Random states for robust quantum metrology - cgogolin.de · Random states for robust quantum metrology 1 / 20 Random states for robust quantum metrology Micha l Oszmaniec, Remigiusz

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

http://arxiv.org/abs/1602.05407

http://www.icfo.eu/

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

http://qurope.eu/projects/siqs

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

http://qurope.eu/projects/siqs

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

?


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



...


Phase estimation

Phase estimation (with local unitary optimization)

1 V1 e− ihϕ

2 V2 e− ihϕ

N VN e− ihϕ


M

⊗ν

=⇒ ϕ



......


Phase estimation


1 V1 e− ihϕ

2 V2 e− ihϕ

N VN e− ihϕ


M

⊗ν

=⇒ ϕ



...

...

...

...


Phase estimation


1 V1 e− ihϕ

2 V2 e− ihϕ

N VN e− ihϕ


M

⊗ν

=⇒ ϕ



...

...

...

...


Phase estimation


1 V1 e− ihϕ

2 V2 e− ihϕ

N VN e− ihϕ


M

⊗ν

=⇒ ϕ



...

...


Phase estimation

Phase estimation (without local unitary optimization)

1 V1 e− ihϕ

2 V2 e− ihϕ

N VN e− ihϕ


M

⊗ν

=⇒ ϕ



...

...


Phase estimation

Phase estimation (without local unitary optimization)

1 V1 e− ihϕ

2 V2 e− ihϕ

N VN e− ihϕ


M

⊗ν

=⇒ ϕ



...

...


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


Cramer Rao bound


∆2ϕ ≥ 1

ν Fcl

({pn|ϕ

})where

pn|ϕ = probability that M yields outcome n given ϕ


Fcl

({pn|ϕ

}):=∑n

1

pn|ϕ

(dpn|ϕ

dϕ

)2






Cramer Rao bound


∆2ϕ ≥ 1

ν Fcl

({pn|ϕ

})where

pn|ϕ = probability that M yields outcome n given ϕ


Fcl

({pn|ϕ

}):=∑n

1

pn|ϕ

(dpn|ϕ

dϕ

)2






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)


[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




j=1 h(j)?


F(ρ,H) := supM

Fcl

({pn|ϕ

})


Fine print:



LU optimization:

FLU (ρ,H) := supV=V (1)⊗···⊗V (N)

F(V ρV †, H)







j=1 h(j)?


F(ρ,H) := supM

Fcl

({pn|ϕ

})


Fine print:



LU optimization:

FLU (ρ,H) := supV=V (1)⊗···⊗V (N)

F(V ρV †, H)







j=1 h(j)?


F(ρ,H) := supM

Fcl

({pn|ϕ

})


Fine print:



LU optimization:

FLU (ρ,H) := supV=V (1)⊗···⊗V (N)

F(V ρV †, H)







j=1 h(j)?


F(ρ,H) := supM

Fcl

({pn|ϕ

})


Fine print:



LU optimization:

FLU (ρ,H) := supV=V (1)⊗···⊗V (N)

F(V ρV †, H)







j=1 h(j)?


F(ρ,H) := supM

Fcl

({pn|ϕ

})


Fine print:



LU optimization:

FLU (ρ,H) := supV=V (1)⊗···⊗V (N)

F(V ρV †, H)





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















[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
















[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
















[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
















[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
















[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322
















[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322











not useful

F /∈ O(N) useful






[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322











not useful

F /∈ O(N) useful






[13] G. Toth, Phys. Rev. A, 85 (2 2012), 022322


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?

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}


Random states

more seriously





We will consider:




Random states

more seriously





We will consider:



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.


Random states

more seriously





We will consider:










Random states

more seriously





We will consider:










Random states

more seriously





We will consider:










Random states

more seriously





We will consider:










Random states more seriously





We will consider:









We will consider:









We will consider:









We will consider:




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, . . .


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, . . .

Random states for robust quantum metrology | Typical QFI of random states 9 / 20

Typical QFI of random states


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

)


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

))

Random states for robust quantum metrology | Typical QFI of random symmetric states 11 / 20

What about symmetric states?


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.


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





(σN ,

1|SN |

)≥ 1/N1/3



PrU∼µ(SN )

(Fk(U) /∈ Θ

(δ22 N

2

kd

))≤ exp

(−Θ

(δ42 N

d−1

k2d

)),









(σN ,

1|SN |

)≥ 1/N1/3



PrU∼µ(SN )

(Fk(U) /∈ Θ

(δ22 N

2

kd

))≤ exp

(−Θ

(δ42 N

d−1

k2d

)),





Random states for robust quantum metrology | Attaining the Heisenberg limit with a fixed measurement 14 / 20

Can we access the “usefulness”?


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







ψN ∼ SN


PrU∼µ(SN )

(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)

)≤ exp (−Θ(N))








ψN ∼ SN


PrU∼µ(SN )

(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)

)≤ exp (−Θ(N))








ψN ∼ SN


PrU∼µ(SN )

(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)

)≤ exp (−Θ(N))








ψN ∼ SN


PrU∼µ(SN )

(∃ϕ∈[0,2π] Fcl(U,ϕ) /∈ Θ(N2)

)≤ exp (−Θ(N))


Random states for robust quantum metrology | Generating random symmetric states 16 / 20

How can we generate random symmetric states?


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

http://arxiv.org/abs/quant-ph/1108.6264







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

















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

















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).
















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

















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

















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










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]








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













U∼µ(H)f(U)− E

U∼µε,t(H)f(U)

∣∣∣∣ ≤ ε/|H|t

Implies that polynomials (like the QFI) also are close









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













U∼µ(H)f(U)− E

U∼µε,t(H)f(U)

∣∣∣∣ ≤ ε/|H|tImplies that polynomials (like the QFI) also are close









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

















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














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


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

Random states for robust quantum metrology | Conclusions 19 / 20

Conclusions




Usefulness is robust against noise and particle loss.



Conclusions







Conclusions







Conclusions







Conclusions






Random states for robust quantum metrology | References 20 / 20

References

Thank you for your attention!slides on www.cgogolin.de

[1] LIGO Collaboration, Nature Phys., 7 (2011), 962–965.

[2] LIGO Collaboration, Nature Photon., 7 (2013), 613–619.

[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.

[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.

[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 inOptics, 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, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek,H. Weinfurter, L. Pezze, and A. Smerzi, 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.

[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.

[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.

[21] M. Oszmaniec, R. Augusiak, C. Gogolin, J. Ko lodynski, A. Acın, andM. Lewenstein, (2016), arXiv: 1602.05407.

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




http://arxiv.org/abs/1602.05407

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Random states for robust quantum metrology - cgogolin.de · Random states for robust quantum metrology 1 / 20 Random states for robust quantum metrology Micha l Oszmaniec, Remigiusz