Pateblen

I have been an electrical engineer in the aerospace industry since 1979 and currently work for NASA as a communications and navigation technology specialist and chairman of NASA's Software Defined Radio Architecture Team. Pateblen 04:50, 1 August 2007 (UTC)

What is measurement?

In regards to HUP, Complementarity, EPR, Bell...

Shouldn’t the notion of “measurement” or “observation” be generalized to one of “manifestation” of a physical property? Measurements and observations are simply special cases of a physical entity manifesting some property to an experimenter or observer. And aren’t these fundamentally the same as one physical entity manifesting a property to another physical entity? Why is a spinning charged particle manifesting a radiating field into free space any different than an experimenter measuring the field? The particle is the manifesting entity, manifesting its field to space, and space is the entity which the field is being “manifested to.” Free space is, in effect, measuring the field.

We would probably all agree that a spinning charged particle radiates a field into space of a certain frequency and amplitude. Why would an experimenter’s refusal to measure the field suppress the manifestation of the field to free space? And how could the experimenter absolutely refuse to “measure” the field in some way anyway – wouldn’t the field cause some minor heating of his skin, thus constituting a sort of measurement?

My contention is this: The properties of physical entities which are inclined to be manifested according to certain physical conditions cannot be suppressed by anyone’s or anything’s “refusal” to measure or observe them. In fact the refusal itself is not really realizable in any absolute sense.

So any EPR or Bell type experiment cannot be realized in the physical world since a thorough and absolute suppression of a manifesting property is required and is not possible. Same with the Schrödinger cat experiment – the box containing the cat is required to be an absolute and perfect suppressor of the cat’s physical properties until it is opened. This impossible-to-realize box leads to the impossible conclusion. Pateblen 04:20, 1 August 2007 (UTC)

If you want a certain system to maintain quantum coherence, it is indeed necessary to keep it from interacting with external degrees of freedom (see decoherence). But it's not all-or-nothing: The more it interacts, the quicker the coherence is lost. For example, the nuclear spin of a hydrogen atom in water can maintain quantum coherence for something on the order of a second (its NMR "T2 time"), just because the spin interacts only a little with the magnetic field created by nearby spins. My guess is that with a small interaction between a system and an external degree of freedom, you'll go from something like (system state) $\otimes$ (external state) to something like [(system state) $\otimes$ (external state) + (small entangled correction)]. Seems like it should be possible to write down a toy model and see what happens, but I couldn't come up with anything successful in five minutes.

Oh yeah, another, better way to look at it is as follows (I seem to recall this from Preskill's online lectures on quantum computing, but maybe somewhere else). Assume for simplicity that the system is a spin-1/2 or other system with a 2D Hilbert space. After an interaction with "external", you have (system state 1) $\otimes$ (some external state "A") + (system state 2) $\otimes$ (some external state "B"). If A=B (up to a phase factor), you have perfect quantum coherence of the system, and if A and B are orthogonal, you have perfect decoherence. During the unitary evolution of the interaction, A and B start out equal, and as time goes on, they almost always become orthogonal. (For example, if you're purposefully measuring it, then A and B would differ in degrees of freedom inside the experimenter's brain, and by that point they're obviously orthogonal.) Anyway, if you can finish your (EPR or Bell or quantum-computing or whatever) experiment before A and B become appreciably non-parellel, decoherence isn't a problem. --Steve 18:30, 1 August 2007 (UTC)

Thanks for the response. It seems that any EPR/Bell experiment requires that decoherence evolution must be frozen in time, not allowed to even begin, until someone decides to make a measurement. Freezing decoherence surely is unrealizable in the real physical world since it would require absolute and perfect isolation from external degrees of freedom. --FP Eblen 20:01, 2 August 2007 (UTC)

I don't see why that's necessary. Using the notation I introduced above, if |<A|B>|=0.9999999999 when you make your measurement of the system, you'll get the same results (to within experimental uncertainty) as if there had been no environmental interaction and decoherence whatsoever. This is true for *any* system observable that you might choose to measure, including the ones in EPR or Bell experiments, and it would probably be a neat exercise in linear algebra to prove it directly (although I haven't done so). Of course, if you wait long enough, |<A|B>| decreases down towards zero, and then the experiment is compromised; but with a careful set up, minimal environmental interaction, and quick measurement, decoherence can have vanishingly small effects on the entangled quantum state of your system. --Steve 06:47, 3 August 2007 (UTC)

Items to Review as of 8-2-07

Quantum Decoherence

In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. Decoherence occurs when a system interacts with its environment, or any complex external system, in such a thermodynamically irreversible way that ensures different elements in the quantum superposition of the system+environment's wavefunction can no longer interfere with each other. Decoherence has been a subject of active research for the last two decades [2].

