ScienceWeek

THEORETICAL PHYSICS: ON QUANTUM MEASUREMENT LIMITS

The following points are made by V. Giovannetti et al (Science 2004 306:1330):

1) Measurement is a physical process, and the accuracy to which measurements can be performed is governed by the laws of physics. In particular, the behavior of systems at small scales is governed by the laws of quantum mechanics, which place limits on the accuracy to which measurements can be performed. These limits to accuracy take two forms. First, the Heisenberg uncertainty relation [1] imposes an intrinsic uncertainty on the values of measurement results of complementary observables such as position and momentum, or the different components of the angular momentum of a rotating object. Second, every measurement apparatus is itself a quantum system: As a result, the uncertainty relations together with other quantum constraints on the speed of evolution [such as the Margolus-Levitin theorem [2]] impose limits on how accurately we can measure quantities, given the amount of physical resources, such as energy, at hand to perform the measurement.

2) One important consequence of the physical nature of measurement is the so-called "quantum back action": The extraction of information from a system can give rise to a feedback effect in which the system configuration after the measurement is determined by the measurement outcome. For example, the most extreme case (the so-called von Neumann or projective measurement) produces a complete determination of the post-measurement state. When performing successive measurements, quantum back action can be detrimental, because earlier measurements can negatively influence successive ones.

3) A common strategy to get around the negative effect of back action and of Heisenberg uncertainty is to design an experimental apparatus that monitors only one out of a set of incompatible observables: "less is more" [3]. This strategy, called "quantum nondemolition measurement" [3-6], is not as simple as it sounds. One has to account for the system's interaction with the external environment, which tends to extract and disperse information, and for the system dynamics, which can combine the measured observable with incompatible ones. Another strategy to get around the Heisenberg uncertainty is to employ a quantum state in which the uncertainty in the observable to be monitored is very small (at the cost of a very large uncertainty in the complementary observable). The research on quantum-enhanced measurements was spawned by the invention of such techniques [3] and by the birth of more rigorous treatments of quantum measurements.

4) Most standard measurement techniques do not account for these quantum subtleties, so that their precision is limited by otherwise avoidable sources of errors. Typical examples are the environment-induced noise from vacuum fluctuations (the so-called "shot noise") that affects the measurement of the electromagnetic field amplitude, and the dynamically induced noise in the position measurement of a free mass (the so-called "standard quantum limit"). These sources of imprecision are not as fundamental as the unavoidable Heisenberg uncertainty relations, because they originate only from a non-optimal choice of measurement strategy. However, the shot noise and standard quantum limits set important benchmarks for the quality of a measurement, and they provide an interesting challenge to devise quantum strategies that can defeat them.

5) In summary: Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits on the precision of measurement. Conventional measurement techniques typically fail to reach these limits. Conventional bounds to the precision of measurements such as the shot noise limit or the standard quantum limit are not as fundamental as the Heisenberg limits and can be overcome using quantum strategies that employ "quantum tricks" such as squeezing and entanglement.

References (abridged):

1. H. P. Robertson, Phys. Rev. 34, 163 (1929)

2. N. Margolus, L. B. Levitin, Physica D 120, 188 (1998)

3. C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980)

4. K. Bencheikh, J. A. Levenson, P. Grangier, O. Lopez, Phys. Rev. Lett. 75, 3422 (1995)

5. G. J. Milburn, D. F. Walls, Phys. Rev. A. 28, 2065 (1983)

Science http://www.sciencemag.org

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QUANTUM PHYSICS: ZERO-POINT FLUCTUATIONS

The following points are made by Miles Blencowe (Nature 2003 424:262):

1) In 1927, Werner Heisenberg (1901-1976) introduced his famous quantum principle, which states that the uncertainties in the position and the velocity of a particle are inversely proportional to each other: a particle's position or its velocity can be known precisely, but not both at once. This principle is one of the cornerstones of quantum mechanics, and is traditionally relevant to the domain of subatomic particles. But what about more familiar macroscopic objects, comprising many atoms, that we think of as possessing simultaneously well-defined positions and velocities of their center-of-mass? If we could be sufficiently precise in our measurements on such objects, would we encounter the quantum uncertainty principle at work?

2) If you clamp one end of a wooden ruler to the edge of a table and then pluck the other, free end, it vibrates with decaying amplitude and eventually returns to apparent rest. But if you were to look at the free end of the ruler under a sufficiently powerful microscope, it would not be at rest at all, but jiggling up and down in a random fashion. This motion is a consequence of the air molecules striking the ruler, as well as of its countless, fluctuating internal defects, and is an example of thermal brownian motion.

3) There are other, quantum fluctuations in the ruler, though, that are completely masked by this classical thermal motion. These quantum "zero-point" fluctuations have much smaller amplitude and arise from the necessary uncertainty in position and velocity stated in Heisenberg's principle. The situation is analogous to the hydrogen atom, which is stable because the attractive electrostatic force that would like to pull the electron into a tighter volume around the proton is balanced by the repulsive effect of the electron's fluctuating velocity. Similarly, for a macroscopic object such as a crystal beam or a ruler, the elastic restoring force on the bent beam balances the repulsive effect of its fluctuating center-of-mass velocity.

4) Because the magnitudes of the zero-point fluctuations in position and velocity are so small, they can only be detected if the structure is cooled down to very low temperatures. As the temperature is lowered, the amplitude of thermal motion decreases. Eventually, there will be no thermal motion, only pure, temperature-independent zero-point fluctuations. At a temperature of about a hundredth of a kelvin, zero-point fluctuations should dominate in a structure with a mechanical vibration frequency of about one billion cycles per second (1 gigahertz, or GHz).

