus, maximal entanglement of the target in the plane wave case precludes ξ-dependence of the DCS -a dependence which is present for nonzero purity (see Fig. 4).
Interestingly, the role Θ k plays in the response is effectively equivalent to a rotation of the spin polarization of the neutron. In other words, in this plane-wave limit and from the standpoint of the DCS, tuning the properties of the beam by manipulation of the entangler is similar to changing the polarization of the incident neutrons. We next analyze the polarization of the scattered neutron in the plane-wave limit. From Eq. (9), withh and c ⊥ = c −κ (κ·c). For example, if the incident polarization is in the y-z plane, i.e.χ x , and we restrict k to the y-z plane (κ x = 0), then the polarization of the scattered neutron for the triplet-to-singlet transition at T = 0 is with c ⊥ /c ⊥ = (0,κ z , −κ y ) and c 2 ⊥ = 1 − |κ·c| 2 . The situation becomes more interesting away from the planewave limit as discussed next.
Returning to the main focus of the entangled probe, the DCS also contains three components:   Thermal stateρt triplet-to-singlet DCSs d 2 σ/dΩ dE k t→s (spherical plots (θ, ϕ)) for the case |d ⊥ | ≈ ξ, φ = 0, k0 ≈ 1.5 × 10 4 µm −1 , and the dimer aligned along the y direction. The DCS for ∆ < ξ is proportional to α (1 − |κ0·c| 2 ) = 2, with c corresponding to |λα . While the ordering of sizes is faithful, plots do not have the same scale for visualization purposes, as increases in ∆ achieve sharp decreases in the DCS magnitudes when |d ⊥ | ≈ ξ.
The largest scattering amplitudes occur when |d ⊥ | ∼ ξ and ∆ is small. As soon as the value |d ⊥ | departs from the length ξ, the DCS starts to attenuate exponentially, at which point a large ∆ is required to counter this effect. Figure 6 compares the angular dependence of DCSs in a triplet-to-singlet transition of a thermal initial target stateρ t for different competing length scales and κ 0 = (k 0 −k )/|k 0 −k |. In each of these, ξ ≈ |d ⊥ | = 0 and so the beam is entangled. The case ∆ > ξ does not differ qualitatively from what one would have obtained with an un-entangled probe (ξ = 0). Indeed, the "flower-shape" DCS when ∆ > ξ is reminiscent of a twoslit-type interference pattern and, as mentioned above, is also obtained in the case of a standard probe (ξ = 0). On the other hand, the behavior of a non-overlapping entangled wave packet (∆ < ξ ∼ |d ⊥ | ) is different than an un-entangled probe: the distinct two-slit interference pattern vanishes, leaving a DCS which is insensitive to (θ, φ) ∈ [0, π] × [0, 2π) and far stronger in magnitude. This spherical symmetry arises from a delicate summation of contributions from the triplet states making up ρ t , which do not individually exhibit this feature (for example, see the |λ x DCS of Fig. 7).
The situation is even more remarkable when the target state is pure. Then, some interference terms are proportional to the purity P su(2)⊕su(2) (|λ t ) of the target state and consequently the DCS can identify entanglement in the target. Figure 7 displays the triplet-to-singlet DCS for three particular target states using the same entangled, ∆ < ξ neutron probe. Maximally entangled Belltype states of the target show a special shape distinct from those of un-entangled or partially entangled states. The latter depict two-slit-type interference patterns with proper characteristics of the particular symmetry of the probe-target system, while Bell-type target states seem to forbid those two-slit-type interference patterns as a result of their non-local correlations, i.e., their entanglement. Such "quantum erasure" of the interference pattern can be understood by realizing that orthogonality of the incident spin states corresponding to the two paths are preserved after scattering if and only if the target state is maximally entangled. Let us expand: In the limit ∆ ξ, |d ⊥ |, the neutron wave packet scatters significantly only from those scattering centers that lie in its trajectory, i.e. the dimer spinŝ j interacts mainly with the neutron spin state |χ α j . Consider, for instance, the scattered neutron spin-dimer triplet-to-singlet transition where |χ sc j is the spin state corresponding to the j-th neutron wave packet after scattering with final momentum k , and c ⊥ = c −κ 0 (κ 0 ·c). The path interference term in the DCS is proportional to For a maximally entangled state, c is real-valued and c * ⊥ ×c ⊥ vanishes identically, thus explaining the observed quantum erasure phenomenon. There remains a question of how to detect the neutron after it has undergone interaction with the target. In principle, the phase, φ, added by the entangler is determined by the experimental setup. While this phase may be present in general, it is ubiquitous in neutron scattering experiments and may vary with neutron wavelength in some cases [1]. In that context, to obviate the need to average the DCS over φ a spin echo technique might be used to remove this phase by placing a disentangler [1,3] after the target. This quantum detection strategy is known as spin-echo (se) measurement.
