port-based teleportation and its applications port-based teleportation and its applications satoshi...

第 2回 QUATUO研究会

Port-based teleportationand

its applications

Satoshi Ishizaka

Graduate School of Integrated Arts and SciencesHiroshima University

Collaborator: Tohya Hiroshima (ERATO-SORST)

Outline

• Port-based teleportation

[SI and T. Hiroshima, PRL 101, 240501 (2008); PRA 79, 042306 (2009)]

• Its applications

- universal programmable processor

- attacking position-based cryptography

[S. Beigi and R. Konig, New J. Phys. 13, 093036 (2011)]

- entanglement recycling and generalized teleportation

[S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, 010505 (2013)]

- relation to no-signaling

[D. Pitalua-Garcıa, arXiv:1206.4836 (2012)]

A reliving machine

EPR

input

Time

BS

M

|φ+⟩

εhistoryteleport

εhistory(|input⟩)

Teleportation enables to “relive” the past events

[C. H. Bennett, private comm.]

[N. Imoto, 7th QCM&C]

[C. Brukner, et. al., Phys. Rev. A (2003)]

“Reliving” succeeds only with Psuccess =1d2

A simple way to increase Psuccess

c.f. [L. Vaidman (2003)]

EPR

BS

M

U

|ψ⟩i P=1/d 2

EPR

BS

M

UσiU−1

j P=(1−1/d 2)/d 2

EPR

BS

M

k P=(1−1/d 2)2/d 2

UσiU−1σjUσiU

−1 U |ψ⟩

• R round · · · P =1− (1−1/d2)R =1− ε

• huge EPR resource · · · 22n22n log(1/ε) (double exp)

• only for reversible operations, complicated

Port-based teleportation

Alice

POVM

select

outcome i

Bob

|χin⟩B1

B2

B3

BN

A1

A2

A3

AN|ψ ⟩

Bi

|χin⟩

• 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

Formulation

• Bob’s operation is fixed

teleportation = discrimination of quantum signals

• e.g. standard teleportation scheme

Bob’s operation is fixed to {I, σx, σy, σz} = {σi}⇓

Alice must (globally) descriminate 4 quantum signals:

{(I ⊗ σi)|ψ〉} = {|φ+〉, |ψ+〉, |ψ−〉, |φ−〉}⇓

∴ Bell state measurement is optimal for Alice

(Alice measures a qubit to be teleported instead of Bob’s qubit)

Formulation

|ψ〉=|φ+〉A1B1 · · ·|φ+〉AN BN

• Alice’s POVM: Π(i)AC

Alice

POVM

outcome i

Bob

Bi1

Ai

P+

σ(i)AB

B

111

11

C

σ(i)AB =

1dN−1

P+AiB

⊗ 11Ai· · · non-commutable & mixed

• Average fidelity: f =(Fd + 1)/(d + 1)

F =1d2

N∑

i=1

trΠ(i)ABσ

(i)AB · · ·

Entanglement fidelitymaximized

• Problem of discriminating {σ(1), σ(2), · · · , σ(N)}perr =1− d2F/N

Duality is broken

• Duality between superdense coding and teleportation

(i) All teleportation schemes

∑

x∈X

tr[(Mx ⊗ Λx)(σ ⊗ ψ)]A = trσA

(ii) All dense coding schemes

tr(Λx ⊗My)ψ = δxy

(i) ⇐⇒ (ii) for “tight” cases [R. F. Werner, (2000)]

• but perr =1− d2F

Nfor port-based teleportation (“non-tight”)

Deterministic scheme — optimal protocol (d=2)

• |ψ〉 is maximally entangled (|ψ〉 = |φ+〉⊗N )

Πi =ρ−12 σ(i)ρ−

12 with ρ=

N∑

i=1

σ(i) · · · SRM is optimal

F =1

2N+3

N∑

k=0

(N−2k−1√k+1

+N−2k+1√

N−k+1

)2(

N

k

)

• |ψ〉 is also optimized

O†ΠiO=∑

s

z(s)ρ− 1

y(s)s σ(i)

s ρ− 1

y(s)s · · · generalized SRM

O†O=∑

j

γ(j)11(j)[N ]A ,

�N−2s+1N+2s+3

� 1y(s)

