The analyser outcome law of the fixed-mode polarisation model: Born probabilities and the Malus ratio

Adopt the state and composition postulate P1, the observable and probability postulate P2 and the ideal conditional-measurement postulate P3 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes), the radiation extension postulate clause (i) (Radiation extension of the framework: transverse mode and photon assignments) and the declared fixed-mode polarisation model The fixed-mode polarisation model of the framework, in the state vocabulary of The state space of the framework: preparations, pure rays and density operators and the observable and measurement vocabularies of Assigned sharp observables and the outcome law of the framework and Declared ideal measurements and instruments of the framework. Assume Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, The Axiom of Choice), including the hypotheses inherited from the radiation postulate and the cited spectral and measurement mathematics. Fix angles $\phi ,\theta ,{\theta }^{\mathrm{\prime }}\in \mathbb{R}$ and write ${\psi }_{\phi }=\cos \phi e_1+\sin \phi e_2$ and $P_{\theta }$ for the analyser projections of the model. Then:

(a) the declared polarisation analyser at angle $\theta$ is a declared ideal sharp measurement with outcome structure $(P_{\theta },P_{\theta }^{\perp })$, the outcome labels $+1$ and $-1$ naming the first and second projection, in the sense of clause (a) of Declared ideal measurements and instruments of the framework: $P_{\theta }$ and $P_{\theta }^{\perp }$ are bounded orthogonal projections with $P_{\theta }{P}_{\theta }^{\perp }=0$ and $P_{\theta }+P_{\theta }^{\perp }=I$;

(b) the spectral projection-valued measure $E$ of the bounded self-adjoint operator $P_{\theta }$ (assigned with unit $1$; its values on the two outcomes are $1$ and $0$, while the analyser labels are $+1$ and $-1$) satisfies $E(\{1\})=P_{\theta },E(\{0\})=P_{\theta }^{\perp },E(\mathbb{R}\setminus \{0,1\})=0,$ consequently the outcome law of the assigned sharp quantity $P_{\theta }$ in a preparation $\rho \in D(V)$ is $p_{+1}(\rho )=\mathrm{tr}(\rho P_{\theta }),p_{-1}(\rho )=\mathrm{tr}(\rho P_{\theta }^{\perp })=1-p_{+1}(\rho ),$ which is exactly the law of the declared analyser;

(c) for the linear-polarisation preparation ${\psi }_{\phi }$ and a general pure preparation with unit vector $\psi =\alpha e_1+\beta e_2$, $p_{+1}({\psi }_{\phi })={\cos }^2(\theta -\phi ),p_{-1}({\psi }_{\phi })={\sin }^2(\theta -\phi ),p_{+1}(\psi )=\mathrm{\mid }\alpha \cos \theta +\beta \sin \theta {\mathrm{\mid }}^2;$ for a statistical mixture $\rho ={\sum }_n{w}_n{\rho }_n$ of preparations the law is affine, $p_{+1}(\rho )={\sum }_n{w}_n{p}_{+1}({\rho }_n)$;

(d) the law is invariant under the phase of the preparation and under the phase of the analysed direction: $p_{+1}$ depends only on the rays $[{\psi }_{\phi }]$ and $[e_{\theta }]$, is $2\pi$-periodic in each argument, and satisfies $p_{+1}(\theta ,\phi )=p_{+1}(\theta -\phi ,0)$;

(e) if $p_{+1}(\rho )>0$, the conditional state after the outcome $+1$ of the analyser is ${\rho }_{+1}=P_{\theta }\rho P_{\theta }\mathrm{/}{p}_{+1}(\rho )$; for the pure preparation $[{\psi }_{\phi }]$ with $\cos (\theta -\phi )\mathrm{\ne }0$ it is the pure state of the analysed ray $[e_{\theta }]$, and a second analyser at angle ${\theta }^{\mathrm{\prime }}$ then gives the conditional probability ${\cos }^2({\theta }^{\mathrm{\prime }}-\theta )$ for the outcome $+1$. If the outcome is discarded, the state is $\mathrm{\Phi }(\rho )=P_{\theta }\rho P_{\theta }+P_{\theta }^{\perp }\rho P_{\theta }^{\perp }$;

(f) the assertions (a)-(e) are conditional consequences of P1, P2, P3 and the adopted model. They assert no dynamics, no laboratory polariser, no detector efficiency, no loss, no sampling model and no empirical result; in particular the fraction ${\cos }^2(\theta -\phi )$ is a model probability of the declared ideal analyser, not a measured count.

• The same number classically. The number ${\cos }^2(\theta -\phi )$ is also the intensity ratio of the declared classical transverse-field model of Malus and Jones ratios of the declared analyser: the same cos^2 from the Born law and from a classical transverse amplitude ↗; the comparison shows that this ratio alone does not distinguish the two declared models.
• Absorption. For a laboratory absorber the "blocked" outcome of the declared analyser is used only through its discarded-outcome law: clause (v) of P3 asserts no post-absorption one-photon state, and no such state is used here.