Hamiltonian in which the first excited state has a much larger energy than the
ground state. Thermal dissipation then brings the system into equilibrium, and
as seen above, the resulting equilibrium state is nearly pure. Those low lying
excited states for which a selection rule suppresses the transition to a lower
energy state can be made nearly pure in the same way.

A filter with transmission matrix $T$ transforms a beam in the pure state
$\psi$ into a beam in the pure state $\psi^{\prime}=T\psi$. If $\psi(t)$
denotes the pure state at time $t$ then the same slicing scenario as above
gives $\psi(t+\Delta t)=T(t)\psi(t)$. Therefore

$\frac{d}{dt}\psi(t)=\frac{\psi(t+\Delta t)-\psi(t)}{\Delta t}+O(\Delta
t)=K(t)\psi(t)+O(\Delta t).$

In the continuum limit $\Delta t\to 0$ we obtain the differential equation

$\frac{d}{dt}\psi(t)=K(t)\psi(t).$

In the special case of a lossless medium this becomes the time-dependent
Schrödinger equation

$i{\mathchar 22\relax\mkern-9.0muh}\frac{d}{dt}\psi(t)=H(t)\psi(t).$ (26)

If the ground state of a quantum system is finitely degenerate, thermal
sources produce mixed states in the subspace of minimal energy, and one needs
more care to prepare a pure state. Now a filter is needed that temporarily
creates by an interaction a modified Hamiltonian that removes the degeneracy.
In the purification of electron beams through idealized Stern–Gerlach magnets
(Busch et al.[25]), the twofold spin degeneracy of electrons is removed by
utilizing the different motion of different spin states in magnetic fields.
This acts as a beam splitter, and upon projection to a single beam, the state
collapses to a pure state. The twofold helicity degeneracy of light is removed
through a similar process in the purification of light through beam-splitting
polarization filters, while in absorptive polarization filters, the degeneracy
is removed through lossy thermal convergence to the ground state of the
modified Hamiltonian.

If a density operator $\rho$ is mixed, it still represents the complete
information about the state. This contrasts with the traditional view where a
density operator $\rho$ represents uncertainty about the true, pure state with
allegedly more complete information. Indeed, if taken at face value, the
latter view is internally inconsistent – see the somewhat self-contradictory
discussion in the second quote by Landau and Lifschitz or a more detailed
discussion in Neumaier [125, Section 15.2]. According to the view presented in
this paper, more complete can at best refer to imagined counterfactual
alternatives.

###  5.3 Energy spectrum and quantum fluctuations

Wohl aber kann dem Elektron auch in der Quantentheorie eine Ausstrahlung
zugeordnet werden; diese Strahlung wird beschrieben erstens durch die
Frequenzen, die als Funktionen zweier Variablen auftreten, quantentheoretisch
in der Gestalt:
(However, even in quantum theory it is possible to ascribe to an electron the
emission of radiation. In order to characterize this radiation we first need
the frequencies which appear as functions of two variables. In quantum theory
these functions are of the form)
$\nu(n,n-a)=\frac{1}{j}[W(n)-W(n-a)]$.

Werner Heisenberg, 1925 [73, p.881]

In the special case of passage through a homogeneous medium, the filters are
independent of their location, hence $T(t)$ is time-independent. In this case,
$K$ and hence the Hamiltonian $H$ are time-independent. In particular, the
Schrödinger equation $i{\mathchar 22\relax\mkern-9.0muh}\dot{\psi}=H\psi$ for
pure states in a lossless medium can be solved by decomposing

$\psi(t)=\sum_{k}e^{-itE_{k}/{\mathchar 22\relax\mkern-9.0muh}}\psi_{k}$ (27)

as a superposition of eigenvectors $\psi_{k}$ of $H$ satisfying the time-
independent Schrödinger equation

$H\psi_{k}=E_{k}\psi_{k}$ (28)

with appropriate boundary conditions. All real eigenvalues $E_{k}$ together
form the energy spectrum of $H$.

If the state-space is finite-dimensional the spectrum is finite. In quantum
information theory, only finite-dimensional state spaces figure. This is
possible since in this context the setting is such that for its building
blocks, only a few states are energetically accessible. Thus the state space
is a tensor product of very low-dimensional state spaces (most typical of
qubits).

But most quantum systems need for their description an infinite-dimensional
state space. This is already the case for an optical beam. While a beam of
classical monochromatic light oscillates with a fixed frequency, the
associated quantum beam oscillates with infinitely many overtones of this
classical frequency, though in dim light most of them are hardly excited.
After the qubit (which ignores in its optical realization the excited states
of a photon), the quantum oscillator is the second simplest quantum system of
fundamental importance. The operators on this state space are generally
expressed in terms of basic operators satisfying canonical commutation
relations. This points to the fundamental importance of Lie algebras in
quantum physics; see the books by Neumaier & Westra [127] and Woit [168] for
Lie algebras and Lie groups in general quantum physics, and more specifically
Leonhardt & Neumaier [106] for Lie algebra techniques for the design of
quantum optical circuits.

In general, each classical degree of freedom of a quantum system determines a
corresponding quantum oscillator with an associated infinite-dimensional state
space. The observation by Dirac [47] that the Poisson bracket is the classical
limit of the scaled commutator implies that coupled quantum oscillators are
described by a tensor product of oscillator state spaces. This is easily
generalized to arbitrary composite systems, and is relevant even in quantum
information theory, which exploits the exponential complexity of tensor
products of many qubits.

If the state-space is infinite-dimensional, the energy spectrum is always
bounded from below but usually not bounded from above. Therefore the number of
eigenvalues is usually infinite and the sum in (27) has infinitely many terms.
In the common case where part of the energy spectrum is continuous, the sum in
(27) must be replaced by an integral to make sense.

In a pure time-dependent state $\psi(t)$, the quantum values take the form

$\langle
A\rangle_{t}=\psi(t)^{*}A\psi(t)=\sum_{j,k}e^{-it(E_{j}-E_{k})/{\mathchar
22\relax\mkern-9.0muh}}\psi_{j}^{*}A\psi_{k}.$

This shows that the quantum system oscillates with angular frequencies

$\omega_{jk}=|E_{j}-E_{k}|/{\mathchar 22\relax\mkern-9.0muh}.$

The same frequencies can be obtained for mixed states from the von Neumann
equations.

An unbounded energy spectrum implies the presence of arbitrarily high
frequencies, forming quantum fluctuations observable statistically as noise
(even with the human eye, Stiles [148]). Lower frequency oscillations can be
probed by spectroscopy and produce absorption or emission lines with
frequencies $\nu_{jk}=\omega_{jk}/2\pi$. With $h:=2\pi{\mathchar
22\relax\mkern-9.0muh}$ we find the Rydberg–Ritz combination principle that
the observed spectral lines are given by differences

$\nu_{jk}=|\nu_{j}-\nu_{k}|,$ (29)

of frequencies $\nu_{j}$ determined from Planck’s relation $E_{k}=h\nu_{k}$
between energy and frequency. This relation was the starting point of
Heisenberg’s matrix mechanics (Heisenberg [73]), which lead to modern quantum
physics.

###  5.4 The emergence of quantum fields

If, without in any way disturbing a system, we can predict with certainty
(i.e., with probability equal to unity) the value of a physical quantity, then
there exists an element of physical reality corresponding to that physical
quantity.

Einstein, Podolsky, and Rosen, 1935 [55, p.777]

In Subsection 4.7 we defined complex sources and detectors as combinations of
a source or detector with one or more filters. With the description of a
medium as a limit of many very narrow filters we may extend our view of
sources and detectors. As in Subsection 4.5 we consider a source as a surface
in space where quantum beams exit and can be in principle observed. Similarly
we consider a detector as a surface in space where quantum beams enter and can
trigger one of the detection elements on the other side. This is equivalent to
wrapping the original source in layers of filters formed by the medium around
the source, and the original detector in layers of filters formed by the
medium around the detector. This gives a new perspective on the interpretation
of our investigations. We can make the delineation of the source and the
detector arbitrary depending on which part of the medium we count to where.

Now suppose that we have a detector that is sensitive only to quantum beams
entering a tiny region in space, which we call the detector’s tip. We assume
that we can move the detector such that its tip is at an arbitrary point $x$
in the medium, and we consider a fixed source, extended by layers of the
medium so that $x$ is at the boundary of the extended source. The measurement
performed in this constellation is a property of the source. The results
clearly depend only on what happens at $x$, hence they may count as a
measurement of a property of whatever occupies the space at $x$. Thus we are
entitled to consider it as a local property of the world at $x$ at the time
during which the measurement was performed. (Using coincidence measurements
analogous to those used in quantum instrument tomography we could even measure
bilocal properties of the world, given as the quantum values of operators
depending on two positions.)

In general, the quantity whose quantum value is measured by the detector
depends on $x$, weakly or strongly depending on whether the quantum beams
interact weakly or strongly with the medium. Therefore our set-up actually
measures the quantum value $\langle A(x)\rangle$ of a position-dependent
quantity $A(x)$. We interpret $A(x)$ as an operator-valued quantum field whose
quantum values describe a property of the world in all points $x$ accessible
to the measurement device. If we idealize the detector as having an infinitely
sharp tip consisting of a single point only and being placed everywhere, we
arrive at the view that all operationally verifiable local properties of the
world are described by quantum values of suitable quantum fields. In
particular, beam properties become field properties at the points where the
beam passes. Thus from the new perspective what we previously called quantum
beams are just quantum fields concentrated along a narrow path. This is
completely analogous to the way optical beams arise already in classical
optics.

Realistic measurements need to take account of the extension of the tip. The
result is therefore a weighted average over the region covered by the tip.
Hence the detector measures the quantum value $\langle\widetilde{A}(x)\rangle$
of a smeared field

$\widetilde{A}(x)=\int dsw(s)A(x+s)$

with a suitable weight function $w(s)$ depending on the tip.

Note that our reasoning is precisely the same as the reasoning used in
classical physics to ascribe hydromechanic, electromagnetic, or gravitational
properties to regions in space by placing detectors at arbitrary points in
these regions. The only difference in the quantum case is the quantum nature
of the measurements, which lead to quantum values expressed by quantum
expectations of fields rather than the fields themselves.

By construction, the quantum values $\langle A(x)\rangle$ of fields are
operationally quantifiable properties in regions where they are sufficiently
stationary, sufficiently slowly varying in space, and accessible by a detector
whose tip is sensitive to these properties. They are elements of reality in
the sense of Einstein, Podolski & Rosen [55].

###  5.5 Quantum dynamics in mixing linear media

It seems that the only possibility of introducing an irreversible behaviour in
a finite system is to avoid the unitary time evolution altogether by
considering non-Hamiltonian systems.

Göran Lindblad, 1976 [108, p.119]

It is fairly clear that although these measurements can be predicted by
quantum theory they do not fall within the usual description of measurement
theory.

