Port-based teleportation and its applications

Transcription

1 2 QUATUO Port-based teleportation and its applications Satoshi Ishizaka Graduate School of Integrated Arts and Sciences Hiroshima University Collaborator: Tohya Hiroshima (ERATO-SORST)

2 Outline Port-based teleportation [SI and T. Hiroshima, PRL 101, (2008); PRA 79, (2009)] Its applications - universal programmable processor - attacking position-based cryptography [S. Beigi and R. König, New J. Phys. 13, (2011)] - entanglement recycling and generalized teleportation [S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, (2013)] - relation to no-signaling [D. Pitalúa-García, arxiv: (2012)]

3 A reliving machine EPR teleport input ε history Time BSM φ + ε history ( input ) Teleportation enables to relive the past events [C. H. Bennett, private comm.] [N. Imoto, 7th QCM&C] [Č. Brukner, et. al., Phys. Rev. A (2003)] Reliving succeeds only with P success = 1 d 2

4 EPR ψ BSM A simple way to increase P success c.f. [L. Vaidman (2003)] i P=1/d 2 EPR EPR U Uσ i U 1 Uσ i U 1 σ j Uσ i U 1 BSM BSM U ψ j P=(1 1/d 2 )/d 2 k P=(1 1/d 2 ) 2 /d 2 R round P =1 (1 1/d 2 ) R =1 ɛ huge EPR resource 2 2n22n log(1/ɛ) (double exp) only for reversible operations, complicated

5 Port-based teleportation Alice χ in POVM A 1 A 2 A 3 A N ψ B 1 B 2 B 3 B N select Bob B i χ in outcome i Such a teleportation scheme is truly possible? What is the optimal success probability or fidelity? What is the optimal ψ and optimal POVM? We answer these questions

6 Formulation Bob s operation is fixed teleportation = discrimination of quantum signals e.g. standard teleportation scheme Bob s operation is fixed to {I, σ x, σ y, σ z } = {σ i } Alice must (globally) descriminate 4 quantum signals: {(I σ i ) ψ } = { φ +, ψ +, ψ, φ } Bell state measurement is optimal for Alice (Alice measures a qubit to be teleported instead of Bob s qubit)

7 Formulation ψ = φ + A1 B 1 φ + AN B N Alice s POVM: Π (i) AC Alice POVM C A i P + 11 σ (i) AB B i Bob B σ (i) AB = 1 d N 1 P + A i B 11 Ā i outcome i non-commutable & mixed Average fidelity: f =(F d + 1)/(d + 1) F = 1 d 2 N i=1 trπ (i) ABσ (i) AB Entanglement fidelity maximized Problem of discriminating {σ (1), σ (2),, σ (N) } p err =1 d 2 F/N

8 Duality is broken Duality between superdense coding and teleportation (i) All teleportation schemes tr[(m x Λ x )(σ ψ)]a = trσa x X (ii) All dense coding schemes tr(λ x M y )ψ = δ xy (i) (ii) for tight cases [R. F. Werner, (2000)] but p err =1 d2 F N for port-based teleportation ( non-tight )

9 Deterministic scheme optimal protocol (d = 2) ψ is maximally entangled ( ψ = φ + N ) N Π i =ρ 1 2 σ (i) ρ 1 2 with ρ= σ (i) SRM is optimal F = 1 2 N+3 N k=0 ψ is also optimized ( N 2k 1 k+1 i=1 O Π i O = z(s)ρ 1 y(s) s σ (i) s s O O = γ(j)11(j) [N] A, j F =cos 2 ( π N +2 ) + N 2k+1 N k+1 ) 2 ( N k y(s) s ρ 1 ) generalized SRM N 2s+1 N+2s+3 1 y(s) = s s+1 sin 2π(s+1) N+2 sin 2πs N+2

10 Deterministic scheme optimal fidelity Average fidelity ( f ) N d =2 classical limit d =3 classical limit 2 cos 2π N best max. entangle & SRM Number of ports (N) f exceeds f cl =2/3 for N > 2 and approaches to 1 for N This scheme certainly works as quantum teleportation Three EPR-pairs are enough to demonstrate this scheme φ + N +SRM nearly achieves the best f around small N

11 When the number of ports is small... Alice performs an orthogonal measurement { 0, 1, 2, } Alice orthogonal C separable N 1 select B i Bob B outcome i This M&P protocol is optimal for N d This is not quantum teleportation f f cl = 2 d + 1 N > d is necessary to exceed f cl Can fidelity exceed the classical limit by using entanglement?

