From Bell’s Inequality to Sound Intensity

The temptation is to treat these six topics as a family of loose resemblances. They are not. They are consequences of one algebraic fact, and several of the connections between them are precise theorems rather than metaphors.

Quantum observables are represented by operators that need not commute. For two observables \(\hat A,\hat B\) with commutator \([\hat A,\hat B]=\hat A\hat B-\hat B\hat A\), the impossibility of assigning both sharp values is the seed of everything that follows. From it grow, in order: the uncertainty relations (a statement about a single system at an instant); entanglement and the Bell violations (a statement about separated systems); and the entropic and informational reformulations that let the first two be measured in the common currency of bits. The programme of this report is to make each of those growths explicit, then to show that the whole structure casts a sharp classical shadow in acoustic signal analysis, where the same mathematics reappears with the quantum constant \(\hbar\) simply set aside.

A compact statement of the destination: non-commutativity forces uncertainty; uncertainty is most naturally measured by entropy; entropy connects to information; entanglement is what allows one party’s information to beat the other party’s local uncertainty; and the strength of that uncertainty is precisely what caps how non-local — how Bell-violating — the world is allowed to be. Schrödinger’s equation supplies the reversible, information-conserving stage on which all of it plays out.

Bell’s theorem (Bell, 1964) concerns separated systems. Any local hidden-variable theory — in which measurement outcomes are fixed locally by pre-existing “elements of reality”, as Einstein, Podolsky and Rosen wished — must obey an inequality. In the experimentally convenient CHSH form (Clauser, Horne, Shimony & Holt, 1969), one forms the combination of correlation functions

where \(a,a'\) and \(b,b'\) are two measurement settings on each side. The three regimes stack cleanly:

Quantum mechanics violates the classical bound, reaching \(2\sqrt2\) (Cirel’son, 1980), and loophole-free experiments have confirmed the violation directly (Hensen et al., 2015; Giustina et al., 2015; Shalm et al., 2015). Local realism is therefore untenable. Two facts set up the rest of the report. First, achieving the violation requires measuring non-commuting observables (spin along different axes) on each side — so uncertainty is already present at the scene. Second, quantum theory stops at \(2\sqrt2\) and not at the algebraic ceiling of \(4\); the reason it stops precisely there is the subject of §4.

The familiar relation is a statement about variances. Robertson (1929) gave the general form for any two observables,

and Schrödinger (1930) sharpened it by restoring the covariance term that Robertson had dropped,

For position and momentum, \([\hat x,\hat p]=i\hbar\), and both reduce to \(\sigma_x\sigma_p\ge\hbar/2\). Note already that this inequality is kinematic: it is a property of the state in Hilbert space at an instant, a consequence of the operator structure, and not of any dynamics. This will matter when we come to Schrödinger’s equation.

The entropic reformulation

Variance is a weak and sometimes ill-defined measure — it misbehaves for heavy-tailed or multimodal distributions. The modern statement replaces it with entropy. For two observables with eigenbases \(\{|x_j\rangle\},\{|z_k\rangle\}\), Maassen & Uffink (1988) proved

where \(H\) is the Shannon entropy of the measurement outcomes and \(c\) quantifies complementarity. For a single qubit measured along \(x\) and \(z\), \(c=1/\sqrt2\), giving \(H(X)+H(Z)\ge\ln 2\) — exactly one bit of unavoidable uncertainty. For the continuous position–momentum pair, the Białynicki-Birula–Mycielski relation (1975) uses differential entropies,

and, because the Gaussian maximises entropy at fixed variance, this entropic statement actually implies the Heisenberg relation as a corollary. The significance is that uncertainty is now phrased in Shannon and von Neumann entropy — it speaks the language of information directly, which is what makes the bridges of §4 and §5 possible.

The one genuinely dynamical uncertainty

There is a second uncertainty relation whose lineage is the equation of motion. Taking \(\hat A\) as any observable, the Heisenberg equation (itself a consequence of the Schrödinger equation, see §6) gives \(\tfrac{d}{dt}\langle\hat A\rangle=\tfrac{i}{\hbar}\langle[\hat H,\hat A]\rangle\). Feeding this into Robertson’s relation with \(\hat B=\hat H\) yields, after defining the characteristic time \(\tau_A=\sigma_A/|d\langle\hat A\rangle/dt|\),

This is the Mandelstam–Tamm relation (1945), the honest energy–time uncertainty and the quantum speed limit. It is the point at which the dynamics, rather than the kinematics, is bound to uncertainty.

