Page 1
Random states for robust quantum metrology 1 / 20
Random states for robust quantum metrology
Micha l Oszmaniec, Remigiusz Augusiak, Christian Gogolin,Janek Ko lodynski, Antonio Acın, and Maciej Lewenstein
arXiv: 1602.05407
ICFO - The Institute of Photonic Sciences
SIQS Workshop at San Servolo Venice2016-03-14
Page 2
Random states for robust quantum metrology | Motivation 2 / 20
Quantum metrology is an emerging quantumtechnology. . .
[that] aims to use strong quantum correlationsin order to develop systems, involving large-scaleentanglement, that outperform classical systemsin a series of relevant applications.”
— http://qurope.eu/projects/siqs
. . . “
[1] LIGO Collaboration, Nature Phys., 7 (2011), 962–965
[2] LIGO Collaboration, Nature Photon., 7 (2013), 613–619
Page 3
Random states for robust quantum metrology | Motivation 2 / 20
Quantum metrology is an emerging quantumtechnology. . .
[that] aims to use strong quantum correlationsin order to develop systems, involving large-scaleentanglement, that outperform classical systemsin a series of relevant applications.”
— http://qurope.eu/projects/siqs
. . . “
[1] LIGO Collaboration, Nature Phys., 7 (2011), 962–965
[2] LIGO Collaboration, Nature Photon., 7 (2013), 613–619
Page 4
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation
1
V1 e− ihϕ
2
V2 e− ihϕ
N
VN e− ihϕ
N-k
VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
?
Page 5
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation
1 V1 e− ihϕ
2 V2 e− ihϕ
N VN e− ihϕ
N-k VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
...
Page 6
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation (with local unitary optimization)
1 V1 e− ihϕ
2 V2 e− ihϕ
N VN e− ihϕ
N-k VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
......
Page 7
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation (with local unitary optimization)
1 V1 e− ihϕ
2 V2 e− ihϕ
N VN e− ihϕ
N-k VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
...
...
...
...
Page 8
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation (with local unitary optimization)
1 V1 e− ihϕ
2 V2 e− ihϕ
N VN e− ihϕ
N-k VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
...
...
...
...
Page 9
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation (with local unitary optimization)
1 V1 e− ihϕ
2 V2 e− ihϕ
N VN e− ihϕ
N-k VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
...
...
Page 10
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation (without local unitary optimization)
1 V1 e− ihϕ
2 V2 e− ihϕ
N VN e− ihϕ
N-k VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
...
...
Page 11
Random states for robust quantum metrology | Phase estimation 3 / 20
Phase estimation
Phase estimation (without local unitary optimization)
1 V1 e− ihϕ
2 V2 e− ihϕ
N VN e− ihϕ
N-k VN−k e− ihϕ
M
⊗ν
=⇒ ϕ
ρ LU encoding POVM estimator
How good isthe estimation?Mean squarederror: ∆2ϕ
...
...
Page 12
Random states for robust quantum metrology | Phase estimation 4 / 20
Cramer Rao bound
Cramer-Rao Bound [3, 4] for unbiased estimators:
∆2ϕ ≥ 1
ν Fcl
({pn|ϕ
})
wherepn|ϕ = probability that M yields outcome n given ϕ
and the classical Fischer information (FI)
Fcl
({pn|ϕ
}):=∑n
1
pn|ϕ
(dpn|ϕ
dϕ
)2
Moreover: Tight in the limit of many repetitions ν →∞ [5].
[3] H. Cramer, Princeton Univ. Press., 1946
[4] C. R. Rao, Bulletin of the Calcutta Mathematical Society, 37 (1945), 81
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
Page 13
Random states for robust quantum metrology | Phase estimation 4 / 20
Cramer Rao bound
Cramer-Rao Bound [3, 4] for unbiased estimators:
∆2ϕ ≥ 1
ν Fcl
({pn|ϕ
})where
pn|ϕ = probability that M yields outcome n given ϕ
and the classical Fischer information (FI)
Fcl
({pn|ϕ
}):=∑n
1
pn|ϕ
(dpn|ϕ
dϕ
)2
Moreover: Tight in the limit of many repetitions ν →∞ [5].