Decoherence does not provide a mechanism for the actual wave function collapse; rather it provides a mechanism for the appearance of wavefunction collapse. The quantum nature of the system is simply "leaked" into the environment so that a total superposition of the wavefunction still exists, but exists beyond the realm of measurement.

Decoherence represents a major problem for the practical realization of quantum computers, since these heavily rely on the undisturbed evolution of quantum coherences.

Mechanisms Decoherence is not a new theoretical framework, but instead a set of new theoretical perspectives in which the environment is no longer ignored in modeling systems. To examine how decoherence operates we will present an "intuitive" model (which does require some familiarity with the basics of quantum theory) making analogies between visualisable classical phase spaces and Hilbert spaces before presenting a more rigorous derivation of how decoherence destroys interference effects and the "quantum nature" of systems, in Dirac notation. Then the density matrix approach will be presented for perspective (there are many different ways of understanding decoherence).

Phase space picture

An N-particle system can be represented in non-relativistic quantum mechanics by a wavefunction, $\psi (x_{1},x_{2},...,x_{N})$, which has analogies with the classical phase space. A classical phase space contains a real-valued function in 6N dimensions (each particle contributes 3 spatial coordinates and 3 momenta), whereas our "quantum" phase space contains a complex-valued function in a 3N dimensional space (since the position and momenta do not commute) but can still inherit much of the mathematical structure of a Hilbert space. Aside from these differences, however, the analogy holds.

Different previously isolated, non-interacting systems occupy different phase spaces, or alternatively we can say they occupy different, lower-dimensional subspaces in the phase space of the joint system. The effective dimensionality of a system's phase space is the number of degrees of freedom present which --in non-relativistic models -- is 3 times the number of a system's free particles. For a macroscopic system this will be a very large dimensionality. When two systems (and the environment would be a system) start to interact, though, their associated state vectors are no longer constrained to the subspaces, but instead the combined state vector time-evolves a path through the "larger volume", whose dimensionality is the sum of the dimensions of the two subspaces. (Think, by analogy, of a square (2-d surface) extended by just one dimension (a line) to form a cube. The cube has a greater volume, in some sense, than its component square and line axes.) The relevance of this is that the extent that two vectors interfere with each other is a measure of how "close" they are to each other (formally, their overlap or Hilbert space scalar product together) in the phase space. When a system couples to an external environment the dimensionality of, and hence "volume" available, to the joint state vector increases enormously -- each environmental degree of freedom contributes an extra dimension.

The original system's wavefunction can be expanded as a sum of elements in a quantum superposition, in a quite arbitrary way. Each expansion corresponds to a projection of the wave vector onto a basis, and the bases can be chosen at will. Let us choose any expansion where the resulting elements interact with the environment in an element-specific way; such elements will -- with overwhelming probability -- be rapidly separated from each other by their natural unitary time evolution along their own independent paths -- so much in fact that after a very short interaction there is almost no chance of any further interference and the process is effectively irreversible; the different elements effectively become "lost" from each other in the expanded phase space created by the coupling with the environment. The elements of the original system are said to have decohered. The environment has effectively selected out those expansions or decompositions of the original state vector that decohere (or lose phase coherence) with each other. This is called "environmentally-induced-superselection", or einselection. The decohered elements of the system no longer exhibit quantum interference between each other, as might be seen in a double-slit experiment. Any elements that decohere from each other via environmental interactions are said to be quantum entangled with the environment. (Note the converse is not true: not all entangled states are decohered from each other.)

Any measuring device, in this model, acts as an environment since any measuring device or apparatus, at some stage along the measuring chain, has to be large enough to be read by humans; it must possess a very large number of hidden degrees of freedom. In effect the interactions may be considered to be quantum measurements. As a result of an interaction, the wave functions of the system and the measuring device become entangled with each other. Decoherence happens when different portions of the system's wavefunction become entangled in different ways with the measuring device. For two einselected elements of the entangled system's state to interfere, both the original system and the measuring in both elements device must significantly overlap, in the scalar product sense. As we have seen if the measuring device has many degrees of freedom, it is very unlikely for this to happen.

