Pauli-based fermionic teleportation with free massive particles by electron-exchange collisions

Fundamental non-locality in quantum mechanical scattering processes is investigated by means of Fermionic teleportation. Our scenario employs free massive particles, electrons and atoms, within a twofold electron-exchange collision approach whereby, in the first collision, also generating the necessary spin–spin entanglement between initially unpolarized electrons and atoms. It is shown that in a second collision an arbitrary spin polarization state of a free electron can be teleported onto the other electron which has never been in contact with the first one. The underlying scheme relies on and is a direct consequence of the Pauli-principle and makes use of a high symmetry in spin space thereby avoiding restoration of the teleported states. The scattering process (teleportation) is highly symmetric allowing for interchanging the electronic and atomic constituents without loss of generality. Re-interpretation of Li and Na data reveals the feasibility and capability of Fermionic teleportation demonstrated by numerical and experimental data for the teleportation fidelity.

elastically scattered and their spins become entangled [15]. This entanglement can be found, and fixed in advance, by choosing appropriate scattering angles and/or energies. By varying the scattering parameters, the full range of spin-spin correlations can be generated, and in particular Bell-correlated spin pairs. Thus, the final spin system represents a tunable EPR resource [15,16]. This is in contrast to realized entanglement generation between electrons as its creation is nonetheless photon-assisted [17]. Subsequently, a second electron with arbitrary (unknown) spin polarization is scattered from the atom and it is shown that its polarization state is non-locally teleported onto the first electron (still unpolarized after the first scattering). The experiment represents a single-outcome Bell measurement. Hence, restoration of the teleported state by unitary transformation [2], typical for the standard scheme, becomes obsolete in our approach.
The success of teleportation (fidelity) depends on the spin-spin correlations and again is directly related to the Pauli-principle. Re-interpretation of existing experimental and numerical sodium and lithium data [18][19][20][21], as well as recent calculations [22] underpins our results, revealing the feasibility and capability of such experiments. As an essential role is played by the Pauli-principle, we will refer to our approach as Pauli-based Fermionic teleportation (PBFT).
The paper is organized as follows. In the next section we will elucidate the underlying physics by considering a simple example first, assuming a maximally entangled EPR resource prepared in the first collision. The basic principles will be explained concentrating on pure states. In the following sections we will generalize the discussion. Firstly, we will consider the case where the electrons to be teleported are in a mixed state but still assuming the initially scattered particles to form the maximally entangled singlet state. Secondly, we will account for the general case in terms of density matrix where also the entangled particles can be in arbitrary states. In the last section we will introduce and adapt the concept of fidelity as a measure of success for teleportation and compare our predictions with pre-existing experimental sodium and lithium data. It will be shown that lessentangled mixed states can also be used as a resource for PBFT with fidelity beyond the boundary set by classical physics. For illustrative purposes, we will also provide videos of the numerical and experimental data. Finally, in the appendix we will present detailed derivations of the general formulas.

