Chiral Majorana interference as a source of quantum entanglement

PHYSICAL REVIEW B 97, 155416 (2018) Chiral Majorana interference as a source of quantum entanglement Luca Chirolli,1,*José Pablo Baltanás,2and Diego Frustaglia2 1Fundación IMDEA Nanoscience, E-28049 Cantoblanco Madrid, Spain 2Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain (Received 18 January 2018; published 17 April 2018) Two-particleHanburyBrown–TwissinterferometrywithchiralMajoranamodesproduces maximallyentangled electron-hole pairs. We promote the electron-hole quantum number to an interferometric degree of freedom and complete the set of linear tools for single- and two-particle interferometry by introducing a key phase gate that, combined with a Mach-Zehnder, allows full electron-hole rotations. By considering entanglement witnesses built on current cross-correlation measurements, we find that the possibility of independent local-channel rotations in the electron-hole subspace leads to a significant boost of the entanglement detection power. DOI: 10.1103/PhysRevB.97.155416 I. INTRODUCTION Entanglement is at the core of quantum theory and representsa keyresourcefor quantuminformation andcomputation. Generation,manipulation,anddetectionof entangledelectrons is at the basis of quantum computing with integrated solid-state devices. A great amount of attention has been devoted to entanglement generation in multiterminal mesoscopic conductors, with most noticeable schemes relying on Cooper pair emission from superconducting contacts [1,2], correlated electron-hole (e-h) entangled states by tunnel barriers [3], and integrated single-particle emitters [4] (see Refs. [5,6]forareview). Detection of entangled fermions in the context of quantum transport was first proposed as a particular consequence of antibunching in current cross-correlation measurements for a subclass of states (spin-entangled particles propagating along different channels) by using a beam splitter (BS) analyzer [7–9]. This was later generalized to the case of multiple mode and occupancy entanglement [10–12], where current cross correlations can provide entanglement witnesses [13,14]. Here,we suggesttheuseofchiral Majoranamodes (χMMs) as a tool for the generation, manipulation, and detection of entanglement in the e-h and channel degree of freedom (DoF) in multiterminal platforms. This is done by integrating setups proposed in the literature, such as Mach-Zehnder (MZ) [15,16] and Hanbury Brown–Twiss (HBT) [17,18] interferometers, together with a phase gate. The latter permits the implementation ofanenergy-independentphaseshiftbetweenelectronandhole states. Most importantly, the combined action of a MZ and a phase gate allows for arbitrary rotations in the e-h DoF and to perform local operations in each propagating channel. TheadvantageofusingchiralMajoranachannelsistwofold. On the one hand, two-particle interferometry in topological superconductors (TSCs) hosting χMMs at their boundary [19] produces superpositions of maximally entangled states. In particular, as pointed out in Ref. [17], postselecting states with one fermion per lead yields maximally entangled pairs in the *[email protected] electron-hole space. This guarantees the production of exactly maximally entangled states. On the other hand, the possibility to arbitrarily rotate the state in the e-h DoF independently in each channel allows for measurements of the e-h state in any basis, boosting the power of the entanglement witness far beyond the limits of the proposals with ordinary particle currents. Our approach makes it possible to exploit single- and two-particle interferometry in the e-h DoF as a platform for quantum computation in dual-rail architectures [20,21]. Our proposal can be easily generalized to ordinary electrons and holes in hybrid normal/superconductor heterostructures. Interferometry in the e-h DoF is in principle possible with chiral channels in contact with a superconductor [22–25], where e-h superpositions copropagate in the same channel, in analogy with copropagating spin-resolved edge states in the integerquantumHalleffect(IQHE)[26–29].Recentexperiments with chiral one-dimensional (1D) channels in contact with s-wave superconductors opened the way to exploring Andreev reflection on 1D chiral channels [23,25]. At the same time, no proposal for controlled BS, MZ, or HBT e-h interferometers is currently available with ordinary superconductors. The paper is structured