Nature http://www.nature.com/nature

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ON THE UNCERTAINTY PRINCIPLE: WHO NEEDS HEISENBERG?

The following points are made by Harry S. Lipkin (a commentary to ScienceWeek 2001 June):

1) There is all this fuss about Heisenberg and the Uncertainty Principle and the spookiness of quantum mechanics. But I knew all about the uncertainty principle when I was a young electronic engineer working on radar and had never heard of Heisenberg or quantum mechanics. I knew that if you sent a radar pulse at a target and measured the time between the transmission of the pulse and the reception of the echo, you could measure the distance to the target. But to measure the distance to the target precisely you needed a very short pulse in time and a very wide band receiver. The shorter the pulse, the better the distance measurement, but also the wider band you needed in frequency in order to use this information. If the source was a delta function in time, you needed infinite bandwidth in order to use the information. If the source was a gaussian, the frequency spectrum was a gaussian and there was an uncertainty principle between the two gaussians. One was the fourier transform of the other.

2) You can play the same game in space. The radar pulse occupies a finite length in space as it travels to its destination and back. But the wave number spectrum of this wave packet is just the fourier transform of the amplitude of this electromagnetic wave packet in space. Wave number and position are complementary just like time and frequency. So what's the big deal about complementarity? The time of a radar pulse and its frequency are complementary. If you design the wave packet so that you know one very precisely, you destroy the precision of the other. So what's the big deal and where is the spookiness?

3) The big deal and the spookiness were provided by Einstein in his 1905 explanation of the photoelectric effect. Long before Heisenberg, Einstein explained the photoelectric effect by saying that light was not simply an electromagnetic wave; it was also a shower of photons. Here is the wave-particle duality and the spookiness of quantum mechanics. Light travels as a wave in accordance with Maxwell's equations. But when it interacts with electrons in an atom, it does so as a photon which exchanges energy and momentum with the electron according to billiard ball mechanics. But where is the photon before it interacts? It is a wave described by Maxwell's equations. And where will it hit an electron? Nobody knows. We only can know the probability that it will hit an electron if there is an electron at a given point in space. We can repeat this experiment many times by shining light on material and looking at the electrons coming out. We can make our beam of light so weak that only one photon comes out every second. What will the next photon do? Nobody knows. But Einstein says that he can tell you the probability that it will hit an electron and give the spectrum of energy and momentum of the electrons that are knocked out of his target.

4) Einstein said that I cannot believe what Maxwell's equations tell me about the signal I will receive when I send a one microsecond radar pulse to a distant target and should be able to detect a very weak radar echo. Einstein said that the weakest signal I can detect in my receiver has an energy (hf) where (h) is Planck's constant and (f) is the frequency of my radar. So if Maxwell tells me that I should receive a one microsecond echo pulse with an energy of (1/2)(hf) after every pulse, Einstein says that I should receive no echo after half of the pulses, and an echo signal with the full energy of (hf) after the other half. But nobody will ever be able to tell me which outgoing pulse will produce an echo and which one will not.

5) This is already spooky enough, but it is even worse. Einstein tells me that the echoes I receive will look nothing like Maxwell's predicted one-microsecond pulse. If I can build a detector with a time resolution of one picosecond, each echo will appear in a one-picosecond bin somewhere among the million bins in the one microsecond pulse predicted by Maxwell. And no one can predict in which bin it will appear. But if we send out many millions of pulses, we will find that the histogram of the number of pulses in each picosecond bin will end up giving exactly the pulse predicted by Maxwell with the same total energy. And a century after Einstein's prediction it still stands with the spookiness that no one will ever predict better.

6) This is enough to make me understand the spookiness of quantum mechanics and believe that it describes the real world. Many physicists much better than myself tried to do better and failed. That's enough for me. I don't need Heisenberg.

7) Now back to the uncertainty principle. Einstein's literal use of Planck's (E = hf) means that my uncertainty between frequency and time is now an uncertainty between energy and time, and the radar pulse with the minimum energy also has a minimum momentum which is (hk), where (k) is the wave number. So the uncertainty between position and wave number suddenly becomes an uncertainty between position and momentum. Here is the uncertainty principle. And the culprit is Einstein in 1905, long before Heisenberg invented his uncertainty principle.

8) There's the spookiness. There's the uncertainty. There's why Planck's quantum changes the way we look at everything on the atomic scale. God was playing dice with photons long before Bohr and Heisenberg, Einstein did it all. He didn't like it. He refused to believe that God would do such terrible things. But he taught us that God did them.

9) A simple "do-it-yourself" version of Einstein's spookiness can be seen by looking through two pairs of polarized sun glasses. When the two pairs are oriented parallel, one behind the other, parallel light comes through. When the lens of one is turned by 90 degrees so that the two lenses are perpendicular to one another, no light gets through. But when one lens is rotated by 45 degrees, half of the light that gets through the first lens gets through the second. This is all classical electromagnetism.

10) But Einstein tells us that light is a shower of photons, and therefore that half of the photons that get through one lens will get through the second. But if the light is so weak that only one photon gets through the first lens every minute, nobody can know whether or not the next photon will get through the second lens. We only know that half of them will get through.

11) Einstein opened the Pandora's box of quantum-mechanical spookiness. He tried vary hard to close it afterwards. But he never succeeded. It is still there in the next millennium.

[Editor's note: Professor Lipkin received his Ph.D. in physics from Princeton University in 1950. His doctoral dissertation was the first experimental test of the validity of the Dirac equation for relativistic positrons. He was one of the founders of the Physics Department at the Weizmann Institute of Science. He is the author of /Quantum Mechanics: New Approaches to Selected Topics/.]

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