Brian Davies, 1969 [42, p.278]

We can also find the correct dynamical laws under assumptions where the
Schrödinger equation or the von Neumann equation is inappropriate. It is well-
known that in practice most physical processes are dissipative. This means
that depending on the situation, energy decreases or entropy increases, due to
interaction with the medium through which a quantum system passes. This cannot
happen for processes described by the Schrödinger equation or the von Neumann
equation, hence a more complicated description is needed.

We may repeat the derivation of the quantum Liouville equation in the more
general situation of a mixing linear medium. We assume again that the medium
is modeled by means of many slices consisting of very thin filters. But these
are now mixing, hence described by a multi-term operator sum expansion. Only
one of the terms can be close to the identity; the others are small. The
assumption that leads to a sufficiently general dynamics for the state is that

$\rho(t+\Delta
t)=T(t)\rho(t)T(t)^{*}+\sum_{\ell}\gamma_{\ell}T_{\ell}(t)\rho(t)T_{\ell}(t)^{*},$

with a primary transmission operator $T(t)$ of the form (22), secondary
transmission operators $T_{\ell}(t)$ of the form

$T_{\ell}(t)=\Delta t^{1/2}L_{\ell}(t)+O(\Delta t),$

and nonnegative constants $\gamma_{\ell}$. (As in Jagadish & Petruccione [87,
Section 7], whose argument we follow, these could be absorbed into the
$L_{\ell}(t)$ by rescaling the latter; but it is more convenient in the
applications to have them separately.) Then

$\frac{d}{dt}\rho(t)=\frac{\rho(t+\Delta t)-\rho(t)}{\Delta t}+O(\Delta
t)=K(t)\rho(t)+\rho(t)K(t)^{*}+\sum_{\ell}\gamma_{\ell}L_{\ell}(t)\rho(t)L_{\ell}(t)^{*}+O(\Delta
t^{1/2}).$

Suppressing the time argument and taking the continuum limit $\Delta t\to 0$
we obtain the quantum master equation

$\frac{d}{dt}\rho=K\rho+\rho K^{*}+\sum_{\ell}\gamma_{\ell}L_{\ell}\rho
L_{\ell}^{*}.$ (30)

In the lossless case, the intensity $I={\mathop{\rm tr}\,}\rho(t)$ remains
constant, hence

$0={\mathop{\rm tr}\,}\frac{d}{dt}\rho={\mathop{\rm tr}\,}K\rho+{\mathop{\rm
tr}\,}\rho K^{*}+\sum_{\ell}\gamma_{\ell}{\mathop{\rm tr}\,}L_{\ell}\rho
L_{\ell}^{*}={\mathop{\rm
tr}\,}\Big{(}K+K^{*}+\sum_{\ell}\gamma_{\ell}L_{\ell}^{*}L_{\ell}\Big{)}\rho.$

This must hold for all positive semidefinite $\rho$ and then by linearity for
all $\rho$. Therefore

$K+K^{*}+\sum_{\ell}\gamma_{\ell}L_{\ell}^{*}L_{\ell}=0.$

We use again the decomposition (24) and find that

$\frac{2}{{\mathchar
22\relax\mkern-9.0muh}}V=-K-K^{*}=\sum_{\ell}\gamma_{\ell}L_{\ell}^{*}L_{\ell},$
$K=-\frac{i}{{\mathchar 22\relax\mkern-9.0muh}}H-\frac{1}{{\mathchar
22\relax\mkern-9.0muh}}V=-\frac{i}{{\mathchar
22\relax\mkern-9.0muh}}H+\frac{1}{2}\sum_{\ell}\gamma_{\ell}L_{\ell}^{*}L_{\ell}.$

Inserting this into the quantum master equation (30) results in the Lindblad
equation

$\frac{d}{dt}\rho=\frac{i}{{\mathchar
22\relax\mkern-9.0muh}}[\rho,H]+\sum_{\ell}\gamma_{\ell}\Big{(}L_{\ell}\rho
L_{\ell}^{*}-\frac{1}{2}L_{\ell}^{*}L_{\ell}\rho-\frac{1}{2}\rho
L_{\ell}^{*}L_{\ell}\Big{)}$ (31)

(Lindblad [108], Gorini et al. [67]), the standard equation for describing
dissipative, memoryless quantum systems.

The situation is quite different for the passage through nonlinear media
characterized by multiple scattering, which destroys the linearity. The thin
slices would now have to be described by nonlinear filters, for which the
theory developed here is inadequate. Here one needs to start from quantum
field theory and derive the appropriate nonlinear properties by perturbation
theory; see, e.g., Akhmanov et al. [2] for the case of nonlinear quantum
optics.

###  5.6 Quantum phase space

Another consequence of the present approach is the natuaral emergence of a
phase space structure for quantum dynamics. To motivate the concept we first
review the basics of classical Hamiltonian mechanics.

Conservative classical mechanics. In the Hamiltonian formulation of
conservative classical mechanics, $N$ distinguishable particles are modelled
in terms of their position and momentum. These constitute phase space
variables $q_{k}$ and $p_{k}$ ($k=1,\ldots,N$) which take values in
$\mathbb{R}^{3}$ and are usually combined into vectors $(p,q)$ with values in
the phase space $\mathbb{R}^{6N}$. Their value at a given time $t_{0}$,
togegther with the Hamiltonian $H(p,q)$, a twice continuously differentiable
function of the phase space variables, uniquely determine through the ordinary
differential equations

$\frac{d}{dt}q(t)=\frac{\partial}{\partial
p}H(p,q),~{}~{}~{}\frac{d}{dt}p(t)=-\frac{\partial}{\partial q}H(p,q)$ (32)

their value $p(t)$ and $q(t)$ at any time in the past or future,

For a system of free particles, these phase space variables are in principle
measurable to arbitrary accuracy by probing their motion with appropriate
measurement devices. For a system of interacting particles, measurement
accuracy is restricted to an accuracy depending on how fast the phase space
variables change with time. Classical observables are all sufficiently nice
complex-valued functions $f=f(p,q)$ of the phase space variables and include
the Hamiltonian $H$, the observable describing the total energy of the system.
Classical observables change in time according to the equations

$\frac{d}{dt}f=H\,\angle\,f$ (33)

with the Lie product (the negative Poisson bracket)

$f\,\angle\,g:=\partial_{p}f\cdot\partial_{q}g-\partial_{p}g\cdot\partial_{q}f.$

The Lie product is antisymmetric and satisfies the Jacobi identity and the
Leibniz rule, hence turns $C^{\infty}(\mathbb{R}^{6N})$ into a commutative
Poisson algebra. Applied to the phase space variables themselves (wich are
simple linear functions of $p$ and $q$), we recover (32) and find for their
Lie product the well-known canonical relations

$p_{ka}\,\angle\,q_{ka}=1,~{}~{}~{}q_{ka}\,\angle\,p_{ka}=-1,~{}~{}~{}(k=1,\ldots,N,~{}a=1,2,3)$

and vanishing other Lie products, defining the Heisenberg Lie algebra $h(3N)$.
The Poisson algebra $C^{\infty}(\mathbb{R}^{6N})$ is canonically associated to
this Heisenberg algebra as its Lie–Poisson algebra. For details see, e.g.,
Neumaier & Westra [127, Chapter 12].

Many other classical conservative systems have a formulation in terms of
commutative Poisson algebras constructed from phase spaces forming Poisson
manifolds; see, e.g., Marsden & Ratiu [114]. But Poisson algebras form an even
more comprehensive framework than phase space manifolds. For example,
indistinguishable particles cannot be modelled in terms of observable phase
space variables since the only observables are the permutation invariant
functions in $C^{\infty}(\mathbb{R}^{6N})$. However, these form again a
commutative Poisson algebra. Based on this, classical equilibrium statistical
mechanics and equilibrium thermodynamics can be developped in a purely
algebraic way; see Neumaier & Westra [127, Part II].

Conservative quantum mechanics. We now show that conservative quantum
mechanics can also be cast into the framework of commutative Poisson algebras.
As we have seen, quantum systems can be modelled in terms of their normalized
density operators $\rho$. For a stationary quantum system, the components of
the density operator in a fixed basis (and from these by computation all nice
functions of the density operator) are in principle measurable to arbitrary
accuracy through quantum tomography, by probing the system with appropriate
measurement devices. For a nonstationary quantum system, measurement accuracy
is restricted to an accuracy depending on how fast the density operator
changes with time.

This close analogy to the classical situation just discussed suggests that we
may treat the components of the normalized density operator in a fixed basis
as the phase space variables of a quantum system. Following this analogy, the
quantum phase space consists of all Hermitian positive semidefinite operators
$\rho$ of trace $1$, and the quantum observables are all sufficiently nice
complex-valued functions $f=f(\rho)$ of the phase space variables. In
particular, this includes the linear functionals

$\langle A\rangle={\mathop{\rm tr}\,}\rho A.$

Thus all quantum values and all sufficiently nice functions of quantum values
are observable. The commutator Lie product for linear operators

$A\,\angle\,B:=\iota(AB-BA),~{}~{}~{}\iota:=\frac{i}{{\mathchar
22\relax\mkern-9.0muh}}$ (34)

defines a Lie algebra whose Lie–Poisson algebra is the algebra of all
infinitely often differentiable functions$f=f(\rho)$ of a compact operator
$\rho\in{\mathop{\rm Lin}\,}\mathbb{H}$, with induced Lie product

$f\,\angle\,g:={\mathop{\rm
tr}\,}\rho(\partial_{\rho}f\,\angle\,\partial_{\rho}g).$ (35)

If we restrict the arguments $\rho$ to phase space variables, i.e., Hermitian
positive semidefinite operators $\rho$ of trace $1$, we obtain a commutative
Poisson algebra of quantum observables, which equips our quantum phase space
with the structure of a Poisson manifold. The Poisson structure of the space
of normalized density operators exhibited below, which justifies treating this
space as quantum phase space, was first recognized by Man’ko et al. [113];
symplectic reductions to coadjoint orbits are discussed, e.g., in Requist
[140] and Bonet-Luz & Tronci [19], This is quite different from the far more
common approach, pioneered by Strocchi [151], to treat the projective space of
wave functions up to phase as a symplectic quantum phase space manifold, which
covers only pure states.