[3] H. Cramer, Princeton Univ. Press., 1946
[4] C. R. Rao, Bulletin of the Calcutta Mathematical Society, 37 (1945), 81
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
Page 14
Random states for robust quantum metrology | Phase estimation 4 / 20
Cramer Rao bound
Cramer-Rao Bound [3, 4] for unbiased estimators:
∆2ϕ ≥ 1
ν Fcl
({pn|ϕ
})where
pn|ϕ = probability that M yields outcome n given ϕ
and the classical Fischer information (FI)
Fcl
({pn|ϕ
}):=∑n
1
pn|ϕ
(dpn|ϕ
dϕ
)2
Moreover: Tight in the limit of many repetitions ν →∞ [5].
[3] H. Cramer, Princeton Univ. Press., 1946
[4] C. R. Rao, Bulletin of the Calcutta Mathematical Society, 37 (1945), 81
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
Page 15
Random states for robust quantum metrology | Phase estimation 5 / 20
Quantifying the usefulness of ρ
How to quantify the usefulness of ρ for a given H =∑N
j=1 h(j)?
Quantum Fischer information (QFI) [6]
F(ρ,H) := supM
Fcl
({pn|ϕ
})
Again: Tight in the limit of many repetitions ν →∞.
Fine print:
local estimation regime [5]
adaptive measurements [7]
LU optimization:
FLU (ρ,H) := supV=V (1)⊗···⊗V (N)
F(V ρV †, H)
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
[6] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett., 72 (1994), 3439–3443
[7] A. Fujiwara, J. Phys. A: Math. Gen., 39.40 (2006), 12489
Page 16
Random states for robust quantum metrology | Phase estimation 5 / 20
Quantifying the usefulness of ρ
How to quantify the usefulness of ρ for a given H =∑N
j=1 h(j)?
Quantum Fischer information (QFI) [6]
F(ρ,H) := supM
Fcl
({pn|ϕ
})
Again: Tight in the limit of many repetitions ν →∞.
Fine print:
local estimation regime [5]
adaptive measurements [7]
LU optimization:
FLU (ρ,H) := supV=V (1)⊗···⊗V (N)
F(V ρV †, H)
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
[6] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett., 72 (1994), 3439–3443
[7] A. Fujiwara, J. Phys. A: Math. Gen., 39.40 (2006), 12489
Page 17
Random states for robust quantum metrology | Phase estimation 5 / 20
Quantifying the usefulness of ρ
How to quantify the usefulness of ρ for a given H =∑N
j=1 h(j)?
Quantum Fischer information (QFI) [6]
F(ρ,H) := supM
Fcl
({pn|ϕ
})
Again: Tight in the limit of many repetitions ν →∞.
Fine print:
local estimation regime [5]
adaptive measurements [7]
LU optimization:
FLU (ρ,H) := supV=V (1)⊗···⊗V (N)
F(V ρV †, H)
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
[6] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett., 72 (1994), 3439–3443
[7] A. Fujiwara, J. Phys. A: Math. Gen., 39.40 (2006), 12489
Page 18
Random states for robust quantum metrology | Phase estimation 5 / 20
Quantifying the usefulness of ρ
How to quantify the usefulness of ρ for a given H =∑N
j=1 h(j)?
Quantum Fischer information (QFI) [6]
F(ρ,H) := supM
Fcl
({pn|ϕ
})
Again: Tight in the limit of many repetitions ν →∞.
Fine print:
local estimation regime [5]
adaptive measurements [7]
LU optimization:
FLU (ρ,H) := supV=V (1)⊗···⊗V (N)
F(V ρV †, H)
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
[6] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett., 72 (1994), 3439–3443
[7] A. Fujiwara, J. Phys. A: Math. Gen., 39.40 (2006), 12489
Page 19
Random states for robust quantum metrology | Phase estimation 5 / 20
Quantifying the usefulness of ρ
How to quantify the usefulness of ρ for a given H =∑N
j=1 h(j)?
Quantum Fischer information (QFI) [6]
F(ρ,H) := supM
Fcl
({pn|ϕ
})
Again: Tight in the limit of many repetitions ν →∞.
