Quantum weak and modular values in enlarged Hilbert spaces

We introduce an enlarged state, which combines both pre- and postselection states at a given time $t$ in between the pre- and postselection. Based on this form, quantum weak and modular values can be completely interpreted as expectation values of a linear combination of given operators in the enlarged Hilbert space. This formalism thus enables us to describe and measure the weak and modular values at any time dynamically. A protocol for implementing an enlarged Hamiltonian has also been proposed and applied to a simple example of a single spin under an external magnetic field. In addition, the time-dependent weak and modular values for pre- and postselection density matrices mapping onto an enlarged density matrix are also discussed.

Figure 1

Weak values of ${\mathbf{\sigma }}_x$ for the preselection state in (17) and various postselection states as in (18), which are the same in both cases of normal Hilbert space and enlarged Hilbert space.

Figure 2

A graphical illustration of the enlarged density matrix on the Bloch spheres. This representation is equivalent to the Majorana geometric representation in a sphere.

Figure 3

A scheme of gates on both the extra spin ($e$) and the system single-particle spin ($s$).