The Lie product (35) induces a Lie product

$\langle A\rangle\,\angle\,\langle B\rangle:=\langle A\,\angle\,B\rangle$ (36)

for the quantum values. The quantum Hamiltonian $H$ is an operator, hence not
a quantum observable in the present sense. However, its quantum value

$\overline{H}:=\langle H\rangle$

qualifies as a quantum observable, and again describes the total energy of the
system. Quantum observables change in time according to the equations

$\frac{d}{dt}f=\overline{H}\,\angle\,f$ (37)

Applied to time-dependent quantum values

$\langle A\rangle_{t}:={\mathop{\rm tr}\,}\rho(t)A$

we find

$\frac{d}{dt}\langle A\rangle_{t}=\langle H\,\angle\,A\rangle_{t}$ (38)

as differential equation for arbitrary quantum values. We call (38) the
Ehrenfest equation since Ehrenfest [53] found in 1927 the special case of this
equation where $A$ is a quantum position or momentum variable and
$H=\displaystyle\frac{p^{2}}{2m}+V(q)$ is the sum of kinetic and potential
energy. Indeed, due to the canonical commutation rules, we have in this case

$\frac{d}{dt}\langle q\rangle_{t}=\langle
H\,\angle\,q\rangle_{t}=\frac{\langle
p\rangle_{t}}{m},~{}~{}~{}\frac{d}{dt}\langle p\rangle_{t}=\langle
H\,\angle\,p\rangle_{t}=\langle-\nabla V(q)\rangle_{t}.$ (39)

The Ehrenfest equation implies that the collection of all quantum values at a
given time $t_{0}$ uniquely determines all quantum values at any time in the
past or future. In the literature, this dynamics for quantum values is
referred to as the Ehrenfest picture (Man’ko et al. [113]) and plays a role in
various quantum-classical approaches to chemical and physical systems where
the fast degrees of freedom are treated by quantum mechanics while the low
ones are approximately handled as classical variables. See, e.g., Kapral &
Ciccotti [92], Marx & Hutter [115], Kapral [91]. The Ehrenfest picture is more
thoroughly discussed in Neumaier [125, Section 2.2].

By specializing $A$ to a basis of the space of operators we see that the
Ehrenfest equation is equivalent to the quantum Liouville equation

$i{\mathchar 22\relax\mkern-9.0muh}\dot{\rho}(t)=[H,\rho(t)].$ (40)

Thus the phase space setting for the quantum dynamics is equivalent to the
traditional setting for conservative quantum dynamics.

##  6 Measurement uncertainty

Some hypotheses are dangerous, first and foremost those which are tacit and
unconscious. And since we make them without knowing them, we cannot get rid of
them. Here again, there is a service that mathematical physics may render us.
By the precision which is its characteristic, we are compelled to formulate
all the hypotheses that we would unhesitatingly make without its aid.

Henri Poincaré, 1902 [135, p.151]

This section gives a thorough, precise discussion of various aspects of
uncertainty in quantum measurements.

###  6.1 Statistical uncertainty

Aus diesen Gründen ist eine gleichzeitige genaue Beobachtung von $q$ und $p$
prinzipiell ausgeschlossen. […] Man kann aber auch beide Größen in einer
einzigen Beobachtung messen, also wohl gleichzeitig, aber nur mit beschränkter
Genauigkeit. Bei einer solchen Beobachtung fragt man in der klassischen
Theorie nach dem ’Fehler’ des gemessenen Wertes. […] Die ’Beobachtungsfehler’
erscheinen in der neuen Theorie als mit der statistischen Unbestimmtheit
selbst zusammengeschmolzen.
(For these reasons, a simultaneous exact observation of $q$ and $p$ is
impossible in principle. […] However one can measure both quantities in a
single measurement, hence simputaneously, but only with limited accuracy. In
such an observation one asks in the classical theory for the ’error’ of the
value measured. […] The ’observation errors’ appear in the new theory as being
melted together with the statistical uncertainty itself.)

Earle Kennard 1927 [94, p.340f]

Results of measurements cannot be absolutely accurate. This unavoidable
imperfection of measurements is expressed in their inaccuracy.

Semyon Rabinovich, 2005 ([139, p.2])

We write $|x|:=\sqrt{x^{*}x}$ for the Euclidean norm of a vector
$x\in\mathbb{C}^{m}$, and generalize it to vectors $A\in({\mathop{\rm
Lin}\,}\mathbb{H})^{m}$ with operator components by defining171717 Actually we
never need $|A|$ but only the notation $|A|^{2}$ for $A^{*}A$, especially when
$A$ is some composite formula.  the operator

$|A|:=\sqrt{A^{*}A}.$

In this section we add the adjective ”quantum” to all theoretical quantum
notions that suggest by their traditional name a statistical interpretation
and hence might confuse the borderline between theory and measurement. This is
in the spirit of the convention of Allahverdyan et al. [5] who add instead the
prefix ”q-”. Since we use statistical language, we adopt the trace 1
normalization of states, so that mean rates reduce to probabilities. Thus the
state of the source is the positive linear mapping that assigns to each
$X\in{\mathop{\rm Lin}\,}\mathbb{H}$ its quantum expectation

$\overline{X}=\langle X\rangle:={\mathop{\rm Tr}\,}\rho X.$

More generally, given a fixed state, the q-expectation of a vector
$X\in({\mathop{\rm Lin}\,}\mathbb{H})^{m}$ with operator components is the
vector $\overline{X}=\langle X\rangle\in\mathbb{C}^{m}$ with components
$\overline{X}_{j}=\langle X_{j}\rangle$. Its quantum uncertainty or
q-uncertainty is the nonnegative number

$\sigma_{X}:=\sqrt{\langle(X-\overline{X})^{*}(X-\overline{X})\rangle}=\sqrt{\langle
X^{*}X\rangle-|\overline{X}|^{2}}.$ (41)

We may also define the quantum covariance matrix

$C_{X}:=\langle(X-\overline{X})(X-\overline{X})^{*}\rangle\in\mathbb{C}^{n\times
n},$

in terms of which

$\sigma_{X}=\sqrt{{\mathop{\rm tr}\,}C_{X}}.$

###### 6.1 Proposition.

(Robertson [141])
Any two Hermitian operators $A,B\in{\mathop{\rm Lin}\,}\mathbb{H}$ satisfy the
uncertainty relation

$\sigma_{A}\sigma_{B}\geq\frac{1}{2}|\langle[A,B]\rangle|$ (42)

with the commutator

$[A,B]:=AB-BA.$

###### Proof.

The relation remains unchanged when subtracting from $A$ and $B$ its quantum
expectation, hence it suffices to prove it for the case where both quantum
expectations vanish. In this case, $\langle A^{2}\rangle=\sigma_{A}^{2}$ and
$\langle B^{2}\rangle=\sigma_{B}^{2}$, and the Cauchy–Schwarz inequality gives

$|\langle AB\rangle|^{2}\leq\langle A^{2}\rangle\langle
B^{2}\rangle=\sigma_{A}^{2}\sigma_{B}^{2},$

hence $|\langle AB\rangle|\leq\sigma_{A}\sigma_{B}$. On the other hand, one
easily checks that $i\mathop{\rm Im}\langle
AB\rangle=\frac{1}{2}\langle[A,B]\rangle$, so that
$\frac{1}{2}|\langle[A,B]\rangle|=|\mathop{\rm Im}\langle
AB\rangle|\leq|\langle AB\rangle|$. Combining both inequalities gives the
assertion. ∎

We now relate these theoretical quantum notions to the statistical notions
introduced in Subsection 2.3.

###### 6.2 Proposition.

The inequality

$\min_{\xi\in\mathbb{C}^{m}}{\bf E}(|a_{k}-\xi|^{2})={\bf
E}(|a_{k}-\overline{A}|^{2})\geq\langle|A-\overline{A}|^{2}\rangle=\sigma_{A}^{2}$
(43)

bounds the statistical uncertainty of the measurement results $a_{k}$ in terms
of the theoretical quantum uncertainty of the quantity $A$ measured.

For $m=1$, this is due to Holeveo [82, (9.8), p.88]; it was later redicovered
by de Muynck & Koelman [44] and Werner [165, Proposition 3(2)].

###### Proof.

This follows by observing that

${\bf E}(|a_{k}-\xi|^{2})-{\bf E}(|a_{k}-\overline{A}|^{2})={\bf
E}(|a_{k}-\xi|^{2}-|a_{k}-\overline{A}|^{2})={\bf
E}(|\overline{A}-\xi|^{2})=|\overline{A}-\xi|^{2}$ (44)

is minimal for $\xi=\overline{A}$. The claim now follows from the following
proposition. ∎

###### 6.3 Proposition.

Given a detector measuring $A=P[a_{k}]\in\mathbb{H}^{m}$. Then:

(i) For every Hermitian positive semidefinite $G\in\mathbb{C}^{m\times m}$,
the operator $P[a_{k}^{*}Ga_{k}]-A^{*}GA$ is positive semidefinite.

(ii) For any $\xi\in\mathbb{C}^{m}$ and any state $\langle\cdot\rangle$,

${\bf E}(|a_{k}-\xi|^{2})\geq\langle|A-\xi|^{2}\rangle.$ (45)

The case $m=1$ is already in Holevo [81, Lemma 13.1].

###### Proof.

Using the quantum measure $P_{k}$ ($k\in K$) of the detector, we write

$\Delta:=\sum_{k\in K}(A-a_{k})^{*}GP_{k}(A-a_{k}).$

(i) For any $\psi\in\mathbb{H}$, we define the vectors
$\psi_{k}:=(A-a_{k})\psi$ and find
$\psi^{*}N\psi=\sum\psi_{k}^{*}GP_{k}\psi_{k}\geq 0$. Hence $\Delta$ is
positive semidefinite. Since

$\begin{array}[]{lll}\Delta&:=&\displaystyle\sum\Big{(}A^{*}GP_{k}A-a_{k}^{*}GP_{k}A-A^{*}GP_{k}a_{k}+a_{k}^{*}GP_{k}a_{k}\Big{)}\\\
&=&A^{*}GP[1]A-A^{*}GP[a_{k}]-P[a_{k}]^{*}GA+P[a_{k}^{*}Ga_{k}]\\\
&=&A^{*}GA-A^{*}GA-A^{*}GA+P[a_{k}^{*}Ga_{k}]=P[a_{k}^{*}Ga_{k}]-A^{*}GA,\end{array}$

the first part follows.

(ii) In the special case where $G$ is the identity matrix, we find from (8)
that

${\bf E}(|a_{k}|^{2})=\langle P[|a_{k}|^{2}]\rangle=\langle
P[a_{k}^{*}a_{k}]\rangle=\langle N+A^{*}A\rangle=\langle\Delta:\rangle+\langle
A^{*}A\rangle\geq\langle A^{*}A\rangle=\langle|A|^{2}\rangle.$

Since $P[a_{k}-\xi]=P[a_{k}]-\xi=A-\xi$ we may apply this with $a_{k}-\xi$ in
place of $a_{k}$ and $A-\xi$ in place of $A$ and find (45). ∎

In the special case of projective measurements, $A=P[a_{k}]$ satisfies

$P_{k}(A-a_{k})=P_{k}A-a_{k}P_{k}=\sum_{j}a_{j}P_{k}P_{j}-a_{k}P_{k}=0,$

so that $\Delta:=0$ in the proof of Proposition 6.3. Therefore inequality (43)
holds in this case with equality for all states. A converse was proved in
Kruszyński & De Muynck [98, Proposition 2]. Thus the difference in (43)
describes the lack of projectivity. We may view it as a measure of excess
uncertainty of a real detector, as opposed to an idealized projective one.
Busch et al. [26, (16)] view this difference as a measure of intrinsic noise.

