Resetting uncontrolled quantum systemsTime travel with wormholes K. Thorne, Black Holes and Time...
Transcript of Resetting uncontrolled quantum systemsTime travel with wormholes K. Thorne, Black Holes and Time...
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Resetting uncontrolled quantum systems
Institute for Quantum Optics and Quantum Information (IQOQI), Vienna
Miguel NavascuΓ©s
MN, arXiv:1710.02470
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I invented a time-warping device,
ask me how!
Institute for Quantum Optics and Quantum Information (IQOQI), Vienna
Miguel NavascuΓ©s
MN, arXiv:1710.02470
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Definition of TIME-WARP
noun | \ ΛtΔ«m ΛwΘ―rp \
Time-warp
1. an anomaly, discontinuity, or suspension held to occur in the progress of time
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π‘ = 0
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π‘ = 1 hour
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π‘ = 2 hours
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π‘ = 0
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π‘ = 0
Time warp for two hours
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π‘ = 2 hours
What should we find in the box if we are to claim a time warp experience?
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Not expected
π‘ = 2 hours
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π‘ = 2 hours
Not expected
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π‘ = 2 hours
Not expected
π‘ = 2 hours, no intervention
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Expected
π‘ = 2 hours
Not expected
π‘ = 2 hours, no intervention
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Expected
π‘ = 2 hours
Not expected
π‘ = 2 hours, no intervention
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π‘ = 2 hours
ExpectedNot expected
π‘ = 2 hours, no intervention
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{π π‘ : π‘}
Time warp: operational definition
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{π π‘ : π‘}
π 0
π 1π 2
Time warp: operational definition
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Time warp protocol for t β [0, π]
Time warp: operational definition
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π πβ²
πβ² β π
Time warp: operational definition
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
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A brief history of time warp
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King Raivata and princess Revati
(400BC?)
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Peter Damian(1007-1072)
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(first edition in 1895)
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Warping time physically
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π πβ²
0 < πβ² < π
Time warp in special relativity
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
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GΓΆdel spacetime
M. Buser, E. Kajari and W. P. Schleich, New J. Phys. 15 013063 (2013).
K. GΓΆdel, Rev. Mod. Phys. 21 447 (1949).
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Time travelwith
wormholes
K. Thorne, Black Holes and Time Warps: Einstein'sOutrageous Legacy, Commonwealth Fund Book Program(1994).
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π πβ²
πβ² β π
Time warp with time machines
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
ΰ΅Ώπβππ»0π|π(0)
Evolution after time T
π
|π(0)
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
π
ΰ΅Ώπβππ»0πΎπ π|π(0) |π
Evolution after time T
|π = |π
|π(0)
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ΰ΅Ώπβππ»0πΎπ π|π(0) |π
Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
π
Evolution after time T
πΎπ = 1 β2πΊπ
π π2
|π(0)
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
π
ππ πβππ»0πΎπ π |π(0) |π
Evolution after time T
|π =
π
ππ |π
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
Evolution after time T
|π =
π
ππ |π
Measurement
π
|π
π
ππ πβππ»0πΎπ π |π(0) |π
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
π
ππ πβππ»0πΎπ π |π(0)
Evolution after time T
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
π
ππ πβππ»0πΎπ π |π(0) β πβππ»0πΌπ |π(0)
Evolution after time T βͺ 1
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
π
ππ πβππ»0πΎπ π |π(0) β πβππ»0πΌπ |π(0)
Evolution after time T βͺ 1
πΌ β« 1
πΌ β€ 0Interesting cases
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π πβ²
πβ² β π
Time warp with the time translator
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
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Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman Phys. Rev. Lett. 64, 2965 (1990).
The time translator
π
π
|π =
π
ππ |π
Measurement
π
|π
Problem: astronomicallysmall probability of
postselection
π
ππ πβππ»0πΎπ π |π(0) β πβππ»0πΌπ |π(0)
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βThe time translator has the same chances of succeeding as I have of delocalizing and
relocalizing somewhere elseβ
L. Vaidman
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Time warp, the lame way
π 0
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π 0
Time warp, the lame way
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π 0
Time warp, the lame way
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π 0
(0.3,0.2,0.9),(0.4,0.7,0.1),(0.5,0.6,0.3),
Time warp, the lame way
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π 0
Time warp, the lame way
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π β5732
Time warp, the lame way
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π
π
|π =
π
ππ |π
These proposals do not require control or knowledge of the physical system that we influence
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π
π
|π =
π
ππ |π
All interesting proposals to achieve time warp rely on special or general relativity
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π πβ²
πβ² < 0
Main result
π 0Time warpfor t β [0, π]
Uncontrolled system
Non-relativisticquantum physics
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π πβ²
πβ² < 0
Main result
π 0Time warpfor t β [0, π]
Uncontrolled system
Non-relativisticquantum physics
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π πβ²
πβ² < 0
Main result
π 0Time warpfor t β [0, π]
Uncontrolled system
Non-relativisticquantum physics
reasonable probability of success
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Scenario
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Goal
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π > 0
π
|π(0)
π‘ = π + Ξ
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Obvious solution
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
|π(0) = ΰ΅Ώπ+ππ»0π|π(π)
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
|π(0) = ΰ΅Ώπ+ππ»0π|π(π)
Obvious solution
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
S, uncontrolled
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π»0?S, uncontrolledπ π
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π»0?S, uncontrolled
π π
π π
π»ππ(π)?π π
π π
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π»0?S, uncontrolled
π π
π π
π»ππ(π)?π π
π π
|π(0) = ΰ΅Ώπ+ππ»0π|π(π) ?
