Quantum metrology vs. quantum information science

I. Practical phase estimation

II. Phase-preserving linear amplifiers III. Force detection

IV. Probabilistic quantum metrology

Carlton M. Caves Center for Quantum Information and Control, University of New Mexico

Center for Engineered Quantum Systems, University of Queensland http://info.phys.unm.edu/~caves

Noise, Information, and Complexity @ Quantum Scale, Erice, October 11, 2013

Collaborators: J. Combes, C. Ferrie, Z. Jiang, S. Pandey, M. D. Lang, M. Piani

Center for Quantum Information and Control

A new way of thinking

Quantum information science

Computer science Computational complexity depends on physical law.

Old physics Quantum mechanics as nag.

The uncertainty principle restricts what can be done.

New physics Quantum mechanics as liberator. What can be accomplished with quantum systems that can’t be

done in a classical world? Explore what can be done with quantum systems, instead of

being satisfied with what Nature hands us.

Quantum engineering

Metrology Taking the measure of things

The heart of physics

Old physics Quantum

mechanics as nag. The uncertainty

principle restricts what can

be done.

New physics Quantum mechanics as

liberator. Explore what can be done with quantum systems, instead of being satisfied with

what Nature hands us. Quantum engineering

Old conflict in new guise Asking better questions Getting better answers

Tent Rocks Kasha-Katuwe National Monument

Northern New Mexico

I. Practical phase estimation

Optimal practical phase estimation

M. D. Lang and C. M. Caves, PRL, to be published, arXiv:1306.2677 [quant-ph].

Straightforward application of Quantum Cramér-Rao Bound (QCRB).

Optimal practical phase estimation

Straightforward application of Quantum Cramér-Rao Bound (QCRB).

Quantum metrology making a difference

The LIGO Scientific Collaboration, Nat. Phot. 7, 613 (2013).

Squeezed light in the LIGO Hanford detector

~ 2 dB of shot-noise reduction

Entanglement in linear optical networks

Z. Jiang, M. D. Lang, and C. M. Caves, PRA, to be published, arXiv:1308.6346 [quant-ph].

Use Bargmann-Fock representation.

Holstrandir Peninsula overlooking Ísafjarðardjúp Westfjords, Iceland

II. Phase-preserving linear amplifiers

Phase-preserving linear amplifiers

Ball-and-stick (lollipop) diagram.

Phase-preserving linear amplifiers

Refer noise

to input

Added noise number

Noise temperature

added-noise

operator

output noise

input noise

added noise gain

Zero-point noise

C. M. Caves, PRD 26, 1817 (1982). C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, PRA 86, 063802 (2012).

Ideal phase-preserving linear amplifier

The noise is Gaussian. Circles are drawn here at half the standard deviation of the Gaussian.

A perfect linear amplifier, which only has the (blue) amplified input noise, is not physical.

NonGaussian amplification of initial coherent state

Which of these are legitimate linear amplifiers?

What is a phase-preserving linear amplifier? Immaculate amplification of

input coherent state

Smearing probability distribution. Smears out the amplified coherent state and includes amplified input noise and added noise. For coherent-state input, it is the P function of the output.

THE PROBLEM What are the restrictions on the smearing

probability distribution that ensure that the amplifier map is physical (completely positive)?

THE ANSWER Any phase-preserving linear amplifier is

equivalent to a two-mode squeezing paramp with the smearing function being a rescaled Q function of a physical initial state σ of the ancillary mode.

Attacking the problem

NonGaussian amplification of initial coherent state

To IV

When does the ancilla state have to be physical?

(orthogonal) Schmidt operators

Z. Jiang, M. Piani, and C. M. Caves, QIP 12, 1999 (2013).

When does the ancilla state have to be physical?

