Energy Cost of Localization and Measurement

Localizing a field measurement more sharply usually requires finer spatial structure, broader momentum support, or faster control. None of those facts by itself gives a universal energy price. An energy cost belongs to a complete physical protocol: the measured observable, allowed input states, instrument, controller, accuracy criterion, duration, and endpoint energy reference must all be fixed before a lower bound has operational meaning.

Required background. Measurement-induced energy, noise, and backreaction supplies the instrument-level definitions of error, disturbance, and switching work.

Helpful background. Quantum energy inequalities explain why a bound on sampled stress energy is not automatically a bound on apparatus work.

A localized measurement is a specified task

Consider a real scalar field in $n+1$ dimensions and a normalized spatial profile $F_\ell$ of characteristic width $\ell$. A concrete target observable is

$$X_\ell(t_0)=\int d^n x\,F_\ell(\mathbf x)\phi(t_0,\mathbf x).$$

This is a smeared observable, not a point value of the field. An indirect measurement couples it to a probe with pointer coordinate $Q$ and conjugate momentum $P$, reads an outcome $y$, and thereby implements an instrument $\{\mathcal I_y\}$. A useful comparison holds fixed at least

$$\mathcal T=(F_\ell,\tau,\mathcal S, \varepsilon,p_{\rm fail},D),$$

where $\tau$ is the interaction duration, $\mathcal S$ is the allowed finite-energy input-state set, $\varepsilon$ is a declared error measure, $p_{\rm fail}$ is any inconclusive-outcome probability, and $D$ measures state disturbance. Changing any entry changes the task.

Spatial and temporal localization also have separate Fourier consequences. If $F_\ell(\mathbf x)=\ell^{-n}F(\mathbf x/\ell)$, its Fourier transform samples momenta of order $1/\ell$. A switching function $\chi_\tau(t)=\chi(t/\tau)$ samples frequencies of order $1/\tau$. A nonzero profile cannot be both exactly compactly supported and exactly band limited, so a finite-bandwidth implementation must declare a tail tolerance rather than quietly identify the two notions.

Localized QFT measurements require a system–probe coupling, not just an abstract projection. The algebraic construction of Fewster and Verch 2020, §§ 3–5 makes this separation explicit: the coupling region, probe observable, induced system observable, and state update are different parts of the model.

The operational energy account

When the controller is represented by prescribed time dependence, let the effective field–probe Hamiltonian be

$$H(t)=H_\phi+H_P+H_{\rm int}(t),$$

with $H_{\rm int}$ zero at the initial and final accounting times. The work supplied through that time dependence is

$$W_C=\int_{t_i}^{t_f}dt\, \operatorname{tr}\!\left[\rho(t)\,\partial_tH(t)\right].$$

For a field–probe model with no other energy exchange this equals $\Delta E_\phi+\Delta E_P$. A laboratory cycle generally also contains a physical controller, record formation, amplification, cooling, and reset. One should therefore report the signed endpoint changes and the nonnegative resources separately:

$$W_{\rm cycle}=W_C+W_{\rm read}+W_{\rm reset}, \qquad (\Delta E_\phi,\Delta E_P,\Delta E_C) \text{ separately}.$$

The field-energy increase is not the whole measurement cost. Nor is every positive term necessarily irrecoverable: recoverable probe energy and heat dumped during reset are thermodynamically different quantities.

Reproducible benchmark: the same readout, two probes

Take a massless scalar field in $1+1$ dimensions at one instant and the Gaussian

$$F_\ell(x)=\frac{e^{-x^2/(2\ell^2)}}{\sqrt{2\pi}\,\ell}, \qquad \int dx\,F_\ell(x)^2=\frac{1}{2\sqrt\pi\,\ell}.$$

Use the impulsive unitary

$$U=\exp(-igX_\ell P).$$

It gives $Q_{\rm out}=Q+gX_\ell$. If the initial pointer is a centered minimum-uncertainty Gaussian with $\operatorname{Var}Q=s^2$ and $\operatorname{Var}P=1/(4s^2)$, the unbiased estimator $Q_{\rm out}/g$ has additive RMS noise $\varepsilon=s/g$. The same unitary shifts the field momentum by $gPF_\ell$. For a centered product input, the mean field-energy injection is

$$\Delta E_\phi =\frac{g^2\langle P^2\rangle}{2} \int dx\,F_\ell^2 =\frac{1}{16\sqrt\pi\,\ell\varepsilon^2}.$$

Thus $\ell=1$ and $\varepsilon=0.25$ give $\Delta E_\phi=0.5641896$ in the benchmark’s natural energy units. This $1/(\ell\varepsilon^2)$ scaling is a result for the chosen dimension, normalization, impulsive coupling, state, and error definition. It is not a general localization theorem.

