Three ideal analysers in series: the quarter-intensity model computation

Adopt the following finite polarization assignment and measurement rules as additional model assumptions for this example, using the declared-instrument vocabulary of Declared ideal measurements and instruments of the framework. These are stipulated for the finite polarization space, beyond the massive-matter sector of P1; they are not consequences of its massive-particle assignment. Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the mathematical suppliers. Fix a propagation label $k_0\in {\mathbb{R}}^3\setminus \{0\}$ in ${\mathrm{m}}^{-1}$ relative to fixed Euclidean spatial coordinates, and a real orthonormal transverse pair $v_x,v_y\in {\mathbb{R}}^3$ with $k_0\cdot v_x=k_0\cdot v_y=0$. Identify $H={\mathbb{C}}^2$, with its standard inner product and orthonormal basis $(e_x,e_y)$, unitarily with $\mathbb{C}{v}_x\oplus \mathbb{C}{v}_y$ by $e_x\mapsto v_x$, $e_y\mapsto v_y$. Its state space is $D(H)$ and a unit ray $[\psi ]$ represents $T_{\psi ,\psi }$ for $\mathrm{\parallel }\psi \mathrm{\parallel }=1$ (The state space: density operators, pure and mixed states, and purity, Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space). The identification of these unit rays as polarization preparations is the adopted finite assignment, motivated by the source account in Binney and Skinner, Section 1.3.6, printed pages 14-15. It supplies no photon number or radiation dynamics. Angles are dimensionless, read in radians. For a real angle $\alpha$ put $e_{\alpha }:=\cos \alpha e_x+\sin \alpha e_y,e_{\alpha }^{\perp }:=-\sin \alpha e_x+\cos \alpha e_y\mathrm{.}$ The ideal analyser $A(\alpha )$ is the declared ideal sharp measurement with the two outcomes transmit and block, whose projections are $T_{e_{\alpha },e_{\alpha }}$ and $T_{e_{\alpha }^{\perp },e_{\alpha }^{\perp }}$ respectively, with outcome and update rules stipulated in [F2] below (Declared ideal measurements and instruments of the framework clause (c), Rank-one operator calculus).

The example takes the preparation $[e_x]$, the chain of analysers $A(0)$ with $\alpha =0^{\circ }$, $A(\pi \mathrm{/}4)$ with $\alpha ={45}^{\circ }$ and $A(\pi \mathrm{/}2)$ with $\alpha ={90}^{\circ }$, and computes:

(i) The analyser $A(\alpha )$ transmits the preparation $[e_{\beta }]$ with probability $\mathbb{P}(\text{transmit})=\mid ⟨e_{\alpha },e_{\beta }⟩{\mid }^2={\cos }^2(\alpha -\beta ),$ and, when ${\cos }^2(\alpha -\beta )>0$, after transmission the conditional state is the pure state of $e_{\alpha }$.

(ii) Consequently the chain $0^{\circ },{45}^{\circ },{90}^{\circ }$ transmits with joint probability ${\cos }^2(0-0)\cdot {\cos }^2(\frac{\pi }{4}-0)\cdot {\cos }^2(\frac{\pi }{2}-\frac{\pi }{4})=1\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4},$ whereas the chain $0^{\circ },{90}^{\circ }$ alone transmits with probability ${\cos }^2(\pi \mathrm{/}2)=0$.

(iii) Removing the middle analyser therefore changes the transmission of the chain from $\frac{1}{4}$ to $0$ within the declared model: the intervening declared measurement changes the state, and the later analyser sees the state selected by the earlier one, not the original preparation.

(iv) Model comparison only. A declared classical amplitude model for a transverse field in the same plane, in which the analyser transmits the component along $e_{\alpha }$ and the transmitted intensity is stipulated to be proportional to the squared modulus of that component, gives the same ratio ${\cos }^2(\alpha -\beta )$ for a preparation along $e_{\beta }$. The agreement of these two model computations does not establish that either describes nature, and it does not by itself establish field quantization (What the worked instruments of the companion page establish); no classical Maxwell theorem is invoked, and the field model is a stipulated finite-dimensional comparison. No transmission counts, detector efficiencies, backgrounds or sampling statistics are reported.

• Idealisation. Each analyser is represented by the declared ideal sharp measurement of clause (c) of Declared ideal measurements and instruments of the framework: the blocked component is not available for later experiments and the transmitted component leaves the analyser in the corresponding pure state. A physical Polaroid or polarizing beam splitter has its own absorption, reflection, efficiency and background behaviour, which this example does not model.
• No radiation model. The two-level space rather than a quantized radiation mode is the declared model; no photon number, mode function, field quantization or Maxwell dynamics is asserted, and no position or arrival-time information is described.
• Sampling. The numbers $1,\frac{1}{2},\frac{1}{4}$ are single-trial model probabilities. To describe a run of $N\ge 1$ trials one must additionally specify a joint sampling law. For example, independent Bernoulli variables $X_1,\dots ,X_N\in \{0,1\}$ with $\mathbb{P}(X_j=1)=q$ encode identical transmit trials at a stated probability $q$; their frequency is $f_N=N^{-1}{\sum }_j{X}_j$. Even for this sampling model a particular finite frequency need not be close to $q$: at $q=1\mathrm{/}4$ the all-blocked and all-transmitted runs have probabilities $(3\mathrm{/}4{)}^N>0$ and $(1\mathrm{/}4{)}^N>0$. Identical single-trial probabilities alone specify no joint sampling law. No trial data or inference is reported here.