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The Essence of Quantum Computing

4.4 Consequences of the quantum postulates


A desire to understand Nature is an attempt to minimise an information gap – the information we try to prise from Nature and Nature’s ability to withhold by making us run around in circles.

Physics gives rise to observer-participancy; observer-participancy gives rise to information; and information gives rise to physics46. – John Wheeler

What we call the laws of physics is the information we extract from Nature, compact and package within a falsifiable conjecture which we have so far failed to refute. We are forced to conjecture (or hypothesize) and endlessly refine, discard, and amend our conjectures by strenuously trying to refute them because Nature appears determined to be secretive.47 Since our present best understanding of the Universe is enshrined in quantum mechanics, it appears natural to agree with David Deutsch: “Our best theories are not only truer than common sense, they make more sense than common sense.”48 The consequences of the postulates are strikingly non-intuitive. Here are some examples.

Heisenberg’s uncertainty principle.       In 1927, Heisenberg showed that both the position and the velocity of a quantum object cannot be simultaneously measured exactly, even in theory. Further, this restriction cannot be overcome by refining the measuring apparatus or techniques. The classical concept of exact position and exact velocity being measured simultaneously is barred in quantum mechanics. It is a limit imposed by Nature as we penetrate into the subatomic world. Heisenberg originally stated his uncertainty principle, based on how waves behave in classical physics, as follows: If q is the error in the measurement of any coordinate and p is the error in its canonically conjugate momentum, then

Δp Δq≥ h/2.

The modern and correct interpretation of Heisenberg’s uncertainty principle is in terms of the standard deviation and the commutator operator rather than the earlier misconceived notion that the uncertainty principle is about measuring a quantity C to some ‘accuracy’ Δ(C) which then causes the value of D to be ‘disturbed’ by an amount Δ(D) in such a way that some sort of inequality is satisfied. Mathematically, given a large number of identical quantum systems each in state |Ψ〉, suppose we measure C on some of those systems, and D in others, where C and D are represented by two observables (represented by operators), then the standard deviation Δ(C) of the C results times the standard deviation Δ(D) of the results for D will satisfy the inequality49

                                                                            Δ(C)Δ(D)≥  |〈Ψ|[C,D]|Ψ〉| / 2

This is the content of Heisenberg’s Uncertainty Principle50. It imposes a fundamental limit on the sharpness of the dynamics (courtesy the Planck’s constant) by preventing us from even talking about initial conditions since we cannot know the exact position and velocity of a particle simultaneously. Hence, it prevents us from even talking about trajectories. Thus, indeterminism is built into the very structure of matter where certain pairs of variables (such as position and momentum) cannot even exist simultaneously with perfectly defined values.
In the most extreme case, absolute precision of one variable in a pair would entail absolute imprecision regarding the other. This is in sharp contrast to classical physics where “if we know the present exactly, we can calculate the future”. The Uncertainty Principle says the opposite: it is not the conclusion that is wrong but the premise. One cannot calculate the precise future motion of a particle, but only a range of possibilities for its future motion. (However, the probabilities of each motion, and the distribution of many particles following these motions, can be calculated exactly from Schrödinger’s wave equation.) It is thus not possible to make a complete mental picture of events in the quantum world. Heisenberg concluded:

The mathematically formulated laws of quantum theory show clearly that our ordinary intuitive concepts cannot be unambiguously applied to the smallest particles. All the words or concepts we use to describe ordinary physical objects, such as position, velocity, color, size, and so on, become indefinite and problematic if we try to use them of elementary particles.51

At the mental level, we feel we understand the large-scale world only when we have developed a mental picture of it. Newtonian mechanics pandered to that mental level by dealing with measurable and observable quantities such as position, velocity, etc. Quantum events cannot be pictured in our mind, and therefore it leaves us with the queasy feeling that it is unreal, further strengthened by the fact that quantum mechanics deals with not real numbers but complex numbers.

Complementarity. In classical physics, particles and waves behave distinctly differently. In quantum mechanics, an object may behave in either way! In fact, Bohr’s complementarity principle (or wave-particle duality) states that a quantum object can behave either as a particle or as a wave, but never as both simultaneously.52 At the quantum level, both particle and wave aspects of reality are equally significant. However, the wave character is not a physical vibration but rather a wavelike function that encodes information about where a quantum entity will be found once it is detected.

Position is a precisely locatable particle property; waves have no precise location, but they have precise momentum. Heisenberg’s uncertainty principle seems to say that the observer can choose which aspect of Nature he wants to observe and with what precision.
About complementarity, John Wheeler remarked,

Bohr’s principle of complementarity is the most revolutionary scientific concept of this century and the heart of his fifty-year search for the full significance of the quantum idea.53

[46] Wheeler (1989), pp. 313-314.
[47] Popper (1963).
[48] Deutsch (2009).
[49] For a proof, see Nielsen & Chuang (2000), p. 89.
[50] Heisenberg (1927); Hilgevoord & Uffink (2016).
[51] Heisenberg (1974), p. 114. (Quoted in Zukav (1979), p. 47.)
[52] Bohr (1928). See also: Passon (Undated); Bohr (1934); Bohr (1937).
[53] Wheeler (1963).

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