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. Nondeterministic linear amplifiers 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 Center for Quantum Information and Control

2. I. What’s the problem? View from Cape Hauy Tasman Peninsula Tasmania

3. Phase-preserving linear amplifiers Refer noise to input Added noise number Noise temperature output noise input noise added noise gain added-noise operator C. M. Caves, PRD 26, 1817 (1982). C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, arXiv:1208.5174 [quant-ph]

4. Ideal phase-preserving linear amplifier Parametric amplifier

5. The noise is Gaussian. Circles are drawn at half the standard deviation of the Gaussian. A perfect linear amplifier only has the (blue) amplified input noise.

6. 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).

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 Are these 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? 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 Given that the amplifier map must be physical (completely positive), what are the quantum restrictions on the smearing probability distribution?

13. Addressing the problem Tack 1 This is hopeless.

14. Addressing the problem Tack 2 But we have no way to get from this to the standard input-output relation and thus no way to untangle the primary mode from the ancilla and to derive constraints on the smearing distribution. Our definition of a linear amplifier appears to be strictly weaker than the standard input-output relation.

15. Addressing the problem Tack 3

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?

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. Addressing the problem Tack 3

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

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

25. Nondeterministic 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. 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.. 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. Nondeterministic linear amplifier phase-preserving symmetry Projector onto subspace of first N + 1 number states

27. Nondeterministic linear amplifier independent of k

28. Nondeterministic linear amplifier

29. Nondeterministic linear amplifiers The bottom line

30. Lessons from the bottom line ● These should not be thought of as amplifiers. They are useful for other jobs, where something more immaculate than perfection, even though rarely attained, is valued more highly than partial achievement, but they are not useful for amplification. ● Results on nondeterministic devices should always be assessed in the light of success probabilities. Nondeterministic linear amplifiers The bottom line

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