###  6.2 Imperfect measurements

The fact that actual measurements are always imprecise is well-known and led
Poincare to distinguish carefully the ”mathematical continuum” from the
”physical continuum.” In the mathematical continuum the notion of identity
satisfies the usual transitivity condition […] By contrast, this property
cannot be assumed for the notion of ”indistinguishability” in the physical
continuum attached to the raw data of experiments. […] To pass from the
physical continuum to the mathematical continuum requires an idealization,
namely that infinitely precise measurements are in principle, if not in fact,
attainable.

Ali and Emch, 1974 [6, p.1545]

The discrete nature of the reading scale entails that a given measuring
apparatus allows only a measurement of a discrete version of the observable
under consideration. With this we do not, however, deny the operational
relevance of continuous observables. On the contrary, their usefulness as
idealisations shows itself in the fact that they represent the possibility of
indefinitely increasing the accuracy of measurements by choosing increasingly
refined reading scales.

Busch, Lahti and Mittelstaedt, 1996 ([30, p.81])

The spectral norm of a quantity measured by an arbitrary detector is bounded
since the sum (9) is finite and $\|P_{k}\|\leq 1$ for all $k$. Hence the
components of $A$ are bounded linear operators.181818 To extend the notion of
measurement to unbounded quantities such as position or momentum – which is
outside the scope of the present paper –, one would need to proceed in an
idealized fashion, using continuous POVMs for idealized measurements with
infinite precision. Then quantum expectations are defined only for
sufficiently regular density operators. For a proper treatment of the
unbounded case see, e.g., the books mentioned at the beginning of this paper.
In particular, (11) is defined for all density operators $\rho$. Note that the
same bounded linear operator $A$ can be decomposed in many ways into a linear
combination of the form (9); thus there may be many different detectors with
different scales measuring quantities corresponding to the same operator $A$.

In practice, one is often interested in measuring a given (bounded or
unbounded) operator $X$ of interest. Designing realistic detectors that allow
a high quality measurement corresponding to theoretically important operators
is the challenge of high precision experimental physics. Due to experimental
limitations, this generally involves both statistical and systematic errors.

If we approximate a (possibly vector-valued) quantity $X$ by a measurable
substitute quantity $A$ – some such approximation is unavoidable in practice
–, we make a systematic error that may depend on the state of the system
measured. (44) implies the formula

${\bf E}(|a_{k}-\overline{X}|^{2})={\bf
E}(|a_{k}-\overline{A}|^{2})+|\overline{A}-\overline{X}|^{2}$ (46)

for the mean squared error of $a_{k}$ as an approximation of $\overline{X}$.
In particular, the term

$\Delta:=|\overline{A}-\overline{X}|=|\langle X\rangle-\langle A\rangle|,$

the bias due to the substitution of $A$ for $X$, is a lower bound for the root
mean squared error (RMSE)

$\varepsilon_{X}:=\sqrt{{\bf E}(|a_{k}-\overline{X}|^{2})}.$

Unlike the RMSE but like quantum expectations, the bias is a theoretical
property of a state, independent of measurement. We say that $A$ is an
unbiased approximation of $X$ in all states such that the bias vanishes. If
the bias vanishes in all states from an open subset of the set of all states,
we necessarily have $X=A$. In particular, there are no everywhere unbiased
approximations of unbounded operators.191919 For unbounded operators $A$ (for
example the position operator vector $q$ in a particular coordinate system),
formula (11) and hence the bias is well-defined only for sufficiently regular
density operators $\rho$. This reflects a slight deficiency of the standard
textbook presentation of expectations. However, measurement equipment for
unbounded operators such as position, say, only produces results in a bounded
range. Hence it corresponds in fact to a measurement device for a clipped
version $X=F(q)$ of the position operator $q$ with bounded $F$, resulting in a
bounded $X$.  Note that in practice, $\overline{A}\neq\overline{X}$ due to
imperfections. In particular, unbiased measurements are necessarily
idealizations.

The RMSE $\varepsilon_{X}$ measures the uncertainty in the value assigned to
$\overline{X}$. But because of the broken watch effect,202020 Measuring time
with a broken watch shows twice a day the exact time, whereas a watch that is
slow 1 second per day shows the correct time at most once in a century.  it
cannot be regarded as the measured uncertainty in the value assigned to $X$.
Thus we need to add a systematic uncertainty correction that corrects for the
possibility that the uncertainty of $A$ is less than the uncertainty of $X$.
The latter has nothing to do with measurement and hence must be a theoretical
quantity computable from quantum uncertainties. Now

$\langle|A-\overline{X}|^{2}\rangle=\langle|A-\overline{A}|^{2}\rangle+|\overline{A}-\overline{X}|^{2}=\sigma_{A}^{2}+|\overline{A}-\overline{X}|^{2}$
(47)

in analogy to (46), proved by expanding all squares. Comparing this with the
formula

$\langle|X-\overline{A}|^{2}\rangle=\langle|X-\overline{X}|^{2}\rangle+|\overline{X}-\overline{A}|^{2}=\sigma_{X}^{2}+|\overline{X}-\overline{A}|^{2}$

obtained by interchanging the role of $A$ and $X$, we see that the natural
uncertainty correction to the mean squared error is
$(\sigma_{X}^{2}-\sigma_{A}^{2})_{+}$, where $x_{+}:=\max(x,0)$ denotes the
positive part of a real number $x$. We therefore regard

$\Delta_{X}[a_{k}]:=\sqrt{{\bf
E}(|a_{k}-\overline{X}|^{2})+(\sigma_{X}^{2}-\sigma_{A}^{2})_{+}},$ (48)

the square root of the corrected mean squared error, as the measurement
uncertainty when measurements of $A$ are performed in place of measurements of
$X$. Such imperfect measurements make sense even for unbounded quantities $X$
in states with finite quantum uncertainty $\sigma_{X}$. By (45) and (47),

$\begin{array}[]{lll}\Delta_{X}[a_{k}]^{2}&=&{\bf
E}(|a_{k}-\overline{X}|^{2})+(\sigma_{X}^{2}-\sigma_{A}^{2})_{+}\\\
&\geq&\langle|A-\overline{X}|^{2}\rangle+\sigma_{X}^{2}-\sigma_{A}^{2}\\\
&=&\sigma_{X}^{2}+|\overline{A}-\overline{X}|^{2}\end{array}$

hence the measurement uncertainty is bounded from below by theoretical error
measures,

$\Delta_{X}(a_{k})\geq\sqrt{\sigma_{X}^{2}+|\overline{A}-\overline{X}|^{2}}=\sqrt{\langle|X-\overline{A}|^{2}\rangle}.$

In particular, $\Delta_{X}(a_{k})$ is always at least as large as the quantum
uncertainty of $X$, and larger if there is a nonzero bias. This gives an
operational interpretation to these theoretical terms.

By changing the scale of a detector we may define measurements of many
different quantities $A$ based on the same quantum measure. By picking the
scale carefully one can in many cases choose it such that $A$ approximates a
particular operator $X$ of interest with small measurement uncertainty
$\Delta_{X}[a_{k}]$ for the collection of states of interest. There are
several alternative ways to quantify what constitutes adequate approximations;
see, e.g., Appleby [8, 9], Barcielli et al. [11], Busch et al. [26, 31], and
references there.

Finding a good match of $A$ and $X$ by choosing a good scale $a_{k}$ is the
process called tuning. It corresponds to the classical situation of labeling
the scale of a meter to optimally match a desired quantity. If the detector
can also be tuned by adjusting parameters $\theta$ affecting its responses,
the operators $P_{k}=P_{k}(\theta)$ depend on these parameters, giving

$A=A(\theta)=\sum a_{k}P_{k}(\theta).$

Now both the labels $a_{k}$ and the parameters $\theta$ can be tuned to
improve the accuracy with which the desired $X$ is approximated by
$A(\theta)$, perfecting the tuning.

###  6.3 Measurement errors

Von dem neuen Standpunkt schwebt nun zunächst dieser Fehlerbegriff in der
Luft, denn er setzt doch nicht nur den Begriff eines beobachteten, sondern
auch den Begriff eines ’wahren’ Wertes voraus, und letzteren gibt es im
physikalischen Sinne nicht mehr.
(From the new point of view, theis notion of error floats at first in the air,
bor it presupposes not only the notion of an observed but only that of a
’true’ value, and the latter no longer exists in a physical sense.)

Earle Kennard 1927 [94, p.340]

This mean value $x_{0}$ locates the wave packet in the crude sense that an
observing apparatus must be placed near $x_{0}$ if it is to have a significant
chance of interacting with the particle.

Carl Helstrom, 1974 [76, p.454]

Measurement is the process of determinating the value of a physical quantity
experimentally with the help of special technical means called measuring
instruments. The value of a physical quantity […] is found as the result of a
measurement. The true value of a measurand is the value of the measured
physical quantity, which, being known, would ideally reflect, both
qualitatively and quantitatively, the corresponding property of the object.

Semyon Rabinovich, 2005 ([139, p.1f])

Measurement errors are in principle unavoidable, because a measurement is an
experimental procedure and the true value of the measurable quantity is an
abstract concept.

Semyon Rabinovich, 2005 ([139, p.11])

Measurement errors are ubiquitous in physical practice; their definition
requires, however, some care. A single measurement produces a number, the
measurement result. The splitting of the measurement result into the sum of an
intended result (the true value) and the deviation from it (the measurement
error) depends on what one declares to be the true value. Thus what can be
said about measurement errors depends on what one regards as the true value of
something measured.

In general, the true value is necessarily a theoretical construct, an
idealization arrived at by convention. Since measured are only actual results,
never the hypothesized true values, there is no way to determine
experimentally which convention is the right one. Both the quantum formalism
and the experimental record are independent of what one declares to be the
true value of a measurement. Different conventions only define different ways
of bookkeeping, i.e., different ways of splitting the same actual measurement
results into a sum of true values and errors, in the communication about
quantum predictions and experiments. Nothing in the bookkeeping changes the
predictions and the level of their agreement with experiment.

Thus the convention specifying what to consider as true values is entirely a
matter of choice, an interpretation. The convention one chooses determines
what one ends up with, and each interpretation must be judged in terms of its
implications for convenience and accuracy. Like conventions about defining
measurement units [23], interpretations can be adjusted to improvements in
theoretical and experimental understanding, in order to better serve the
scientific community.

In Subsections 6.1 and 6.2 we did not assume a notion of true value in the
quantum case. However, (43) implies that the least uncertain value $\xi$ is
the quantum expectation $\overline{A}$. Thus in a statistical sense, the best
possible value that can be assigned is $\overline{A}$. This suggests that in
the unbiased case, $\overline{A}$ should be (in analogy to classical
statistics) considered as being the true value of $A$, which is measured
approximately.