Return to the question left open in §2: why does quantum mechanics stop at \(2\sqrt2\) and not at \(4\)? Oppenheim & Wehner (2010) answered it, and the answer is the deepest link in the whole picture. They recast the CHSH scenario as a game of steering from one party to the other, in which the achievable violation is bounded jointly by two ingredients: how strongly one party can steer the other’s state, and how certain the statistics of the other party’s complementary measurements can be. The latter is a fine-grained uncertainty relation. Their result, stated plainly, is that quantum mechanics cannot be more non-local with measurements that respect the uncertainty principle.

An independent route reaches the same wall from a purely informational axiom. Information Causality (Pawłowski et al., 2009) postulates that \(m\) transmitted bits should grant access to no more than \(m\) bits of a distant party’s data; imposing it recovers \(2\sqrt2\) and excludes the stronger PR-box correlations. Two very different starting points — local uncertainty on one hand, a bound on information gain on the other — both single out the quantum boundary. (Later work, e.g. Coles et al., 2012, and studies by Ramanathan and co-workers, refined exactly how far the uncertainty–nonlocality correspondence extends across the full quantum set.)

The result that gathers all of the threads into a single line is the entropic uncertainty relation in the presence of quantum memory (Berta, Christandl, Colbeck, Renes & Renner, 2010). Let a system \(A\) be measured in one of two complementary bases while a memory \(B\), possibly entangled with \(A\), is held aside. Then

The remarkable object is the conditional von Neumann entropy \(S(A|B)\). For entangled states it can go negative — a signature of entanglement itself, impossible for any classical correlation. When \(A\) and \(B\) are maximally entangled, \(S(A|B)=-\log_2 d\) exactly cancels the complementarity term \(\log_2(1/c)\), and the bound collapses to zero: with a suitable entangled memory one could, in principle, predict both complementary outcomes, apparently defeating uncertainty.

This is precisely where Bell’s resource reaches into Heisenberg’s domain, and it is quantified in the currency of entropy. Entanglement (the negative conditional entropy) is what lets one side’s information overcome the other side’s local uncertainty. Ekert’s protocol (1991) closes the loop in practice: a Bell violation certifies a secure cryptographic key precisely because no local, pre-existing information could have reproduced the observed correlations. Uncertainty, entanglement, entropy and information are here not four subjects but one.

It is easy to over-claim here, so the boundary is worth drawing carefully. The uncertainty relations are kinematic; they are properties of Hilbert-space states, not consequences of the dynamics. Schrödinger’s equation does not, strictly, produce \(\sigma_x\sigma_p\ge\hbar/2\) — the non-commuting operator structure does. Yet the equation, and the man, are threaded through the whole picture in three concrete ways.

• Schrödinger’s name sits on the tightest general relation. The Robertson–Schrödinger inequality of §3 is his, and it was his 1935 reply to EPR that named Verschränkung — entanglement — calling it the characteristic trait of quantum mechanics. The entire subject of Bell’s theorem was christened in that paper (Schrödinger, 1935).
• The equation is where the genuinely dynamical uncertainty lives. The Mandelstam–Tamm relation \(\Delta E\,\Delta t\ge\hbar/2\) of §3 is derived straight from the Schrödinger equation through \([\hat H,\hat A]\). This, not the kinematic \(\sigma_x\sigma_p\), is the honest place where the dynamics is tied to uncertainty.
• The equation conserves information while manufacturing entanglement entropy. Schrödinger evolution is unitary, so it exactly conserves the von Neumann entropy of a closed system — total information is preserved. Yet the entanglement entropy of a subsystem grows under that same unitary flow. That is how a closed quantum system thermalises and how decoherence proceeds. The equation therefore conserves global information while generating local entropy — binding the dynamics once more to the entropy–information side of the story.

So the correct statement is not “Schrödinger’s equation gives the uncertainty principle”, but rather that it supplies the reversible, information-conserving stage — and, through Mandelstam–Tamm, the one uncertainty relation that is genuinely about time evolution.