As a consequence, the system behaves as a classical statistical ensemble of the different elements rather than as a single coherent quantum superposition of them. From the perspective of each ensemble member's measuring device, the system appears to have irreversibly collapsed onto a state with a precise value for the measured attributes, relative to that element.

Wave Function Collapse

The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable is called "collapse", or "wavefunction collapse". The final eigenstate appears randomly with a probability equal to the square of its overlap with the original state. The process of collapse has been studied in many experiments, most famously in the double-slit experiment. The wavefunction collapse raises serious questions of determinism and locality, as demonstrated in the EPR paradox and later in GHZ entanglement.

In the last few decades, major advances have been made toward a theoretical understanding of the collapse process. This new theoretical framework, called quantum decoherence, supersedes previous notions of instantaneous collapse and provides an explanation for the absence of quantum coherence after measurement. While this theory correctly predicts the form and probability distribution of the final eigenstates, it does not explain the randomness inherent in the choice of final state.

Quantum superposition is the application of the superposition principle to quantum mechanics. The superposition principle is the addition of the amplitudes of waves from interference. In quantum mechanics it is the amplitudes of wavefunctions, or state vectors, that add. It occurs when an object simultaneously "possesses" two or more values for an observable quantity (e.g. the position or energy of a particle).

More specifically, in quantum mechanics, any observable quantity corresponds to an eigenstate of a Hermitian linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the projection postulate states that the state will be randomly collapsed onto one of the values in the superposition (with a probability proportional to the square of the amplitude of that eigenstate in the linear combination).

The question naturally arose as to why "real" (macroscopic, Newtonian) objects and events do not seem to display quantum mechanical features such as superposition. In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger's cat, which highlighted the dissonance between quantum mechanics and Newtonian physics.

In fact, quantum superposition results in many directly observable effects, such as interference peaks from an electron wave in a double-slit experiment.

If two observables correspond to noncommutative operators, they obey an uncertainty principle and a distinct state of one observable corresponds to a superposition of many states for the other observable.

The measurement problem is the key set of questions that every interpretation of quantum mechanics must address. The wavefunction in quantum mechanics evolves according to the Schrödinger equation into a linear superposition of different states, but the actual measurements always find the physical system in a definite state. Any future evolution is based on the state the system was discovered to be in when the measurement was made, meaning that the measurement "did something" to the process under examination. Whatever that "something" may be does not appear to be explained by the basic theory.

The best known example is the "paradox" of the Schrödinger's cat: a cat is apparently evolving into a linear superposition of basis vectors that can be characterized as an "alive cat" and states that can be described as a "dead cat". Each of these possibilities is associated with a specific nonzero probability amplitude; the cat seems to be in a "mixed" state. However, a single particular observation of the cat does not measure the probabilities: it always finds either an alive cat, or a dead cat. After that measurement the cat stays alive or dead. The question is : how are the probabilities converted to an actual, sharply well-defined outcome?

Different interpretations of quantum mechanics propose different solutions of the measurement problem.

• The Copenhagen interpretation is rooted in the philosophical positivism. It claims that quantum mechanics deals only with the probabilities of observable quantities; all other questions are considered "unscientific" (meta-physical). Copenhagen regards the wavefunction as a mathematical tool used in the calculation of probabilities with no physical existence (not an element of reality). Waveform collapse is therefore a meaningless concept; the waveform only describes a specific experiment.

• Consciousness causes collapse proposes that the presence of a conscious being causes the wavefunction to collapse. However, this interpretation depends on a definition of "consciousness".

• The measurement apparatus is a macroscopic object. Perhaps, it is the macroscopic character of the apparata that allows us to replace the logic of quantum mechanics with the classical intuition where the positions are well-defined quantities.

Some interpretations claim that the latter approach was put on firm ground in the 1980s by the phenomenon of quantum decoherence. It is claimed that decoherence allows physicists to identify the fuzzy boundary between the quantum microworld and the world where the classical intuition is applicable. Quantum decoherence was proposed in the context of the many-worlds interpretation, but it has also become an important part of some modern updates of the Copenhagen interpretation based on consistent histories ("Copenhagen done right"). Quantum decoherence does not describe the actual process of the wavefunction collapse, but it explains the conversion of the quantum probabilities (that are able to interfere) to the ordinary classical probabilities.

