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Any CPTP map can be understood in terms of a unitary transformation acting in a Hilbert space larger than the Hilbert space H1 associated with the relevant statistical operator. In particular, for a state ρ ∈ H1 in the larger space H1 ⊗ H2 , there is a state ρ ∈ H2 such that ρ = tr2 (ρ ⊗ ρ ) . 32) for some unitary operator, U [261]. Some other examples of CPTP maps are the nonselective measurement ρ → i P (|ψi )ρP (|ψi ) and the composition of quantum systems ρ → ρ ⊗ ρ , such as when an ancilla is added to a system.

Purposes, it is also convenient that this provides a (dual rail) realization of a qubit, as it makes use of two quantum field modes to represent a single qubit. The apparatus in Figs. 4–5 represent two such mutually exclusive experimental arrangements for a single qubit. The class of experiments interpolating between these is not shown here, but corresponds to a single wing of the apparatus shown in Fig. 2 in Chapter 3. 39 Two distinct kinds of measurement apparatus can be placed away from the beam-splitter, allowing measurements of spatial path and interference to be made.

In one apparatus, the features of the pattern of interference between beams A and A that are allowed to merge, as in Fig. 5, enable measurement of the visibility, v, of the interference pattern arising ¯ a variable phaseshifter introduces a phase shift similarly to the way given P; that different path-lengths from diaphragm slits to a given point on the detection screen do in the double-slit experiment of Fig. 3. The other apparatus consists of detectors placed in beams A and A before they reach a common point, and enables measurement of the distinguishability, D, of path, as shown in Fig.