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 πβ²
πβ² β π
Time warp: operational definition
{π π‘ : π‘}
π 0Time warpfor t β [0, π]
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
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
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)
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) ?
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
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
π π π
ππ
π,π unknown
π₯ = 0
(βπ)
ππ
π
π
ππ
πQuantum resetting
protocol
π₯ = 0
Desideratum: π(π₯ = 0|π) β 0, for all π
(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
π
π
π
π
ππ
π
|πβ
|πβ
Singlet state preparation
|πβ =1
2( |01 β |10 )
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΅ =
π=0,1,2,3
ππππ
ππ, Pauli matrices
π΄, π΅, 2 Γ 2 matrices
π0 = 0Tr( π΄, π΅ ) = 0
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
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
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
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
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
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
| 0 |0Ϋ¦ π ππ§ {0,1}
Measurement resultson target qubit
IBM QX4: Raven
Theoretical simulation
π
π
π
π
ππ
|πβ
|πβ
|π |π |π |πΰ΅Ώ|π(π‘π)
|π(0)
Experimental implementation: Prototype II
IBM QX2: Sparrow
Measurement resultson target qubit
| 0 |0Ϋ¦ π ππ§ {0,1}
Theoretical simulation
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.
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