Hugh Everett's relative state interpretation, also referred to as the many-worlds interpretation, attempts to avoid the problem by suggesting it is an illusion. Under this system there is only one wavefunction, the superposition of the entire universe, and it never collapses -- so there is no measurement problem. Instead the act of measurement is actually an interaction between two quantum entities, which entangle to form a single larger entity, for instance living cat/happy scientist. Everett also attempted to demonstrate the way that in measurements the probabilistic nature of quantum mechanics would appear; work later extended by Bryce DeWitt and others and renamed the many-worlds interpretation. Everett/DeWitt's interpretation posits a single universal wavefunction, but with the added proviso that "reality" from the point of view of any single observer, "you", is defined as a single path in time through the superpositions. That is, "you" have a history that is made of the outcomes of measurements you made in the past, but there are many other "yous" with slight variations in history. Under this system our reality is one of many similar ones.

The Bohm interpretation tries to solve the measurement problem very differently: this interpretation contains not only the wavefunction, but also the information about the position of the particle(s). The role of the wavefunction is to create a "quantum potential" that influences the motion of the "real" particle in such a way that the probability distribution for the particle remains consistent with the predictions of the orthodox quantum mechanics. According to the Bohm interpretation combined with the von Neumann theory of measurement in quantum mechanics, once the particle is observed, other wave-function channels remain empty and thus ineffective, but there is no true wavefunction collapse. Decoherence provides that this ineffectiveness is stable and irreversible, which explains the apparent wavefunction collapse.

Planck Time and Planck Units

In physics, the Planck time (t_P), is the unit of time in the system of natural units known as Planck units. It is the time it would take a photon travelling at the speed of light in a vacuum to cross a distance equal to the Planck length. The unit is named after Max Planck.

It is defined as

$t_{P}={\sqrt {\frac {\hbar G}{c^{5}}}}\approx 5.39121(40)\times 10^{-44}{\mbox{ s}}$

where:

$\hbar =h/2\pi$ is the reduced Planck constant

$G$ is the gravitational constant

$c$ is the speed of light in a vacuum

$t_{P}$ is in seconds.

The two digits between the parentheses denote the uncertainty in the last two digits of the value.

Significance

According to the Big Bang theory nothing is known about the universe at time=0, though it is presumed that all fundamental forces coexisted and that all matter, energy, and spacetime expanded outward from an extremely hot and dense singularity. One Planck time after the event is the closest that theoretical physics can get to it, and at that time it appears that gravity separated from the other fundamental forces.

One second is about 1.855×10⁴³ Planck times. The estimated age of the Universe in the Big Bang theory (4.3×10¹⁷ s) would be roughly 8×10⁶⁰ Planck times. The average life expectancy of a human is approximately 3.9×10⁵² Planck times.

As of 2006, the smallest unit of time that has been directly measured is on the attosecond (10⁻¹⁸ s) timescale, or around 10²⁶ Planck times.

Derivation

Ignoring a factor of $\pi$, the Planck mass is roughly the mass of a black hole with a Schwarzschild radius equal to its Compton wavelength. The radius of such a black hole would be, roughly, the Planck length.

The following thought experiment illuminates this fact. The task is to measure an object's position by bouncing electromagnetic radiation, namely photons, off it. The shorter the wavelength of the photons, and hence the higher their energy, the more accurate the measurement. If the photons are sufficiently energetic to make possible a measurement more precise than a Planck length, their collision with the object would, in principle, create a minuscule black hole. This black hole would "swallow" the photon and thereby make it impossible to obtain a measurement. A simple calculation using dimensional analysis suggests that this problem arises if we attempt to measure an object's position with a precision greater than one Planck length.

This thought experiment draws on both general relativity and the Heisenberg uncertainty principle of quantum mechanics. Combined, these two theories imply that it is impossible to measure position to a precision greater than the Planck length, or duration to a precision greater than the time a photon moving at c would take to travel a Planck length. Hence in any theory of quantum gravity combining general relativity and quantum mechanics, traditional notions of space and time will break down at distances shorter than the Planck length or times shorter than the Planck time.

See also

• 1 E-44 s
• Planck epoch

References

Status Summary as of 8-1-07

"What is measurement?" removed from HUP talk, EPR paradox talk, Bell talk and placed on my user page. "The key to EPR is this:" remains on EPR paradox talk. "A profound aspect..." remains on Complementarity (physics) main.