as follows: in Sec. II we describe the system under study, characterized by Dirac and Majorana edge channels; in Sec. III we describe the interferometric setup, by reviewing interferometric elements proposed in the literature and by introducing a fundamental phase gate; in Sec. IV we assess the presence of entanglement in the output channels by studying the current cross correlations and their relation to entanglement witnesses, showing how the possibility to perform independent local-channel rotations in the e-h subspace greatly enhances the entanglement detection power. In Sec. Vwe conclude the work with a summary of the results. II. THE SYSTEM The main ingredients are chiral Dirac modes (χDMs) and χMMs in quantum anomalous Hall insulator/SC structures, as those proposed in Refs. [15,16]. The recent experimental detection of χMMs in these systems [30,31] makes the present 2469-9950/2018/97(15)/155416(5) 155416-1 ©2018 American Physical Society CHIROLLI, BALTANÁS, AND FRUSTAGLIA PHYSICAL REVIEW B 97, 155416 (2018) proposal feasible and particularly appealing. The system consists in the two-dimensional (2D) surface of a topological insulator (TI) on top of a substrate divided in ferromagnetic (FM) and SC regions. The TI surface hosts a single 2D fermionic Dirac cone described by H0=−iv(∇×s)z, with sa vector of spin Pauli matrices. A FM domain wall acts as Hfm =M(r)sz and gaps the system everywhere apart from a line where the domain wall changes sign. Along this line a 1D χDM forms, analogous to the edge states of the IQHE, and can be used as an electronic waveguide. Similarly, SC proximity induces singlet pairing described by HSC =kc† k,↑c† −k,↓+H.c that opens a topological gap, thus realizing a 2D TSC. Gapless χMMs form along the border between the SC and the FM regions. By properly arranging these regions it is possible to realize an interferometric setup composed by several linear elements. We now characterizethetransportinterms ofscattering matricesin the Landauer-Büttiker formalism adapted to describe χMMs [32]. III. INTERFEROMETRIC SETUP The setup is illustrated in Fig. 1. We follow the notation of Ref.[17]anddenoteχDMswithdoublearrowlinesandχMMs with single arrow lines. Given that χDMs have sy=−1, we can regard the fundamental excitations as spinless Dirac fermionsdescribedbyfermionicoperatorsa()atenergy.For energies 0 <we define electron- and holelike states in channel iat energy as ai,+()=ai() and ai,−()=a† i(−), and introduce an e-h DoF τ=±1=e,h. Analogously, at the boundary of the TSC a single χMM flows, either clockwise γ1() at the boundary between the TSC and the M↑magnetic domain, or anticlockwise γ2() at the boundary between the TSC and the M↓magnetic domain. Current is injected in the system upon biasing contacts 1 and 2 and it is collected in contacts 3 and 4, that are kept grounded. The resulting current and noise in contacts 3 and SC 1 3 4 eV 2 QPC 4-terminal Mach Zehnder M↑ M↓ M↑ ϕ(Vg) gate voltage eV SC1 SC2 FIG. 1. Setup: a domain wall on top of a TI 2D surface state generates χDMs (double arrows) and an s-wave SC opens a topological gap, at whose boundary with the domain wall χMMs form (single arrow). Carriers are injected into the system by biasing contacts 1 and 2 and go first through the four-terminal device, which acts as a two-particle interferometer. The outgoing carriers then undergo local operations through phase shifts and MZs and collide into a QPC that mixesthechannels.Currentsandcorrelationsaremeasuredincontacts 3 and 4. The MZs and phase shifts can be also placed after the QPC. 4 can be obtained in terms of unitary scattering matrices Srelating incoming Dirac modes aj,σ to outgoing Dirac modes bi,τ , bi,τ ()= j=1,2;σ=± Si,τ;j,σ ()aj,σ ().(1) Particle-hole symmetry implies S()=τxS∗(−)τx. The setup in Fig. 1is characterized by four kinds of linear elements: (i) a BS, (ii) a MZ interferometer, (iii) a phase shifter, and (iv) a four-terminal element. Additionally, an ordinary quantum point contact (QPC) is introduced to mix χDMs, in analogy with interferometry in the IQHE edge states [33]. Some of these elements have been suggested in the literature, so that we here review them in the spirit of two-particle interferometry. A. Beam splitter In this context, we denote a BS as an element that takes an incoming χDM and produces two outgoing χMMs. The trijunction between a magnetic domain wall and a TSC forces the incoming electron a+and hole a−states to split into two χMMs, γ1and γ2[15,16]. For energies much smaller than the SC gap, we can assume the BS scattering matrix to be energy independent, reading [15,16] γ1() γ2()=1 √211 i−ia+() a−().