Now realize exactly the same $Q$ distribution and coupling with two probe families during a sufficiently short pulse:

1. a free pointer, $H_P^{\rm free}=P^2/(2M)$;
2. a trapped pointer, $H_P^{\rm trap}=P^2/(2M)+M\Omega^2Q^2/2$.

Their initial mean energies are

$$E_P^{\rm free}=\frac{1}{8Ms^2}, \qquad E_P^{\rm trap}=\frac{1}{8Ms^2}+\frac{M\Omega^2s^2}{2}.$$

For $M=\Omega=1$ and $s=0.25$, these are $2$ and $2.03125$. Relative to the trapped probe’s ground state, however, its preparation energy is $2.03125-0.5=1.53125$. The outcome noise and the field injection are unchanged. Even this elementary comparison shows why “the cost” depends on the apparatus Hamiltonian and on whether one counts absolute internal energy or energy above a reusable reference state. Varying $M$ changes the free-pointer preparation energy without changing the idealized readout statistics; a physical pulse generator may then acquire the missing cost, which is why $W_C$ must be measured rather than inferred from the POVM.

A conditional conservation-law bound

Some cross-protocol statements do exist once their hypotheses are fixed. Let an apparatus measure $A$ while a unitary conserves $L_S+L_P$. With noise operator

$$N=M_{\rm out}-A_{\rm in}, \qquad \varepsilon_\psi(A)^2=\langle N^2\rangle_{\psi\otimes\xi},$$

and with the Yanase condition $[M,L_P]=0$, the Robertson relation gives the WAY–Ozawa bound

$$\varepsilon_\psi(A)^2\ge \frac{|\langle[A,L_S]\rangle_\psi|^2} {4(\Delta_\psi L_S)^2+4(\Delta_\xi L_P)^2}.$$

This is Ozawa 2002, Eqs. (3)–(14), pp. 050402-1–050402-3. For a position proxy $A=Q_S$, conserved momentum $L_S=P_S$, and $\hbar=1$,

$$(\Delta P_P)^2\ge \frac{1}{4\varepsilon^2}-(\Delta P_S)^2.$$

If, additionally, $H_P=P_P^2/(2M)$ and $\langle P_P\rangle=0$, then

$$E_P\ge \frac{1}{2M} \left[\frac{1}{4\varepsilon^2}-(\Delta P_S)^2\right]_+.$$

For $\varepsilon=0.25$, $\Delta P_S=0.5$, and $M=10$, this gives $(\Delta P_P)^2\ge3.75$ and $E_P\ge0.1875$. At $M=40$ the energy bound is only $0.046875$, although the required momentum variance is unchanged. The theorem constrains an asymmetry resource; converting that resource into energy requires an apparatus Hamiltonian.

Exact position measurements under momentum conservation and the Yanase condition are ruled out even for continuous unbounded generators by Kuramochi and Tajima 2023, Theorem 1 and the position application. The Yanase condition is consequential: Ozawa 1991 constructs a momentum-conserving position-measurement countermodel outside the familiar restriction. It does not refute the conditional theorem; it identifies which premise carries the limitation.

What the benchmark does not prove

The impulse model has Gaussian tails, ideal time switching, and no ultraviolet regulator. A QFT calculation intended as a continuum claim must replace the impulse by a smooth switching, impose a common finite-energy state domain, and show convergence as the regulator is removed. A position proxy in a one-particle sector is not a covariant point-local field observable.

A quantum energy inequality can reject a proposed negative sampled stress-energy history, but it does not determine $W_{\rm read}$ or $W_{\rm reset}$. The free-field inequalities of Fewster and Eveson 1998, for example, depend on the field, worldline, state class, and sampling function.

