Bell-like non-locality from Majorana end-states

Transcription

1 Bell-like non-locality from Majorana end-states Alessandro Romito with Yuval Gefen (WIS) , Daejeon, Workshop on Anderson Localiation in Topological Insulators

2 Outline Topological superconductors and Majorana end-states Quantum non-locality Quantum non-locality with Majorana end-states

3 Edge-states in superconductors BCS superconductor H = 1 2 Z dr dr 0 (r) H0 H 0 (r 0 )= X >0 (r) T =(c (r),c (r)) Δ is the paring potential For spinless fermions At ero energy Zero energy ecitations which are their own antiparticles: Majorana bound-states

4 Edge-states in superconductors Majorana bound states i = i { i, j} =2 i,j Spatially separated X 1 X 2 X 1 End-states of p-wave superconductors, topologically protected from local fluctuations Vortices in p+ip superconductors, 5/2 quantum Hall states [Moore & Read (1991),] Read &Green (2000), Kitaev (2001)... ]

5 Anyons Majorana bound states are non-abelian anyons U# U Braiding corresponds to a unitary evolution of the ground state i!û i ψ U#U# ψ ψ U#U# ψ γ ) Γ + Γ + γ - U# Is fied by the echange statistics Robust quantum information processing [Wilcek (1982), ] [Moore & Read (1991),] Read &Green (2000), Kitaev (2001)... ] [Nayak et al. (2008)]

6 The basics: Kitaev chain p-wave spinless superconductor in 1D in the limit gap d ) / ψ 1 ψ 1 d 1 / ψ 1 [A. Yu. Kitaev (2001)]

7 The basics: Kitaev chain p-wave spinless superconductor in 1D Observable signatures G V = 0 = 2ne > /h TP Braiding γ 2,4 γ 5,6 γ 5,6 γ 2,4 V gap d ) / ψ 1 ψ 1 d 1 / ψ 1

8 Searching for Majorana bound states Engineering topological phases in nanostructures [L. Fu and C.L. Kane (2008)] Semiconductor wires with spinorbit coupling eperiments [Y. Oreg et al. (2010), J. D. Sau et al. (2010)] V A V Eperiments in nanostructures InSb wires Delft (2012) InAs wires Weimann (2012) InAs wires Copenhagen (2014) Fe atomic chains Princeton (2014)

9 Non-local signatures Transport gap t = 0 Teleportation [L. Fu (2009)] Majorana bound states SC M C V QSH V Coulomb blockade is essential Vg edge states g L t L t R g R Correlated noise Zocher et al. (2014)] Left Lead L Nanowire s wave SC R Right Lead Finite-sie Majorana cross talk is essential I L V I R

10 Non-local signatures Can we detect genuine quantum nonlocal correlations, e.g. entanglement, in these systems?

11 Detecting quantum non-locality Non-local quantum correlations (entanglement) can be operatively identified via correlation measurements CHSH inequality a a 1/ 2( + ) b b C = ab ab H + a H b + a b Local (hidden variables) C 2 Quantum entangled states 2 < C 2 2 a =, a H =, b = 1 2 +, bh = 1 2 C = 2 2 measurements optimied for the given state [J. Bell (1964), J.F. Clauser et al. (1969) ]

12 Minimal model Total fermion parity is conserved NY ( i) N/2 j = 1 j=1 γ 4 γ > gap ψ 1 d 1 / ψ 1 a γ Y γ 4 iγ 4 iγ 4 i a γ Z iγ 4 6 Majoranas: Minimal viable setup dim (H) = 2 4 Majoranas: Insufficient algebra dim (H) = 1 2 Majoranas: 2 states with different parity No local Hilbert space [D. Clarke, Das Sarma (2015)]

13 Local operators γ 4 γ > γ Y γ Z 6 Majoranas: Minimal viable setup Spatially separable algebras of measurable operators L 4WX { iγ 4 = σ c, -i = σ d, i γ 4 = σ e } R Y>Z { iγ Y γ > = σ c, -iγ > γ Z = σ g, iγ Z γ Y = σ h }

14 Local operators γ 4 γ > y -y γ Y γ Z L 4WX { iγ 4 = σ c, -i = σ d, i γ 4 = σ e } R Y>Z { iγ Y γ > = σ c, -iγ > γ Z = σ g, iγ Z γ Y = σ h }