PBFT of a definite spin state
Quantum teleportation is a three-particle process where the quantum information of particle 1 is transferred onto particle 3 with the assistance of particle 2 in the absence of any direct quantum communication channels between them. In a first step two observers, ALICE and BOB, perform a specific elastic scattering process between quasi one-electron atoms (Na, Li) (particles 2) and electrons (particles 3), both initially unpolarized. Dealing with light atoms, all explicit spin-dependent interactions can be neglected in such an approach and, besides the Coulomb potential, electron-exchange is taken into account, only [23]. After the first scattering S I , with energy E 1 and scattering angle θ 1 (see figure 1), the final spin state of particles 2 and 3 is generally described by a spin-density matrix ρ 23 =T ρ in T + where T denotes the transition operator and ρ in describes the twoparticle density matrixof the uncorrelated initial state. With the help of Clebsch-Gordan coefficients the T matrix elements can be coupled to their total spin S and magnetic component M s which are both conserved during the collision. The T matrix does not depend on the magnetic components and hence, we are able to express the density matrix ρ 23 in terms of scattering amplitudes f (S) , (S=0, 1). The spin-density matrix ρ 23 can be completely characterized in terms of two individually measured polarization vectors, referring to particles 2 and 3, and the nine direct product components s s á´ñ ( ) 3 of the spin-spin correlation tensor (i, j=x, y, z). As shown in our previous research [15,16], the polarization vectors of both particles remain zero under such conditions, as well as all non-diagonal components of the spin-spin correlation tensor, and the only nonvanishing parameters are the spin-spin correlation parameters Required combinations of energy and angle can be read from the published data [15,16,22]. Choosing appropriate values of E 1 and θ 1 for ensuring f (1) =0 in the first scattering, the Pauli-principle requires particles 2 and 3 to be found in the maximally entangled singlet state where ñ | and ñ | denote spin up and down, respectively, with respect to the laboratory z-axis. Equation (2) describes the spin part of the total wave function of the two-electron system after the first elastic collision S I , where particles 2 refer to the bound valence shell s-electrons of the quasi one-electron atoms (Na, Li) while particles 3 describe a free electron beam in the sense of scattering theory; e.g.see [25]. As we are dealing with light atoms, spin-orbit coupling must not be accounted for. For a detailed proof and discussion we refer to the book by Kessler [23, particularly see ch4]. Thus, one obtains information on the joint spin state of the pair but not on the individual spins. The described experiment must therefore be interpreted as a simple case of a Bell measurement. It is important to note that choosing spatial observables (θ 1 , E 1 ) only, one can read off the spinspin correlation parameter without performing a spin measurement. Since all explicitly spin-dependent interactions, and in particular spin-orbit coupling, can be neglected [15], the connection between the spatial and spin variables is entirely due to the Pauli-principle. Changing energy E 1 and/or scattering angle θ 1 changes the symmetry of the spatial wave function and the spin part must adapt accordingly leaving the total wave function antisymmetric. It is because of the central role of the Pauli-principle why we call the experiment described below as PBFT. Figure 1. Scheme of PBFT with massive free particles. ALICE wants to teleport a certain spin state yñ | 1 of her independently prepared electron e 1 , onto BOBʼs initially unpolarized electron e 3 . The degree of spin polarization ( ) P 1 of e 1 is unknown to both ALICE and BOB. ALICE also has the use of atom A 2 at her disposal, which is part of the entangled atom-electron (A 2 +e 3 ) state ñ |00 23 prepared at an earlier stage in the first elastic electron-exchange scattering, governed by the unitary S-matrix S I , together with BOBʼs electron e 3 . Subsequently, ALICE performs a second elastic electron-exchange scattering (S II ) between her particles e 1 and A 2 . Thereby, the second scattering, driven by the unitary S-matrix S II , generates interference between ALICE electron e 1 and atom A 2 which allows for teleporting the degree of spin polarization ( ) P 1 of e 1 onto BOBʼs electron e 3 (spin polarization ( ) P 3 ) which therefore emerges with the original spin state y y ñ = ñ | | ; 3 1 see (14) and main text. The coincidence measurement between electrons e 1 and e 3 ensures that only those pairs out of the same sub-ensemble are counted. Eventually, to give proof of PBFT, the overlap between initial and final state, commonly known as fidelity F, must be measured; see (16) and subsequent main text. We stress the point that, besides performing PBFT via the path   e A e electron atom electron 1 2 3 , we can, due to the underlying high symmetry in spin space, simply interchange the phrases electron and atom in the figure in order to undertake the teleportation sequence   A e A atom electron atom   where the last row in (9) has been rotated back and refer to the laboratory z-axis again. The definite quantum state (3) of ALICE has been teleported onto BOBʼs state, that is y y ñ = ñ | | 3 1 , and hence, PBFT succeeds. By choosing appropriate pairs of scattering angle and energy in order to ensure f (1) (θ 2 , E 2 )=0 one can perfectly separate the ñ |00 12 -part from all other Bell states in (7). Thus, the described process represents a single-outcome Bell measurement.