Fine print:
local estimation regime [5]
adaptive measurements [7]
LU optimization:
FLU (ρ,H) := supV=V (1)⊗···⊗V (N)
F(V ρV †, H)
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
[6] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett., 72 (1994), 3439–3443
[7] A. Fujiwara, J. Phys. A: Math. Gen., 39.40 (2006), 12489
Page 20
Random states for robust quantum metrology | Phase estimation 5 / 20
Quantifying the usefulness of ρ
How to quantify the usefulness of ρ for a given H =∑N
j=1 h(j)?
Quantum Fischer information (QFI) [6]
F(ρ,H) := supM
Fcl
({pn|ϕ
})
Again: Tight in the limit of many repetitions ν →∞.
Fine print:
local estimation regime [5]
adaptive measurements [7]
LU optimization:
FLU (ρ,H) := supV=V (1)⊗···⊗V (N)
F(V ρV †, H)
[5] O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen., 30 (2000), 4481
[6] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett., 72 (1994), 3439–3443
[7] A. Fujiwara, J. Phys. A: Math. Gen., 39.40 (2006), 12489
Page 21
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 22
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 23
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 24
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 25
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 26
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 27
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 28
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
not useful
F /∈ O(N) useful
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Page 29
Random states for robust quantum metrology | Phase estimation 6 / 20
How useful are different states?
Standard quantum limit (SQL): F ∈ O(N)
Achieved by product states
Verified by all separable states [8–10]
Heisenberg limit (HL): F ∈ Θ(N2)
Achieve by GHZ state (= N00N state) [11]
Multipartite entanglement is necessary [12, 13] . . .
. . . but not sufficient [14]
not useful
F /∈ O(N) useful
[8] G. Toth and I. Apellaniz, J. Phys. A: Math. Theor., 47 (2014), 424006
[9] R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Ko lodynski, in: Progress in Optics, ed. by E. Wolf, vol. 60, Elsevier, 2015,pp. 345–435
[10] V. Giovannetti, S. Lloyd, and L. Maccone, Science, 306 (2004), 1330–1336
[11] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett., 96 (2006), 010401
[12] P. Hyllus et al., Phys. Rev. A, 85 (2 2012), 022321
[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
[14] P. Hyllus, O. Guhne, and A. Smerzi, Phys. Rev. A, 82 (1 2010), 012337
Questions:
How generic is usefulness?
Which types of states are useful?
And for which measurements?
How robust is the usefulness?
What is the role of entanglement?
Page 30
Random states for robust quantum metrology | Random states 7 / 20
Random states
more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Page 31
Random states for robust quantum metrology | Random states 7 / 20
Random states
more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Page 32
Random states for robust quantum metrology | Random states 7 / 20
Random states
more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Early summary of results
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Are robust against depolarization and particle loss.
They can be generated with random optical circuits.
Page 33
Random states for robust quantum metrology | Random states 7 / 20
Random states
more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Early summary of results
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Are robust against depolarization and particle loss.
They can be generated with random optical circuits.
Page 34
Random states for robust quantum metrology | Random states 7 / 20
Random states
more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Early summary of results
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Are robust against depolarization and particle loss.
They can be generated with random optical circuits.
Page 35
Random states for robust quantum metrology | Random states 7 / 20
Random states
more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Early summary of results
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Are robust against depolarization and particle loss.
They can be generated with random optical circuits.
Page 36
Random states for robust quantum metrology | Random states 7 / 20
Random states
more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Early summary of results
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Are robust against depolarization and particle loss.
They can be generated with random optical circuits.
Page 37
Random states for robust quantum metrology | Random states 7 / 20
Random states more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Page 38
Random states for robust quantum metrology | Random states 7 / 20
Random states more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Page 39
Random states for robust quantum metrology | Random states 7 / 20
Random states more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Page 40
Random states for robust quantum metrology | Random states 7 / 20
Random states more seriously
Haar measure:µ (H) is the unique uniform measure on SU (H)
Random isospectral states:Fix density matrix σ on some Hilbert space H, then
U σ U † with U ∼ µ (H)
is a random isospectral state.
We will consider:
d-level spins HN := (Cd)⊗N
symmetric (bosonic) subspace SN := span{|ψ〉⊗N : |ψ〉 ∈ Cd}
Page 41
Random states for robust quantum metrology | Random states 8 / 20
Measure concentration
Levy’s lemma
Let f : SU (H)→ R be L-Lipschitz, then
PrU∼µ(H)
(∣∣∣∣f (U)− EU∼µ(H)
f
∣∣∣∣ ≥ ε) ≤ 2 exp
(−|H| ε
2
4L2
).
