Quantum limits on linear amplifiers

I. What’s the problem? II. Quantum limits on noise in phase-preserving linear amplifiers.

The whole storyIII. Completely positive maps and physical ancilla states

IV. Immaculate linear amplifiers. The bad newsV. Immaculate linear amplifiers. The good news

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

Centre for Engineered Quantum Systems, University of Queenslandhttp://info.phys.unm.edu/~caves

Co-workers: Z. Jiang, S. Pandey, J. Combes; M. PianiCenter for Quantum Information and Control

I. What’s the problem?

Pinnacles National ParkCentral California

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 Parametric amplifier

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.

Phase-preserving linear amplifiers

What about nonGaussian added noise? What about higher moments of added noise?

THE PROBLEMWhat are the quantum limits on the entire distribution of added noise?

Microwave-frequency amplifiers using superconducting technology are approaching quantum limits and are being used as linear detectors in photon-

coherence experiments. This requires more than second moments of amplifier noise.

Initial coherent state

Ideal amplification of initial coherent state

NonGaussian amplification of initial coherent state

Which of these are legitimate linear amplifiers?

Per-gyr falcon Jornada del Muerto

New Mexico

II. Quantum limits on noise in phase-preserving linear amplifiers. The

whole story

Harris hawkNear Bosque del Apache

New Mexico

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 PROBLEMWhat are the restrictions on the smearing

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

This is hopeless.

If your problem involves a straightforward determination of when a class of linear operators

is positive,

FIND YOURSELF A NEW PROBLEM.

Attacking the problem. Tack 1

But we have no way to get from this to general statements about the smearing distribution, because

the joint unitary and ancilla state are too general.

Attacking the problem. Tack 2

Attacking the problem. Tack 3, the

last tack

THE PROBLEM TRANSFORMEDGiven that the amplifier map must be physical (completely positive), what are the quantum

restrictions on the ancillary mode’s initial “state” σ?

Attacking the problem. Tack 3, the

last tack

THE ANSWERAny 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. Tack 3, the

last tack

NonGaussian amplification of initial coherent state

To IV

The problem of characterizing an amplifier’s performance, in absolute terms and relative to quantum limits, becomes a species of “indirect

quantum-state tomography” on the effective, but imaginary ancillary-mode state σ.

Quantum limits on phase-preserving linear amplifiers

Moment constraints vs. indirect quantum-state tomography to reconstruct σ?

CW version?

To EndTo Immaculate

Western diamondback rattlesnakeMy front yard, Sandia Heights

III. Completely positive maps and physical ancilla states

When does the ancilla state have to be physical?

Z. Jiang, M. Piani, and C. M. Caves, arXiv:1203.4585 [quant-ph].

(orthogonal) Schmidt operators

When does the ancilla state have to be physical?

Why does the ancilla state for a linear amplifier have to be

physical?

To End

IV. Immaculate linear amplifiers. The bad news

On top of Sheepshead Peak, Truchas Peak in background Sangre de Cristo Range

Northern New Mexico

Immaculate linear amplifierOriginal idea (Ralph and Lund): When presented with an input coherent state, a nondeterministic

linear amplifier amplifies immaculately with probability p and punts with probability 1 – p.

.

T. C. Ralph and A. P. Lund, in QCMC, edited by A. Lvovsky (AIP, 2009), p. 155.

This is an immaculate linear amplifier, more perfect than perfect; it doesn’t even have the amplified input noise.

Immaculate linear amplifier

If the probability of working is independent of input and the amplifier is described by a phase-preserving linear-

amplifier map when it does work, then the success probability is zero, unless when it works, it is a standard

linear amplifier, with the standard amount of noise.

S. Pandey, Z. Jiang, J. Combes, and C. M. Caves, ``Quantum limits on probabilistic amplifiers,’’ arXiv:1304.3901 [quant-ph].

Probabilistic, approximate, phase-insensitive, immaculate linear amplifier

Probabilistic, approximate, phase-insensitive,

immaculate linear amplifier

Phase-insensitive immaculate amplifiers don’t do the job of linear amplification as well as deterministic linear amplifiers or, indeed, even as well as doing nothing. Perhaps, by dropping the requirement of phase insensitivity, they can find a natural home as probabilistic, phase-sensitive, amplitude-specific amplifiers.

V. Immaculate linear amplifiers. The good news

Moo Stack and the Villians of UreEshaness, Shetland

Phase-sensitive immaculate amplification of M coherent states

on a circle

State discrimination

I’m going to hand you one of two quantum states. You need to decide which one I handed you.

If you get it right, I will give you a one-week, all-expenses-paid vacation in Canberra.

If you get it wrong, I will give you a two-week, all-expenses-paid vacation in Canberra.

To avoid spending two weeks in Canberra, you will minimize your error probability.

Unambiguous state discrimination

I’m going to hand you one of two quantum states. You need to decide which one I handed you.

If you get it right, I will give you a six-month, all-expenses-paid trip around the world to any destinations of your choosing.

If you get it wrong, I will pull out my gun and shoot you dead on the spot.

I’ll let you opt out after you’ve examined the state.

You perform the USD measurement, which never gets it wrong, but has an extra outcome where you make no decision.

Reality check: We must be in the United States.

For input coherent states this far apart, you could do perfect immaculate amplification with arbitrarily large gain and with a working probability of 1/2. You would get higher working probabilities for initial states further apart.

Phase-sensitive immaculate

amplification of M coherent states on a

circle

Echidna Gorge Bungle Bungle Range

Western Australia

That’s it, folks! Thanks for your

attention.

Ideal phase-preserving linear amplifierModels

● Parametric amplifier with ancillary mode in vacuum

● Simultaneous measurement of x and p followed by creation of amplified state

● Negative-mass (inverted-oscillator) ancillary mode in vacuum

● Master equation

E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965).

R. J. Glauber, in New Techniques and Ideas in Quantum Measurement Theory, edited by D. M. Greenberger (NY Acad Sci, 1986), p. 336.

C. W. Gardiner and P. Zoller, Quantum Noise, 3rd Ed. (Springer, 2004).

● Op-amp: another kind of linear amplifierA. A. Clerk et al., Rev. Mod. Phys. 82, 1155 (2010).