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41) Now consider the broader class of preparations, the correlated preparations, consisting of situations where measurements of correlated ancillae may be useful for predicting the path. ” Let |Θ ∈ HI ⊗ HII be entangled, that is, not factorable into a state-vector in HI and a state-vector in HII ; for a discussion of entanglement, see Chapter 6. 41 An 41 See Sect. 5 for a discussion of reduced statistical operators and the partial-trace operation. 6 The Mach–Zehnder interferometer 23 ensemble formed by mixing several pure correlated cases, each with a distinct |Θk in HI ⊗HII with respective proportion wk , is the case of mixed correlated preparations.

3 Expectation values and the von Neumann projection 35 V 1 H 2 H Fig. 2. Examples of apparatus performing first kind and second kind measurements, respectively, according to the Pauli classification—light polarization: (1) refracted by a Rochon prism, and (2) selectively absorbed using a dichroic polarization analyzer. photons are destroyed is another example of a second-kind measurement (cf. Footnote 1). Pauli and von Neumann both emphasized that, when the interaction between a measurement apparatus and a system being measured is analyzed, the linearity of the Schr¨ odinger equation describing the state evolution of composite system formed by these two (sub)systems provides consistency between alternative descriptions of system behavior in which the division (or “cut”) between measuring system and measured system is made differently.

26–28). 37 Nonunitary transformations of qubit states have been thoroughly investigated as well, including those involved in the decoherence process; for example, see [207]. 37 For a thorough review of the practical creation, characterization, and manipulation of single qubits in optics, see [333]. 38 Consider the doubleslit diaphragm and opaque-screen detector arrangement shown in Fig. 3. Take a1 (x) to be the (complex) quantum probability amplitude corresponding (via retrodiction, cf. the introduction to Ch.