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What is quantum information? Information symmetry and mechanical motion

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This paper reviews arguments for the necessity of complex numbers in any quantum theory.

Laurence WhiteSeguir

Senior Project Manager at Dycom Industries, IncPublicidad

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- Complex Numbers in Quantum Theory PHY4910 – Spring 2010 – L. White 1 Abstract. This paper reviews arguments by Saxon 1 and Stueckelberg 2 for the necessity of complex numbers in any quantum theory. It concludes that the twoarguments are complementary, and the use of complex numbersin quantum theory isessential. As there is ongoing and promising research on the use of quaternions as a construct for quantum theory 3 , it would be more correct to say that quantum theory is impossible without the imaginary unit i. (Quaternions may be regarded as an extension of the complex numbers.) It is which is woven into the fabric of quantum descriptions, in a way not seen in other theories. For simplicity, this paper refers only to complex numbers, but it should be remembered that these may beinterpreted as a limiting case of quaternions. Introduction. Complex numbers permeate quantum physics. Whereas in classical physics they are used as a powerful notational shorthand, they are unavoidable in quantum theory. Schrödinger was following established practice when he introduced a complex phase factor into his wave equation for the quantum state. Ψ(x,t) = Aei(k.x – ωt) (1) But as quantum theory developed and matured, it became clear that complex numbers playeda critical role at the heart of the theory. This raised the question, “Can you have a quantum theory without complex numbers?” The answer from both Saxon and Stueckelberg was an emphatic “no”. This sets quantum theory apart; so what is it about quantum theory that requires i? Saxon. Saxon‟s argument is concise and may be summarized as follows. All points in space must be physically equivalent – the choice of origin should be irrelevant. If, for example, we shift the origin left by an arbitrary distance b, the function Ψ from equation (1) is multiplied by phase factor eipb/ħ . This leaves the wave function in the same general form Ψ‟ (x,t) = Aei(k’.x’ – ωt) , with no physically significant reference to the origin. Unlike the case of a classical wave, the phase factor eipb/ħ is physically undetectable; therefore, under an arbitrary transformation, Ψ reduces to a multiple of itself. This essential aspect of the state function can only be implemented in a complex framework. So what began as a notational convenience has taken on the character of something more fundamental. When determining probability density, for example, we multiply Ψ by its complex conjugate, giving |Ψ|2 . The result, being a measure of probability, is – as you would expect – a dimensionless number, without phase or direction. So while the state function itself is described by a wave equation that includes a complex phase factor, there is no phase involved in its physical manifestation |Ψ|2 . (We do not observe electrons, for example, as having an absolute phase, although their spatial distribution about the nucleus depends, albeit indirectly, on Ψ.) This is a critical difference from classical physics, and for good reason: in classical physics, the wave equation describes the physical phenomenon (e.g. an electromagnetic wave); in quantum theory, the wave equation describes nothing directly; it serves only as a predictor of physical phenomena when „squared‟ through multiplication by its complex conjugate.
- Complex Numbers in Quantum Theory PHY4910 – Spring 2010 – L. White 2 Stueckelberg. Stueckelberg‟s argument is more technical than Saxon‟s. Finkelstein, Jauch, and Speiser4 had already shown that the only viable Hilbert spaces which could underpin quantum theory were Real Hilbert Space (RHS), Quaternion* . Hilbert Space (QHS), and Complex Hilbert Space (CHS).Assuming nothing that wasn‟t obvious, axiomatic, or trivial, Stueckelbergwas able to rule out RHS as a mathematical structure for quantum theory. He starts with a generic classical definition of error operators and arrives at two possible definitions of mean square error. These allow him to calculate the uncertainty in simultaneous measurements of two observables. One of these gives the trivial result <∆F2 >ψ<∆G2 >ψ ≥ 0 (2) In other words, the uncertainty is greater than zero, unless it isn‟t. The second mean square calculation leads to the result <∆F2 >ψ<∆G2 >ψ ≥ 1/4 <J[F,G]>2 ψ (3) This cannot be manipulated to give the canonical