Entanglement in Particle Physics: Bell Inequalities, Geometry and Relativistics

Transcription

1 Entanglement in Particle Physics: Bell Inequalities, Geometry and Relativistics Colloquium at PSI Reinhold A. Bertlmann Faculty of Physics, University of Vienna June 10 th, 2010

2 Outline

3 Part I

4 Quantum Information Quantum entanglement as source of quantum information theory quantum communication Alice Bob quantum teleportation: transfer of quantum states Alice Bob quantum cryptography: safe secret keys quantum computation: quantum gates, quantum computer R. Ursin, M. Aspelmeyer,..., A. Zeilinger Nature 430, 849, (2004)

5 Entanglement Outline Schrödinger 1935: Trilogy On the present situation in QM... entangled states verschränkte Zustände... quantum system with physically distant parts E. Schrödinger: The whole is in a definite state, the parts taken individually are not. Illustration of entanglement: Schrödinger s cat box, decaying atom, hammer, cyanide, cat entangled state separable state convex combination of product states ψ = 1 2 ( e alive + g dead )

6 EPR-Paradox Outline Einstein Podolsky Rosen 1935 Completeness of theory: Every element of physical reality must have a counter part in physical theory. Quantum system of 2 distant particles 1 2 very peculiar correlations between 1 2 one can predict outcome of measurement on 2 by measuring 1 EPR: 1, 2 predetermined values in a local and complete theory = QM incomplete! Bohr 1935: QM is complete! Furry 1935: QM model: spontaneous factorization of wavefunction

7 Bell s Theorem Bell s Theorem 1964 J.S. Bell:... in a certain experimental situation all LRT (local realistic theories) are incompatible with QM. Alice EPR-type experiment Bob

8 Expectation Value A(n, λ) values of observable A QM (n) σ (1) n B(m, λ) B QM (m) σ (2) m λ... hidden variable n, m... quantization directions A, B 1 8 >< +1 A(n, λ) = 0 no detection >: 1 particle 1 at Alice 8 >< +1 B(m, λ) = 0 no detection >: 1 particle 2 at Bob expectation value for combined spin measurement E(n, m) = R dλ ρ(λ)a(n, λ)b(m, λ) with R dλ ρ(λ) = 1 independent of m n Bell s locality hypothesis

9 Bell Inequalities E(n, m) E(n, m ) dλ ρ(λ) A(n, λ)b(m, λ) { 1 ± A(n, λ)b(m, λ) } + dλ ρ(λ) A(n, λ)b(m, λ) { 1 ± A(n, λ)b(m, λ) } = Bell inequality CHSH-type Clauser, Horne, Shimony, Holt S = E(n, m) E(n, m ) + E(n, m ) + E(n, m) 2 LRT in terms of probabilities E(n, m) = P(n, m ) + P(n, m ) P(n, m ) P(n, m ) = P(n, m ) Bell inequality of Wigner-type P(n, m) P(n, n ) + P(n, m)

10 Bell Inequality QM Quantum Mechanics quantum mechanical expectation value in spin singlet state ψ = 1 2 ( ) E(n, m) = ψ A QM (n)b QM (m) ψ = ψ σ n σ m ψ = cos(n m) for choice n m = 3π 4, n m = n m = n m = π 4 Bell angles = QM S QM = 2 2! > 2 LRT contradiction!

11 Experiments Outline Experiments: polarization of entangled photons Freedman and Clauser 1972 Fry and Thompson 1976 Aspect group 1982 Zeilinger group 1998 Results: Experiments violate BI! e.g. Weihs,..., Zeilinger: polarizers random time flip = S exp = 2.73 ± 0.02 experiment in agreement with QM Conclusion: QM and Nature are nonlocal!

12 Spooky Outline

13 Literature Introduction to BI s J.S. Bell: Speakables and Unspeakables in QM, Cambr.Uni.Press 1987 R.A. Bertlmann, A. Zeilinger: Quantum [Un]speakables, Springer 2002 A. Einstein, B. Podolsky, N. Rosen: Phys.Rev. 47, 777 (1935) E. Schrödinger: Naturwissenschaften 23, 807, 823, 844 (1935)

14 Part II

15 Strange Mesons Strange mesons selected by Nature to demonstrate fundamental principles of QM such as : superposition principle oscillation and decay property regeneration mechanism quasi-spin property K-meson bound state of quarks (q q) : u... up flavor d... down s... strange mass: MeV J P = 0 pseudoscalar strong interactions: S conservation weak interactions: S violation