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π»0?S, uncontrolled
π π
π π
π»ππ(π)?π π
π π
|π(0) = ΰ΅Ώπ+ππ»0π|π(π) ?
We donβt know π»0.
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π»0?S, uncontrolled
π π
π π
π»ππ(π)?π π
π π
Even if we knew π»0, we wouldnβt know how to
implement πβππ»0π on S.
We donβt know π»0.
|π(0) = ΰ΅Ώπ+ππ»0π|π(π) ?
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π
Controlled lab
S, uncontrolled
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Resetting
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
MN, arXiv:1710.02470
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Resetting
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
MN, arXiv:1710.02470
Exact past state
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Resetting
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
We ignore how S evolves (unitarily) by itself and with other quantum systems
MN, arXiv:1710.02470
We know ππ = dim(π»π )
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Resetting
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
We ignore how S evolves (unitarily) by itself and with other quantum systems
MN, arXiv:1710.02470
We know ππ = dim(π»π )
Imposible if we drop anyof the two assumptions
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Sketch of a quantum resetting protocol
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)LAB
π‘ = π
Initial conditions
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(a) Probe interaction
π
LAB
π‘ β (π, π + πΏ)
π
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π
LAB
π‘ = π + πΏ
π
(a) Probe interaction
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π
LAB
π‘ β (π + πΏ, 2π + πΏ)
π
(b) Rest
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π
LAB
π‘ β (2π + πΏ, 2 π + 2πΏ)
π
(a) Probe interaction
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π
LAB
π‘ = 2(π + πΏ)
π
πβ²
(a) Probe interaction
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π
LAB
π‘ β (2π + 2πΏ, 3π + 2πΏ)
(b) Rest
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π
LAB
π‘ = π(π + πΏ)
π returned probes
(a) Probe interaction
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Heralding measurement
π
|π(0)LAB
π‘ = π(π + πΏ)
(c) Probe processing
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In the language of process diagrams
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)LAB
π‘ = π
Initial conditions
π = πβππ»0π π
π
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(a) Probe interaction
π = πβππ»0π
π
LAB
π‘ β (π, π + πΏ)
π Unitary interaction
π
π
π π
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(b) Rest
π = πβππ»0π
π
LAB
π‘ β (π + πΏ, 2π + πΏ)π
π
π
π π
π
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π = πβππ»0π π
π
π π
π
(a) Probe interaction
π
π
π
π
LAB
π‘ = π(π + πΏ)
πreturned probes
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π = πβππ»0π
(c) Probe processing
π
|π(0)LAB
π‘ = π(π + πΏ)
π
π
π π
π
π
π
π
π
π₯ = 0
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π
π
π π
π
π
π
π
π
π,π unknown
π₯ = 0
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π
π
π π
π
π
π
π
π
π
π
π π π
ππ
π,π unknown
π₯ = 0
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(βπ)
ππ
π
π
ππ
πQuantum resetting
protocol
π₯ = 0
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(βπ)
ππ
π
π
ππ
πQuantum resetting
protocol
π₯ = 0
Desideratum: π(π₯ = 0|π) β 0, for all π
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π
π
π
ππ
ππ = ππβ¨ππ
π
ππ
ππ
ππ
π₯ = 0
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π = ππβ¨ππ Resetting protocol will fail with probability 1
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(βπ)
ππ
π
π
ππ
πQuantum resetting
protocol
π₯ = 0
Desideratum: π(π₯ = 0|π) β 0, for all π
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π
π
π
ππ
π₯ = 0
(βπ)
π
πQuantum resetting
protocol
π(π₯ = 0|π) β 0, except for a subset of unitaries of zero measure
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Do quantum resetting protocols exist?
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π
LAB
Scenario
ππ = 2
ππ = 2
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π
LAB
π‘ = 4(π + πΏ)
4 returned probes
Scenario
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π
π
π
π
ππ
π
|πβ
|πβ
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π
π
π
π
ππ
π
|πβ
|πβ
Singlet state preparation
|πβ =1
2( |01 β |10 )
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π
π
π
π
ππ
π
|πβ
|πβ
Projection onto the spacespanned by
π₯ = 0
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Why does this work?