(2) B. Mach-Zehnder By combing two BSs separated by a region where the two χMMs propagate along two paths of different length one can realize a MZ interferometer for electrons and holes described by [15,16,34] a+ a−R=S† BSeiπnv+ikL10 0eikL2SBSa+ a−L ,(3) where k(L1−L2)=δL/vMis the phase difference gathered at energy and nvis the number of vortices in the SC, with vM the velocity of the χMMs. The scattering matrix thus mixes incoming chiral electron and hole states in the left (L) lead to outgoing chiral electron and hole states in the right (R) lead. At finite energy, by varying the path length difference δL it is possible to perform arbitrary rotations around the τxaxis between incoming and outgoing states in a given channel. At zero energy this element is expressed as a τxscattering matrix in the e-h space, that represents a Z2MZ interferometer being able only to change an electron into a hole, and vice versa. C. Phase shifter A fundamental ingredient appearing in the setup of Fig. 1 is the phase shifter between electrons and holes in a given Dirac channel. This can be easily accomplished by a top gate that locally shifts the chemical potential. For a gate voltage Vg such that |Vg|M, where Mis the magnitude of the Zeeman splitting of the magnetic domains, electrons and holes will in general acquire an opposite phase, a+()→eiϕga+(),a −()→e−iϕga−(),(4) 155416-2 CHIRAL MAJORANA INTERFERENCE AS A SOURCE OF … PHYSICAL REVIEW B 97, 155416 (2018) with ϕg=(e/v)L 0dx Vg(x). Importantly, this phase is energy independent so that also carriers at =0 acquire it. The scattering matrix associated to the phase shift realizes a phase gate P=exp(iϕgτz).(5) This element is of paramount importance in that, combined with the MZ, it provides a way to rotate the states in the e-h space and generate any superposition state. Moreover, a channel-dependent shift can be obtained by modifying the path length of a given channel. This can be achieved by moving the domain wall through a magnetic field. D. Four-terminal element Finally, the core of the setup in Fig. 1is a four-terminal device that mixes two incoming χDMs into two outgoing χDMs. In terms of electron and hole channels, the element mixes four incoming states into four outgoing states. This four-terminal element was introduced in Refs. [15,17] and it is described by the scattering matrix ⎛ ⎜ ⎜ ⎜ ⎝ b1+ b1− b2+ b2− ⎞ ⎟ ⎟ ⎟ ⎠=1 2 ⎛ ⎜ ⎜ ⎜ ⎝ 11 1 1 11−1−1 1−1−ηη 1−1η−η ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ a1+ a1− a2+ a2− ⎞ ⎟ ⎟ ⎟ ⎠ ,(6) with η=(−1)nveiδL/¯hvMa phase due to the propagation of the χMMs in the interferometer (here δL is the path length difference between the two-particle trajectories [15,17]). At zero energy the phase can be only η=±1, depending on the number of vortices nvin the system. A more generic fourterminal element can be obtained by allowing for a cross talk between the χMMs in the SC region. IV. CROSS CORRELATIONS AS ENTANGLEMENT WITNESSES The general idea developed in Refs. [7–11] is that, given an unknown initial state that is possibly entangled in the e-h and channel DoF, it is possible to establish the presence of entanglement via measuring current cross correlations after mixing the channels through a QPC and relate the cross correlator to an entanglement witness. We now show that witnessing entanglement in an e-h system is not only possible, but also much more effective, thanks to the possibility to insert MZs before and after the QPC. The most general unknown two-particle state at the input of the QPC can be cast in the generic form |=sin θ(cos φ|11+sin φ|22)+cos θ|12,(7) with θ,φ ∈[0,π/2] and |ij two-particle states at energy  in leads iand j,|ij =α,β (ij ) α,β a† i,α(E)a† j,β(E)|0, where |0is the grounded Fermi sea. The states satisfy (jj) α,β = −(jj) β,α and the normalization conditions α,β |(12) α,β |2=1 and α,β |(jj) α,β |2=1/2. The state described by Eq. (7)displaysentanglementintheoccupation number andchannel DoF [35,36]. Before the QPC, we induce a phase difference between the electron and hole in each channel by local gate voltages and domain-walldisplacement. Forthe moment wedonot consider the MZs that are present in the setup of Fig. 1. The QPC mixes the channel of the incoming particles without changing the electron/hole character of the particle injected. The outgoing states after the combined system phase shifter plus QPC is given by bj,τ =j=1,2Sτ j,jaj,τ , where bj,τ are the outgoing states after the QPC in lead jwith electron/hole character τ, where the scattering matrix is given by Sτ=rt eiϕτ tr eiϕτ.