We apply this to
F(U) := F(U σ U †, H) with σ,H fixed
and similar functions FLU, Fcl, . . .
Page 42
Random states for robust quantum metrology | Random states 8 / 20
Measure concentration
Levy’s lemma
Let f : SU (H)→ R be L-Lipschitz, then
PrU∼µ(H)
(∣∣∣∣f (U)− EU∼µ(H)
f
∣∣∣∣ ≥ ε) ≤ 2 exp
(−|H| ε
2
4L2
).
We apply this to
F(U) := F(U σ U †, H) with σ,H fixed
and similar functions FLU, Fcl, . . .
Page 43
Random states for robust quantum metrology | Typical QFI of random states 9 / 20
Typical QFI of random states
Page 44
Random states for robust quantum metrology | Typical QFI of random states 10 / 20
Typical QFI of distinguishable particles
Random states of distinguishable particles are not useful for metrology:
Theorem
Pr
U∼µ(HN )
(random state is useful
)≤ exp
(−large in N
)
Page 45
Random states for robust quantum metrology | Typical QFI of random states 10 / 20
Typical QFI of distinguishable particles
Random states of distinguishable particles are not useful for metrology:
Theorem
PrU∼µ(HN )
(FLU(U) /∈ O (N)
)≤ exp
(−Θ
(|HN |N2
))
Page 46
Random states for robust quantum metrology | Typical QFI of random symmetric states 11 / 20
What about symmetric states?
Page 47
Random states for robust quantum metrology | Typical QFI of random symmetric states 12 / 20
Typical QFI of symmetric states
Random (bosonic) symmetric states are useful for metrology:
Theorem
PrU∼µ(SN )
(F(U) /∈ Θ(δ2N2)
)≤ exp
(−Θ(δ3Nd−1)
)
Where δ = dB
(σN ,
1|SN |
)is the Bures distance from the max. mixed state.
Page 48
Random states for robust quantum metrology | Typical QFI of random symmetric states 13 / 20
Robustness against noise
Example (Robustness against depolarization (d = 2))
Provided that δ = dB
(σN ,
1|SN |
)≥ 1/N1/3
the QFI typically scales at least like N4/3.
Theorem (Robustness against loss of k particles)Let Fk be the QFI after k particles were lost, then
PrU∼µ(SN )
(Fk(U) /∈ Θ
(δ22 N
2
kd
))≤ exp
(−Θ
(δ42 N
d−1
k2d
)),
where δ2 := ‖σN − 1|SN |‖HS.
Caveat: If k ∝ N , usefulness is lost, consistent with [15, 16].
[15] B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nature Phys., 7 (2011), 406–411
[16] R. Demkowicz-Dobrzanski, J. Ko lodynski, and M. Guta, Nat. Commun., 3 (2012), 1063
Page 49
Random states for robust quantum metrology | Typical QFI of random symmetric states 13 / 20
Robustness against noise
Example (Robustness against depolarization (d = 2))
Provided that δ = dB
(σN ,
1|SN |
)≥ 1/N1/3
the QFI typically scales at least like N4/3.
Theorem (Robustness against loss of k particles)Let Fk be the QFI after k particles were lost, then
PrU∼µ(SN )
(Fk(U) /∈ Θ
(δ22 N
2
kd
))≤ exp
(−Θ
(δ42 N
d−1
k2d
)),
where δ2 := ‖σN − 1|SN |‖HS.
Caveat: If k ∝ N , usefulness is lost, consistent with [15, 16].
[15] B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nature Phys., 7 (2011), 406–411
[16] R. Demkowicz-Dobrzanski, J. Ko lodynski, and M. Guta, Nat. Commun., 3 (2012), 1063
Page 50
Random states for robust quantum metrology | Typical QFI of random symmetric states 13 / 20
Robustness against noise
Example (Robustness against depolarization (d = 2))
Provided that δ = dB
(σN ,
1|SN |
)≥ 1/N1/3
the QFI typically scales at least like N4/3.