Alternatively, the problem of true values can be avoided if one regards not
the quantum values of operators as the basic observables but a vector of
unknown parameters characterizing a quantum state. In this case, techniques of
quantum estimation theory apply. For recent results, see, e.g., Sidhu et al.
[146].

###  6.4 The approximate nature of Born’s rule

A student has read such and such a number on his thermometer. He has taken no
precautions. It does not matter; he has read it, and if it is only the fact
which counts, this is a reality […] Experiment only gives us a certain number
of isolated points. They must be connected by a continuous line, and this is a
true generalisation. But more is done. The curve thus traced will pass between
and near the points observed; it will not pass through the points themselves.
Thus we are not restricted to generalising our experiment, we correct it; and
the physicist who would abstain from these corrections, and really content
himself with experiment pure and simple, would be compelled to enunciate very
extraordinary laws indeed.

Henri Poincaré, 1902 [135, p.142f]

The Born measure is a mathematical construction; what is its relationship to
experiment? This relationship must be the source of the (alleged) randomness
of quantum mechanics, for the Schrödinger equation is deterministic.

Klaas Landsman, 2019 [103, p.20]

In the standard formulation of Born’s statistical interpretation of quantum
mechanics, based on projective measurements, the measurement results of an
observable represented by a bounded operator $A$ are quantized: the measured
result will be one of the eigenvalues $\lambda_{k}$ of $A$. Multiple
repetition of the measurement results in a random sequence of values
$\lambda_{k}$, with probabilities computed from (13) if the system is in a
pure state. In the limit of arbitrarily many repetitions, the mean value of
this sequence approaches $\overline{A}$ and the standard deviation approaches
$\sigma_{A}$.

According to Born’s statistical interpretation in the standard formulation
(i.e., for projective measurements), each actual measurement result $\lambda$
is claimed to be one of the eigenvalues, which is exactly (according to the
literal reading212121 The formulation appearing in Wikipedia [167] is ”the
measured result will be one of the eigenvalues”. Griffiths & Schroeter [70,
p.133] declare, ”If you measure an observable […] you are certain to get one
of the eigenvalues”. Peres [132, p.95] defines, ”each one of these outcomes
corresponds to one of the eigenvalues of $A$; that eigenvalue is then said to
be the result of a measurement of $A$”. The only exceptions seem to be
textbooks such as Nielsen & Chuang [129, p.84f] that start the formal
exposition with the POVM approach rather than Born’s interpretation. But in
their informal introduction of qubits, even they give priority to projective
measurements!  of most formulations) or approximately (in a more liberal
reading) measured, with probabilities computed from $A$ and the density
operator $\rho$ by the probability form of Born’s rule. Since we saw that
deviations from the eigenvalues in realistic experiments must be interpreted
as measurement errors, we conclude that the eigenvalues are the true values of
Born’s statistical interpretation.

Let us consider the measurement of an unknown quantity
$A\in\mathbb{C}^{2\times 2}$ of a qubit. In a quantum measurement tomography
context $A$ has unknown eigenvalues $\lambda_{1},\lambda_{2}$. The prediction
made by Born’s rule is that the observed distribution is not just bimodal but
has two-point support at the eigenvalues $\lambda_{1}$ and $\lambda_{2}$. But
in most cases, e.g. in realistic Stern–Gerlach experiments, these eigenvalues
are irrational and cannot be exactly measured. Thus deviations from the
eigenvalues are unavoidable, and Born’s rule cannot be exactly valid; it
describes measurements up to some approximation error. This means that Born’s
rule is no longer about real measurements but a postulate about idealized true
values representing the results of hypothetical measurements whose
observations are theoretical numbers, not actual results.

Indeed, according to the experimental record, the response of a sufficiently
good detector produces measurement results concentrated near two spots (or
parallel lines) of the detector. This results in a bimodal distribution with
two more or less sharp peaks, with details depending on the detection method
used and its resolution – just as what one gets when measuring a classical
diffusion process in a double-well potential (see, e.g., Hongler & Zheng
[85]). For example, this happens in the original Stern-Gerlach experiment; cf.
the quote of Busch et al. [25] at the beginning of Section 2.

The approach introduced in this paper handles the situation correctly, since
it gives projective measurements (and hence eigenvalues) no special status. In
a realistic measurement, the possible values obtained when measuring a
particular quantity $A$ depend – according to our discussion in Section 2 – on
the decomposition $A=P[a_{k}]$ used to construct the scale. That this
decomposition is ambiguous follows from Subsection 4.3, where it is shown that
the scale is not determined by the quantity $A$ measured. Since the scale
determines the measurement results, this means that one can with equal right
ascribe different results to the measurement of the same quantity $A$. Thus,
in general, different detectors measuring the same quantity $A$ have different
sets of possible measurement results. In other words, the same quantity $A$
can be measured by detectors with different mathematical characteristics and
in particular different measurement results that generally have nothing to do
with the eigenvalues of $A$.

##  7 On the foundations of quantum mechanics

More than a century after the birth of quantum mechanics, physicists and
philosophers are still debating what a ’measurement’ really means.

DiVincenzo and Fuchs, 2019 [51, p.2]

This section discusses philosophical considerations related to the
development. We discuss

$\bullet$ the nature of the basic substance of quantum matter – particles or
fields?

$\bullet$ the nature of the fuzzy boundary (the Heisenberg cut) between the
macroscopic (observable) and microscopic (only inferable) part of the
universe;

$\bullet$ the thermal interpretation of quantum physics, which extends the
discussion of observable sources and detectors in terms of quantum values
represented by quantum expectations to arbitrary quantum systems even in their
unobservable aspects;

$\bullet$ the quantum measurement problem and what the present paper
contributes towards its solution; and

$\bullet$ questions of reproducibility and objectivity, and the ole of
knowledge in the description of physical phenomena.

###  7.1 Particles and fields

The use of observations concerning the behaviour of particles in the atom
rests on the possibility of neglecting, during the process of observation, the
interaction between the particles, thus regarding them as free. […]
The wave mechanical solutions can be visualised only in so far as they can be
described with the aid of the concept of free particles. […]
Summarising, it might be said that the concepts of stationary states and
individual transition processes within their proper field of application
possess just as much or as little ’ reality’ as the very idea of individual
particles.

Niels Bohr, 1927 [15, pp.587–589]

It seems to me certain that we have to give up the notion of an absolute
localization of the particles in a theoretical model. This seems to me to be
the correct theoretical interpretation of Heisenberg’s indeterminacy relation.
[…] Only if this sort of representation of the atomistic structure be obtained
could I regard the quantum problem within the framework of a continuum theory
as solved.

Albert Einstein, 1934 [54, p.169]

For many purposes the quantization of the electromagnetic field is not
necessary at all, and the response of the photodetector can be understood even
if we continue to picture the field in terms of classical electromagnetic
waves, provided the photoelectrons are treated by quantum mechanics. The field
then simply behaves as an external potential that perturbs the bound electrons
of the photocathode. […] As we shall see, for those electromagnetic fields for
which an adequate classical description exists, the semiclassical and the
fully quantized treatments of the photodetection problem yield virtually
identical answers.

Mandel and Wolf, 1995 [112, p.439]

In its mature form, the idea of quantum field theory is that quantum fields
are the basic ingredients of the universe, and particles are just bundles of
energy and momentum of the fields.

Steven Weinberg, 1999 [163, p.242]

If QFT is about fields, how can its restriction to nonrelativistic phenomena
be about particles?

Art Hobson, 2013 [80, p.212]

What happens in the medium between source and detector is unmodelled in the
pure measurement context assumed in our approach. Without further theoretical
assumptions, it remains a matter of subjective belief.

The present approach works independent of the nature or even the presence of a
mediating substance: What is measured are properties of the source, and this
has a well-define macroscopic existence. We never needed to make an assumption
on the nature of the medium passed from the source to the detector. Thus the
present approach is indifferent to the microscopic cause of detection events.
It does not matter at all whether one regards such an event as caused by a
quantum field or by the arrival of a particle. In particular, a microscopic
interpretation of the single detection events as arrival of particles is not
needed, not even an ontological statement about the nature of what arrives.
Nor would these serve a constructive purpose.

In the present stochastic description of quantum measurement it is sufficient
that something passes from a source through the medium to a detector. Nothing
at all depends on the properties of what flows. In quantum field theory, what
flows are excitations of quantum fields, while in particle physics and in the
quantum mechanics of atoms and molecules, what flows is taken to be a stream
of particles, though the latter have counterintuitive nonclassical properties.

It is therefore questionable whether a single response of a detection element
measures anything beyond the macroscopic quantity that defines the responce.
This is reinforced by the observation that models of photodetection that model
the electromagnetic field as a classical external field but the detector as a
quantum system show the typical effects of photodetection. Indeed, in the
particle-free model discussed by Mandel & Wolf [112] in Sections 9.1–9.3, a
Poisson-distributed stochastic response pattern is found that is
quantitatively identical with the effect predicted by their full quantum field
theoretic treatment of photodetection in Section 14.2 with light beams in a
coherent state.

However, observable properties are associated with arbitrary sufficiently
slowly changing macroscopic regions of spacetime. Thus unlike the particle
view, the field view emerged naturally from our discussion.

The quantum field point of view is indeed quite coherent (Neumaier [125]) and
leaves very little room for mystery. On the other hand, a particle description
makes physical sense only if the microscopic constituents of matter in some
region of space can be considered to be free for a sufficiently long time, so
that one can use a semiclassical approximation – corresponding to geometric
optics for light. This is discussed in much more detail in Chapter 12 of my
book [125].

Thus from the perspective of the present considerations, quantum particles
appear to be ghosts in the beams. This explains their spooky properties in the
quantum physics literature!

###  7.2 Macroscopic quantum systems

Um zur Beobachtung zu gelangen, muss man also irgendwo ein Teilsystem aus der
Welt ausschneiden und über dieses Teilsystem eben ’Aussagen’ oder
’Beobachtungen’ machen. Dadurch zerstört man dort den feinen Zusammenhang der
Erscheinungen und an der Stelle, wo wir den Schnitt zwischen dem zu
beobachtenden System einerseits, dem Beobachter und seinen Apparaten
andererseits machen, müssen wir Schwierigkeiten für unsere Anschauung
erwarten. […] Jede Beobachtung teilt in gewisser Weise die Welt ein in
bekannte und unbekannte oder besser: mehr oder weniger genau bekannte Grössen.
(To arrive at an observation one must somehow cut a subsystem out of the world
and make ’statements’ or ’observations’. This destroys the fine connection of
appearances, and we must expect difficulties for our intuition at the place
where the cut between the system to be observed and the observer and his
equipment. […] Every observation splits in some sense the world into known and
unknown, or better: more or less known, quantities.)