12 Formulation ψ =(O A 11) φ + A1 B 1 φ + AN B N with troo =d N POVM: {Π 0, Π i } teleportation fails for Π 0 Faithful teleportation for Π 1 Π N OΠ i O =P + BA i Θ Āi Λ(σ in )= N i=1 [ ] tr AC Π i (O 11)σ (i) AB (O 11) σc in Success probability of entanglement swapping p=tr(λ 11)P + CD = 1 d N+1 N tro Π i O maximized i=1

13 Probabilistic scheme optimal protocol (d = 2) ψ is maximally entangled ( ψ = φ + N ) Π i =P + BA i 4 N +3+2s 11(s)[N 1] Ā i Π 0 =11 s p= 1 (2s+1) 2 ( ) N +1 2 N (N +1) s N+3 2 +s N i=1 Π i teleportation fails ψ is optimized O Π i O =P + BA i s O O = γ(j)11(j) A j β(s)11(s) [N 1] Ā i Π 0 =11 N i=1 Π i teleportation fails p= N N + 3 = 1 3 N + 3

14 Optimal success probability Probability ( p ) KLM = 1 1 N+1 d =2 1 3 N π N best max. entangle Number of ports (N) Optimizing ψ considerably enhances p Non-maximally entangled ψ provides considerably larger p N must be just 3 times larger than KLM to achive the same p = the cost for removing Bob s unitary transformation

15 Example (N =2, p=2/5) ψ = + [ spin-triplet A 00 B + 11 A 11 B + ψ + A ψ + B ] spin-singlet ψ A ψ B E =1.895 ebits Alice C A 1 A 2 ψ B 1 B 2 Bob Alice C A 1 A 2 ψ res B 1 B 2 Bob Π 1 ψ (α 0 +β 1 ) C = ψ res = ψ res (α 0 +β 1 ) B1 [ ] ψ + ψ + A 1 A 2 CB ψ A1 A 2 ψ CB E =1 ebits Entanglement consumption E = ebits < 1 ebit

16 Average entanglement consumption If teleportation fails... E =0.31 ebits for N = 2 Best case: e-swapping log(n +1) ebits remains Worst case: give up E =E ini 3 N+3 < 3 ebits Entanglement (ebit) ebit Number of ports worst case Number of ports (N) Entanglement initial residual (average) initial residual (average) Only a few ebits are consumed on average even for large N

17 Outline Port-based teleportation [SI and T. Hiroshima, PRL 101, (2008); PRA 79, (2009)] Its applications - universal programmable processor - attacking position-based cryptography [S. Beigi and R. König, New J. Phys. 13, (2011)] - generalized teleportation and entanglement recycling [S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, (2013)] - relation to no-signaling [D. Pitalúa-García, arxiv: (2012)]

18 Application: universal programmable processor Device to manipulate a state via program states ( ε ) program state ε unknown input χ in G PP output ε ( χ in ) DVD Copyright 2008 NEC Corp Vine Linux 2.6 Kernel login: NEC program programmable processor state G fixed operation (indep of ε and χ in ) Universal PP can deal with arbitrary ε Unitary, measurements, trace-nonpreserving, etc... No-go theorem: [M. A. Nielsen and I. L. Chuang (1997)] Faithful & deterministic UPP cannot be realized Port-based teleportation evades the no-go theorem

19 No-go theorem of UPP [M. A. Nielsen and I. L. Chuang, PRL 79, 321 (1997)] G χ in U =(U χ in ) U for all χ in deterministic 8 < : G χ 1 U =(U χ 1 ) U χ 1 χ 2 = χ 1 χ 2 U U G χ 2 U =(U χ 2 ) U ) U is input indep 8 < : G χ U =(U χ ) U U V = χ U V χ U V G χ V =(V χ ) V ) χ U V χ is input indep If U V, then U V = 0 orthogonal ε can only store limited number of operations deterministic & approximate, probabilistic & faithful, or infinite resource

20 Port-based teleportation as a reliving machine SRM Vulcan EPR ε Earth ε Vulcan ε Klingon Time

21 Outline Port-based teleportation [SI and T. Hiroshima, PRL 101, (2008); PRA 79, (2009)] Its applications - universal programmable processor - attacking position-based cryptography [S. Beigi and R. König, New J. Phys. 13, (2011)] - generalized teleportation and entanglement recycling [S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, (2013)] - relation to no-signaling [D. Pitalúa-García, arxiv: (2012)]