Initially, the electrons 1 have been in the definite (but unknown) state yñ | 1 , independently prepared in another lab in advance. After the first scattering S I , particles 2 and 3 were prepared maximally entangledin the singlet state and had no definite spin state of their own. After the second scattering S II , particles 1 and 2 become maximally entangled. ALICEʼs electrons 1 have therefore completely lost their initial spin properties in accordance with the no-cloning theorem [3,26] and, their initial state has been completely transferred to particles 3. BOB, receiving the classical information that ALICE obtained a count in her detector 1, while his detector 3 registers coincidently, knows that his electron is in the state yñ | 3 . The coincidence measurement between ALICEʼs electrons e 1 and BOBʼs e 3 ensures that only those pairs out of the same sub-ensemble are counted. A scheme of the full PBFT is shown in figure 1. As can be seen from (9), based on the Pauli-principle which overall governs the PBFT, and due to its high symmetry in spin space, no restoration of BOBʼs state yñ | 3 has become necessary. Particles 1 and 3 have never interacted directly during PBFT. It is important to note, that no spin measurement is and must be performed during PBFT. Any spin analysis at this stage disrupts the entanglement and PBFT cannot succeed. All that is known is that BOBʼs sub-ensemble of electrons, selected by the coincidence measurement, is in the same states as particles 1. BOB could use his beam for further applications, e.g.as input in a second PBFT or other teleportation experiment. In addition, as a side effect, the initial entanglement has been completely transferred from the pair 2-3 to the pair 1-2. This is known as entanglement-swapping [27,28]. In this respect the elastic electron-exchange collision can be considered as a unitary quantum gate.-Adapting the quantum marginal problem [29][30][31] to such a three-particle Fermionic system might yield a generalized scheme of PBFT. However, this query is beyond the scope of the present paper.-

Interchanging atoms and electrons
The discussed PBFT is highly symmetric. Particularly, the high symmetry in spin space enables us to simply interchange the words electrons and atoms in figure 1. Thereby, instead of teleporting the electron spin, we are able to teleport an unknown spin state of ALICEʼs atom A 1 onto BOBʼs atom A 3 employing the entanglement of ALICEʼs electron e 2 and a classical channel (coincidence measurement). For convenience, the atom  A 1 electron  e 2 atom A 3 PBFT is illustrated in figure 3. Both schemes have certain advantages in terms of fidelity measurement (spin analysis) or separation of the elastically scattered sub-ensembles of electrons or atoms after the first scattering process.

PBFT for mixed spin states
It is interesting that the derived results still hold if ALICEʼs electrons 1 are initially not in a definite but in an arbitrary spin state described by a density matrix ρ 1 . In diagonal form we have where a | | 2 and b | | 2 are the respective probabilities. Since the overlap of the two terms in (10) is incoherent we can discuss them separately and add the results. As before, we assume particles 2 and 3 being in a singlet state, and scatteringangle and energy in the second scattering, again, are chosen such that the triplet amplitude f (1) vanishes. Using (8) we obtain 3 . ALICE wants to teleport a certain spin state yñ | 1 of her independently prepared atom A 1 , onto BOBʼs initially unpolarized atom A 3 . The degree of spin polarization ( ) P 1 of A 1 is unknown to both ALICE and BOB. ALICE also has the electron e 2 at her disposal, which is part of the entangled electron-atom (e 2 +A 3 ) state ñ |00 23 prepared at an earlier stage in the first elastic electron-exchange scattering, governed by the unitary S-matrix S I , together with BOBʼs atom A 3 . Subsequently, ALICE performs a second elastic electron-exchange scattering (S II ) between her particles, A 1 and e 2 . Thereby, the second scattering, driven by the unitary S-matrix S II , generates interference between ALICEʼs atom A 1 and electron e 2 which allows for teleporting the degree of spin polarization ( ) (14) and main text. The coincidence measurement between atoms A 1 and A 3 ensures that only those pairs out of the same sub-ensemble are counted. Eventually, to give proof of PBFT, the overlap between initial and final states (fidelity F) must be measured; see (16)  where the spin density matrix ρ 3 of BOBʼs selected electrons is given by the expression a  a  3  2  3  3  2  3  3 As a result, PBFT is faithful independently of the initial state of ALICEʼs electrons, provided the triplet amplitude vanishes in both scattering processes.