Werner Heisenberg, 1927 [74, p.593f]

A quantum mechanical measuring apparatus consists of a macroscopic system
prepared in a metastable state.

Nicolas van Kampen, 1988 [159, Theorem V, p.102]

Our exposition clarified the meaning of the mathematical concepts used in
terms of measurement, hence defining their operational meaning. Everything is
formulated in terms of objects and observations that we think we understand,
though they themselves are in fact macroscopic quantum objects. Everything
related to measurements is necessarily macroscopic. The microscopic
information is only in the states, quantities, and quantum values used in the
models – these talk about particles or fields, depending on how one models the
situation at hand.

We call a quantum system macroscopic or classical if its observable, coarse-
grained aspects can be described by local equilibrium thermodynamics. This
additional assumption beyond our only postulate (DRP) is based on well-known
results from statistical mechanics. Local equilibrium is the state of matter
where the constituents have a mean free path small compared to the system
size, hence no sensible asymptotic particle picture exists. This contrasts to
the case where the system is very small (mesoscopic or microscopic) or the
mean free path is so large that the system can be considered as a quantum
beam. In the intermediate regime – for very dilute bulk matter, or for
studying details of collision experiments during the interaction of colliding
quantum beams – a microlocal (kinetic) picture is needed.

The computational machinery of statistical mechanics provides a quantitatively
correct deterministic dynamics for the quantum values of smeared fields
describing matter in local equilibrium, Thus the classical deterministic
behavior of macroscopic systems is an automatic consequence of local
equilibrium. This is discussed in more detail in the books by Calzetta & Hu
[34], Neumaier & Westra [127], and Neumaier [125]. The quantum state and the
quantum measure are collective properties of macroscopic objects, the source
and the detector, measurable through quantum tomography by observing
approximations to quantum values. Detection elements are given by all
metastable regions of space with a macroscopically visible binary output,
corresponding to the breaking of metastability in the associated nonlinear
dynamics. Detectors and instruments therefore produce classical, irreversible
results.

The Heisenberg cut is the fuzzy boundary between the macroscopic (observable)
and microscopic (only inferable) part of the universe. The output of the
source (hence the state of the quantum beam) and the input of the detector
(hence the observed relation between input and detection events) depend on the
details of this cut, but no matter how the cut is drawn, the results are
consistent (as long as everything is linear as we assumed).

###  7.3 The thermal interpretation of quantum physics

We assume that (i) the density matrix is observable and (ii) any observable is
a function of the density components.

Lajos Diosi, 1988 [46, p.2887]

The idea of unsharp objectification arises if one intends to leave quantum
mechanics intact and still tries to maintain a notion of real and objective
properties.

Busch, Lahti and Mittelstaedt, 1996 ([30, p.127])

We shall study a simple measurement problem – the measurement of the diameter
of a disk. […] It may happen that the difference of the measurements in
different directions exceeds the permissible error of a given measurement. In
this situation, we must state that within the required measurement accuracy,
our disk does not have a unique diameter, as does a circle. Therefore, no
concrete number can be taken, with prescribed accuracy, as an estimate of the
true value of the measurable quantity.

Semyon Rabinovich, 2005 ([139, p.11])

The analysis in Sections 2–6 suggests that we should reject the traditional
convention that declares the eigenvalues of operators to be the true values in
a measurement. This is done explicitly in the thermal interpretation of
quantum physics introduced in Neumaier [122, 123, 124]; a detailed, definitive
account is in my recent book Coherent quantum physics (Neumaier [125]); see
especially Section 9.2. In the preceding we essentially derived the main
features of the thermal interpretation, including everything that had only
been postulated in my book.

The thermal interpretation gives a new foundational perspective on quantum
mechanics and suggests different questions and approaches than the traditional
interpretations. It extends the discussion of observable sources and detectors
in terms of quantum values represented by quantum expectations to arbitrary
quantum systems even in their unobservable aspects. According to the thermal
interpretation, the objective (reproducible) properties of any quantum system
are the quantum values encoded mathematically in the quantum expectations, and
what can be computed from these. In particular, the Born–von Neumann link of
measurement results to eigenvalues (which did not show up anywhere in our
treatment except for highly idealized projective measurements) is replaced by
a link of reproducible measurement results to quantum values, computable as
quantum expectations. This link is amply corroborated by the fact that the
extensive quantities featured in equilibrium thermodynamics appear in
statistical mechanics as the quantum expectations of conserved quantities. As
a consequence, there are no obstructions to treating large systems including
observers, and even the whole universe, as quantum systems.

The thermal interpretation generalizes the well-known fact that in equilibrium
statistical thermodynamics, all extensive quantities are represented by
quantum expectations. It proclaims – in direct opposition to the tradition
created in 1927 by Jordan, Dirac, and von Neumann – the alternative convention
that the true values are the quantum expectations rather than the eigenvalues.
In the thermal interpretation of quantum physics, given a particular instance
of a quantum system described by a model at a given time, the state defines
all its properties, and hence what exists222222 This gives a clear formal
meaning to the notion of existence. Whether something that exists in this
model sense also exists in Nature depends on how faithful the model is to the
corresponding aspect of Nature.  in the system at that time. The objective
properties of the system are given by quantum expectations and what is
computable from these. All properties of a quantum system at a fixed time
depend on the state $\langle\cdot\rangle$ of the system at this time and are
expressed in terms of definite but uncertain values of the quantities. As
discussed in detail in my book (Neumaier [125]), the identification of these
formal properties with real life properties is done by means of

(CC) Callen’s criterion (cf. Callen [33, p.15]): Operationally, a system is in
a given state if its properties are consistently described by the theory for
this state.

This is a concise version of the principle of identification suggested by
Eddington [52, p.222]; cf. the initial quote of this paper. Callen’s criterion
is enough to find out in each single case how to approximately determine the
uncertain value of a quantity of interest.

As we have seen, the simplest quantum system, a qubit, was already described
by Stokes [150] in 1852, in terms essentially equivalent to the thermal
interpretation – except that the intrinsic uncertainty was not yet an issue,
being at that time far below the experimentally realizable accuracy. This
alternative convention also matches the actual practice in quantum information
theory, where the states are manipulated, transmitted, measured, hence their
properties (i.e., whatever is computable from the state) are treated as
objective properties. The density operator may be viewed simply as a
calculational tool for obtaining these objective properties, in particular
quantum expectations and their uncertainties.

In the thermal interpretation, every observable scalar or vector quantity $X$
has an associated intrinsic state-dependent uncertainty $\sigma_{X}$ within
which it can be (in principle) determined. The idea is that the quantum
expectation $\overline{X}$ itself has no direct operational meaning; only the
fuzzy region of $\xi\in\mathbb{C}^{m}$ with $|\xi-\overline{X}|$ bounded by
the uncertainty $\sigma_{X}$ or another small multiple of $\sigma_{X}$ is
meaningful. Statistics enters whenever a single value has too much
uncertainty, and only then. In this case, the uncertainty can be reduced – as
within classical physics – by calculating statistical means.

This is standard engineering practice when considering the diameter of a disk
that is not perfectly circular. The uncertainty is in the imprecise
definition, just as that in the position of an extended object such as a
doughnut. In particular, the description of a quantum particle as having
momentum $p$ and being at position $q$ is as unsharp as the description of a
classical signal as having frequency $\nu$ at time $t$. Even formally, the
concepts are analogous and share the uncertainty relation, known in signal
analysis as the Nyquist theorem, and discovered in the quantum context by
Heisenberg. The analogy is especially clear in quantum field theory, where on
the one hand position and time and on the other hand momentum and energy,
related to frequency by the Rydberg–Ritz formula (29), are described on an
equal footing. When measuring $X$ it is meaningless to ask for more accuracy
than the uncertainty $\sigma_{X}$, just as meaningless as to ask for the
position of a doughnut to mm accuracy.

Thus the foundations invoked by the thermal interpretation are essentially the
foundations used everywhere for uncertainty quantification, just slightly
extended to accommodate quantum effects by not requiring that quantities
commute.

###  7.4 Measurement in the thermal interpretation

A satisfactory theory of the measuring process must start from a
characterization of the macroscopic properties of a large body. Such
properties must have an objective character.

Daneri, Loinger and Prosperi, 1962 [39, p.305]

Quantum mechanics has often been classified as a merely statistical ensemble
theory, with not much bearings on the individual members of the ensembles. Yet
there is an increasing variety of experiments exhibiting individual quantum
processes which were conceived, devised and explained on the basis of this
very theory.

Busch and Lahti, 1996 [27, p.5899]

Since the interactions between macroscopic systems are effective at
astronomical distances, the only ”closed system” is the universe as a whole.

H.D. Zeh, 1970 [170, p.73]

In the thermal interpretation, measurement outcomes are defined as quantum
expectations of macroscopic detector quantities $X^{\prime}$ strongly
correlated – in the way phenomenologically discussed in Section 2 – with the
microscopic quantities $X$ to be measured. According to the thermal
interpretation, a measurement $a_{k}$ of $X$ is treated as an approximation of
the quantum expectation $\overline{X}$ of $X$. $\overline{X}$ is (in
principle, in general only inaccurately) observable if it varies sufficiently
slowly with $t$ and has a sufficiently small uncertainty $\sigma_{X}$. But it
may require considerable experimental ingenuity to do so with an uncertainty
close to $\sigma_{X}$. The uncertain value $\overline{X}$ is considered
informative only when its quantum uncertainty $\sigma_{X}$ is much less than
$|\overline{X}|$.

Since quantum expectations are always single-valued, this immediately resolves
the unique outcome problem of quantum measurement theory. The thermal
interpretation makes direct sense of individual events even at the theoretical
level. It naturally gives an ontology for individual quantum systems – not
only for thermal systems but also for microscopic systems and even the whole
universe – the only truly closed quantum system.

In particular, because of the single-valuedness of the true values in the
thermal interpretation, probabilities are not intrinsic to quantum physics but
are emergent imperfections. In contrast, Born’s statistical interpretation
needs probabilities in the very foundations, due to the multi-valuedness of
the true values.

Both interpretations are in agreement with the experimental record. The same
number $a_{k}$ obtained by a measurement may be interpreted in two ways,
depending on the convention used: (i) It measures the quantum expectation to
some accuracy. (ii) It measures some random eigenvalue to a possibly higher
(in the idealization even infinite) accuracy. In both cases, the measurement
involves an additional uncertainty related to the degree of reproducibility of
the measurement, given by the standard deviation of the results of repeated
measurements. Tradition and the thermal interpretation agree in that this
uncertainty is – by (43) – at least $\sigma_{X}$. If the eigenvalues $X_{k}$
of $X$ (assumed to have discrete spectrum) are exactly known beforehand, one
can calibrate the pointer scale to make $a_{k}=X_{k}$ for all detector
elements $k$. As long as one ignores the idealization error, the thermal
interpretation and Born’s interpretation become experimentally
indistinguishable. However, this is no longer so in the more realistic case
where eigenvalues are only approximately known – the common situation in
spectroscopy – and must therefore be inferred experimentally. In this case we
saw in Subsection 6.4 that Born’s statistical interpretation paints an
inadequate, idealized picture only. The thermal interpretation, however, still
gives a correct account of the actual experimental situation. Measurements are
regarded as fluctuating discrete readings of detector properties, defined as
quantum expectations of macroscopic pointer variables whose statistical mean
agrees with the quantum expectation of the system quantity measured, as
discussed in Section 2.

