Abstract
We propose a general framework of the quantum/quasi-classical transformations by introducing the concept of quasi-joint-spectral distribution (QJSD). Specifically, we show that the QJSDs uniquely yield various pairs of quantum/quasi-classical transformations, including the Wigner–Weyl transform. We also discuss the statistical behaviour of combinations of generally non-commuting quantum observables by introducing the concept of quantum correlations and conditional expectations defined analogously to the classical counterpart. Based on these, Aharonov’s weak value is given a statistical interpretation as one realisation of the quantum conditional expectations furnished in our formalism.
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Notes
- 1.
A self-adjoint operator defined on infinite dimensional Hilbert spaces may sometimes fail to have any eigenvalues in the sense that \(A |\psi \rangle = a |\psi \rangle \) holds for some non-zero vector \(|\psi \rangle \in \mathscr {H}\), as most famously exemplified by the position operator \(\hat{x}\) and the momentum operator \(\hat{p}\) of a free particle.
- 2.
We say that a pair of self-adjoint operators A and B strongly commutes, if and only if they commute in the level of spectral measures. In a laxer notation, this is to say that
$$\begin{aligned} E_{A}(a)E_{B}(b) = E_{B}(b)E_{A}(a), \quad a, b \in \mathbb {R} \end{aligned}$$holds. Strictly speaking, strong commutativity is generally stronger than mere commutativity when unbounded operators are concerned, but we do not intend to delve into the intricacies, which are not essential for our discussion.
- 3.
The space of density functions on \(\mathbb {R}^{n}\) is a proper subspace of the space of complex measures. An example of this is the delta measure, which is well-defined as a measure but not as a density function. The space of complex measures on \(\mathbb {R}^{n}\) is also a proper subspace of the space of tempered distributions. An example for this is the derivative of the delta measure, which is well-defined as a tempered distribution but not as a complex measure.
- 4.
Note that the image of a Schwartz function under the functional calculus (14) is a bounded operator with the operator norm \(\Vert f(A)\Vert \le \sup _{x \in \mathbb {R}^{n}}|f(x)|\), hence the map is well-defined.
- 5.
The image of a nuclear operator under the Born rule (15) belongs to the space of complex measures, which can be uniquely embedded into the space of tempered distributions. In this sense, we extend the codomain of the map (15) and understand their images to be tempered distributions, rather than complex measures, for later convenience.
- 6.
The reason for our choice of the nomination quasi-joint-spectral distributions, rather than measures, lies in the fact that, in contrast to the JSMs, QJSDs does not necessarily lie in the space of operator valued measures (OVMs). In fact, we understand them as members of the operator valued distributions (OVDs), which is an operator analogue of generalised functions (distributions). The space of OVDs is larger than the space of OVMs, and the latter can be embedded into the former.
- 7.
Here, we adopt the convention
$$\begin{aligned} \int _{\mathbb {K}^{n-k}} \,\#_{\varvec{A}}(\varvec{a})\ dm_{n=k}(\varvec{b}^{c}) = \#_{\varvec{A}}(\varvec{a}) \end{aligned}$$for the case \(k=n\).
- 8.
The quantisation of a Schwartz function f is formally defined as a unique bounded map \(f_{\#_{\varvec{A}}}\) such that the equality
$$\begin{aligned} \mathrm {Tr}\left[ f_{\#_{\varvec{A}}} \rho \right] := \int _{\mathbb {R}^{n}} \check{f}(\varvec{s}) \mathrm {Tr}\left[ \hat{\#}_{\varvec{A}}(\varvec{s}) \rho \right] \, dm_{n}(\varvec{s}) \end{aligned}$$holds for all \(\rho \in N(\mathscr {H})\), where \(\check{f}\) denotes the inverse Fourier transform of f. The r. h. s of the above equation is well-defined, since
$$\begin{aligned} \left| \int _{\mathbb {R}^{n}} \check{f}(\varvec{s}) \mathrm {Tr}\left[ \hat{\#}_{\varvec{A}}(\varvec{s}) \rho \right] \, dm_{n}(\varvec{s}) \right| \le \Vert \check{f}\Vert _{1} \cdot M \Vert \rho \Vert _{\mathrm {nuk}} \end{aligned}$$is finite. Here, \(\Vert \cdot \Vert _{1}\) and \(\Vert \cdot \Vert _{\mathrm {nuk}}\) respectively denote the \(L^{1}\) norm of integrable functions and the nuclear norm (alias trace norm) of nuclear operators, and \(\Vert \hat{\#}_{\varvec{A}}(\varvec{s}) \Vert \le M\), \(\varvec{s} \in \mathbb {K}^{n}\) is the upper bound of the operator norm of the hashed operator. The existence and uniqueness of such an operator \(f_{\#_{\varvec{A}}} \in L(\mathscr {H})\) is due to the fact that the space of continuous linear functionals on \(N(\mathscr {H})\) is isomorphic to the space of bounded operators \(N(\mathscr {H})^{\prime } \cong L(\mathscr {H})\). The continuity of the linear functional \(\rho \mapsto \mathrm {Tr}[f_{\#_{\varvec{A}}} \rho ]\) follows directly from the above evaluation.
- 9.
Here, we adopt the convention
$$\begin{aligned} \int _{\mathbb {K}^{n-k}} h(\varvec{a})\ dm_{n-k}(\varvec{b}^{c}) = h(\varvec{a}) \end{aligned}$$for the case \(k=n\).
- 10.
The proof for the equivalence of the conditions can be carried out by applying the Hahn-Banach theorem on locally convex spaces.
- 11.
Do not confuse the subfamily introduced above with that of (25). We have used different superscript characters \(\kappa \), \(\alpha \) as parameters to make the distinction more easier.
- 12.
We say that a tempered distribution \(u \in \mathscr {S}^{\prime }(\mathbb {K}^{n})\) admits representation by a density function if u(x) is actually an integrable function.
- 13.
Here, the equality \(f=0\) in the second line of (89) is meant to hold \(\rho _{\#_{\varvec{A}}}\)-almost everywhere.
- 14.
The spectrum of a self-adjoint operator A is defined as the largest closed subset \(J \subset \mathbb {R}\) such that \(E_{B}(J) = \mathrm {Id}\) holds.
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Lee, J., Tsutsui, I. (2018). A General Framework of Quasi-probabilities and the Statistical Behaviour of Non-commuting Quantum Observables. In: Ozawa, M., Butterfield, J., Halvorson, H., Rédei, M., Kitajima, Y., Buscemi, F. (eds) Reality and Measurement in Algebraic Quantum Theory. NWW 2015. Springer Proceedings in Mathematics & Statistics, vol 261. Springer, Singapore. https://doi.org/10.1007/978-981-13-2487-1_9
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