This paper some kind of picks up the discussion about the limited domain of validity of Born’s rule started in Foundations of quantum physics I. A critique of the tradition. To go beyond the projective measurements for which Born s rule applies, so-called POVMs (Positive Operator Valued Measures) are presented to describe imperfect real world measurements as actually done in research laboratories. This allows one to consider experimental issues such as imperfect measurements, limited detection accuracy, dark detector counts, and the simultaneous measurement of position and momentum. The description of the measurement process by POVMs and the corresponding framework fits smoothly together with the thermal interpretation as described in Foundations of quantum physics II. The thermal interpretation and Foundations of quantum physics IV. More on the thermal interpretation.
The first chapter explains how real-world measurements can be described by the POVM framework. As motivated by the example of classical polarized light, an arbitrary quantum system is described in terms of a complex Hilbert space of the source and a positive definite Hermitian operator $\rho$ with finite trace which describes the state of a stationary source. The macroscopic intensity of the source is given by the trace of $\rho$. Certain Hermitian operators $X$ (with details depending on the quantum system) define key quantities of the source. The corresponding q-expectations $\langle X\rangle=Tr(\rho X)$ describe the state of the source. It is called pure, if $\rho$ has rank $1$. The usual uncertainty relations for non-commuting quantities $X$ apply. The detector is decomposed into binary detector elements, that respond to a state of the system with a certain probability. To each detector element $k$ a positive semidefinite hermitian operators $P_k$ is assigned. These operators sum up to unity and constitute the POVM. Probabilities of response as measured for each detector element $k$ are described by $p_k=Tr(\rho P_k)$. If the probabilities of response for the detector elements sum up to $1$, there is a unique POVM $P_k$, such that $p_k = Tr(\rho P_k)$. To actually describe what is measured in the POVM setting, a scale $a$ is defined to be an assignment of distinct complex numbers or vectors $a_k$ to the possible detection elements $k$. Statistical expectations of a function can then be calculated using the probabilities of response for each detection element $k$ as
$$E(f(a_k))\doteqdot \sum_{k \in K}p_kf(a_k) $$
assuming $K$ detector elements. One can then say that a detector given by a scale and a POVM is defined to measure a quantity A as given by
$$A \doteqdot P[a_k] = \sum a_kP_k$$
and the usual quantum statistics traces formula $E(a_k)=Tr(\rho A)=\langle A \rangle$ holds. However, the scale is not uniquely determined by the detector elements and the quantity $A$ measured. This is only the case for minimal informationally complete POVMs with $|K| \ge d^2$ where $K$ is the number of detector elements and $d$ the dimension of the Hilbert space of the source. In this case, the POVM and the quantity $A$ measured uniquely determine the scale $a$.
In the second chapter, the POVM framework to describe measurements is then illustrated by several examples, starting with polarization state measurements of classical light. An idealized joint measurement of position and momentum can be described by a coherent state POVM, see the coherent states paper series. For operators with a continuous spectrum, such as for example realistic measurements of position, coherent state POVMs are needed, too. As a more complex concrete example, measuring particle tracks in particle physics by means of Time Projection Chambers (TPC) is explained. For projective measurements, the conventional Born rule emerges as a special case. Based on the considerations in the first two chapters, a new way to introduce quantum mechanics to beginning students is then outlined. The author suggests to start with the POVMs setting to describe quantum measurements right from the start. The quantum Liouville equation for the density operator can then be derived in the context of classical optics. In the lossless case, one obtains the quantum von Neumann equation. As usual, the Lindblad equation is needed for the general dynamics of a dissipative quantum system. Moving towards QFT, discussing the anharmonic oscillator would already need infinite dimensional Hilbert spaces. It is also suggested to introduce Lie algebras in introductory quantum mechanics, which is not usually done in courses targeted at physicists.
The third chapter discusses various aspects of uncertainty in quantum measurements. The statistical uncertainty of the measurement results $a_k$ is shown to be bounded from below by the theoretical q-uncertainty of the quantity $A$ considered. When measurements of $A$ are performed in place of measurements of $X$, the measurement uncertainty is at least as large as the q-uncertainty of $X$, and larger if there is a nonzero bias. The conventional notion that measurement results are eigenvalues of the operator measured is said to pose a problem with reproducibility, as only the statistics of the measurement can be reproduced. Splitting the measurement result into true value and measurement error, it is suggested to take the q-expectation as the true values, as already done in the paper Foundations of quantum physics III. Measurement. As in the thermal interpretation, the POVM description of measurements presented in this paper uses the density oparator as a tool to obtain q-expectations, that describe the state of the system. These q-expectations then follow classical (Ehrenfest) equations of motion, see Foundations of quantum physics II. The thermal interpretation. Because of the single-valuedness of the true values, probabilities are not intrinsic to quantum physics in the thermal interpretation, but emerge as imperfections of the measurement process. In contrast, Born’s statistical interpretation needs probabilities in the very foundations, due to the multi-valuedness of the eigenvalues as true values. The thermal interpretation attributes discreteness to the detection setup, not to a true, discrete value of the quantity measured. For example, bimodal distributions of the measurement results may be explained by environment-induced randomness and dissipation, as in the case of a classical environment-induced diffusion process in a double well potential.
Even though the Robertson uncertainty relation
$$ \sigma_A \sigma_B \ge \frac{1}{2}|\langle[A,B]\rangle|$$
is shown to hold for non-commuting quantities describing the source
$$ \overline{X}=\langle X \rangle \doteqdot Tr(\rho X), $$
it is not quite clear to me at a glance if the usual Heisenberg uncertainty relation $\sigma_{x}\sigma_{p} \geq \frac{\hbar}{2}$ that describes a trade-off between the accuracy of measurements of position and momentum still holds in the usual sense. Particularly confusing in this context seem the remarks on the top of page 30 that the uncertainty is in the imprecise definition, instead of the uncertainty relation being one of the fundamental intrinsic ingredients defining quantum mechanics.
As said in the second paragraph of p31, by using the single-valued q-expectations as true values, the thermal interpretation rejects the intrinsically probabilistic nature of QM. If one instead takes the conventional point of view that quantum mechanics predicts only probabilities for the outcome of specific measurements, there are no problems with reproducibility as what needs to be reproducible are just the probabilities themselves.
It is not quite clear to me how in the framework outlined in this paper, (classical or quantum) scattering experiments would be described, as they involve in addition to just sources and detectors certain targets. Also, concerning quantum mechanic scattering experiments, how would “virtual” processes that only indirectly impact the processes measured be included?
I like it how this paper generalizes Born’s rule for projective measurements by means of POVMs to include some real-world experimental issues that occur in actual laboratory measurements. And there is some cute mathematically rigorous discussion not seen elsewhere. However, I personally would keep teaching introductory (theoretical) quantum mechanics as is, then include the topics of this paper in an advanced course with some experimental focus. Considering just projective measurements (without “dirty” real-world issues such as the efficiency of detectors etc) is often enough for students who want to focus on theoretical physics.
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