Here the report crosses from quantum foundations to signal analysis — and the crossing is by identity, not analogy.

The time–frequency uncertainty relation is Heisenberg’s relation with \(\hbar\) removed. For a finite-energy signal \(g(t)\) with Fourier transform \(\hat g(\omega)\), defining the spreads \(\Delta t,\Delta\omega\) as the standard deviations of \(|g|^2\) and \(|\hat g|^2\) about their means,

This is the same theorem as \(\sigma_x\sigma_p\ge\hbar/2\), because momentum is \(p=\hbar k\) and the pair \((x,k)\) is exactly the Fourier-conjugate pair \((t,\omega)\) wears in signal analysis. Gabor (1946) saw this explicitly: he imported the mathematical apparatus of quantum theory wholesale, coined the “logon” as the elementary cell of information, and identified the minimum-uncertainty signal that saturates the bound — the Gaussian-windowed sinusoid, the Gabor atom,

which is the classical counterpart of the quantum coherent state, and the emblem on the title page of this report. The finite record therefore carries a countable budget of independent degrees of freedom. A signal essentially confined to a time interval \(T\) and a bandwidth \(B\) has about

independent real degrees of freedom — the Shannon number — which is exactly the count of minimum-uncertainty cells that tile a time–frequency region of area \(BT\). This is Slepian, Landau & Pollak’s prolate-spheroidal result (1961): the discrete prolate spheroidal sequences are the eigenfunctions of the joint time- and band-limiting operator, and they are the same extremal-concentration solution that produces the Gabor atom in the continuous case. Thomson’s multitaper spectral estimator (1982) is nothing other than the optimal use of that basis. The uncertainty principle has become, without changing form, an estimation constraint.

Sound intensity is the vector describing net acoustic energy flow. The two-microphone (p–p) method estimates it entirely from the cross-spectrum of the two pressures, and that object is where the entropy and information threads surface in acoustics. Linearised Euler’s equation relates particle velocity to the pressure gradient, \(\rho_0\,\partial u/\partial t=-\partial p/\partial x\); in the frequency domain the gradient is approximated by a finite difference over the spacing \(\Delta r\),

The time-averaged active intensity is \(I=\tfrac12\,\mathrm{Re}\{\hat p\,\hat u^{*}\}\). Carrying the algebra through, the pressure product collapses onto the cross-spectral density \(G_{12}(\omega)\), and the result is the classic estimator (Fahy, 1977; Chung, 1978)

with \(G_{12}\) the one-sided cross-spectrum of the two microphone signals (the overall sign following the convention for \(G_{12}\) and the probe’s orientation). It is worth dwelling on why the imaginary part appears. Velocity is the time-integral of the gradient, so it divides by \(i\omega\) and rotates the phase by ninety degrees; the in-phase pressure–velocity product that carries net energy therefore picks out the quadrature part of the cross-spectrum. The real part — the co-spectrum — gives the reactive intensity: stored, oscillating energy with no net transport.

The resolution–bandwidth corridor

The practical limits of the method are an uncertainty corridor set by the spacing \(\Delta r\). At low frequency the finite-difference approximation is excellent, but the true phase difference \(k\Delta r\) one is trying to measure is tiny, so residual instrumentation phase mismatch swamps it; coherence falls, and the variance of \(\operatorname{Im}\{G_{12}\}\) explodes, demanding more averaging — which costs time–bandwidth. The severity is governed by a single field indicator, the pressure–intensity index \(\delta_{pI}=L_p-L_I\) (Jacobsen, 1991), which measures how much reactive energy overlies the net flux. At the high end the wall is the opposite: when \(\Delta r\) approaches a wavelength, \(k\Delta r\) is no longer small and the gradient approximation itself fails. Choosing \(\Delta r\) is choosing one’s seat on this corridor; probe design is optimisation within an uncertainty relation.

The ordinary coherence between the two microphone signals,

is the frequency-domain correlation coefficient — the everyday acoustic cousin of the separated-measurement correlations that Bell’s theorem concerns. But it is more than a cousin of information; for Gaussian signals it is information, and the derivation is short enough to give in full — this is the identity that closes the entropy–information loop inside two-microphone acoustics.