Three controls are decisive:

• hold $(\ell,\tau,\varepsilon,D,\mathcal S)$ fixed while changing the probe Hamiltonian and switching profile;
• repeat the calculation with the bandwidth cutoff, tail tolerance, and input-energy ceiling varied independently;
• break one theorem hypothesis at a time, especially exact conservation or the Yanase condition, and move the controller that absorbs the conserved quantity into the energy account.

The chapter orientation, claim-validity table, and failure controls place these tests beside the other energy–information results.

Common pitfalls

Turning bandwidth into work. A Fourier lower bound specifies spectral support. Work also depends on dispersion, coupling strength, controller dynamics, and the initial state.

Comparing different tasks. Equal spatial width is insufficient if the instruments have different error, duration, failure probability, or disturbance.

Calling a conditional WAY bound universal. The conservation law, Yanase condition, error definition, and conversion from conserved-quantity variance to energy are all additional assumptions.

Exercises

1. Gaussian field-energy injection

Derive the impulsive benchmark for the Gaussian $F_\ell$. Evaluate it at $\ell=2$ and $\varepsilon=0.1$.

Solution

The commutator $[X_\ell,\pi(x)]=iF_\ell(x)$ gives $U^\dagger\pi(x)U=\pi(x)-gPF_\ell(x)$, up to the irrelevant sign convention for $g$. The cross term averages to zero for a centered product input. Hence

$$\Delta E_\phi=\frac{g^2\langle P^2\rangle}{2}\|F_\ell\|_2^2.$$

Using $g=s/\varepsilon$, $\langle P^2\rangle=1/(4s^2)$, and $\|F_\ell\|_2^2=1/(2\sqrt\pi\ell)$ gives $1/(16\sqrt\pi\ell\varepsilon^2)$. At $\ell=2$ and $\varepsilon=0.1$ the result is

$$\Delta E_\phi=\frac{1}{0.32\sqrt\pi}=1.76309.$$

2. What does the WAY bound actually price?

For $\varepsilon=0.2$, $\Delta P_S=1$, and $M=5$, compute the minimum probe momentum variance and the derived kinetic-energy bound. Repeat for $M=50$.

Solution

The variance bound is

$$(\Delta P_P)^2\ge\frac{1}{4(0.2)^2}-1=6.25-1=5.25.$$

Thus $E_P\ge5.25/(2M)$: it is $0.525$ for $M=5$ and $0.0525$ for $M=50$. Conservation fixes a required momentum spread in this model, while the Hamiltonian fixes its energy price.

3. Diagnose a false universal claim

An analysis finds $\Delta E_\phi\propto\ell^{-1}$ using the impulsive $1+1$-dimensional benchmark and concludes that every localization measurement costs at least $C/\ell$. Give three independent reasons the conclusion does not follow.

Solution

First, the exponent comes from $\|F_\ell\|_2^2\propto\ell^{-1}$ in one spatial dimension with an $L^1$-normalized Gaussian; another dimension or normalization changes it. Second, the derivation fixes a von Neumann impulse, a minimum-uncertainty pointer, and RMS additive noise; another instrument need not inject the same field energy. Third, field injection omits probe preparation, switching, readout, and reset. A valid cross-protocol bound would optimize the complete account over an admissible instrument class while holding error, duration, disturbance, state domain, and tail tolerance fixed.

References

• Fewster, Christopher J., and Simon P. Eveson. “Bounds on Negative Energy Densities in Flat Spacetime.” Physical Review D 58 (1998): 084010. DOI.
• Fewster, Christopher J., and Rainer Verch. “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378 (2020): 851–889. DOI.
• Kuramochi, Yui, and Hiroyasu Tajima. “Wigner–Araki–Yanase Theorem for Continuous and Unbounded Conserved Observables.” Physical Review Letters 131 (2023): 210201. DOI.
• Ozawa, Masanao. “Conservation Laws, Uncertainty Relations, and Quantum Limits of Measurements.” Physical Review Letters 88 (2002): 050402. DOI.
• Ozawa, Masanao. “Does a Conservation Law Limit Position Measurements?” Physical Review Letters 67 (1991): 1956–1959. DOI.