PHASE ESTIMATION WITH THERMAL GAUSSIAN STATES Emilio Bagan with - - PowerPoint PPT Presentation

Tokyo

Emilio Bagan

March 3, 2009

PHASE ESTIMATION WITH THERMAL GAUSSIAN STATES

with J. Calsamiglia, R. Muñoz-Tapia and M. Aspachs

• Improvement of frequency standards
• gravitational-wave detection
• Clock synchronization
• Quantum computation
• Quantum cryptography
• General framework
• One copy
• Coherent thermal
• Squeezed thermal
• Many copies
• Coherent thermal
• Squeezed thermal
• Non-optimal schemes
• Heterodyne
• Homodyne
• Canonical Phase-measurement
• Frequency estimation

Motivating Phase estimation Layout of talk

General framework ⇓ Figures of Merit f(φ, φχ) =

∞

• l=0

alfl(φ, φχ); al ≥ 0 F =

• l

alFl

Parameterization

• f thermal

Gaussian states Thermal Coherent states P-function representation

Thermal squeezed states a b |0 S(r0)|0 U(φ) ρβr(φ)

a b S(r0)|0 U(φ) ρβr(φ) |0 Thermal squeezed states a b |0 S(r0)|0 U(φ) ρβr(φ) Equivalent to

ρβr U(φ)ρβrU †(φ) r(r0, T) β(r0, T) a b S(r0)|0 U(φ) ρβr(φ) |0 Thermal squeezed states a b |0 S(r0)|0 U(φ) ρβr(φ) Equivalent to

General framework

Observations

⇒ Optimal measur.

• Max. Fidelity

General framework Optimal measur.

• Max. Fidelity

χ − → θ ∈ [0, 2π) Attained by the Canonical Phase Measurement          For a single-component figure of merit: al fl Attained by the Canonical Phase Measurement if

• A. S. Holevo,
• Prob. & Stat. Aspects
• f Q. T., (1982)

General framework Optimal measur.

• Max. Fidelity

If

• phases can be absorbed
• Gaussian states OK
• Only one non-zero al OK
• for d > 2 there are counterexamples

⇒ General framework

Summary

Thermal Coherent states

Results

P-function representation ρβα “Schwinger parametrization ”

Thermal Coherent states

Results

Thermal Coherent states

Results

Thermal Coherent states

Assymptotics

Large α Small α In agreement with BMR-T PRA 78, 043829 (2008) for nβ = 0

• π nα

2(2nβ + 1) for nβ ≫ 1 √nα

• 1 − (2 −

√ 2)nβ

• for nβ ≪ 1

Thermal squeezed states

results

ρβr ρβr λ = tanh r “Schwinger parametrization ” +

Thermal squeezed states

results

ρβr |r0 = 2r through a 50/50 BS same n as ρβr

Thermal squeezed states

results

x p φ φ + π For large r0, λ0 → 1 and dominant contribution comes from w ≈ 1 For small r0, λ0 ≈ √n0 and Large squeezing ρβr |r0 = 2r through a 50/50 BS same n as ρβr

Many copies For N (uncorrelated) copies:

• difficult
• for pure states solved in BMM-T, PRA 78, 043829 (2008)
• simplifies for asymptotic N

Asymptotic regime Cramér-Rao bound Braustein and Caves (involves optimization over measurements) in our case Thermal Squeezed Thermal Coherent Recall that Zero temp. agree with: PRA 33, 4033 (1986) and PRA 73, 033821 (2006)

Many copies Asymptotic regime Thermal squeezed Thermal Squeezed (Lossy channel picture)

Many copies Asymptotic regime Thermal squeezed Thermal Squeezed (Lossy channel picture) Heisenberg limited precision not attained! High and low squeezing limits

More realistic schemes Heterodyne measurements Equivalent to a POVM measurement: Thermal squeezed Thermal squeezed Thermal Coherent Optimal bad! 2 X opt. Var Optimal at high temp

Covariant Phase- measurements Known to be suboptimal for squeezed vacuum, but HL scaling Thermal squeezed Thermal Coherent as heterodyne

• ptimal

Thermal Coherent Homodyne measurements No dependence on temperature! Maximum achieved at Equivalent to a POVM measurement: eigenstates of Thermal squeezed

• Opt. only for pure states

Optimal 1/2 X opt. Var 2 X opt. Var Optimal Adaptivity required Adaptivity required

Frequency estimation Optimal time t? Fixed number of copies N optimize t min

t

Var[ω] = e2 16N η2 |α|2 t∗ = 2/η e−iωˆ

nt e−ηˆ nt

Coherent input

Nη2Var[φ] r ηt∗

1 2 3 4 5 6 10-4 0.01 1 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Squeezed input At high squeezing t∗ = [2 + W(−2e−2)] ≈ 1.59/η

Nη2Var[ω]

Squeezed input

Squeezed State Coherent State Homodyne

Nη2Var[φ] n

50 100 150 200 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10

Nη2Var[ω] n0 Frequency estimation

• Coherent state and optimal POVM
• Squeezed state
• Optimal POVM
• Homodyne
• Heterodyne

small n0

Summary and Conclusions

• Canonical phase-measurements not always optimal (OK with Gaussian states)
• Temperature may improve sensitivity
• (Adaptive) Homodyne measurements
• optimal for thermal coherent states
• suboptimal for thermal squeezed states (temperature independent)
• Heterodyne measurements optimal for very mixed states
• Homodyne and heterodyne perform better than canonical phase-measuremnt
• Squeeze states provide little or no improvement in frequency estimation
• General framework
• One copy
• Coherent thermal
• Squeezed thermal
• Many copies
• Coherent thermal
• Squeezed thermal
• Non-optimal schemes
• Heterodyne
• Homodyne
• Canonical Phase-measurement

Frequency estimation

THE END