22 Position-verification in position-based cryptography V 0 P V 0 P measurement outcome x θ =0 θ =45 Quantum tagging & entangle attack [A. Kent, et. al. (2011)] entanglement Generic entanglement attack [H. Buhrman, et. al. (2011)] classical com Time using instantaneous measurement [L. Vaidman (2003)]

23 Measurement & causality σ x A von Neumann measurement B M Time Standard von Neumann measurement contradicts causality [Landau, Peierls, (1931)] How to measure nonlocal variables using entangled probes [Aharonov, Albert, (1980)] Instantaneous measurerability of all (non-local) variables?

24 Instantaneous measurement & port-based teleportation A EPR 1 round classical com B Time Vaidman s scheme [L. Vaidman, PRL 90, (2003)] C Success probability: P = 1 ɛ EPR resource: O(2 4n24n log(1/ɛ) ) SRM ST without correction B EPR A B port based teleportation BSM σ i Time Using port-based teleportation [S. Beigi, R. König, NJP 13, (2011)] port # 1 round classical com C outcomes of every port Success probability: P = 1 ɛ EPR resource: O(2 4n log(1/ɛ)) exponentially efficient!

25 Applying CNOT to yesterday s qubit Yesterday Time Today EPR BSM k EPR σ k SRM i σ k σ k

26 Outline Port-based teleportation [SI and T. Hiroshima, PRL 101, (2008); PRA 79, (2009)] Its applications - universal programmable processor - attacking position-based cryptography [S. Beigi and R. König, New J. Phys. 13, (2011)] - generalized teleportation and entanglement recycling [S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, (2013)] - relation to no-signaling [D. Pitalúa-García, arxiv: (2012)]

27 Generalized teleportation [S. Strelchuk, M. Horodecki, J. Oppenheim, PRL 110, (2013)]... From a group-theoretic perspective all currently known teleportation protocols can be classified into two kinds: those that exploit the Pauli group [BBCJPW93] and those, which use the symmetric permutation group [IH08]. Such a simple change of the underlying group structure lead to two protocols with striking differences in the properties:... The former teleportation scheme was used in the celebrated result of Gottesman and Chuang [GC99] to perform universal Clifford-based computation using teleportation over the Pauli group. The generalized teleportation protocol introduced in this Letter embraces both known protocols, and paves the way for protocols which lead to programmable processors capable of executing new kinds of computation beyond Clifford-type operations.

28 Generalized teleportation [S. Strelchuk, M. Horodecki, J. Oppenheim, PRL 110, (2013)] Generalized teleportation scheme Alice input POVM outcome A 1 A 2 A 3 A N ψ g G B 1 B 2 B g B N U g Bob output p err =1 d 2 F/N A sufficient condition for reliable teleportation tr η 2 signal 1 (1 ɛ)d N+1 Standard teleportation tr η 2 signal = 1 d 2 Port-based teleportation tr η 2 signal = 1 d N+1 (1 + d2 1 N ) What novel forms of computation might lead from here?

29 Entanglement recycling [S. Strelchuk, M. Horodecki, J. Oppenheim, PRL 110, (2013)] Alice B 1 Bob input A 1 A 2 B 2 output POVM A 3 A N ψ B 3 3 outcome i B N Discarded N 1 ports still work for port-based teleportation! After teleporting k qubits, F 1 11k 2N Performance simultaneous transmission ( N! (N k)! outcomes)

30 Outline Port-based teleportation [SI and T. Hiroshima, PRL 101, (2008); PRA 79, (2009)] Its applications - universal programmable processor - attacking position-based cryptography [S. Beigi and R. König, New J. Phys. 13, (2011)] - generalized teleportation and entanglement recycling [S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, (2013)] - relation to no-signaling [D. Pitalúa-García, arxiv: (2012)]

31 No-signaling & port-based teleportation P success N 4 n + N 1 [D. Pitalúa-García, arxiv: (2012)] from no-cloning and no-signaling Alice Bob input POVM ψ maximally mixed (no cloning) 2 outcome i Standard teleportation is still possible with 1 4 n (P sucsess q j ) q j n (P sucsess q j ) 1 4 n (no-signaling)