PBFT for the general case We will now consider the general case where the correlation parameters P 12 and P 23 of the first and second elastic scattering processes can have arbitrary values within their allowed range -[ ] 1, 1 3 . The derivation is straightforward but tedious (see appendix for calculational details) and the final expression for the reduced density matrix of BOBʼs selected particles 3 is given by This results in the (unknown) polarization vector ( ) P 3 being always parallel to ( ) P 1 for P 12 P 23 >0 and antiparallel for P 12 P 23 <0. Comparing (9) and (14) it follows that the efficiency of the polarization teleportation is reduced by the factor P 12 P 23 .

Success of PBFT and fidelity
In order to get some idea of how faithful PBFT might be, we assume ALICEʼs original electron beam 1, which is to be teleported, as being in the definite (but unknown) state  ñ | a . In this case (13) reduces to (see appendix) where we have again used the a-representation. The fidelity F, defined by the expression , is a common and frequently used measure of success of quantum teleportation and hence, of PBFT. It yields the probability that a subset of BOBʼs selected electrons ρ 3 would pass a test for being in the state y ñ | 1 . From (15) we obtain For the general case with < | | ( ) P 1 1 , we refer to (A.37) and the related section in the appendix. Hence, the fidelity F is directly related to the product P 12 P 23 and is shown for Na in figure 4 as a 3D-plot versus the scattering angles θ 1 and θ 2 of the first and second scattering, respectively (also see supplementary video1, available online at stacks.iop.org/ NJP/21/033025/mmedia). The scattering energies are fixed in both collisions to E 1 = E 2 = 10 eV. Here, we used and re-interpreted existing experimental [18] and numerical [20] data (see figure A1 (in the appendix) and supplementary video 3 for the latter, only). The Li experimental [19] and numerical data [21] are presented in figure 5 in the same manner plotted versus the scattering energies E 1 and E 2 of the first and second scattering, respectively (also see supplementary video 2). The scattering angles of both collision processes have been fixed to θ 1 = θ 2 = 107.5°(see figure A2 (in the appendix) and supplementary video 4 for the numerical data, only). It is amazing that (13) and (14) apply to the full range of allowed values for the correlation parameter products, well outside the Bell region and even outside the entangled region; see figures 4 and 5 and figures A1 and A2 in the appendix (also see supplementary videos 1-4). Bennett et al [2] and Popescu [32] have already noted that teleportation could still be possible outside the Bell region but only with reduced efficiency. For the process considered here even separable mixtures can contribute but with even smaller efficiency.
If both scattering processes are pure singlet collisions, that is = = -P P 1 12 23 , we have F=1 indicating a faithful teleportationfrom electrons 1 to electrons 3. This is achieved in both cases, Li and Na, at the maxima of the highest (blue) peaks; see figures 4 and 5 and supplementary videos1 and 3. However, the overall structure of the two cases is different. In the sodium case there are single isolated peaks, only one of them showing large values of P 12 P 23 and, hence, F up to one. For lithium there is a broad (blue-coloured) plateau with values of F>0.9 having peaks atop where F=1 is clearly reached. F decreases steadily with decreasing values of P 12 and P 23 (also see the Li and Na supplementary videos). Now, suppose that ALICE and BOB share no entangled state after the first scattering S I but a separable mixture. Its density matrix ρ 23 can always be written as a combination of direct product states. Physically, they are equivalent to mixtures, generated classically by ALICE and BOB agreeing, e.g. over the phone, on the local preparation of their respective states [16]. The electron pairs are still correlated but all correlations are of pure classical nature [15,16] and therefore useless for quantum teleportation [32]. In the sodium case, this occurs for most of the angle combinations (θ 1 , θ 2 ) of first and second scattering, respectively, as can be seen from the broad yellow coloured plane around F;1/2 displayed in figure 4 (also see figure A1 and supplementary videos1  and 3).