= ss+1

sin2π(s+1)

N+2sin 2πs

N+2

F =cos2(π

N+2)

Deterministic scheme — optimal fidelity

1 2 3 4 5 6 7 8 9 10 11 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of ports (N)

Ave

rage

fide

lity

( f )

classical limit

classical limit

bestmax. entangle & SRM

d = 2

d = 3

1− 12N

cosN+22π

32

31+

• f exceeds fcl =2/3 for N > 2 and approaches to 1 for N→∞B This scheme certainly works as quantum teleportation

B Three EPR-pairs are enough to demonstrate this scheme

B |φ+〉⊗N+SRM nearly achieves the best f around small N

When the number of ports is small...

• Alice performs an orthogonal measurement {|0〉, |1〉, |2〉, · · · }Alice

orthogonal

select

outcome i

Bob| 0 ⟩

|separable⟩

BiC

B|0⟩ |1⟩ |2⟩… |N−1⟩

| 1 ⟩

| 2 ⟩

• This M&P protocol is optimal for N ≤ d

• This is not quantum teleportation · · · f ≤ fcl =2

d + 1• N > d is necessary to exceed fcl

Can fidelity exceed the classical limit by using entanglement?

Formulation

• |ψ〉=(OA⊗11)|φ+〉A1B1 · · ·|φ+〉AN BN with trOO†=dN

• POVM: {Π0, Πi} · · · teleportation fails for Π0

• Faithful teleportation for Π1 · · ·ΠN

OΠiO†=P+

BAi⊗ΘAi

• Λ(σin)=N∑

i=1

trACΠi

[(O⊗11)σ(i)

AB(O†⊗11)]⊗σin

C

• Success probability of entanglement swapping

p= tr(Λ⊗ 11)P+CD =

1dN+1

N∑

i=1

trO†ΠiO · · · maximized

Probabilistic scheme — optimal protocol (d=2)

• |ψ〉 is maximally entangled (|ψ〉 = |φ+〉⊗N )

Πi =P+BAi

⊗∑

s

4N+3+2s

11(s)[N−1]

AiΠ0 =11−

N∑

i=1

Πi

teleportation fails

p=1

2N

∑s

(2s+1)2

(N+1)

(N+1

N+32 +s

)

• |ψ〉 is optimized

O†ΠiO=P+BAi

⊗∑

s

β(s)11(s)[N−1]

AiΠ0 =11−

N∑

i=1

Πi

teleportation failsO†O=

∑

j

γ(j)11(j)A

p=N

N + 3= 1− 3

N + 3

Optimal success probability

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.10.20.30.40.50.60.70.80.9

1

Number of ports (N)

Pro

babi

lity

( p )

bestmax. entangled =

2

1− 8π N√

1− 3N+3

1− 1N+1KLM =

• Optimizing |ψ〉 considerably enhances p

B 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

Example (N =2, p=2/5)

• |ψ〉=√

310

[ spin-triplet

|00〉A|00〉B+|11〉A|11〉B+|ψ+〉A|ψ+〉B]

+√

110

spin-singlet

|ψ−〉A|ψ−〉B . . . . . . . . . . . . . . . . . . . E =1.895 ebits

Alice BobB1

B2

A1

A2

|ψ ⟩C

Alice BobB1

B2

A1

A2 |ψres⟩C

• √Π1|ψ〉 ⊗(α|0〉+β|1〉)C =√

15 |ψres〉 ⊗(α|0〉+β|1〉)B1

• |ψres〉=√

16

[|00〉|00〉+|11〉|11〉+|ψ+〉|ψ+〉

]A1A2CB2

+√

12 |ψ−〉A1A2 |ψ−〉CB2 . . . . . . . . . . . . . . . . . . E =1 ebits

Entanglement consumption ∆E = 0.895 ebits < 1 ebit

Average entanglement consumption

• If teleportation fails ... ∆E =0.31 ebits for N = 2Best case: e-swapping · · · log(N+1) ebits remains

Worst case: give up · · · 〈∆E〉=Eini · 3N+3 < 3 ebits

1 2 3 4 5 6 7 8 9 100123456789

10

10 20 30 40 500

10

20

30

Number of ports (N)

