Quantum limits on linear amplifiers

Whats the problem? Quantum limits on noise in phase-preserving linear amplifiers. The whole storyCompletely positive maps and physical ancilla statesImmaculate linear amplifiers. The bad newsImmaculate linear amplifiers. The good news

Carlton M. CavesCenter for Quantum Information and Control, University of New MexicoCentre for Engineered Quantum Systems, University of Queenslandhttp://info.phys.unm.edu/~caves

Co-workers: Z. Jiang, S. Pandey, J. Combes; M. Piani

I. Whats the problem?Pinnacles National ParkCentral California

Phase-preserving linear amplifiersBall-and-stick (lollipop) diagram.

Phase-preserving linear amplifiersadded-noise operatorC. 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 amplifierThe 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.

Phase-preserving linear amplifiersWhat 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 stateWhich of these are legitimate linear amplifiers?

Per-gyr falcon Jornada del MuertoNew MexicoII. Quantum limits on noise in phase-preserving linear amplifiers. The whole storyHarris hawkNear Bosque del ApacheNew Mexico

What is a phase-preserving linear amplifier?Immaculate amplification of input coherent stateTHE 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 modes initial state ? 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 of the ancillary mode.Attacking the problem. Tack 3, the last tack

NonGaussian amplification of initial coherent stateTo IV

The problem of characterizing an amplifiers 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 .Quantum limits on phase-preserving linear amplifiersMoment constraints vs. indirect quantum-state tomography to reconstruct ? CW version?

Western diamondback rattlesnakeMy front yard, Sandia Heights 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].

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 newsOn top of Sheepshead Peak, Truchas Peak in background Sangre de Cristo RangeNorthern 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 doesnt even have the amplified input noise.

Immaculate linear amplifierIf 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.

Probabilistic, approximate, phase-insensitive, immaculate linear amplifier

Probabilistic, approximate, phase-insensitive, immaculate linear amplifierPhase-insensitive immaculate amplifiers dont 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.

V. Immaculate linear amplifiers. The good newsMoo Stack and the Villians of UreEshaness, Shetland

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

State discriminationIm 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.

Unambiguous state discriminationIm 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. Ill let you opt out after youve examined the state.Reality check: We must be in the United States.

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

Echidna Gorge Bungle Bungle RangeWestern AustraliaThats it, folks! Thanks for your attention.

Ideal phase-preserving linear amplifierModels

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