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π
π
π
π
ππ
π
|πβ
|πβ
Projection onto the spacespanned by
π₯ = 0
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π
π
π
π
ππ
|πβ
|πβ
|0 |0 |0 |0ΰ΅Ώ|π(π‘π)
|π(0)
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π
π
π
π
ππ
|πβ
|πβ
|0 |0 |0 |0
ΰ΅Ώ|π(π‘π) =1
2[π0,0, π0,1]
2 |π(0)
ΰ΅Ώ|π(π‘π)
|π(0)
πππ = (ππβ¨Ϋ¦π|π)π(ππβ¨| π π)
2x2 complex matrices
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π΄, π΅, 2 Γ 2 matrices
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π΄, π΅ =
π=0,1,2,3
ππππ
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
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π΄, π΅ =
π=0,1,2,3
ππππ
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
π0 = 0Tr( π΄, π΅ ) = 0
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π΄, π΅ =
π=1,2,3
ππππ
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
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π΄, π΅ 2 =
π=1,2,3
ππππ
2
=
π=1,2,3
ππ2 π
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
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π΄, π΅ 2 =
π=1,2,3
ππππ
2
=
π=1,2,3
ππ2 π
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matricesCentral polynomial for dimension 2
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π
π
π
π
ππ
|πβ
|πβ
|0 |0 |0 |0
ΰ΅Ώ|π(π‘π) =1
2[π0,0, π0,1]
2 |π(0)
ΰ΅Ώ|π(π‘π)
|π(0)
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π
π
π
π
ππ
|πβ
|πβ
|0 |0 |0 |0
ΰ΅Ώ|π(π‘π) =1
2[π0,0, π0,1]
2 |π(0) β |π(0)
ΰ΅Ώ|π(π‘π)
|π(0)
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π
π
π
π
ππ
|πβ
|πβ
ΰ΅Ώ|π(π‘π) β |π(0)
ΰ΅Ώ|π(π‘π)
|π(0)
Similarly,
|ππ
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π
π
π
π
ππ
π
|πβ
|πβ
π π₯ = 0 π β [0, 1]
π₯ = 0
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π
π
π
π
ππ
π
|πβ
|πβ
π π₯ = 0 π β [0, 1]
π₯ = 0
E.g.: π = ππβ¨ππ
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π
π
π
π
ππ
π
|πβ
|πβ
π π₯ = 0 π β [0, 1]
π₯ = 0
E.g.: π =ππ₯β¨ππ§+πππ¦β¨ππ₯
2
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π
π
π
π
ππ
π
|πβ
|πβ
ΰΆ±πππ π₯ = 0 π β 0.2170
Average probability of success for completey unknown π
π₯ = 0
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Generalization
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π
π
π
ππ
(βπ)
π
n = O ππ3
[O ππ2 log ππ ]
For all ππ
π₯ = 0
π(π₯ = 0|π) β 0, except for a subset of unitaries of zero measure
ππ = 2
MN, arXiv:1710.02470
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Characterization of all quantum resetting protocols (practical for n=4)
MN, arXiv:1710.02470
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Heuristics to identify optimal strategies for high numbers of probes
π²8, ΰ·©π²8,
π²9
π = 8, ππ = 2
π = 9, ππ = 3
MN, arXiv:1710.02470
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What if the protocol fails?
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π
π
π
π
π
π
π
|πβ
|πβ
π₯ = 0
π
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π
π
π
π
π
π
π
|πβ
|πβ
π₯ = 1
π
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π
π
π
π
π
π
π
|πβ
|πβ
π₯ = 1
π
Suppose that thelast measurement
is a non-demolition one π΄
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π
π
π
π
π
π
π
|πβ
|πβ
|πβ
π₯ = 1
π¦ = 0
π
π
π
πβ²
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π
π
π
π
π
π
π
|πβ
|πβ
π₯ = 1
|π ππ΄ =
π
ππ(π) |π π |π π΄
ππ π , non-central matrix polynomialsof degree 4 on the variables
πππ = (ππβ¨Ϋ¦π|π)π(ππβ¨| π π)
π΄
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π
π
π
π
π
π
π
|πβ
|πβ
|πβ
π₯ = 1
π¦ = 0
π
π
πβ²
ΰ΅Ώ|π(π‘π) π=
π
ππ(π)ππ(π) |π(0) π
ππ π , matrix polynomials of degree2, such that Οπ ππ(π)ππ(π) is a central
polynomial
ΰ΅Ώ|π(π‘π)
|π(0)
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ΰΆ±πππ π₯ = 0 π β 0.6585
Average probability of success for completey unknown π
Undoing six possible failures
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Experimental implementation?
