I. What s the problem? Quantum limits on noise in (deterministic) phase-preserving linamps. The whole story III. Nondeterministic immaculate linamps
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1 Diving Deep into a Quantum-Information-Processing Problem: Quantum Limits on the Performance of Linear Amplifiers II. I. What s the problem? Quantum limits on noise in (deterministic) phase-preserving linamps. The whole story III. Nondeterministic immaculate linamps Carlton M. Caves Center for Quantum Information and Control, University of New Mexico Co-workers: J. Combes, Z. Jiang, A. P. Lund, S. Pandey, M. Piani, T. C. Ralph, N. Walk Center for Quantum Information and Control
2 Gratuitous advice Human reasoning problems 1. Making probabilistic judgments 2. Reducing problems to dichotomies 3. Confusing correlation with causation 4. Assuming the converse 5. Goal displacement Among these tribal idols of human nature itself, we must prominently include both our legendary difficulty in acknowledging, or even conceiving, the concept of probability, and also the motivating theme of this article: our lamentable tendency to taxonomize complex situations as dichotomies of conflicting opposites. S. J. Gould, Science 287, 253 (2000). 1. Inadequate risk assessment. Overweighting of anecdotal evidence. Letting ideology, religion, or prejudice trump evidence. 2. In human affairs, there are no hard boundaries all arguments have a slippery slope and essentially all problems are more than a bit and generally multi-parameter.
3 Gratuitous advice Human reasoning problems 1. Making probabilistic judgments 2. Reducing problems to dichotomies 3. Confusing correlation with causation 4. Assuming the converse Almost all discussion in the public sphere founders on some combination of 1 to 4. As scientists, we can and should do better!
4 Gratuitous advice Human reasoning problem 5. Goal displacement Campbell s Law The more any quantitative social indicator is used for social decisionmaking, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor. D. T. Campbell, Journal of MultiDisciplinary Evaluation 7(15), 3 (2011); originally published as Paper #8, Occasional Paper Series, Public Policy Center, Dartmouth College, December Gaming and goal displacement are inevitable with any single indicator. For us, the result is high-impact-factor syndrome, the judging of people in terms of number of publications in high-impact-factor journals.
5 Gratuitous advice Human reasoning problems 1. Using probabilities 2. Reducing problems to dichotomies 3. Confusing correlations with causation 4. Implying or assuming the converse Special advice to physicists (from John Wheeler) 1. Guess, by any method, the answer to a problem before you start. 2. Make mistakes as fast as possible.
6 I. What s the problem? Pinnacles National Park Central California
7 (Deterministic) phase-preserving linear amplifiers Ball-and-stick (lollipop) diagram
8 (Nondeterministic) immaculate linear amplifiers Ball-and-stick (lollipop) diagram This is a distinguishability argument that turns out to be radically inadequate.
9 II. Quantum limits on noise in (deterministic) phasepreserving linamps. The whole story Holstrandir Peninsula overlooking Ísafjarðardjúp Westfjords, Iceland
10 Phase-preserving linear amplifiers Ball-and-stick (lollipop) diagram
11 Phase-preserving linear amplifiers added-noise operator output noise gain input noise added noise Zero-point noise Refer noise to input C. M. Caves, PRD 26, 1817 (1982). C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, PRA 86, (2012). Added noise number Noise temperature
12 To other models Ideal phase-preserving linear amplifier. Parametric amplifier
13 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. An immaculate linear amplifier, which only has the (blue) input noise, is not physical.
14 Phase-preserving linear amplifiers Microwave-frequency amplifiers using superconducting technology are at quantum limits and are being used as linear detectors in photon-coherence experiments. This requires more than second moments of amplifier noise. What about nongaussian added noise? What about higher moments of added noise? THE BIGGER PROBLEM What are the quantum limits on the entire distribution of added noise?
15 Initial coherent state
16 NonGaussian amplification of initial coherent state Which of these are legitimate linear amplifiers?
17 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)?
18 Attacking the problem. Tack 1 This is hopeless. If your problem requires a determination of when a class of linear operators is positive, FIND YOURSELF A NEW PROBLEM.
19 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.
20 Attacking the problem. Tack 3, the right tack
21 Attacking the problem. Tack 3, the right 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?
22 Attacking the problem. Tack 3, the right 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. C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, PRA 86, (2012).
23 NonGaussian amplification of initial coherent state
24 Quantum limits on phase-preserving linear amplifiers 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.
25 Western diamondback rattlesnake My front yard, Sandia Heights III. Nondeterministic immaculate linamps
26 Immaculate linear amplifier 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 This is an immaculate linear amplifier, more perfect than perfect; it doesn t even have the amplified input noise.
27 Immaculate linear amplifier If the amplifier works immaculately on any input circle of coherent states, then its probability of working is zero. Theoretical tool: Unambiguous state discrimination applied to uniformly spaced states on the circle.
28 Probabilistic, approximate, phase-insensitive, immaculate linear amplifier S. Pandey, Z. Jiang, J. Combes, and C. M. Caves, PRA 88, (2013).
29 Probabilistic, approximate, phase-insensitive, immaculate linear amplifier
30 Probabilistic, approximate, phase-insensitive, immaculate linear amplifier Phase-insensitive immaculate amplifiers don t do the job of linear amplification as well as an ideal linear amplifier or, indeed, even as well as doing nothing. Interpolating between ideal and immaculate linamps by concatenating the two such an amplifier adds less noise than an ideal linamp and works more often than an immaculate linamp, perfect linamps being an example never works as well as an ideal linear amplifier. J. Combes, N. Walk, A. P. Lund, T. C. Ralph, and C. M. Caves, PRA 93, (2016).
31 That s it, folks! Thanks for your attention. Echidna Gorge Bungle Bungle Range Western Australia
32 Ideal phase-preserving linear amplifier Models 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 C. W. Gardiner and P. Zoller, Quantum Noise, 3 rd Ed. (Springer, 2004). Back
33 Tent Rocks Kasha-Katuwe National Monument Northern New Mexico Completely positive maps and physical ancilla states
34 When does the ancilla state have to be physical? (orthogonal) Schmidt operators
35 When does the ancilla state have to be physical? Z. Jiang, M. Piani, and C. M. Caves, Quantum Information Processing 12, 1999 (2013).
36 When does the ancilla state have to be physical? Bloch sphere Cone of positive operators
37 Why does the ancilla state for a linear amplifier have to be physical?
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