The strongest correlation, obtainable for separable states, is = | | P 1 3 23 , which yields fidelity F=2/3, (with P 12 =−1, and P 23 =−1/3 in our scheme), shown as a horizontal plane in figures 4 and 5 (also in figures A1 and A2 and in the supplementary videos1-4), representing the upper bound for classical transfer of information [32,33]. In general, one says that any state is useful for teleportation if F>2/3. In our scheme, this requires P 12 P 23 >1/3. Assuming P 12 =−1 it follows that any entangled state ρ 23 , with P 23 <−1/3, is useful for  [18] plotted on top the re-interpreted numerical data [20], both adapted to our present case of interest as described in [15]. See [15,16,18] for details and information on the error bars. (Numerical data only are shown in figure A1; also see supplementary videos1 and 3.)  [19] plotted on top the re-interpreted numerical data [21], both adapted to our present case of interest as described in [15]. See [15,16,19] for details and information on the error bars. (Numerical data only are shown in figure A2; also see supplementary videos2 and 4.) teleportation in the sense of outperforming classical strategies; see figures 4 and 5 (also see figures A1 and A2, and supplementary videos1-4). In particular, this result holds for states ρ 23 with spin correlation parameters -< < -P 1 2 1 3 23 . Though entangled, these states do not violate the CHSH-Bell inequality [24]. Notwithstanding, they provide results F>2/3 for the PBFT with fidelity up to F=0.854 and hence, better than could ever be obtained by classical correlations and local operations.

Acknowledgments
The authors are grateful to ProfsDrsM Drescher and G Alber, to PDs DrsA Dorn and R Moshammer and to DrsB Langer and A Azima for helpful discussions and remarks during the preparation of the manuscript. We are most thankful to DrB Langer for helpful assistance in the preparation of the figures and for preparing the videos. BLis thankful to ProfDrM Drescher and The Hamburg Centre for Ultrafast Imaging (CUI) of the University of Hamburg for inviting him as Guest Scientist and for the hospitality extended to him during his stay. Generous financial support by the CUI which enabled for the finalization of the manuscript is gratefully acknowledged. The authors are thankful to DrD Thompson for proof-reading the manuscript. We acknowledge support from the Open Access Publication Fund of the Westfälische Wilhelms-Universität Münster.

Appendix: Numerical fidelity of PBFT for Na and Li
For convenience we also included figures and supplementary videos of the numerical fidelity, only. For the sodium case in figure A1 (supplementary video3) and for lithium in figure A2 (supplementary video4), respectively.

PBFT for the general case
We will now consider the general case in detail where the spin-spin correlation parameters P 23 =P 23 (θ 1 , E 1 ) and P 12 =P 12 (θ 2 , E 2 ) of the first and second scattering can take on any value inside the allowed range [−1, 1/3] depending on scattering angle and energy.
In the first collision unpolarized atoms and unpolarized electrons are scattered. As we have shown [15,16], the final state after the first scattering is given by the density matrix where particles 2 (ALICEʼs atom) and 3 (BOBʼs electron) are written in first and second position, respectively. ALICEʼs electrons (particles 1) are assumed to be in a definite spin state where the amplitudes are unknown to both, ALICE and BOB. Here, ñ | and ñ | describes states spin up and spin down, respectively, with respect to the laboratory z-axis which may even be unknown to ALICE. The state (A.2) describes a beam completely polarized with respect to a certain direction, say a. (That is, all electrons in yñ | 1 have spin up with respect to a.), and we can write A . 4 a 1 1 1 In the following, it is convenient to use a as quantization axis. It is essential in this respect that the matrix ρ 23 remains invariant under rotations in spin space, that is, ρ 23 keeps its form in any rectangular system, the laboratory system x, y, z as in (A.1) or in the a-system with axes ¢ ¢¢ a a a , , . Therefore, we can write the initial three-particle system in the a-representation, simply by substituting ñ   ñ | | a and ñ   ñ | | a in ρ 23 for particles 2 and 3, respectively. Thus, we explicitly write the initial three particle state before the second scattering as  12  12  3  3  3  3   12  12  3  3  3  3   12  12  3  3  12  12  3  3 where ⊗denotes the direct product. Here and in the following, we abbreviate the correlation parameter of the first scattering as P=P 23 if not causing ambiguities.