commutation relation [a,b] = iħ (4) or the Heisenberg Uncertainty Principle ΔxΔp ≥ ħ/2 (5) So the assumption of a real Hilbert space takes us to a dead end which cannot be reconciled with some of the most basic aspects of quantum theory. Stueckelberg‟s proof is laid out in detail in Appendix A (below). Complementary and Related Arguments, and the Wheeler Question. Saxon‟s argument centers on arbitrary transformations of the state function Ψ; Stueckelberg shows that Real Hilbert spaces lead to trivial or unusable uncertainty principles. Stueckelberg‟s argument is more austere but both arguments are compelling.They do not appear to be equivalent (in the sense, for example, that Schrödinger‟s wave formulation turned out to be equivalent to Heisenberg‟s matrix mechanical approach5 ).They clearly complement each other. Saxon cautions us that, if we accept that the de Broglie wave is the physical embodiment of the state function Ψ, we are also implicitly accepting the hard-wiring of into the core of quantum physics. Stueckelberg demonstrates the mathematical necessity of with convincing rigor. * Quaternions (denoted ℍ) have a rich history. Developed in the nineteenth century by Sir William Rowan Hamilton, they fell out of favor for decades, but have recently enjoyed a renaissance in computer-based models of spatial rotations.
- Complex Numbers in Quantum Theory PHY4910 – Spring 2010 – L. White 3 In a 1980 paper6 , Wootters says the following: “By requiring that the distance between states of a general quantum system be determined by the number of distinguishable intermediate states, could we conclude that the set of states as a whole must have the geometric structure of the set rays in a complex vector space? [...] if statistical distance is involved in determining the geometry of this set, it is not the whole story.” While Wootters' primary focus is not on complex numbers in quantum theory, he nevertheless comments on the intimate connection between quantum theory and complex space. More recently, Fuchs7 also references this connection as follows: “... it is interesting to note that the quantumde Finetti theorem and the conclusions just drawn from it work only within the framework of complex vector-space quantum mechanics. For quantum mechanics based on real Hilbert spaces, the connection between exchangeable density operators and unknown quantum states does nothold.” Here again, Real Hilbert space cannot accommodate the needs of active development in quantum theory. Before concluding, it should be noted that neither Saxon nor Stueckelberg addresses Wheeler‟s question “Why the quantum?” – i.e. they show that there is no quantum theory possible without complex numbers but they do not attempt to explain why quantum physics has this curious aspect. Wootters makes it clear that a statistical/information-theoretic approach hints at some underlying necessity but, as he puts it, statistical distance “is not the whole story.” Conclusion. Uncertainty principles did not arrive with quantum mechanics; they were well known and studied in classical physics. For example, the position and wave number of a classical wave are related by ∆x∆k ≥ 1/2 (6) It is common to picture this in relation to electromagnetic waves but the same is true in principle for waves in the ocean; if we know the precise position of the crest of an ocean wave, we cannot know its wave number with absolute precision, and vice-versa. There is an inbuilt lower limit to the quality of information we can gather for these two observables at the same instant. A glance at the Heisenberg uncertainty principle (in this example, for position and momentum) shows an immediate difference ∆x∆p≥ħ/2 (7)
- Complex Numbers in Quantum Theory PHY4910 – Spring 2010 – L. White 4 Unity in the numerator is now replaced with the reduced Planck constant h/2π; and the difference is not trivial. What has changed? In this regard, Saxon is instructive. When we choose to equate the de Broglie wave with the state function Ψ, we are accepting an uncertainty associated with the wave equation iħ ∂/∂t[Ψ(x,t)] = - ħ2 /2m ∇2 [Ψ(x,t)] + V(x)Ψ(x,t) (8) Setting aside the formalism, the core point is this: quantum uncertainty follows unequivocallyfrom wave-particle duality. If a particle is also a wave then, like all waves, it has an uncertainty. Saxon‟s view might be better characterized as „wave-particle equivalence‟ rather than wave-particle duality. There is nothing „dual‟ about the quantum state; it is a wave and it is a particle. Hence Saxon‟s caution about accepting equation (1) above lightly; there is a lot that comes with it. And, more closely to both Saxon‟s and Stueckelberg‟s