(8) The phase ϕτaccumulated before the QPC has two contributions: the gate contribution ϕg, which is opposite for electrons andholes, andadynamical phasedifferencedue tothe different path length between the four-terminal scattering region and the QPCalongthetwopossiblepaths.Thiscontributionisthesame for electrons and holes, so that we can write ϕτ=ϕL+τϕg, with ϕL=δL/v and vthe velocity of Dirac modes. We now consider the dimensionless current cross correlator between the output channels 3 and 4, C34 ≡h2ν2 2e2lim T→∞T 0 dt1dt2 T2I3(t1)I4(t2),(9) where Tis the measurement time. The current operator of chiral fermions in lead iis written in terms of electron and hole contributions as Ii(t)=e hν  ,ω,τ=± e−iωtτb† i,τ ()bi,τ (+¯hω),(10) and the average ··· is taken over the incoming state by assuming a discrete spectrum characterized by a density of states νin each lead. Importantly, electrons and holes contribute with different sign to the current, which is accounted for by the τin Eq. (10). In each lead there are incoming and outgoingstates.However,duetothechiralityofstateslocalized at the domain-wall boundary, there is no backscattering and the current can be described only in terms of outgoing channels. ThequantityC34 hastheadvantageofbeingalinearfunction of the input state [11]. Assuming a QPC characterized by r=r=√1−Tand t=t=i√T, with Tthe transmission probability of the QPC we find C34()=T(1 −T)[wcos2θ−sin2θ+vsin2θsin(2φ)] +1 2cos2θ τσ τσ(12) τ,σ  2,(11) where vand ware real quantities satisfying |v|,|w|⩽1 that can be expressed in terms of the phases ϕτas v=2Re  τ,σ (11) τ,σ ∗(22) σ,τ eiϕτ+iϕσ,(12) w= τ,σ στ(12) τ,σ ∗(12) σ,τ ei(ϕτ−ϕσ).(13) The correlator C34 is very similar to that of Ref. [11]. Antisymmetry of the |jj states implies that ϕgdrops from v. Analogously, the phase ϕLdrops from w. By further redefining ϕg→π/2+ϕgwehave thatw=τ,σ ((12) τ,σ )∗(12) σ,τ eiϕg(τ−σ). 155416-3 CHIROLLI, BALTANÁS, AND FRUSTAGLIA PHYSICAL REVIEW B 97, 155416 (2018) As a particular case we consider incoming states with singly occupied channels 1 and 2 by choosing θ=0 in the generic input state (7). The current correlator is found to be C34(12)=1 2 τσ τσ|(12) τ,σ |2+T(1 −T)w. (14) The quantity wcaptures all the relevant information on the input state. One can show that wis non-negative for separable input states [10]. The first term in Eq. (14) is nothing but the cross correlator before the QPC, C12(12)=1 2τσ τσ|(12) τσ |2.Experimentally one can act on the QPC and switch the tunneling between counterpropagating states on and off (by setting T=0orT=1) and measure separately C34(12) and C12(12). It then follows that the case C34(12)−C12(12)< 0 witnesses the presence of entanglement in the state |12. For θ= 0, i.e., incoming channels 1 and 2 with fluctuating local occupancy in (7), C34() can be related to the entanglement of formation Ef()[37]: generalized Werner states [38,39] are defined by introducing a joint orthonormal basis for ports 1 and 2 formed by states |χk1⊗|χk2and |(±) kk=(|χk1⊗|χk2±|χk1⊗|χk2)/√2 with k<k , where kenumerates all configurations with two or fewer particles per port. It then follows that the entanglement of formation of a state ρcan be lower bounded by the quantity W(ρ)=kk(−) kk|ρ|(−) kk/2.Analogously,wecan relatethe net correlator δC34()≡C34(T=1/2) −C34(T=0) (15) =C34()−C12(),(16) which depends only on the quantities v,w, and the angle θ,to a lower bound to the entanglement of formation through W()=−2δC34()+cos2(θ)/2.(17) By noticing that W()>−2δC34() we find that W()> 1/2 and, consequently, Ef()>0 whenever δC34()< −1/4 (see Refs. [10,11]). Thus, the sign of −2δC34()−1/2 is sufficient to witness the presence of entanglement in the initial state . We can further postprocess the data and obtain full information about the state. First of all, we use that v(ϕL+π/2) =−v(ϕL). This allows one to define δC(±) 34 = [δC34(ϕL+π/2) ±δC34(ϕL)]/2 such that δC(+) 34 (ϕg)=[w(ϕg) cos2θ−sin2θ]/4,(18) δC(−) 34 (ϕL)=v(ϕL)sin2θsin(2φ)/4.