# Quantum mechanics via quantum tomography

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Referee this paper: arXiv:2110.05294 by Arnold Neumaier

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Abstract: Starting from first principles inspired by quantum tomography rather than Born's rule, this paper gives a new, elementary, and self-contained deductive approach to quantum mechanics. A suggestive notion for what constitutes a quantum detector and for the behavior of its responses leads to a logically impeccable definition of measurement. Applications to measurement schemes for optical states, position measurements and particle tracks demonstrate that this definition is applicable without any idealization to complex realistic experiments.

The various forms of quantum tomography for quantum states, quantum detectors, quantum processes, and quantum instruments are discussed. The traditional dynamical and spectral properties of quantum mechanics are derived from a continuum limit of quantum processes. In particular, the Schrödinger equation for the state vector of a pure, nonmixing quantum system and the Lindblad equation for the density operator of a mixing quantum system are shown to be consequences of the new approach. A slight idealization of the measurement process leads to the notion of quantum fields, whose smeared quantum expectations emerge as reproducible properties of regions of space accessible to measurements.

requested Oct 12
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paper authored Oct 11 to quant-ph
edited Oct 15

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The paper Quantum mechanics via quantum tomography generalizes the traditional setting of quantum mechanics for pure states and projective measurements to include real-world measurements, mixing measurements, and quantum field theory in a single framework.

In this approach to describe quantum measurement processes, only the macroscopic source and detector are relevant. The quantum source is described by a Hermitian, positive semidefinite density operator $\rho$, whose trace corresponds to the intensity of the source. The state of the source is the linear mapping that assigns to each linear operator on the state space of the quantum system its quantum value Tr$\rho A$, calculated from the density operator of the source. The source is called pure, if the rank of its density operator is $1$, and mixed otherwise.

A measurement device is given by a finite number of detection elements $k$, with corresponding quantum measure consisting of a familiy of Hermitian positive semidefinite operators $P_k$ that sum up to $1$. The response probablity for a certain detection element $k$ to respond is given by $p_k=Tr\{\rho P_k\}$. Normalizing the trace of the density operator to $1$, the $p_k$ can be interpreted as the response probability of detector element $k$ given some response, or as the detection probability for the kth detection event. This gives an intuitive meaning for the $p_k$ in case of low intensity measurements. A detector characterized by the quantum measure $P_k$ and a freely choosable scale $a_k$ measures the quantity $A,$ if

$A := \sum a_k P_k$

It can be shown that the statistical expectation of the measurement results equals the quantum expectation of the measured quantity. This corresponds to Born´s rule in expectation form.

To illustrate this new framework to describe quantum measurements, the quantum measures for particular detectors are discussed. The setting of quantum optical networks consisting of optical filters such as beam splitters and polarizers is nicely explained and the quantum measures for the corresponding non-mixing and linear optical components is derived explicitly. Using coherent states, measurements of operators with a continuous spectrum or simultaneous measurements of non-commuting operators can be described using quantum measures. Basic components such as sources, filters, and detectors can be combined to highly complex quantum circuits. A quantum instrument acts as a combined system of a detector and filters that puts the output beam after measurement into a definite state. The infamous collapse state of a quantum system can be described by such a projective instrument. More complicated indirect measurements, such as the derivation of position, momentum, and energy from particle tracks in a TPC can be done using quantum measures too. Projecjtive measurements as described in conventional introductory courses on QM are then derived as those described by projective quantum measures. The squared amplitude rule is retrieved if one assumes in addition pure states and non-degenerate operators.

Theorem 2.1 allows for a number of quantum tomography procedures that are very helpful in calibrating the different components of a quantum measurement. Quantum detector tomography uses

$$p_k = \langle P_k \rangle = Tr \{\rho P_k \}$$

to derive the quantum measure of a given detector. For this a sufficient number of sources with known density matrices is needed, the corresponding response probabilities have to be detected, and the quantum measures calculated. In quantum state tomography, the density operator of a source can be found from measuring a sufficient number of response probabilities $p_k$ of a detector with known informationally complete quantum measure. Filters between the source and the detector can be described by a completely positive map, that transforms the density operator of the input state to the density operator of the output state. Experimentally, this map can be found by process tomography using sources with known density operators as input and measuring the output by state tomography.

Many items that are introduced in conventional introductions to quantum mechanics can be derived in the new framework. By modelling the passage of a quantum beam through an inhomogenous nonmixing linear medium by a slicing procedure, the quantum Liouville equation is obtained. Assuming a lossless medium results in von Neumann s equation.  Additionally assuming pure states leads to the time dependent Schroedinger equation. For  a homogeneous medium, the Rydberg-Ritz combination principle is obtained. The wave or quantum field point of view is more natural in this picture than the particle description. Meassurements of quantum fields are obtained by narrowing the region of sensitivity of a detector to a point and putting it to the same point as the source. Modelling the passage of a quantum beam through an inhomogeneous mixing linear medium gives the Lindblad equation for dissipative systems.

To round up things, some topics concerning measurement uncertainty are discussed. The lack of projectivity of a measurement can be related to the excess uncertainty of a real detector, as opposed to an idealized projective one. Imperfect measurements are characterized by a corrected root mean square error to include the systematic error. Not the individual observations but only their statistical properties - probabilities and expectation values – are reproducible. Measurement errors are defined with respect to some (idealized) true values, which in the Thermal interpretation correspond to the quantum values of quantum measurements.

Some limitations of the new framework mentioned in the paper are for example that quantum beams can only be used if geometric optics is valid. Assuming stationarity allows only to describe the semiclassical regime. The fact that measurement theory is insensitive to the nature of what goes between the source and the detector still sits not very comfortable with me, as conventionally it is exactly the things that flow or sit between source and detectors one wants to investigate by doing measurements in experimental quantum or particle physics. As mentioned concerning an earlier paper series of Arnold Neumaier about quantum foundations, it has to be said here that there are many physicists who are satisfied with the traditional theoretical framework based on projective measurements, as they see the task of theoretical QM in a narrower way and defer concerns pertaining to real-world measurements to experimental QM and engineering issues.
However, it is somehow intriguing that the POVM or quantum measure approach accounts for things like losses, imperfect measurements, limited detection accuracy, dark detector counts, and the simultaneous measurement of position and momentum which goes beyond what is tractable in the conventional point of view. Some derivation, for example the one of the Lindblad equation are rather cute and the description of experiments including the functioning of optical networks is very nice.
What would interest me is the question if all simultaneous measurements of non-commuting operators can be described by quantum measures using certain coherent states?

reviewed Oct 30 by (6,140 points)
edited Nov 14 by Dilaton

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