The result is classical and rigorous (Gelfand & Yaglom, 1959; Pinsker, 1964; regularity conditions in Komaee, 2020), and its use as a lower bound on information transfer is standard practice in sensory coding (Bialek et al.; Gabbiani, 1996). Its meaning for the acoustician is direct: when the coherence drops, the mutual information between the channels drops, and it is this that inflates the error bars and forces more averaging. The practitioner’s remark that “the coherence fell, so I cannot trust this band” is, exactly, an information-theoretic statement.

None of this is ornamental. The same structure is doing daily work across measurement and acoustics.

Sound power without a special room

Because the surface integral \(\oint \mathbf I\cdot d\mathbf S\) returns radiated power regardless of the surrounding field, intensity lets one rank noise sources on a running machine in situ, rather than hauling it into an anechoic or reverberant chamber (the basis of ISO 9614). The pressure–intensity index is not a nuisance but the go/no-go criterion that certifies whether a given measurement can be trusted — uncertainty appearing as a field indicator.

The resolution corridor as a design specification

Choosing the microphone spacing \(\Delta r\), or the window in a spectral estimate, is spending the time–bandwidth budget of §7–8. Small \(\Delta r\) lifts the high-frequency wall but worsens the low-frequency phase-mismatch wall; a flat-top window buys amplitude accuracy at the cost of frequency resolution. These are seats on the same corridor.

Multitaper estimation, everywhere

Thomson’s method (1982), built on the prolate spheroidal basis, is the optimal answer to “extract the most independent spectral information from a finite record”, and it is now standard in vibration diagnostics, seismology and any low-SNR spectral estimate.

Array acoustics: the density-matrix analogy made industrial

In beamforming and noise-source imaging the central object is the array cross-spectral matrix — the Hermitian, positive-semidefinite density-matrix-like object of §8. Its eigenstructure does the work: subspace methods such as MUSIC (Schmidt, 1986) separate a signal subspace from a noise subspace, and modern aeroacoustics explicitly denoises the cross-spectral matrix by truncating its small eigenvalues. Nearfield acoustic holography goes further still — Williams & Maynard (1980), with the full theory in Maynard, Williams & Lee (1985) — reconstructing the entire field, evanescent components included, from a pressure hologram, and thereby beating the wavelength resolution limit precisely by capturing the near-field information that ordinary far-field measurement discards. (Jacobsen later extended holography to particle-velocity probes.) The eigenvalue “entropy” of the matrix is the honest count of how many independent sources are radiating.

Coherence as the universal workhorse

Ordinary, multiple and partial coherence underpin transfer-path analysis, modal testing and structural health monitoring. By §9, low coherence is low information transfer, which sets the averaging budget and the error bars directly.

Wavelets: escaping the fixed tiling

Because the Gabor atom fixes a single cell shape, constant-Q and wavelet or matching-pursuit methods were developed to let the tiling adapt — fine time resolution at high frequency, fine frequency resolution at low. One cannot beat the uncertainty area, but one can reshape the cells; that reshaping is a whole branch of audio signal processing.

The uncertainty principle as a perceptual boundary

The Gabor limit shows up directly in hearing: the briefer a sound, the broader its spectrum, which bounds how precisely the pitch of a very short sound can be encoded. Strikingly, human listeners appear to beat the naive time–frequency bound (Oppenheim & Magnasco, 2013 — a different Oppenheim from §4), a finding tested and refined in the psychoacoustics of extremely brief tones (Psychonomic Bulletin & Review, 2015). Whether and how the auditory system does so remains an open experimental question, phrased in precisely these terms.

The genuine quantum floor

Everything above is classical Fourier uncertainty, with no \(\hbar\). The quantum relation proper re-enters only at the ultimate sensitivity limit. Optical and optomechanical acoustic sensors — laser Doppler vibrometry, membrane-optomechanical microphones, photonic pressure sensors — meet the standard quantum limit, the same balance of measurement back-action against imprecision noise that constrains gravitational-wave interferometers. There one beats the naive floor with squeezed light, redistributing uncertainty between quadratures. That is the entropic trade-off of §3 turned into an instrument technique — and it is the point at which the analogy stops being an analogy and becomes the same quantum mechanics with which this report began.