resetting uncontrolled quantum systemstime travel with wormholes k. thorne, black holes and time...
TRANSCRIPT
Resetting uncontrolled quantum systems
Institute for Quantum Optics and Quantum Information (IQOQI), Vienna
Miguel NavascuΓ©s
MN, arXiv:1710.02470
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
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
π‘ = 0
π‘ = 1 hour
π‘ = 2 hours
π‘ = 0
π‘ = 0
Time warp for two hours
π‘ = 2 hours
What should we find in the box if we are to claim a time warp experience?
Not expected
π‘ = 2 hours
π‘ = 2 hours
Not expected
π‘ = 2 hours
Not expected
π‘ = 2 hours, no intervention
Expected
π‘ = 2 hours
Not expected
π‘ = 2 hours, no intervention
Expected
π‘ = 2 hours
Not expected
π‘ = 2 hours, no intervention
π‘ = 2 hours
ExpectedNot expected
π‘ = 2 hours, no intervention
{π π‘ : π‘}
Time warp: operational definition
{π π‘ : π‘}
π 0
π 1π 2
Time warp: operational definition
Time warp protocol for t β [0, π]
Time warp: operational definition
π πβ²
πβ² β π
Time warp: operational definition
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
A brief history of time warp
King Raivata and princess Revati
(400BC?)
Peter Damian(1007-1072)
(first edition in 1895)
Warping time physically
π πβ²
0 < πβ² < π
Time warp in special relativity
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
GΓΆdel spacetime
M. Buser, E. Kajari and W. P. Schleich, New J. Phys. 15 013063 (2013).
K. GΓΆdel, Rev. Mod. Phys. 21 447 (1949).
Time travelwith
wormholes
K. Thorne, Black Holes and Time Warps: Einstein'sOutrageous Legacy, Commonwealth Fund Book Program(1994).
π πβ²
πβ² β π
Time warp with time machines
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
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)
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)
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)
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
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
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
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
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
π πβ²
πβ² β π
Time warp with the time translator
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
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)
βThe time translator has the same chances of succeeding as I have of delocalizing and
relocalizing somewhere elseβ
L. Vaidman
Time warp, the lame way
π 0
π 0
Time warp, the lame way
π 0
Time warp, the lame way
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
π 0
Time warp, the lame way
π β5732
Time warp, the lame way
π
π
|π =
π
ππ |π
These proposals do not require control or knowledge of the physical system that we influence
π
π
|π =
π
ππ |π
All interesting proposals to achieve time warp rely on special or general relativity
π πβ²
πβ² < 0
Main result
π 0Time warpfor t β [0, π]
Uncontrolled system
Non-relativisticquantum physics
π πβ²
πβ² < 0
Main result
π 0Time warpfor t β [0, π]
Uncontrolled system
Non-relativisticquantum physics
π πβ²
πβ² < 0
Main result
π 0Time warpfor t β [0, π]
Uncontrolled system
Non-relativisticquantum physics
reasonable probability of success
Scenario
Goal
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π > 0
π
|π(0)
π‘ = π + Ξ
Obvious solution
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
|π(0) = ΰ΅Ώπ+ππ»0π|π(π)
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
|π(0) = ΰ΅Ώπ+ππ»0π|π(π)
Obvious solution
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
S, uncontrolled
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π»0?S, uncontrolledπ π
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π»0?S, uncontrolled
π π
π π
π»ππ(π)?π π
π π
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) ?
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.
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) ?
π
Controlled lab
S, uncontrolled
Resetting
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
MN, arXiv:1710.02470
Resetting
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
MN, arXiv:1710.02470
Exact past state
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 )
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
Sketch of a quantum resetting protocol
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)LAB
π‘ = π
Initial conditions
(a) Probe interaction
π
LAB
π‘ β (π, π + πΏ)
π
π
LAB
π‘ = π + πΏ
π
(a) Probe interaction
π
LAB
π‘ β (π + πΏ, 2π + πΏ)
π
(b) Rest
π
LAB
π‘ β (2π + πΏ, 2 π + 2πΏ)
π
(a) Probe interaction
π
LAB
π‘ = 2(π + πΏ)
π
πβ²
(a) Probe interaction
π
LAB
π‘ β (2π + 2πΏ, 3π + 2πΏ)
(b) Rest
π
LAB
π‘ = π(π + πΏ)
π returned probes
(a) Probe interaction
Heralding measurement
π
|π(0)LAB
π‘ = π(π + πΏ)
(c) Probe processing
In the language of process diagrams
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)LAB
π‘ = π
Initial conditions
π = πβππ»0π π
π
(a) Probe interaction
π = πβππ»0π
π
LAB
π‘ β (π, π + πΏ)
π Unitary interaction
π
π
π π
(b) Rest
π = πβππ»0π
π
LAB
π‘ β (π + πΏ, 2π + πΏ)π
π
π
π π
π
π = πβππ»0π π
π
π π
π
(a) Probe interaction
π
π
π
π
LAB
π‘ = π(π + πΏ)
πreturned probes
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
π
π
π π
π
π
π
π
π
π,π unknown
π₯ = 0
π
π
π π
π
π
π
π
π
π
π
π π π
ππ
π,π unknown
π₯ = 0
(βπ)
ππ
π
π
ππ
πQuantum resetting
protocol
π₯ = 0
(βπ)
ππ
π
π
ππ
πQuantum resetting
protocol
π₯ = 0
Desideratum: π(π₯ = 0|π) β 0, for all π
π
π
π
ππ
ππ = ππβ¨ππ
π
ππ
ππ
ππ
π₯ = 0
π = ππβ¨ππ Resetting protocol will fail with probability 1
(βπ)
ππ
π
π
ππ
πQuantum resetting
protocol
π₯ = 0
Desideratum: π(π₯ = 0|π) β 0, for all π
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
Do quantum resetting protocols exist?