(19) We then notice that cos2θ=2C12()/(2 ¯w−1), where ¯w= dϕg 2πw(ϕg). Upon introducing δ¯ C(+) 34 =dϕg 2πδC(+) 34 (ϕg)=1 4[(1 +¯w) cos2θ−1],(20) we can express cos2θ=2 3(4δ¯ C(+) 34 −C12 +1),(21) allowingonetoestablishtheoccupationandchanneladmixture as a function of measurable quantities. Finally, we notice that the states |jj can only be e-h singlets, so that v depends only on the relative phase difference between 11 and 22. By varying ϕLone can then access sin(2φ)= 2(maxϕLδC− 34 −minϕLδC− 34)/(1 −cos2θ).The analysis allows one to fully access the occupation-number (e-h) and channel DoF entanglement by further exploiting the phase before the QPC[11].This resultcanbegeneralized togenericmixedinput states ρ, as the combination of the maps C12 and C34 preserves the linearity of δC34. A. HBT state Having established the general entanglement witnessing protocol via current cross-correlation measurements, we now apply it to the output state of the two-particle interferometer in the setup of Fig. 1. Upon biasing only the contacts 1 and 2 the incoming state at energy reads |in = a† 1+()a† 2+()|0and the outgoing state reads |out = Sj,μ;1,+Sk,ν;2,+b† j,μ()b† k,ν()|0(summed over repeated indexes), with Sthe scattering matrix in Eq. (6). In Ref. [17]the system was studied as a HBT two-particle interferometer and it was recognized that postselecting states with a single fermion per lead yields maximally entangled states. In this case the current cross correlations allows one to access v=−Re[ηe2iϕL] and w={[1 +cos(2ϕg)] +Re[η][1 −cos(2ϕg)]}/2, and the associated witness for T=1/2 and ηreal is W(HBT)=1 8[3 −η+2ηcos(2ϕL)−(1 −η) cos(2ϕg)]. (22) In particular, for η=−1, ϕL=π/2, and ϕg=π/2 one can reach W=1, which corresponds to Ef=1. This means that the phases are means to rotate the initial state to have maximum overlap with the generalized Werner states, confirming that the state coming out from the HBT interferometer is a superposition of maximally entangled states. B. Local operations The quantity C34 differs from that of Ref. [11]inthe measured observable: particle current in the original case in contrast to charge current in the present case. This grants a much more powerful characterization of the incoming states. By inserting MZs before the QPC we can rotate the e-h state on each channel and measure any linear combination of the three Pauli matrices τi, with i=1,2,3. This operation only affects the state |12(the states |jj are e-h singlets, so that any single-particle rotation can only affect the global phase of the state |jj ) and C12 measurements can assess every local single-particle observable. The insertion of MZs and phase shifters together with the possibility of switching the QPC on and off give us the opportunity to cross correlate the local operation and to perform a full tomography of the input state. V. CONCLUSIONS In this work we suggest to use interferometry with chiral Majorana modes to generate, manipulate, and detect quantum entanglement in the electron-hole and channel degrees of freedom in multiterminal hybrid normal-superconductor platforms. A fundamental two-particle Hanbury Brown–Twiss interferometer previously proposed in the literature allows one to generate maximally entangled pairs in the electron-hole degree of freedom. We integrate a set of single-particle elements, suchasbeamsplittersandMach-Zehnderinterferometers, with 155416-4 CHIRAL MAJORANA INTERFERENCE AS A SOURCE OF … PHYSICAL REVIEW B 97, 155416 (2018) a fundamental phase gate that allows one to generate a phase difference between electron and hole at zero energy and that, combined with the Z2Mach-Zehnder interferometer, allows full rotation in the e-h space. Entanglement in the output states is detected through the assessment of entanglement witnesses by means of current cross-correlation measurements, extended to the case of particle-hole current carrying states. The completion of the set of single-particle and two-particle linear elements allows one to perform any operation in the e-h degree of freedom and to fully manipulate quantum states encoded in this platform. The possibility to perform independent local-channel rotations in the e-h subspace greatly enhances the entanglement detection power and makes electron-hole systemsan idealplatform forquantum computationin dual-rail architectures. Note added. Recently, B. Lian et al. [40] independently studied chiral Majorana modes on a similar Corbino geometry. ACKNOWLEDGMENTS The authors are thankful to F. Taddei, V. Giovannetti, and C. W. J. Beenakker for very useful discussions. 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