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IBM Q Experience
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π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
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π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
Implementation with single-qubit gates and CNOTS?
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π
π
π
π
π
π π
2
π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
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π
π
π
π
π
π π
2
π βπ
2
π
π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
π (π) π (βπ)π (β2π)
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π
π
π
π
π
π π
2
π βπ
2
π
π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
π (π) π (βπ)
π
π (β2π) π
π
π (π) π (βπ)π (β2π)
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π
π
π
π
π
π π
2
π βπ
2
π
π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
π (π) π (βπ)
π
π (β2π) π
π
π (π) π (βπ)π (β2π)
π βπ
2π
π
2
π βπ
2 π π
2
π
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IBM QX4: Raven
Largest circuit depth allowed!
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IBM QX4: Raven
Target system
π
Preparation of | π+ β¨2
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Theoretical simulation
IBM QX4: Raven
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Successful post-selection codes
IBM QX4: Raven
Theoretical simulation
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| 0 |0Ϋ¦ π ππ§ {0,1}
Measurement resultson target qubit
IBM QX4: Raven
Theoretical simulation
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IBM QX4: Raven
Crude reality
![Page 144: Resetting uncontrolled quantum systemsTime travel with wormholes K. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy, Commonwealth Fund Book Program (1994). ... The](https://reader034.fdocuments.in/reader034/viewer/2022050110/5f4818f4f2e67650c94c8f82/html5/thumbnails/144.jpg)
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π
π
π
π
ππ
|πβ
|πβ
|π |π |π |πΰ΅Ώ|π(π‘π)
|π(0)
Experimental implementation: Prototype II
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IBM QX2: Sparrow
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IBM QX2: Sparrow
Target system
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Successful post-selection codes
IBM QX2: Sparrow
Theoretical simulation
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IBM QX2: Sparrow
Measurement resultson target qubit
| 0 |0Ϋ¦ π ππ§ {0,1}
Theoretical simulation
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IBM QX2: Sparrow
Crude reality
![Page 151: Resetting uncontrolled quantum systemsTime travel with wormholes K. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy, Commonwealth Fund Book Program (1994). ... The](https://reader034.fdocuments.in/reader034/viewer/2022050110/5f4818f4f2e67650c94c8f82/html5/thumbnails/151.jpg)
IBM QX2: Sparrow
![Page 152: Resetting uncontrolled quantum systemsTime travel with wormholes K. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy, Commonwealth Fund Book Program (1994). ... The](https://reader034.fdocuments.in/reader034/viewer/2022050110/5f4818f4f2e67650c94c8f82/html5/thumbnails/152.jpg)
Conclusion
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Do there exist protocols with average probability of success (withprior ππ) arbitrarily close to 1?
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
Can we shorten resetting protocols?
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
Ξ β₯ 3π
Can we shorten resetting protocols?
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π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
Ξ βͺ π?
Can we shorten resetting protocols?
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Can we fast-forward?
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = Ξ
π
|π(0)
π‘ = 0
Ξ βͺ π?
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LAB
π
Can we shorten resetting protocols?/Can we fast-forward?
Hint: use which-path superpositions.
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Simple experimental implementation?
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Refocusing
I. S. B. Sardharwalla, T. S. Cubitt, A, W. Harrow, N. Linden, arXiv:1602.07963
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
We ignore π»0, but any operation on S is allowed
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(βπ)
π
π
π
π
β
unitaries
π
I. S. B. Sardharwalla, T. S. Cubitt, A, W. Harrow, N. Linden, arXiv:1602.07963
π = πβππ»0π
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Refocusing (obvious solution)
π
π
π = πβππ»0ππ
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
I. S. B. Sardharwalla, T. S. Cubitt, A, W. Harrow, N. Linden, arXiv:1602.07963
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Refocusing (obvious solution)
π
π
π΄π = πβππ»0ππ
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
I. S. B. Sardharwalla, T. S. Cubitt, A, W. Harrow, N. Linden, arXiv:1602.07963
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π
π
π
π
π
I. S. B. Sardharwalla, T. S. Cubitt, A, W. Harrow, N. Linden, arXiv:1602.07963
π
π
Channel tomography
π΄
Refocusing (obvious solution)
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
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π
π
π
π
π
πβ1
π
π
Channel tomography
π΄
Refocusing (obvious solution)
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
I. S. B. Sardharwalla, T. S. Cubitt, A, W. Harrow, N. Linden, arXiv:1602.07963
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π
π
π
π
π
πβ1
π
π
Channel tomography
(βπ)
π
β
π΄
Refocusing (obvious solution)
|π(0)
|π(0)
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
I. S. B. Sardharwalla, T. S. Cubitt, A, W. Harrow, N. Linden, arXiv:1602.07963