π
LAB
Scenario
ππ = 2
ππ = 2
π
LAB
π‘ = 4(π + πΏ)
4 returned probes
Scenario
π
π
π
π
ππ
π
|πβ
|πβ
π
π
π
π
ππ
π
|πβ
|πβ
Singlet state preparation
|πβ =1
2( |01 β |10 )
π
π
π
π
ππ
π
|πβ
|πβ
Projection onto the spacespanned by
π₯ = 0
Why does this work?
π
π
π
π
ππ
π
|πβ
|πβ
Projection onto the spacespanned by
π₯ = 0
π
π
π
π
ππ
|πβ
|πβ
|0 |0 |0 |0ΰ΅Ώ|π(π‘π)
|π(0)
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
π΄, π΅, 2 Γ 2 matrices
π΄, π΅ =
π=0,1,2,3
ππππ
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
π΄, π΅ =
π=0,1,2,3
ππππ
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
π0 = 0Tr( π΄, π΅ ) = 0
π΄, π΅ =
π=1,2,3
ππππ
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
π΄, π΅ 2 =
π=1,2,3
ππππ
2
=
π=1,2,3
ππ2 π
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
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
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)
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)
π
π
π
π
ππ
|πβ
|πβ
ΰ΅Ώ|π(π‘π) β |π(0)
ΰ΅Ώ|π(π‘π)
|π(0)
Similarly,
|ππ
π
π
π
π
ππ
π
|πβ
|πβ
π π₯ = 0 π β [0, 1]
π₯ = 0
π
π
π
π
ππ
π
|πβ
|πβ
π π₯ = 0 π β [0, 1]
π₯ = 0
E.g.: π = ππβ¨ππ
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
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
Generalization
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
Characterization of all quantum resetting protocols (practical for n=4)
MN, arXiv:1710.02470
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
What if the protocol fails?
π
π
π
π
π
π
π
|πβ
|πβ
π₯ = 0
π
π
π
π
π
π
π
π
|πβ
|πβ
π₯ = 1
π
π
π
π
π
π
π
π
|πβ
|πβ
π₯ = 1
π
Suppose that thelast measurement
is a non-demolition one π΄
π
π
π
π
π
π
π
|πβ
|πβ
|πβ
π₯ = 1
π¦ = 0
π
π
π
πβ²
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΄
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)
ΰΆ±πππ π₯ = 0 π β 0.6585
Average probability of success for completey unknown π
Undoing six possible failures
Experimental implementation?
IBM Q Experience
π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
Implementation with single-qubit gates and CNOTS?
π
π
π
π
π
π π
2
π
π
π
π
ππ
π
|πβ
|πβ
π₯ = 0 Experimental implementation: Prototype II
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)
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
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
IBM QX4: Raven
Largest circuit depth allowed!
IBM QX4: Raven
Target system
π
Preparation of | π+ β¨2
Theoretical simulation
IBM QX4: Raven
Successful post-selection codes
IBM QX4: Raven
Theoretical simulation
| 0 |0Ϋ¦ π ππ§ {0,1}
Measurement resultson target qubit
IBM QX4: Raven
Theoretical simulation
IBM QX4: Raven
Crude reality
π
π
π
π
ππ
|πβ
|πβ
|π |π |π |πΰ΅Ώ|π(π‘π)
|π(0)
Experimental implementation: Prototype II
IBM QX2: Sparrow
IBM QX2: Sparrow
Target system
Successful post-selection codes
IBM QX2: Sparrow
Theoretical simulation
IBM QX2: Sparrow
Measurement resultson target qubit
| 0 |0Ϋ¦ π ππ§ {0,1}
Theoretical simulation
IBM QX2: Sparrow
Crude reality
IBM QX2: Sparrow
Conclusion
Do there exist protocols with average probability of success (withprior ππ) arbitrarily close to 1?
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
Can we shorten resetting protocols?
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
Ξ β₯ 3π
Can we shorten resetting protocols?
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = π
π
|π(0)
π‘ = π + Ξ
Ξ βͺ π?
Can we shorten resetting protocols?
Can we fast-forward?
π
|π(π) = ΰ΅Ώπβππ»0π|π(0)
π‘ = Ξ
π
|π(0)
π‘ = 0
Ξ βͺ π?
LAB
π
Can we shorten resetting protocols?/Can we fast-forward?
Hint: use which-path superpositions.
Simple experimental implementation?
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
(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
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
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
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
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
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