Theorem (Robustness against loss of k particles)Let Fk be the QFI after k particles were lost, then
PrU∼µ(SN )
(Fk(U) /∈ Θ
(δ22 N
2
kd
))≤ exp
(−Θ
(δ42 N
d−1
k2d
)),
where δ2 := ‖σN − 1|SN |‖HS.
Caveat: If k ∝ N , usefulness is lost, consistent with [15, 16].
[15] B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nature Phys., 7 (2011), 406–411
[16] R. Demkowicz-Dobrzanski, J. Ko lodynski, and M. Guta, Nat. Commun., 3 (2012), 1063
Page 51
Random states for robust quantum metrology | Attaining the Heisenberg limit with a fixed measurement 14 / 20
Can we access the “usefulness”?
Page 52
Random states for robust quantum metrology | Attaining the Heisenberg limit with a fixed measurement 15 / 20
Attaining the Heisenberg limit
Finding the right measurement can be tricky.
Which measurement is good usually depends on the state.
Adaptive measurements can be necessary.
Not for random states:
ψN ∼ SN
Theorem (Pure random symmetric states typically attain the HL)
PrU∼µ(SN )
(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)
)≤ exp (−Θ(N))
Concretely: ∀ϕ : EFcl ≥ 0.0244N2
Page 53
Random states for robust quantum metrology | Attaining the Heisenberg limit with a fixed measurement 15 / 20
Attaining the Heisenberg limit
Finding the right measurement can be tricky.
Which measurement is good usually depends on the state.
Adaptive measurements can be necessary.
Not for random states:
ψN ∼ SN
Theorem (Pure random symmetric states typically attain the HL)
PrU∼µ(SN )
(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)
)≤ exp (−Θ(N))
Concretely: ∀ϕ : EFcl ≥ 0.0244N2
Page 54
Random states for robust quantum metrology | Attaining the Heisenberg limit with a fixed measurement 15 / 20
Attaining the Heisenberg limit
Finding the right measurement can be tricky.
Which measurement is good usually depends on the state.
Adaptive measurements can be necessary.
Not for random states:
ψN ∼ SN
Theorem (Pure random symmetric states typically attain the HL)
PrU∼µ(SN )
(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)
)≤ exp (−Θ(N))
Concretely: ∀ϕ : EFcl ≥ 0.0244N2
Page 55
Random states for robust quantum metrology | Attaining the Heisenberg limit with a fixed measurement 15 / 20
Attaining the Heisenberg limit
Finding the right measurement can be tricky.
Which measurement is good usually depends on the state.
Adaptive measurements can be necessary.
Not for random states:
ψN ∼ SN
Theorem (Pure random symmetric states typically attain the HL)
PrU∼µ(SN )
(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)
)≤ exp (−Θ(N))
Concretely: ∀ϕ : EFcl ≥ 0.0244N2
Page 56
Random states for robust quantum metrology | Attaining the Heisenberg limit with a fixed measurement 15 / 20
Attaining the Heisenberg limit
Finding the right measurement can be tricky.
Which measurement is good usually depends on the state.
Adaptive measurements can be necessary.
Not for random states:
ψN ∼ SN
Theorem (Pure random symmetric states typically attain the HL)
PrU∼µ(SN )
(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)
)≤ exp (−Θ(N))
Concretely: ∀ϕ : EFcl ≥ 0.0244N2
Page 57
Random states for robust quantum metrology | Generating random symmetric states 16 / 20
How can we generate random symmetric states?
Page 58
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate
, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 59
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)
2 Add a non-trivial non-linear gate
, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 60
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate
, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 61
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 62
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 63
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 64
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Approximate t-designs
An ε-approximate t-design µε,t(H) approximates µ(H)
For any balanced monomial f : SU(H)→ R of degree t, itholds that∣∣∣∣ E
U∼µ(H)f(U)− E
U∼µε,t(H)f(U)
∣∣∣∣ ≤ ε/|H|tImplies that polynomials (like the QFI) also are close
Measure concentration, albeit weaker, also carries over [20]
Page 65
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Approximate t-designs
An ε-approximate t-design µε,t(H) approximates µ(H)
For any balanced monomial f : SU(H)→ R of degree t, itholds that∣∣∣∣ E
U∼µ(H)f(U)− E
U∼µε,t(H)f(U)
∣∣∣∣ ≤ ε/|H|t
Implies that polynomials (like the QFI) also are close
Measure concentration, albeit weaker, also carries over [20]
Page 66
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Approximate t-designs
An ε-approximate t-design µε,t(H) approximates µ(H)
For any balanced monomial f : SU(H)→ R of degree t, itholds that∣∣∣∣ E
U∼µ(H)f(U)− E
U∼µε,t(H)f(U)
∣∣∣∣ ≤ ε/|H|tImplies that polynomials (like the QFI) also are close
Measure concentration, albeit weaker, also carries over [20]
Page 67
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 68
Random states for robust quantum metrology | Generating random symmetric states 17 / 20
Generation with random circuits
States drawn with a unitary design would be sufficientand they can be generated with random circuits!