However, the details of the required spin echo apparatus will depend on the target, both because the latter will rotate the neutron polarization and because the scattering is inelastic. The neutron polarization induced by Figure 7. Pure state triplet-to-singlet DCSs d 2 σ/dΩ dE k t→s (spherical plots (θ, ϕ)) for the case |d ⊥ | ≈ ξ, ∆ < ξ, φ = 0, k0 ≈ 1.5 × 10 4 µm −1 , and the dimer aligned along the y direction. DCSs are to scale with each other, illustrating the effect target-state entanglement has on the DCS interference pattern. The DCS for |λα is proportional to 1 − |κ0·c| 2 . Purity values for the displayed maximally-, partially-and unentangled states are 0, 1/4, and 1, respectively. the target, which will depend on φ, can be found from an expression identical to Eq. (9) with the replacement whereσ se = U † φσ U φ is the unitarily transformed spin operator with U φ = ν=0,1 e i(−1) ν φ |χ α ν χ α ν |, where α is the spin-echo axis.
Only the components of the neutron polarization produced by the target that are perpendicular to the α-axis will be modified by the spin-echo disentangler and this has to be taken into account in order to measure the full DCS with the phase φ eliminated. In general, the final neutron polarization measured after the disentangler will be reduced in magnitude, just as it is for conventional spin echo measurements, and the variation of the polarization with entanglement length ξ will contain information about the electron spin correlations in the target, including their state of entanglement. Calculations of these effects will be the subject of a future communication.

V. OUTLOOK
We formalized a scattering theory for an incident probe whose quantum state is prepared in either a mode-or multiparticle-entangled fashion. The key idea is to control the intrinsic entanglement among the subsystems of the probe, such as its spin and pathways, to learn about the entanglement present in the target. Exploiting such control involves several adjustable length and energy scales that compete with those of the target. This competition may generate amplification or erasure of interference patterns that betray information about the target's entanglement. Together with interferometric methods of quantum detection of the scattered wave, such as measurement in a spin-echo mode, entangled probes promise to become a powerful tool for future investigations. Quantum imaging techniques [15][16][17][18][19] using light exploit the quantum nature of the probe, including entanglement and squeezing, to achieve enhanced precision measurement [20] and sensing, for instance. These interferometric techniques often treat the interaction between the probe and the target in a semiclassical manner. By contrast, the quantum entangled-probe scattering studied in this paper represents a fully quantum-mechanical treatment of the interaction allowing, for example, the determination of spatio-temporal information about correlations in the target's elementary constituents.
In an effort to describe and manipulate the internal degrees of freedom of the probe, one is forced to depart from a plane wave description and consider a full-fledged entangled wave packet formulation wherein the transverse coherence of the probe is an adjustable variable. When applied to magnetic scattering of neutrons, our framework generalizes van Hove's theory. To gain an intuition for the effect of this technique, we analyzed the particular case of magnetic scattering of a neutron by a spin-dimer target state. We find an enhancement in the differential cross section (DCS) when in the regime where the transverse width of the incident wave packets is smaller than the entanglement length (i.e., the separation between packets) and the entanglement length is tuned to match a magnetic correlation length of the target. Remarkably, a maximally entangled target state does not show the typical Young-like interference pattern that is present for a non-maximally entangled dimer. This finding can be interpreted as the quantum erasing by a quantum-entangled double-slit of the interference pattern expected from an un-entangled or classical double-slit. The reason behind such an interesting quantum erasing effect is traced to the effect that, whenever the target state is maximally entangled, the orthogonality between incident and scattered neutron spin states corresponding to each path is preserved. If this is not the case then there is always some interference between the paths' contributions to the DCS.
We summarize our main results in Table II. While the most spectacular effects we have found relate to the situation in which the wave packet size ∆ is smaller than the probe entanglement length ξ, some effects, such as the fact that the DCS does not depend on ξ for maximally entangled target states, also persist when the intrinsic coherence length of the neutron (related to the wave packet size in our calculations) is larger than ξ. Entanglement lengths up to about 25 µm have been achieved [21] but the inherent coherence length of neutrons has been measured to be larger than this value [22,23] in several ex-  Table II. Summary of main results illustrating the DCSs, d 2 σ dΩdE k τ →τ , resulting from a neutron beam (with Hilbert space H probe = H path ⊗ H spin ) scattered from a motionless spin dimer target. Different scenarios emerge depending on the relation between the relevant scales of the problem, with ξ the entanglement length, |d ⊥ | the dimer size,κ0 = (k0 − k )/|k0 − k |, and c characterizing the pure target state as defined in Section IV.
periments, implying that only the first four rows of the table are immediately accessible experimentally. While it is not yet clear to what extent our results are generally applicable to other entangled systems, we hope that this theory and future experiments that it informs may shed light on complex phases exhibited by novel materials such as multiferroics, unconventional superconductors, quantum spin liquids, and frustrated magnets.