Westfjords Iceland

III. Force detection

Standard quantum limit (SQL) for force detection

Back-action force

Langevin force

measurement (shot) noise

Monitor position

Time domain Back-action force

Langevin force

measurement noise

Frequency domain

measurement noise

Back-action force

Langevin force

SQL for force detection

measurement noise

Back-action force

Langevin force

SQL for force detection

SQL for force detection

The right wrong story

SQL for force detection

In an opto-mechanical setting, achieving the SQL at a particular frequency requires squeezing at that frequency, and achieving the SQL over a wide bandwidth requires frequency-dependent squeezing.

Single-parameter estimation: Bound on the error in estimating a classical parameter that is coupled to a quantum system in terms of the inverse of the quantum Fisher information.

Quantum Cramér-Rao Bound (QCRB)

Multi-parameter estimation: Bound on the covariance matrix in estimating a set of classical parameters that are coupled to a quantum system in terms of the inverse of a quantum Fisher-information matrix.

Waveform estimation: Bound on the continuous covariance matrix for estimating a continuous waveform that is coupled to a quantum system in terms of the inverse of a continuous, two-time quantum Fisher-information matrix.

Waveform QCRB. Spectral uncertainty principle

Prior-information term

At frequencies where there is little prior information,

Minimum-uncertainty noise

M. Tsang, H. M. Wiseman, and C. M. Caves, PRL 106, 090401 (2011).

No hint of SQL—no back-action noise, only measurement noise—but can the bound be achieved?

Beating the SQL Quantum noise cancellation (QNC) using oscillator and negative-mass oscillator.

Monitor collective position Q

Primary oscillator Negative-mass oscillator

Conjugate pairs

Oscillator pairs

QCRB

M. Tsang and C. M. Caves, PRL 105,123601 (2010).

Quantum-mechanics-free subsystems

M. Tsang and C. M. Caves, PRX 2, 031016 (2012).

Conjugate pairs

Oscillator pairs

Quantum-mechanics-free subsystems Conjugate variables

Dynamically coupled variables

Quantum-mechanics-free subsystems Conjugate pairs

Oscillator pairs

Paired sidebands about a carrier frequency

W. Wasilewski , K. Jensen, H. Krauter, J. J. Renema, M. V. Balbas, and E. S. Polzik, PRL 104, 133601 (2010).

Paired collective spins polarized along opposite directions

B. Julsgaardd, A. Kozhekin, and E. S. Polzik, Nature 413, 400 (2001).

Bungle Bungle Range Western Australia

IV. Probabilistic quantum metrology

Probabilistic quantum metrology In probabilistic metrology, a selection measurement is made, outcomes unfavorable for estimation are discarded, and parameter(s) are estimated on the favorable outcomes.

The selection measurement can be regarded as a state-preparation procedure that prepares states more sensitive to the parameter(s).

Analysis should ● Use a fixed performance metric applied to all data available after post-selection, with the data collected from optimal measurements and applied to optimal estimators. ● Include success probability properly in the performance metric. This should happen automatically if the problem is set up correctly. ● Compare with performance of optimal deterministic protocol using the same performance metric.

We analyze single-parameter estimation using mean-square-error as the performance metric and employing the quantum Cramér-Rao bound as the quantum limit on estimation accuracy.

Probabilistic quantum metrology? Probably not

MSE(x) ¸1

NI½Q(x)

MSE(x) ¸1

NXI¾QAjX(x)

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, arXiv:1309.6620 [quant-ph].

Probabilistic quantum metrology? Probably not

MSE(x) ¸1

NI½Q(x)

MSE(x) ¸1

NXI¾QAjX(x)

Cherno®bound

: Pr[NXI¾ ¸ NI½] · e¡Np(X)

µe p(X)I¾

I½

¶NI½=I¾

J. Combes, C. Ferrie, Z. Jiang, and C. M. Caves, arXiv:1309.6620 [quant-ph].

Metrology with abstention B. Gendra, E. Ronco-Bonvehii, J. Calsamiglia, R. Muñoz-Tapia, and E. Bagan, NJP 14, 105015 (2012).

Thanks for your attention

Red-backed fairy wren Oxley Common, Brisbane

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