15 Spatially separable operators d 4W = 0, d XZ = 0, d Y> = 0 Ψ = Ad / 4W d / Y> d / XZ + Bd / 4W + Cd / Y> Dd / XZ γ 4 γ > y -y γ Y γ Z C 2 if AD BC=0 Ψ = A + B + C + D L 4WX { iγ 4 = σ c, -i = σ d, i γ 4 = σ e } R Y>Z { iγ Y γ > = σ c, -iγ > γ Z = σ g, iγ Z γ Y = σ h }

16 Detecting entanglement in different basis d 4W = 0, d XZ = 0, d Y> = 0 Ψ = Ad / 4W d / Y> d / XZ + Bd / 4W + Cd / Y> Dd / XZ γ 4 γ > γ 4 γ > y -y γ Y y -y γ Y γ Z γ Z C 2 if AD BC=0 Ψ = A + B + C + D C 2 if AD + BC=0 Ψ = A + B C + D

17 Detecting entanglement in different basis d 4W = 0, d XZ = 0, d Y> = 0 Ψ = Ad / 4W d / Y> d / XZ + Bd / 4W + Cd / Y> Dd / XZ γ 4 γ > γ 4 γ > y -y γ Y y -y γ Y γ Z γ Z C 4WX Y>Z 2 if AD BC=0 C 4>Z Y>X 2 if AD + BC=0 C XZY >4W 2 if AC BD=0 C Y>4 XZW 2 if AB CD=0 γ 4 γ > γ 4 γ > -y y γ Y y -y γ Y γ Z γ Z

18 Non-local signatures Any state of the system will show C > 2 in at least one of the four measurement sets. Any state will look non-local for some choice of measurement sets

19 Is it just algebra? Formally the algebra holds for 2 spin ½ particles generalied entanglement L 4WX { iγ 4 = σ c, -i = σ d, i γ 4 = σ e } R Y>Z { iγ Y γ > = σ c, -iγ > γ Z = σ g, iγ Z γ Y = σ h }, y,, y, γ γ 4 > y -y γ Y γ Z γ ' γ 4 > -y Y γ Y = = ' γ Z The new operators cannot be measured by two spatially separable detectors in the spin system [ Barnum et al. (2002), ]

20 Minimum entanglement 2 2 C C 4WX Y>Z = C XZY >4W = C Y>4 WXZ = 2 2 C 4WX Y>Z = C XZY >4W = C 4WZ Y>X = 2 C ψ = ma C 4WX Y>Z, C XZY >4W, C Y>4 WXZ, C 4WZ Y>X C 1 min t C(ψ) 2 6/5

21 What to measure? w 3,A w 1,A A L g3 g1 g2 A R g4 w 2,A w 4,A H { = w W,} + w 4,} γ 4 c },{ c / },{ + w X, + w W, c,{ c/,{ w 3,B B L B R w 4,B Sensing the dots charge configuration n },{, n,{ = { 0,1, (1,0)} w 5,B g5 g6 w 6,B For weak measurements (short times, weak tunneling) (n },{ = 0, n,{ = 1, n }, = 0, n, = 1) P 1,0,1,0 = 2λ { > λ > Δt Y A { A > A { = 1/λ { [Re w 4,} w W,} + Re w W, w X, Re w 4,} w X, y] We can measure the local spin in any direction C > 2 is a signature of non-local ero-energy fermionic ecitations

22 Outlook Topological superconductors have non-local ero energy ecitations Any low-energy state of a Majorana system is entangled in some measurement configuration! The required measurements are accessible in currenttechnology nanostructures What for non-ising anyons? What beyond the minimal geometry? Non-pure states? Quantum discord?

23 Etra

24 Disclaimer: Majorana fermions vs. end-states Filed theory for fermions requires four-component fields. The four component describe both a particles and its antiparticles The four components can be a redundant description of one particle Ecitations which are their own antiparticles

25 Bogoliubov ecitations in a s-wave superconductor spinless p-wave superconductor Vortices in spinless p-wave superconductors, 5/2 quantum Hall state [N. Read and D. Green (2000)]