With this substitution, we differentiate Eq. (A1) with respect to time in order to express the transition rate of ψ → ψ . This quantity, once → 0 + is taken, we define in the text as This limit may be resolved by using in which P denotes the Cauchy principal value (PV). The parentheses in Eq. (A3) then become −iπ δ(ω(ψ , ψ k1 )) + δ(ω(ψ , ψ k2 )) The difference of PVs will cancel, as described below, so we omit them from now on.
In the wave-packet context, and in contrast to the plane-wave context, the time dependence of the transition rate is to be expected. As the wave packet evolves with time from −∞ to t, the total probability of the specific transition ψ → ψ to occur is given by t −∞ dt W ψ→ψ (t ). Because we want the cross section per final-state energy, we multiply W ψ→ψ by the density of states ρ(E k ) = mk 2 L 2π 3 dΩ k . Performing this evaluation, the probability of transition from ψ to states surrounding ψ after a long time t is where we have used E k1 = E k2 from the first delta function. This same condition is the reason for cancellation of the PVs, as it fixes ω(ψ , ψ k1 ) = ω(ψ , ψ k2 ). The time integration used ∞ −∞ dt e iut = 2πδ(u) for a frequency u. The energy transferred ω = E k1 − E k is equal to the energy acquired by the target, E λ − E λ . Thus Eq. (3) has been recovered, as well as the subsequent form of the energy-conserving delta function in Eq. (5).
Appendix B: Scattering of Multipartite Entangled Probe

Incoming particle-entangled state
The main manuscript deals primarily with a singleparticle probe prepared in a mode-entangled state. This is distinct from a setup involving probe states with multiple entangled particles: the latter may lead to a higherorder correlation function due to the additional instances of single-particle interactions with the target. There are several quantitative adjustments which effect this difference in the case of multiparticle scattering.
First, the Hilbert space of the system is extended to accommodate the momentum and spin states of the second particle. Indistinguishable particles impose an additional superselection rule on the state space, that is, the exchange statistics symmetry. Here, we only focus on fermionic probes. Indexing the two particle orbitals by A and B, we notate a basis for the state of two spin-1 2 particles of momenta k A and k B and entanglement vector ξ: where x i = (r i , σ i ) labels the position and spin of the particle i = A, B, and ν = 0, 1. These basis functions obey the fermionic superselection rule, antisymmetry under exchange x A ↔ x B , and are defined by It is worth noting that b ξ kk = c ν kk = 0 and so are not included in this basis. χ α ν (σ i ) is the spinor of the i particle aligned up (ν = 0) or down (ν = 1) along the α quantization axis.
To illustrate the effect of particle entanglement we define an example initial state from the basis vectors to be is the Hamiltonian of the free probes with momenta k A and k B .
We next describe the final state which is detectable by the set-up of Fig.8: two detectors separated in space are polarized to only detect up or down (ν, ν = 0, 1) as seen along the axes β and γ: −e i(k B ·r A +k A ·r B ) χ γ ν (σ A )χ β ν (σ B ) .(B1) A spinor written in the basis along the axis β = (θ , φ ) is connected to the α = (θ, φ) basis by the rotation χ β ν = R β ν0 χ α 0 + R β ν1 χ α 1 , Figure 8. We have two detectors D1 and D2, spin-resolved along the β and γ spin quantization axis. Neutrons are counted when neutrons hit the different detectors simultaneously.
with rotation matrix and χ α 0 = c θ e iφ s θ , χ α 1 = −s θ e iφ c θ , s θ ≡ sin θ 2 and c θ ≡ cos θ 2 . Substituting these expressions in Eq. (B1) we find that this out state can be written in terms of ξ = 0 basis functions as The interaction potential is now extended to include a symmetric interactionV between the two particles. Additionally, we take it to obey the locality condition withV (x A , x B ) =V (x B , x A ), i.e., a symmetric potential. Calculating the potential matrix element between our initial and final states, The spin components of expressions such as this will be determined by the initial and final configurations, averaged and summed over as usual in the cross-section.

Final two-fermions cross-section
The amplitude for transition from the initially prepared probe to a final basis state, again measuring away from the forward direction, is Following the same procedure as in Appendix A from here, the probability of transition from Ψ in to states surrounding Ψ out per final state energies E k A and E k B after a long time t is and the density of states being for a pair of plane waves of individual energies The flux, now, is the combined contribution to the timeintegrated flux by the two probes A and B averaged over a characteristic area: I A + I B =