Thus the thermal interpretation is in full agreement with the standard recipes
for drawing inferences from inaccurate measurement results. The situation is
precisely the same as in classical metrology, where observable quantities
always have a true value determined by the theoretical description, and all
randomness in measurements is assumed to be due to measurement noise.

The thermal interpretation regards each measurement result $a_{k}$ as an
approximation of the true value $\overline{X}$, with typical error
$|a_{k}-\overline{X}|$ of at least $\sigma_{X}$, by (43). In the limit of
arbitrarily many repetitions, the statistical mean value of the approximations
approaches the quantum expectation $\overline{X}$, and their standard
deviation approaches the quantum uncertainty $\sigma_{X}$. The observed
discreteness of counts in instruments like geiger counters is explained as an
effect due to the discreteness of the recording device. The latter introduces
a systematic error, of the same nature as the errors discussed in Subsection
6.4.

For example, binary responses of the macroscopic detector elements may be
explained as in Neumaier [125, Chapter 11] by a bistability (cf. Bonifacio &
Lugiato [17], Gevorgyan et al. [64]) of their coarse-grained microscopic
dynamics analogous to the bistability that give rise to binary responses in
classical coin tossing. The bimodal distribution of the measurement results
may be due to environment-induced randomness and environment-induced
dissipation, as for a classical, environment-induced diffusion process in a
double-well potential. For a discussion of the dynamical aspects of the
quantum chaos responsible for this see, e.g., Ingraham & Acosta [86], Zhang &
Feng [171], Belot & Earman [12], Gomez et al. [66], and Chapter 11 of my book
(Neumaier [125]). The bimodal distribution may also be due more immediately to
the experimental setup. For example, the arrangement in a Stern–Gerlach
experiment together with simple theory leads to two more or less focussed
beams, which accounts for the approximate 2-valuedness of the response at the
screen. The thermal interpretation attributes this discreteness to the
detection setup, not to a true, discrete value of the spin.

For precision measurements, e.g., those related to the quantum Hall effect,
relevant for preparing calibration states for metrology standards (Lindeley
([108], Kaneko et al. [90]), one needs to be able to prepare states with tiny
quantum uncertainties, so that their measurement can be performed with very
high accuracy. The possibility of achieving tiny uncertainties is linked to
the spectrum. Let $X$ be a scalar or vector quantity whose $m$ components are
defined on a common dense domain. We call the set of $\xi\in\mathbb{C}^{m}$
for which no linear operator $R(\xi)$ exists such that $R(\xi)|X-\xi|^{2}$ is
the identity the spectrum $\mathop{\rm Spec}X$ of $X$. The spectrum is always
a closed set, but it may be empty, as, e.g., for the vector formed by position
and momentum of a particle. The following result implies that the preparation
of states such that $X$ has arbitrarily small uncertainty is possible
precisely when $\overline{X}$ belongs to the spectrum of $X$.

###### 7.1 Theorem.

$\xi\in\mathbb{C}^{m}$ belongs to the spectrum of a scalar or vector quantity
$X$ with $m$ components iff states exist that have arbitrarily small positive
$\langle|X-\xi|^{2}\rangle$.

###### Proof.

By definition, the operator $B:=|X-\xi|^{2}$ is Hermitian and positive
semidefinite, hence essentially self-adjoint with nonnegative spectrum
$\Sigma$. Note that $\xi$ belongs to the spectrum of $X$ iff $0\in\Sigma$. We
consider the spectral projector $P(s)$ to the invariant subspace corresponding
to the spectrum in $[0,s]$. If $0\not\in\Sigma$ then $P(s)=0$ for some $s>0$,
and $\langle|X-\xi|^{2}\rangle=\langle B\rangle\geq s$ cannot be arbitrarily
small. On the other hand, if $0\in\Sigma$ then, for all $s>0$, the projector
$P(s)$ is nonzero. Thus we can find vectors $\phi(s)$ such that
$\psi(s):=P(s)\phi(s)$ is nonzero. By scaling $\phi(s)$ appropriately we can
ensure that $\psi(s)$ has norm one. Then in the corresponding pure state,
$\langle|X-\xi|^{2}\rangle=\langle B\rangle\leq s$. Since this works for any
$s>0$, the claim follows. ∎

Thus though in the thermal interpretation eigenvalues no longer play the role
they traditionally have in measurement, they continue to be essential for high
precision measurements. Elsewhere in quantum physics, they play a role
primarily in the case of the Hamiltonian, where the eigenvalues specify the
possible energy levels of a quantum system. The latter is relevant not only on
the theoretical level, where a spectral representation allows the explicit
solution of the quantum Liouville equation and the time-dependent Schrödinger
equation. It is also important in many contexts with experimentally relevant
consequences: We have seen the Rydberg–Ritz formula (29) from spectroscopy,
which can access a large number of energy levels. In many other cases, only
few energy levels are experimentally accessible, in which case the quantum
system can be modeled as a few level system, drastically simplifying the task
of quantum state tomography. In much of quantum chemistry, only the electronic
ground state is considered relevant. Its energy, parameterized by the nuclear
coordinates, determines the potential energy surface whose properties are
sufficient to describe the shape of all molecules and their chemical
reactions. The energy spectrum is also prominent in thermal equilibrium
physics since it determines the partition function from which all other
thermodynamic properties can be derived. For a treatment of thermal
equilibrium quantum physics in the spirit of the thermal interpretation –
i.e., based on quantum expectations without making reference to statistical
assumptions – see the online textbook by Neumaier & Westra [127].

###  7.5 The quantum measurement problem

The effect of a single scattering is calculated and shown to damp the off-
diagonal elements in the reduced spatial density matrix by a factor that is
simply the Fourier transform of the probability distribution for different
momentum transfers. […] Sec. 5 contains a brief discussion of what
interpretational problems of quantum mechanics the decoherence approach does
and does not solve.

Max Tegmark, 1993 [155, p.2]

Usually the system and its properties – whatever can be said about them – are
regarded as fundamental. Physical systems are considered objectively, outside
the context of measurements, real or potential. Within this traditional
viewpoint the question for physics is this: What are the properties of
physical systems, and what laws do they obey? Notoriously this leads to the
quantum measurement problem. Having considered physical systems to obey
quantum mechanics, what are we to make of measurements? How does measurement
fit into the framework of quantum theory?

Caves, Fuchs, Manne, and Renes, 2004 [35, p.194]

There is no consistent quantum measurement theory, unless an appropriate
reinterpretation of what it means for an observable to have a definite value
can be found.

Busch and Lahti, 2009 [29, p.375]

Decoherence actually aggravates the measurement problem: where previously this
problem was believed to be man-made and relevant only to rather unusual
laboratory situations, it has now become clear that ”measurement” of a quantum
system by the environment (instead of by an experimental physicist) happens
everywhere and all the time: hence it remains even more miraculous than before
that there is a single outcome after each such measurement.

Klaas Landsman, 2017 [102, p.443]

We need to take the existence of measurement outcomes as a priori given, or
otherwise give an account outside of decoherence of how measurement outcomes
are produced, because the property of classicality is ultimately a statement
about measurement statistics.

Maximilian Schlosshauer, 2019 [143, p.72]

All traditional foundations of quantum mechanics depend heavily – far too
heavily – on the concept of (hypothetical, idealized) measurements exactly
satisfying Born’s rule, a nontrivial technical rule far from being intuitive.
This – almost generally assumed – exact validity without a precise definition
of the meaning of the term measurement is probably the main reason why, nearly
100 years after the discovery of the basic formal setting for modern quantum
mechanics, these foundations are still unsettled. No other scientific theory
has such controversial foundations that persisted for so long.

The source of this poor state of affairs is that Born’s rule for projective
measurements, the starting point of the usual interpretations, constitutes a
severe idealization of measurement processes in general. Except in a few very
simple cases, it is too far removed from experimental practice to tell much
about real measurements, and hence about how quantum physics is used in real
applications. Foundations that provide a safe ground for interpreting reality
cannot start with idealized concepts only. Thus it is worthwhile to reconsider
the measurement problem in terms of the new foundations presented in this
paper, which apply without any idealization.

In a measurement we extract within some measurement tolerance an unambiguous
value from an environment and simultaneously make a claim that this value
reveals a property of the system at the measurment time. Here unambiguous
means that the uncertainty is significantly smaller than the error tolerance
claimed for the measurement.

The measurement problem is the problem to show convincingly how Born’s rule
(11) can be justified entirely in terms of the unitary dynamics of a larger
quantum model containing a measured quantum system and a quantum detector
measuring it. For such quantum models of measurements see Busch & Lahti [28],
several chapters in Busch et al. [25, 31]. and the detailed study by
Allahverdyan et al. [3, 4].

Based on the present approach we may give a reasonably precise formulation. A
complete solution of the measurement problem would consist of

1\. A derivation of the states $\rho$ and the operators $P_{k}$ from the
microscopic description of typical macroscopic quantum systems that serve as
sources and detectors. This is the classicality problem of quantum
measurement.

2\. A description of a single measurement, deriving from the state and
dynamics of the composite quantum system – formed by a measured system and a
detector – the measurement result and its accuracy. This is the definite
outcome problem of quantum measurement.

3\. To show that a single particle moving along a beam triggers at most one of
an array of detection elements. This is the unique outcome problem of quantum
measurement.

The first part is solved in principle by decoherence (Zurek [172],
Schlosshauer [143]). In an extremely short decoherence time, density operators
of macroscopic systems tend to become essentially diagonal in a preferred
basis selected by the environment. Mathematically (Tegmark [155]), this is a
consequence of the Riemann–Lebesgue Lemma, which asserts that time integrals
over signals with a purely continuous spectrum vanish in the limit where the
time span gets arbitrarily large.

The second part is solved in principle by the thermal interpretation of
quantum physics (Neumaier [125]). Outcomes of experiments are observations (or
results computed from these) of macroscopic detectors. This are given by the
extensive and intensive quantities of their description in terms of local
equilibrium thermodynamics. The extensive quantities are quantum expectations
of appropriate quantum field operators, while the intensive quantities are
determined by the extensive quantities through the machinery of local
equilibrium thermodynamics. In particular, the outcomes are automatically
definite, determined by the state.