arguments, this particular uncertainty – related as it is to equations (1) and (8) – cannot be derived from the presumption of a real space. It can only emerge from a mathematical setting which includes i. In moving from the well- established formalisms of classical physics to their counterparts in the quantum world, we find that complex numbers cease to be optional; you cannot describe wave or particle behavior at the quantum level without resort to numbers involving . And why is this? Because, as stated above, the quantum wave equation describes nothing directly; it serves only as a predictor of physical phenomena when „squared‟ through multiplication by its complex conjugate. In classical physics, the wave equation is a direct mathematical description of the physical observable; in quantum physics, on the other hand, it cannot be. It is rather, as Born first realized, a mathematical description of something else – the probability amplitude. There is no other useful way to interpret the quantum wave equation, a conclusionunpalatable to many classical physicists but inescapable nonetheless. So, a wave equation with a very classical look to it leads us into some strange territory and demands that we let go not only of classical physics but also our attachment to real numbers. Going back to the ocean wave, it is as if our mathematics cannot describe the wave itself but only permits us to predict its probable shape from variations in the quality of light reflected from clouds overhead. Complexity is a natural offshoot of the fact that – at the quantum level – direct unmediated observation of the wave is no longer a possibility. The thing we observe is rooted in a space we cannot perceive.
- Complex Numbers in Quantum Theory PHY4910 – Spring 2010 – L. White 5 Appendix A. Stueckelberg’s Argument in Detail. Stueckelberg‟s approach to Real Hilbert Space usesreduction ad absurdum – assume a quantum theory built on ℝ and demonstrate a resulting contradiction. In Boolean terms (A => B) ^ (B => ¬A) (1) Here is his reasoning. Observables are symmetric tensors (or symmetric linear operators) in RHS so that Fab = Fba or FT = F, GT = G, ... (2) The criterion for the impossibility of measuring F and G simultaneously is a non vanishing commutator FG – GF = [F,G] ≠ 0 (3) The expectation value <|F,G|> vanishes because [F,G] is an antisymmetric tensor. Only the positive definite observable P = -[F,G]2 = PT (4) can occur in <∆F2 ><∆G2 >≥ λ2 <P> where λ ∈ ℝ (5) In order to express the uncertainty principle we introduce the error operators ∆F = F – 1 <F>ψ ; ∆G = G – 1 <G>ψ (6) from which we form the mean square errors <∆F2 >ψ and <∆G2 >ψ. There are two possibilities λ2 <P>ψ (7.1) <∆F2 >ψ<∆G2 >ψ≥ λ2 <C>2 ψ (7.2) where P is a positive observable of dimension [F]2 [G]2 and C is an observable of dimension [F][G]. Taking equation (7.1) first <P>ψ = (finite‟)2 > 0 (8) Now <∆F2 >ψ = 0 (9) So 0.(finite)2 ≥ λ2 (finite‟)2 (10) {
- Complex Numbers in Quantum Theory PHY4910 – Spring 2010 – L. White 6 which has only the trivial solution λ = 0, corresponding to the trivial statement <∆F2 >ψ<∆G2 >ψ ≥ 0 (11) Taking equation (7.2) next, we introduce an antisymmetric operator J(FG) and its inverse J-1 (FG), which may be normalized to -1 and commutes with F and G. This gives rise to an uncertainty principle of the form <∆F2 >ψ<∆G2 >ψ ≥ λ2 <C(FG)>2 (12) which requires that J2 (FG)< 0. So the operator J begins to look like a matrix stand-in for the imaginary unit i. Furthermore, the uncertainty principle in this second case is of the form <∆F2 >ψ<∆G2 >ψ ≥ 1/4 <J[F,G]>2 ψ (13) which is incompatible with Heisenberg. So, to summarize Stueckelberg: positing a Real Hilbert Space for quantum theory results either in an uncertainty principle that tells us nothing, or an uncertainty principle that is incommensurate with Heisenberg; either way, RHS cannot serve as a basis for a quantum theory. 1 Saxon, D.S. “Elementary Quantum Mechanics,” Holden Day (June 1968). ISBN 978-0070549807. 2 Stueckelberg, E.C.G. (1960) “Quantum theory in real Hilbert Space,” Helv. Phys. Acta.33, 727. 3 Singh, J. P., Prabakaran, S., “Quantum Computing Through Quaternions”, EJTP 5 No. 19, 2008. 4 Finkelstein, D., Jauch, J.M., Speiser, D., “Notes on quaternion quantum mechanics.”CERN report 597. 5 Lanczos, C., “On a Field Theoretical Representation of the New Quantum Mechanics”, CLCPPC, Volume III, p. 2-858. 6 Wootters, W. K., “Statistical Distance and Hilbert Space”, Phys. Rev.D 23 No. 2. 7 Fuchs, C. A. “Quantum Mechanics as Quantum Information (and only a little more).” http://arxiv.org/abs/quant-ph/0205039.

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