I.
What’s the problem?
II.
Quantum limits on noise in phase-preserving linear amplifiers.
The whole story
III.
Completely positive maps and physical ancilla states
IV. Immaculate linear amplifiers. The bad news
V. Immaculate linear amplifiers. The good news
Carlton M. Caves
Center for Quantum Information and Control, University of New Mexico
Centre for Engineered Quantum Systems, University of Queensland http://info.phys.unm.edu/~caves
Co-workers: Z. Jiang, S. Pandey, J. Combes; M. Piani
Center for Quantum Information and Control

Pinnacles National Park
Central California

Ball-and-stick
(lollipop) diagram.

addednoise operator output noise gain input noise added noise
Zero-point noise
Refer noise to input
Added noise number
C. M. Caves, PRD 26, 1817 (1982).
C. M. Caves, J. Combes, Z. Jiang, and
S. Pandey, PRA 86, 063802 (2012).
Noise temperature

The noise is Gaussian. Circles are drawn at half the standard deviation of the Gaussian.
A perfect linear amplifier, which only has the (blue) amplified input noise, is not physical.

Microwave-frequency amplifiers using superconducting technology are approaching quantum limits and are being used as linear detectors in photoncoherence experiments. This requires more than second moments of amplifier noise.
What about nonGaussian added noise?
What about higher moments of added noise?
THE PROBLEM
What are the quantum limits on the entire distribution of added 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
Harris hawk
Near Bosque del Apache
New Mexico

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)?

Attacking the problem.
Tack 1
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 2
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 3, the last tack

Attacking the problem.
Tack 3, the last tack
THE PROBLEM TRANSFORMED
Given 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 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.

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 quantumstate tomography” on the effective, but imaginary ancillary-mode state σ.
Moment constraints vs. indirect quantumstate tomography to reconstruct σ?
CW version?
To Immaculate To End

Western diamondback rattlesnake
My front yard, Sandia Heights

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

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

Original 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.

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

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 home as probabilistic, phase-sensitive amplifiers.

Moo Stack and the Villians of Ure
Eshaness, Shetland

Phase-sensitive immaculate amplification of M coherent states on a circle

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, allexpenses-paid vacation in Canberra.
If you get it wrong, I will give you a two-week, allexpenses-paid vacation in Canberra.
To avoid spending two weeks in Canberra, you will minimize your error probability.

Reality check: We must be in the
United States.
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, allexpenses-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.

Phase-sensitive immaculate amplification of M coherent states on a circle
For coherent states this far apart, you could do perfect immaculate amplification with arbitrarily large gain and with a working probability of 1/2. You could get higher working probabilities for states further apart.

That’s it, folks!
Thanks for your attention.
Echidna Gorge
Bungle Bungle Range
Western Australia

Ideal phase-preserving linear amplifier
Model
● Parametric amplifier with ancillary mode in vacuum
● Simultaneous measurement of x and p followed by creation of amplified state
E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965).
● Negative-mass (inverted-oscillator) ancillary mode in vacuum
● Master equation
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, 3 rd Ed.
(Springer, 2004) .
● Op-amp: another kind of linear amplifier
A. A. Clerk et al.
, Rev. Mod. Phys. 82, 1155 (2010).