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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
• 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)]
- generalized teleportation and entanglement recycling
[S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, 010505 (2013)]
- relation to no-signaling
[D. Pitalua-Garcıa, arXiv:1206.4836 (2012)]
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
• 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)]
- generalized teleportation and entanglement recycling
[S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, 010505 (2013)]
- relation to no-signaling
[D. Pitalua-Garcıa, arXiv:1206.4836 (2012)]
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
• 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)]
- generalized teleportation and entanglement recycling
[S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, 010505 (2013)]
- relation to no-signaling
[D. Pitalua-Garcıa, arXiv:1206.4836 (2012)]
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
[S. Strelchuk, M. Horodecki, J. Oppenheim, PRL 110, 010505 (2013)]
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
[S. Strelchuk, M. Horodecki, J. Oppenheim, PRL 110, 010505 (2013)]
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
• 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)]
- generalized teleportation and entanglement recycling
[S. Strelchuk, M. Horodecki, and J. Oppenheim, PRL 110, 010505 (2013)]
- relation to no-signaling
[D. Pitalua-Garcıa, arXiv:1206.4836 (2012)]
No-signaling & port-based teleportation
[D. Pitalua-Garcıa, arXiv:1206.4836 (2012)]
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)