From a mathematical perspective (cf. Neumaier [125, Section 10,5]),
decoherence tells roughly the same story as the thermal interpretation, but
only in terms of averages, whereas the thermal interpretation refines this to
a different, more detailed story for each single case.

Informally, the third part is a simple consequence of the conservation of
energy. However, due to the highly nonstationary situation alluded to in the
formulation, the problem raised in point 3 is effectively unsolved. Within
quantum field theory, which should provide the setting for answering the
question, the formulated assertion can currently not even be precisely stated.
A proper solution would first require to give, within a model describing an
experiment exclusively in terms of quantum fields, a precise meaning to the
notion of a single particle moving at a particular time in a given single-
particle state along a beam. This is a question on the borderline between
quantum mechanics and quantum field theory touched in discussions on the
preparation of photons on demand (see, e.g., Keller et al. [93] and Neumaier
[121]) but it is nowhere clearly resolved.

###  7.6 Reproducibility and objectivity

The purpose of measurements is the determination of properties of the physical
system under investigation.

Busch, Lahti and Mittelstaedt, 1996 ([30, p.25])

Measurements are regarded metrologically to be better the lower their
uncertainty is. However, measurements must be reproducible, because otherwise
they lose their objective character and therefore become meaningless.

Semyon Rabinovich, 2005 ([139, p.22])

In a fundamentally statistical theory like quantum mechanics the results of
individual measurements tell us almost nothing: It is always the probability
distribution of outcomes for a fixed experimental arrangement which can
properly be called the result of an experiment.

Busch, Lahti and Werner 2014 [32, p.5]

Quantum mechanics is concerned with macroscopic phenomena, which are not
perturbed by observation.

Nicolas van Kampen, 1988 [159, Theorem II, p.98]

In theoretical physics, rhysical systems are described mathematically by means
of models that quantitatively reflect the essence of the system. These models
are formulated within a given theoretical framework, which allows one to fix
the model assumptions. Typically one specifies the phase space (including a
reference coordinate system for fixing the collections of the observables),
the causal rules (Galilean in nonrelativistic physics, Minkowski in special
relativity, local Minkowski in general relativity), and the parameterized
Hamiltonian in conservative mechanics, more elaborate equations of motion in
dissipative mechanics.

A physical model is a template which contains parameters that are usually not
fixed but to be determined by system identification, the process of matching
key properties of the model with experimental results. For example, a
classical quartic oscillator is a model defined by a Hamiltonian

$H=p^{2}/2m+kq^{2}/2+gq^{2}/4$

together with the specification of classical Hamiltonian dynamics for the
conjugate position and momentum variables $q$ and $p$. The coefficients in the
Hamiltonian, the mass $m$ and the coupling constant $g$ are the parameters.
The claim that, within a given accuracy, a particular oscillator is well
described by this model can be decided by making experiments on the oscillator
and comparing it with the predictions of the model.

If the model is correct to some accuracy, there will be a parameter setting
that matches the prediction. In the limit of arbitrarily many and arbitrarily
accurate measurements, the best parameters will usually be determined by the
experiment and standard statistical parameter estimation techniques. This can
be done in a reproducible way. Thus the parameters are operationally
quantifiable, independent of an observer. Reproducibility and observer
independence are the hallmark of scientific investigations. Reproducibility
ensures that the parameters are indeed properties of the system measured,
rather than an artifact of the observational procedure. Observer independence
ensures that these properties are objective.

In the case of quantum tomography, the model is – according to our discussion
so far – a finite-dimensional Hilbert space chosen to model the quantum
system, together with the mathematical machinery deduced from the detector
response principle. The state is an element of the quantum phase space, a
complex positive semidefinite Hermitian matrix of parameters that are not
determined by the model but by experiments, using the traditional, universally
agreed statistical techniques.

When a source is stationary, response rates and probabilities can be measured
in principle with arbitrary accuracy. These probabilities, and hence
everything computsble from them – quantum expectation values and the density
operator but not the individual detector events – are operationally
quantifiable, independent of an observer, in a reproducible way. Thus the
density operator is an objective property of a stationary quantum system, in
the same sense as in classical mechanics, positions and momenta are objective
properties.

Through quantum tomography, the quantum state of a sufficiently stationary
source, the quantum measure of a meansurement device, and the transmission
operator of a sufficiently linear and stationary filter can in principle be
determined with observer-independent protocols. Thus they are objective
properties of the source, the measurement device, or the filter, both before
and after measurement. Measurements only serve to determine or to check their
quantitative form, thereby revealing or confirming preexisting properties of
the source, the detector, and the filter, or demonstrating deviations from
their assumed behavior.

###  7.7 Knowledge and probabilities

The Compton effect has as its consequence that the electron is caused to jump
[…] As our knowledge of the system does change discontinuously at each
observation its mathematical representation must also change discontinuously;
this is to be found in classical statistical theories as well as in the
present theory.

Werner Heisenberg, 1930 [75, p.35f]

This, I see as the line of attack we should pursue with relentless
consistency: the quantum system represents something real and independent of
us. […]
What might have been [Einstein’s] greatest achievment in building general
relativity? […] It was in identifying all the things that are ’numerically
additional’ to the observer-free situation – i.e., those things that come
about purely by bringing the observer (scientific agent, coordinate system,
etc.) back into the picture. This was a breakthrough. […]
The dream I see for quantum mechanics is just this. Weed out all the terms
that have to do with gambling commitments, information, knowledge and belief,
and what is left behind will play the the role of Einstein’s manifold.

Christopher Fuchs, 2003 [63, p.989f]

A physical model (phase space, dynamical law, form of the Hamiltonian)
determines the subject matter of a physical investigation and fixes the
language for its discussion. It therefore plays in physics the same role that
axioms play in mathematics. Although different models can be chosen to discuss
the same real situation we usually do not perceive this choice as a
restriction of objectivity in science since all that is to choose is made
explicit.

Subjectivity in the choice of the model is the kind of subjectivity one has
everywhere in physics. It has nothing to do with subjective knowledge of
details about a particular system under experimental investigation commonly
discussed in the foundations of quantum physics and modelled by subjective
probabilities. Indeed, it seems pointless to try to assign subjective
probabilities for decisions to use a particular model as the subject of
discourse.

In the quartic oscillator model discussed, the only subjectivity is in the
choice of the ansatz for the Hamiltonian and the use of classical mechnaics
rather than quantum mechanics. There is no subjectivity in the parameters: If
the parameters do not agree with the true parameters within the limits of the
model accuracy claimed, more and more independent measurements lead sooner or
later to a statistically arbitrarily significant discrepancy with experiments.

Precisely the same holds in quantum tomography. Again, the only subjectivity
is in the choice of the model – in this case the Hilbert space representing
the quantum system and the assumption inherent in the DRP that states are
modelled by density operators. Everything else can be determined and checked
completely independent of any subjective knowledge. Assuming that a quantum
system is in a state different from the true state simply leads to wrong
predictions that can be falsified by sufficiently long sequences of
measurements. Nothing depends on the knowledge of an observer. The latter can
be close to the objective truth or far away – depending on how well informed
the observer is.

The assignment of states to stationary sources is as objective as any
assignment of properties to macroscopic objects. Thus the knowledge people
talk about when referring to the meaning of a quantum state resides solely in
what is encoded in (and hence ”known to”) the model used to describe a quantum
system – not to any subjective mind content of a knower!

In particular, as quantum values of members of a quantum measure, all
probabilities are objective frequentist probabilities in the sense employed
everywhere in experimental physics – classical and quantum. That the
probabilities are only approximately measurable as relative frequencies simply
says that – like all measurements – probability measurements are of limited
accuracy only. Similarly, as a property of macroscopic instrument, the
collapse and its generalizations is an objective phenomenon – not a subjective
update of knowledge.

Thus by shifting the attention from the microscopic structure to the
experimentally accessible macroscopic equipment (sources, detectors, filters,
and instruments) we have gotten rid of all potentially subjective elements of
quantum theory.

##  8 Conclusion

In conclusion it may be mentioned that the present theory suggests a point of
view for regarding quantum phenomena rather different from the usual ones. One
can suppose that the initial state of a system determines definitely the state
of the system at any subsequent time. […] One can obtain a good deal of
information (of the nature of averages) about the values at the subsequent
time considered as functions of the initial values. The notion of
probabilities does not enter into the ultimate description of mechanical
processes.

Paul Dirac, 1927 [48, p.641]

It is perhaps remarkable that all the complexities of the observational
process can be subsumed under so simple a form.

Carl Helstrom, 1976 [79, p.59]

In this paper, the standard kinematical, statistical, dynamical, and spectral
rules of introductory quantum mechanics were derived with little effort from a
single assumption – the detector response principle (DRP) introduced in
Subsection 2.2. At the same time we found the conditions under which these
standard rules are valid.

Though very elementary, our notion of a detector and the associated formal
notion of a discrete quantum measure is very flexible and accounts for all
basic aspects of practical measurement processes.

The new approach is indifferent to the microscopic cause of detection events.
It cleanly separates the experimentally relevant formal aspects of the quantum
measurement process from the controversial issues of quantum foundations. In
this way, a clear formal definition of measurement is possible without
touching the quantum measurement problem related to the dynamical origins of
the measurement results. No idealization is involved. It is absolutely clear
what a measurement amounts to and how accurately it may represent theoretical
quantum values in operational terms. It was pointed out that to fully solve
the quantum measurement problem, more research is needed on the
characterization of quantum systems that are nonstationary on experimentally
directly accessible time scales.

Quantum theory is much more than a probability theory. It predicts lots of
nonprobabilistic stuff, such as spectra of all sorts, mechanical, optical and
electrical properties of materials, stability of chemical compounds and
chemical reaction rates. These nonprobabilistic predictions of quantum theory
are derived from the deterministic evolution of the quantum state. The quantum
evolution is characterized as Hamiltonian dynamics on the Poisson manifold of
normalized density operators.

Quantum expectations figuring in the derivations of these results have the
meaning of objective quantum values, independent of a probability
interpretation The latter is needed only when averaging actual repeated
measurements of individually unpredictable events. Thus the situation is
completely analogous to that of classical mechanics, and a notion of intrinsic
quantum probability is superfluous.

A link to experimental practice was established through careful definitions in
terms of quantum tomography, replacing the traditional a priori link via
idealized projective measurements. As a result, there is no obstruction to
treating large systems including observers, and even the whole universe, as
quantum systems. A special role for a notion of particles, a sharp
classical/quantum interface, an experimenter/observer, or their knowledge, was
nowhere needed. In particular, the only reasons are gone why, in spite of
their strange features, interpretations such as Bohmian mechanics or Many-
Worlds have become respectable among some physicists.

The new picture is simpler and more general than the traditional foundations,
and closer to actual practice. This makes it suitable for introductory courses
on quantum mechanics. Thus there is no longer an incentive for teaching
quantum mechanics starting with a special, highly idealized case in place of
the real thing.

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