We will now consider the electron-exchange scattering between ALICEʼs electrons (particles 1) and the quasi one-electron atoms (particles 2) described by the scattering matrix S II . The total spin S and its component M s in the direction of the quantization axis a remain constant, and the scattering amplitudes are independent of M s . The collision between particles in the states  ñ  ñ | | a a 1 2 and  ñ  ñ | | a a 1 2 are therefore pure triplet transitions,  . Horizontal plane divides classical (F<2/3 , no teleportation) from the quantum mechanical (F>2/3) correlated regions of PBFT. Distance of lines of altitude is ΔF=0.025. Re-interpreted numerical data [21] adapted to our present case of interest as described in [15]. Figure A1. Numerical fidelity F=F(θ 1 , E 1 , θ 2 , E 2 ) of PBFT with Na as 3D-plot versus electron scattering angles q 1 and θ 2 of the first (S I ) and second (S II ) scattering process, respectively. The scattering energies are fixed at = = E E 10 eV 1 2 . Horizontal plane divides classical (F<2/3 , no teleportation) from the quantum mechanical (F>2/3) correlated regions of PBFT. Re-interpreted numerical data [20] adapted to our present case of interest as described in [15]. a a S a a a a 1 2 The term ( f−g) is known as the interference amplitude. Note, that the minus sign is implicitly taking care of the Pauli-principle. The amplitudes f and g are related to the triplet and singlet scattering amplitudes f (1) and f (0) by the relations, e.g. [23,34,35],  1 2  12  12  3  3  3  3   2  12  12  2  12  12   3  3  3  3   12  12  12  12   3  3  3  3  1  12  12  1  12  12  3  3   1  12  12  1  12  12 3 3

Normalization and differential cross section
Building the trace, we sum over the diagonal elements of (A.10) which results in Applying the identity (9), we remain with where we introduce the differential cross section σ 12 of the second scattering process performed by ALICE alone. Thus, we write the normalized density matrix as  Thus, we have to sum over the elements of (A.10), diagonal in M 1 and M 2 (particles 1 and 2 owned by ALICE) and obtain the normalized reduced density matrix Expressing the direct and exchange amplitudes in terms of the singlet and triplet ones by using (A.9) and applying (A.12), we obtain of their respective states. Hence, the spin correlations are classical, that is they can be considered as prearranged by ALICE and BOB [15,16]. Any direct product (entangled) state is useless for teleportation as for such states manipulation of particle 2 during the second scattering has no effect on what can be predicted on particle 3 . Whereas particles 1 and 2 become entangled, particles 3 are unaffected and keep their state obtained during the first collision. One can still find a certain number of BOBʼs particles in state  ñ | a 3 but this is a classical effect based on classical statistics. The best result for these classical mixtures is obtained for P 23 =−1/3 and P 12 =−1 giving fidelity F=2/3 [32].
Finally, let us consider the spin polarization. Since ρ 3 is diagonal in the a-representation, the spin polarization vector ( ) P 3 of BOBʼs selected electrons is directed along a. Its magnitude is given by the difference of the two diagonal elements of (A.20), that is Since the spin polarization vector of ALICEʼs initial electron beam is also oriented along a, we can write in vector notation which gives (14) in the main text. The spin polarization vector is necessarily parallel or anti-parallel to ( ) P 1 depending on the sign of the product P 12 P 23 . For P 12 P 23 =1, we have = ( ) ( ) P P 3 1 indicating a faithful quantum teleportation. P 12 P 23 can also be used as a measure of success.
We will now transform back to the xyz-coordinate frame (laboratory frame). Using (A.4) where  ñ | a and  ñ | a denote the spin states with respect to direction a, we write BOBʼs density matrix in the form 12 23 12 23 Substituting the relations (24) into (A.25) and expressing the results in explicit matrix form, we obtain BOBʼs density matrix r 3 in the xyz-system where σ i denote the relevant Pauli matrices. We obtain A . 2 8  The result (A.31) shows that the spin polarization vectors of particles 3 (BOBʼs electrons) and that of particles 1 (ALICEʼs electrons) are always parallel or anti-parallel depending on the sign of P 12 P 23 . In the case where the scattering angles and energies of both collisions, S I and S II , are chosen in such a way that P 12 =P 23  Thus, we obtain the fidelity as a function of the two correlation parameters P 12 and P 23 , driving the first and second scattering processes, and the magnitude | |  12 23 For maximally entanglement in both scattering processes, that is P 12 =P 23 =−1, we get F=1 which yields a faithful teleportation. In the case where ALICEʼs electrons are originally in a mixed state, we have < | |