1 Take a good set of gates universal for 2-mode linear optics:
V1 :=1√5
(1 2 i2 i 1
)V2 :=
1√5
(1 −2−2 1
)V3 :=
1√5
(1 + 2 i 0
0 1− 2 i
)2 Add a non-trivial non-linear gate, like VXK := exp (− iπ na nb/3).
3 Combine them to random circuits of length K.
⇒ When K →∞ yields an approximate t-design for any t [17, 18].
! Results for qubits on random circuits efficiently yielding approximate unitarydesigns [18, 19] do not carry over.
Don’t know how K has to depend on N . =⇒ Numerics
[17] J. Bourgain and A. Gamburd, arXiv e-print, (2011), arXiv: quant-ph/1108.6264
[18] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki, arXiv e-print, (2015), arXiv: quant-ph/1208.0692
[19] R. A. Low, PhD thesis, University of Bristol, 2010, arXiv: quant-ph/1006.5227
[20] R. A. Low, Proc. Royal Soc. A, 465 (2009), 3289–3308
Page 69
Random states for robust quantum metrology | Generating random symmetric states 18 / 20
Numerics
N = 100, |ψN 〉 =∑N
n=0√xn|n,N − n〉
x0 = xN = 1/2 (N00N)
xn =(Nn
)1
2N(balanced)
x0 = 1 (polarized)
K = 60
1/ Fcl1/ F
Page 70
Random states for robust quantum metrology | Generating random symmetric states 18 / 20
Numerics
N = 100, |ψN 〉 =∑N
n=0√xn|n,N − n〉
x0 = xN = 1/2 (N00N)
xn =(Nn
)1
2N(balanced)
x0 = 1 (polarized)
K = 60
1/ Fcl1/ F
Page 71
Random states for robust quantum metrology | Conclusions 19 / 20
Conclusions
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Usefulness is robust against noise and particle loss.
They can be generated with random optical circuits.
Page 72
Random states for robust quantum metrology | Conclusions 19 / 20
Conclusions
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Usefulness is robust against noise and particle loss.
They can be generated with random optical circuits.
Page 73
Random states for robust quantum metrology | Conclusions 19 / 20
Conclusions
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Usefulness is robust against noise and particle loss.
They can be generated with random optical circuits.
Page 74
Random states for robust quantum metrology | Conclusions 19 / 20
Conclusions
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Usefulness is robust against noise and particle loss.
They can be generated with random optical circuits.
Page 75
Random states for robust quantum metrology | Conclusions 19 / 20
Conclusions
Random states of distinguishable particles not useful.
Random symmetric states (bosons) typically are useful.
Exists one fixed measurement to access their usefulness.
Usefulness is robust against noise and particle loss.
They can be generated with random optical circuits.
Page 76
Random states for robust quantum metrology | References 20 / 20
References
Thank you for your attention!slides on www.cgogolin.de
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Funded by: the European Research Council (ERC AdG OSYRIS and CoG QITBOX), the Axa Chair in Quantum InformationScience, the John Templeton Foundation, the EU (IP SIQS and QUIC), the Spanish MINECO (SEVERO OCHOA GRANT SEV-2015-0522 and FOQUS FIS2013-46768), and the Generalitat de Catalunya (SGR 874 and 875). M. O.: START scholarshipgranted by Foundation for Polish Science and NCN grant DEC-2013/09/N/ST1/02772; C. G.: MPQ-ICFO and ICFOnest+(FP7-PEOPLE-2013-COFUND); J. K.: EU MSCA IF QMETAPP