16 QM of K-mesons Strangeness eigenstates S K 0 = + K 0 S K 0 = K 0 CP-transformation P... parity C... charge conjugation CP eigenstates CP K 0 = K 0 CP K 0 = K 0 CP K 0 1 = + K 0 1 K 0 1 = 1 2 ( K 0 K 0 ) CP K 0 2 = K 0 2 K 0 2 = 1 2 ( K 0 + K 0 ) strong interactions: CP conservation weak interactions: small CP violation almost conserved

17 K-meson decays K K decay S K1 0 2π τ S sec short-lived K L K2 0 3π τ L sec long-lived physical masses m S, m L with m = m L m S = ev small CP-violation K L 2π small short-lived, long-lived states ε 10 3 CP-violating parameter K S = 1 ( p K 0 q N K 0 ) p = 1 + ε, q = 1 ε K L = 1 ( p K 0 + q K 0 ) N 2 = p 2 + q 2 N decaying states evolve in time Weisskopf-Wigner approximation KS L (t) = e iλ S L t KSL with λs L = ms L i 2 ΓS L and ΓS L τ 1 S L = time evolution for K 0 and K 0

18 K 0 K 0 Outline Oscillation K 0 beam produced at t = 0 probability for finding K 0 or K 0 at t > 0 K 0 K 0 (t) 2 = 1 { e Γ S t + e ΓLt + 2 e Γt cos( mt) } 4 K 0 K 0 (t) 2 = 1 { e Γ S t + e ΓLt 2 e Γt cos( mt) } 4 with m = m L m S and Γ = 1 2 (Γ L + Γ S ) ΓS L τ 1 S L K 0 beam oscillates with frequency m/2π oscillation visible at t τ S, m τ S = 0.47 K 0 beam at t = 0 = K 0 equal to K 0 far away from source through K L

19 Quasi-spin of K-mesons K 0 K 0 2-dim Hilbertspace strangeness +1 up 1 down / operators in quasi-spin space: S σ 3 CP σ 1 CP σ 2 Hamiltonian H = M i 2 Γ = ( ) 1 2 a 1 + h σ with KS L eigenstates Analogy K-meson spin- 1 2 photon K 0 z V K 0 z H K S xy L = 1 2 ( V i H ) K L xy R = 1 2 ( V + i H )

20 Bell Inequality Wigner-type LRT satisfy BI choice: fix quasi-spin freedom in time freedom in quasi-spin fix time choose: fix time vary quasi-spin of K-meson = rotation in quasi-spin space for BI we need 3 different angles quasi-spins: K S, K 0, K 0 1 Bell inequality of Wigner-type P(K S, K 0 ) P(K S, K 0 1 ) + P(K 0 1, K 0 ) P... probability contains unphysical CP-even state K 0 1 But! BI inequality for physical CP-parameter experimentally testable! how does it come?

21 Experiment Outline BI optimal Inequality for weights p, q of K S, K 0, K 0 1 Experiment decay of K-mesons p q experimentally testable! semileptonic decay of strange mesons quark level K 0 (d s) π (dū) l + ν l s ū l + ν l K 0 ( ds) π + ( du) l ν l s u l ν l = l + tags K 0 in K L state p 2... probability for K 0 in K L l tags K 0 l = µ, e q 2... probability for K 0 in K L charge asymmetry δ = Γ(K L π l + ν l ) Γ(K L π + l ν l ) = p 2 q 2 Γ(K L π l + ν l )+Γ(K L π + l ν l ) p 2 + q 2

22 Conclusion Outline p q Bell inequality for δ δ 0 Experiment: δ exp = (3.27 ± 0.12) 10 3 BI violated! consider 2 BI s δ 0 and δ 0 K 0 K 0, p q = δ = 0 CP conservation in contradiction to experiment! Conclusion LRT are only compatible with strict CP conservation in K 0 K 0 mixing! δ 0 K 0 K 0 entanglement CP violation nonlocal contextual

23 Literature BI s for K-mesons R.A. Bertlmann, B.C. Hiesmayr: Phys.Rev.A 63, (2001) R.A. Bertlmann, W. Grimus, B.C. Hiesmayr: Phys.Lett.A 289, 21 (2001) F. Uchiyama: Phys.Lett.A 231, 295 (1997) A. Bramon, M. Nowakowski: Phys.Rev.Lett. 83, 1 (1990) N. Gisin, A. Go: Am.J.Phys. 69 (3), 264 (2001) G.C. Ghirardi, R. Grassi, R. Ragazzon: DAΦNE Physics Handbook, Vol.I, p.283 (1992) J.S. Bell: Speakables and Unspeakables in Quantum Mechanics, Cambridge University Press 1987 R.A. Bertlmann, A. Zeilinger: Quantum [Un]speakables, Springer 2002