Ent

angl

emen

t (eb

it)

initial

residual (average)

worst case

Number of ports

Ent

angl

emen

t

initial

residual (average)

1 ebit

Only a few ebits are consumed on average even for large N

Outline







- generalized teleportation and entanglement recycling




Application: universal programmable processor

• Device to manipulate a state via program states (|ε〉)

unknown input

outputprogram state

| ε ⟩G

|χin⟩ε ( |χin⟩ )

PPNEC

Vine Linux 2.6Kernel 2.4.19login:

DVD

NEC CorpCopyright 2008

program programmableprocessorstate

G · · · fixed operation (indep of |ε〉 and |χin〉)• 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

No-go theorem of UPP

[M. A. Nielsen and I. L. Chuang, PRL 79, 321 (1997)]

G|χin〉|U〉=(U |χin〉)|U ′〉 for all |χin〉 · · · deterministic8<: G|χ1〉|U〉=(U |χ1〉)|U ′〉 → 〈χ1|χ2〉=〈χ1|χ2〉〈U ′|U ′′〉G|χ2〉|U〉=(U |χ2〉)|U ′′〉 ) |U ′〉 is input indep8<: 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 6= V , then 〈U |V 〉 = 0 · · · orthogonal

• |ε〉 can only store limited number of operations

⇒ deterministic & approximate,

probabilistic & faithful, or infinite resource

Port-based teleportation as a reliving machine

EPR

Time

SR

M

εEarth

εVulcan

εKlingon

Vulcan

Outline











Position-verification in position-based cryptography

xV0 V1

Time

P0 P1

entanglement

|0⟩ |1⟩|+⟩ |−⟩

θ =0°θ =45°outcome

measurement

classical com

Quantum tagging & entangle attack

[A. Kent, et. al. (2011)]

Generic entanglement attack

[H. Buhrman, et. al. (2011)]

using instantaneous measurement

[L. Vaidman (2003)]

Measurement & causality

TimeA B

|0⟩|0⟩ |0⟩|1⟩ |1⟩|+⟩ |1⟩|−⟩

von Neumannmeasurement

σx

M

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?

Instantaneous measurement & port-based teleportation

Time1−round classical com

EPRA B

C

Time

1−round classical com

EPRA B

C

BS

M

SR

M σi

port #ou

tcomes

of ev

ery p

ort

ST without correction

port−based teleportation

B’

Vaidman’s scheme

[L. Vaidman, PRL 90, 010402 (2003)]

Success probability: P = 1− ε

EPR resource: O(24n24n log(1/ε))

Using port-based teleportation

[S. Beigi, R. Konig, NJP 13, 093036 (2011)]

Success probability: P = 1− ε

EPR resource: O(24n log(1/ε))

exponentially efficient!

Applying CNOT to yesterday’s qubit

EPR

Time

SR

M

i

EPR

BS

M

k

σk

σk

σk

Yesterday Today

Outline











Generalized teleportation

[S. Strelchuk, M. Horodecki, J. Oppenheim, PRL 110, 010505 (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.

Generalized teleportation


Generalized teleportation scheme

Alice

POVM

outcome

BobB1

B2

Bg

BN

A1

A2

A3

AN|ψ ⟩

input output

g∈G

Ug

perr =1− d2F/N

A sufficient condition

for reliable teleportation

trη2signal ≤

1(1− ε)dN+1

• Standard teleportation · · · trη2signal = 1

d2

• Port-based teleportation · · · trη2signal = 1

dN+1 (1 + d2−1N )

“What novel forms of computation might lead from here?”

Entanglement recycling


Alice

POVM

outcome i

BobB1

B2

B3

BN

A1

A2

A3

AN|ψ ⟩

input

3

output

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)

Outline











No-signaling & port-based teleportation


Psuccess ≤ N

4n + N − 1from no-cloning and no-signaling

Alice

POVM

outcome i

Bob

|ψ ⟩

input

2

maximally mixed(no−cloning)

Standard teleportation is still possible with14n

(Psucsess − qj)

qj +14n

(Psucsess − qj) ≤ 14n

(no-signaling)

port-based teleportation and its applications port-based teleportation and its applications satoshi...

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