1. Quantum limits on linear amplifiers 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

2. I. What’s the problem? Pinnacles National Park Central California

3. Phase-preserving linear amplifiers Ball-and-stick (lollipop) diagram.

4. 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).

5. Ideal phase-preserving linear amplifier Parametric amplifier

6. 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.

7. Phase-preserving linear amplifiers 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? 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.

8. Initial coherent state

9. Ideal amplification of initial coherent state

10. NonGaussian amplification of initial coherent state Which of these are legitimate linear amplifiers?

11. Per-gyr falcon Jornada del Muerto New Mexico II. Quantum limits on noise in phase-preserving linear amplifiers. The whole story Harris hawk Near Bosque del Apache New Mexico

12. 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)?

13. 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

14. 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

15. Attacking the problem. Tack 3, the last tack

16. 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

17. 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. Tack 3, the last tack

18. NonGaussian amplification of initial coherent state To IV

19. 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

20. Western diamondback rattlesnake My front yard, Sandia Heights III. Completely positive maps and physical ancilla states

21. 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

22. When does the ancilla state have to be physical?

23. Why does the ancilla state for a linear amplifier have to be physical? To End

24. IV. Immaculate linear amplifiers. The bad news On top of Sheepshead Peak, Truchas Peak in background Sangre de Cristo Range Northern New Mexico

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

26. 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].

27. Probabilistic, approximate, phase-insensitive, immaculate linear amplifier

28. 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.

29. V. Immaculate linear amplifiers. The good news Moo Stack and the Villians of Ure Eshaness, Shetland

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

31. 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.

32. 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.

33. 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

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

35. Ideal phase-preserving linear amplifier Model s ● 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, 3 rd Ed. (Springer, 2004). ● Op-amp: another kind of linear amplifier A. A. Clerk et al., Rev. Mod. Phys. 82, 1155 (2010).