24 Part III

25 Separability Outline Hilbert space H = H A H B Alice & Bob Set of separable states defined by density matrices ρ i = ψ i ψ i { S = ρ = p i ρ i A ρ i B 0 p i 1, } p i = 1 i i reduced density matrix ρ B = tr Aρ Entangled state if not separable ω S c complement of S S S c = H Separability criterion (2 2 and 2 3 dimensions) Theorem: positive partial transposition Peres, Horodecki (1 A T B ) ρ 0 ρ separable T B (σ i B) kl = (σ i B) lk

26 Bell Inequality CHSH-type consider usual Bell inequalities CHSH inequality A CHSH... CHSH operator Clauser-Horne-Shimony-Holt (ρ A CHSH ) 2 with A CHSH = n σ A ( m m ) σ B + n σ A ( m+ m ) σ B ρ local ρ separ QM (ρ A CHSH ) = 2 2 ρ = ψ ψ Bell singlet with ψ = 1 2 ( A B A B )

27 Werner State Outline shift CHSH-operator (ρ 2 1 A CHSH ) 0 and (ρ 2 1 A CHSH ) < 0 Example Werner state inequality valid for all entangled states? YES for pure entangled states NO for mixed entangled states ρ W = p ρ + (1 p) for 1 3 < p 1 2 satisfies CHSH Bell inequality! usual BI s not good to find entangled states!

28 Entanglement Witness Entanglement witness inequality EWI detects entanglement of state observable A entanglement witness such that (ρ A) 0 ρ S (ω A) < 0 for some ω S c A A t tangent functional if ρ 0 S such that (ρ 0 A) = 0 then ρ 0 S A t characterizes convex set S of separable states Entanglement measure Hilbert Schmidt distance D(ω) E(ω) D(ω) = min ρ S ρ ω 2 2 distance D(ω) of entangled state ω S c to set S of separable states

29 BNT Theorem consider EWI EWI (ρ A) (ω A) 0 ρ S maximal violation of EWI B(ω) = max [ min (ρ A) (ω A) ] A α 2 1 ρ S Theorem Bertlmann Narnhofer Thirring B(ω) = D(ω) min of D ρ 0 ω S c max of B A max A t A max = ρ 0 ω (ρ 0 ρ 0 ω) 1 ρ 0 ω 2

30 Illustration of BNT Theorem maximal violation of EWI B(ω) is equal to distance D(ω) ω set of separable states S optimal operator A max to set S is tangent

31 Geometry of Entangled States 4 Bell states ψ ρ = 1 4 ( 1 σx A σ x B σ y A σy B σz A σ z B) P 0 φ ω = 1 4 ( 1 σx A σ x B + σ y A σy B + σz A σ z B) P 1 φ + ω + = 1 4 ( 1 + σx A σ x B σ y A σy B + σz A σ z B) P 2 ψ + ρ + = 1 4 ( 1 + σx A σ x B + σ y A σy B σz A σ z B) P 3 Density matrix in (c 1, c 2, c 3 ) parameter space spin space ρ c = 1 4 ( i=1 c i σa i σb i ) positivity of density matrix = tetrahedron (P 0, P 1, P 2, P 3 ) simplex partial transposition positivity separable Thm Peres Horodecki

32 Illustration of Geometry intersection of tetrahedrons = separable states double-pyramide (convex set)

33 Entangled Separable States

34 Literature EWI R.A. Bertlmann, H. Narnhofer W. Thirring: Phys.Rev.A 66, (2002) M. Horodecki, P. Horodecki, R. Horodecki: in Quantum Information, eds. G. Alber et al., Springer Tracts in Modern Physics Vol. 173, p. 151 (2001) K.G.H. Vollbrecht, R.F. Werner: quant-ph/ B. Terhal: Phys.Lett.A 271, 319 (2000) and quant-ph/ D. Bruß: J. Math. Phys. 43, 4237 (2002) M. Lewenstein, D. Bruß, J.I. Cirac, B. Kraus, M. Kuś, J. Samsonowicz, A. Sanpera, R. Tarrach: Journal of Modern Optics, 47, 2841 (2000) M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki: Phys.Rev.A 62, (2000) and quant-ph/

