PHASE ESTIMATION WITH THERMAL GAUSSIAN STATES Emilio Bagan with J. Calsamiglia, R. Muñoz-Tapia and M. Aspachs Tokyo March 3, 2009 Tuesday, March 3, 2009
Motivating Phase • Improvement of frequency standards estimation • gravitational-wave detection • Clock synchronization • Quantum computation • Quantum cryptography • General framework • One copy Layout of talk • Coherent thermal • Squeezed thermal • Many copies • Coherent thermal • Squeezed thermal • Non-optimal schemes • Heterodyne • Homodyne • Canonical Phase-measurement • Frequency estimation Tuesday, March 3, 2009
General framework ⇓ ∞ � f ( φ , φ χ ) = a l f l ( φ , φ χ ); a l ≥ 0 Figures of Merit l =0 � F = a l F l l Tuesday, March 3, 2009
Parameterization of thermal Gaussian states Thermal Coherent states P-function representation Tuesday, March 3, 2009
Thermal squeezed states b ρ β r ( φ ) S ( r 0 ) | 0 � U ( φ ) a | 0 � Tuesday, March 3, 2009
Thermal squeezed states b ρ β r ( φ ) S ( r 0 ) | 0 � U ( φ ) a | 0 � Equivalent to b S ( r 0 ) | 0 � ρ β r ( φ ) U ( φ ) a | 0 � Tuesday, March 3, 2009
Thermal squeezed states b ρ β r ( φ ) S ( r 0 ) | 0 � U ( φ ) a | 0 � Equivalent to r ( r 0 , T ) β ( r 0 , T ) ρ β r U ( φ ) ρ β r U † ( φ ) b S ( r 0 ) | 0 � ρ β r ( φ ) U ( φ ) a | 0 � Tuesday, March 3, 2009
General framework Observations ⇒ Optimal measur. Max. Fidelity Tuesday, March 3, 2009
General framework Optimal measur. Max. Fidelity Tuesday, March 3, 2009
General framework Attained by the Canonical Phase Measurement → θ ∈ [0 , 2 π ) χ − Optimal measur. Max. Fidelity A. S. Holevo, Attained by the Canonical Phase Measurement if Prob. & Stat. Aspects of Q. T., (1982) For a single-component figure of merit: a l f l Tuesday, March 3, 2009
General framework Summary ⇒ If • • phases can be absorbed • Gaussian states OK • Only one non-zero a l OK • for d > 2 there are counterexamples Tuesday, March 3, 2009
Thermal Coherent states Results P-function representation ρ βα “Schwinger parametrization ” Tuesday, March 3, 2009
Thermal Coherent states Results Tuesday, March 3, 2009
Thermal Coherent states Results Large α Small α Thermal Coherent states � π n α Assymptotics for n β ≫ 1 2(2 n β + 1) In agreement with √ √ n α � � BMR-T PRA 78, 043829 (2008) 1 − (2 − 2) n β for n β ≪ 1 for n β = 0 Tuesday, March 3, 2009
ρ β r Thermal squeezed states λ = tanh r results ρ β r “Schwinger parametrization ” + Tuesday, March 3, 2009
Thermal squeezed states results same n as ρ β r ρ β r | r 0 = 2 r � through a 50/50 BS Tuesday, March 3, 2009
Thermal squeezed states results same n as ρ β r For small r 0 , λ 0 ≈ √ n 0 and ρ β r | r 0 = 2 r � through a 50/50 BS Large squeezing p φ + π φ For large r 0 , λ 0 → 1 and dominant x contribution comes from w ≈ 1 Tuesday, March 3, 2009
For N (uncorrelated) copies: Many 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 Coherent Thermal Squeezed Recall that Zero temp. agree with: PRA 33, 4033 (1986) and PRA 73, 033821 (2006) Tuesday, March 3, 2009
Many copies Asymptotic regime Thermal squeezed Thermal Squeezed (Lossy channel picture) Tuesday, March 3, 2009
Many copies Asymptotic regime Thermal squeezed Thermal Squeezed (Lossy channel picture) Heisenberg limited precision not attained! High and low squeezing limits Tuesday, March 3, 2009
More realistic schemes Heterodyne Equivalent to a POVM measurement: measurements bad! Thermal squeezed Optimal Thermal squeezed 2 X opt. Var Thermal Coherent Optimal at high temp Tuesday, March 3, 2009
Covariant Phase- Known to be suboptimal for squeezed vacuum, but measurements HL scaling Thermal squeezed as heterodyne Thermal Coherent optimal Tuesday, March 3, 2009
Equivalent to a POVM measurement: eigenstates of Homodyne Thermal squeezed measurements Opt. only for pure states No dependence on temperature! Maximum achieved at Adaptivity 2 X opt. Var required Optimal 1/2 X opt. Var Thermal Coherent Adaptivity required Optimal Tuesday, March 3, 2009
Frequency estimation nt e − η ˆ e − i ω ˆ nt Optimal time t ? Fixed number of copies N optimize t Coherent input e 2 η 2 min Var[ ω ] = t ∗ = 2 / η 16 N | α | 2 t Squeezed input 1.6 1.5 N η 2 Var[ ω ] 1 N η 2 Var [ φ ] 1.4 η t ∗ 1.3 0.01 1.2 10 -4 1.1 1.0 0 1 2 3 4 5 6 r At high squeezing t ∗ = [2 + W ( − 2e − 2)] ≈ 1 . 59 / η Tuesday, March 3, 2009
Frequency Squeezed input estimation 0.030 10 8 0.025 Homodyne N η 2 Var[ ω ] 6 N η 2 Var [ φ ] 0.020 4 small n 0 2 0.015 0 0.0 0.2 0.4 0.6 0.8 1.0 0.010 Coherent 0.005 State Squeezed State 0.000 0 50 100 150 200 n n 0 • Coherent state and optimal POVM • Squeezed state • Optimal POVM • Homodyne • Heterodyne Tuesday, March 3, 2009
• General framework Summary • One copy and • Coherent thermal Conclusions • Squeezed thermal • Many copies • Coherent thermal • Squeezed thermal • Non-optimal schemes • Heterodyne • Homodyne • Canonical Phase-measurement Frequency estimation • 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 Tuesday, March 3, 2009
THE END Tuesday, March 3, 2009
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