35 Part IV

36 Lorentz Transformations for Spin- 1 2 Particles massive spin- 1 2 particle state p, σ p... momentum, σ... spin Lorentz transformation Λ = momentum dependent rotation of spin Wigner rotation: W(Λ, p) := L 1 (Λp) Λ L(p) U(Λ) p, σ = N(p) U(W(Λ, p)) Λp, σ W(Λ, p)) Wigner s little group: Lorentz transformations that leave invariant the chosen standard momentum k here: k... rest frame momentum = little group = SO(3)

37 Wigner Rotations Figure: Wigner rotation of spins for particles moving (anti-)parallel to the z-axis and observer moving along the x-direction, original figure: P. M. Alsing, and G. J. Milburn, Quant. Inf. Comp. 2, No. 6 (2002)

38 Mixing Spin & Momentum consider finite width momentum space distribution: ψ 1 particle = dµ(p) f σ (p) p, σ σ Lorentz transformation = entangles spin- and momentum degrees of freedom = spin entropy not invariant i.e. trace over momentum = mixed spin state Peres Scudo Terno 2 particles: change of spin entanglement Gingrich Adami dependent on boost strength & initial state: width of distributions entanglement (spin spin / spin mom / mom mom)

39 Initial State Outline Initial state: ψ total = ψ mom ψ spin spins and momenta initially separable from each other momenta: entangled ψ mom = cos α p +, p + sin α p, p + momenta sharp single rotation spins: entangled ψ spin = cos β + sin β Bell ψ ± states

40 Boosted State Lorentz transformation Λ: ρ = ψ ψ = ρ Λ = ψ Λ ψ Λ transformed state ψ Λ total = cos α Λp +, Λp (U + U ) ψ spin + sin α Λp, Λp + (U U + ) ψ spin only two } momenta {{} allowed: initially: p+, p after boost: Λp+, Λp momentum qubits total system: 4 qubits: 2 momentum qubits Λp +, Λp & 2 spin qubits, study entanglement between different partitions of 4 qubit Hilbertspace using linear entropy of subsystems ρ i : linear entropy E(ρ) = i ( 1 Tr ρ 2 i )

41 Partition into 4 Individual Qubits subsystems consist of 1 qubit each momentum Λp +, Λp or spin, Entanglement change: E(ρ Λ ) E(ρ) = sin 2 δ sin 2 (2α) cos 2 (2β) Wigner angle δ depends on boost strength and direction Entanglement-Egg-Tray: Difference between linear entropies of initial and δ = ± π 2 Wigner rotated Bell-type state

42 Partition into Spin & Momentum subsystems consist of 2 momentum or 2 spin qubits respectively Entanglement after boost: E(ρ Λ ) = 1 2 sin2 δ sin 2 (2α) (1 sin(2β)) ˆ3 + cos(2δ) + 2 sin 2 δ sin(2β) initially E(ρ) = 0 in this partition Entanglement change between momentum and spin of δ = ± π 4 Wigner rotated spin-bell-type state

43 Results for Initial Bell ψ ± Spin States Entanglement change for 1 vs 3 qubit partition and spin momentum partition maximal Wigner angle δ = π 2 = Entanglement change in these partitions identical Wigner angle: as function of boost velocities v and w in perpendicular directions however: Entanglement in partition Alice Bob is invariant!

44 Initial Spin Triplet States Different spin states: ψ spin = sin θ cos φ + sin θ sin φ 1 2 ( + ) + cos θ parametrization using spherical coordinates in space of spin triplet states completely analogous results to Bell ψ ± spin states Figures: Plots shown for maximally entangled momentum states and Wigner angles π 2 (left) and π 4 (right)

45 Literature A. Peres, P.F. Scudo, and D.R. Terno, Phys.Rev.Lett. 88, (2002) P.M. Alsing, and G.J. Milburn, Quant.Inf.Comp. 2, No. 6 (2002) R.M. Gingrich and Ch. Adami, Phys.Rev.Lett. 89, (2002) I. Fuentes-Schuller and R.B. Mann, Phys.Rev.Lett. 95, (2005) T.F. Jordan, A. Shaji, and E.C.G. Sudarshan, Phys.Rev.A 73, (2006) N. Friis, R.A. Bertlmann, M. Huber, and B.C. Hiesmayr, Phys.Rev.A 81, (2010)

46 Quantum Strange Geometry of Relativistic Outline Entanglement Entanglement Entanglement Entanglement Vienna Quantum Engineers Philipp Krammer, Nicolai Friis, Reinhold A. Bertlmann Walter Grimus Beatrix Hiesmayr Andreas Gabriel, Tanja Traxler, Hatice Tataroglu Reinhold A. Bertlmann Entanglement in Particle Physics: Bell Inequalities, Geometry