Non-representative Quantum Mechanical Weak Values

Abstract

The operational definition of a weak value for a quantum mechanical system involves the limit of the weak measurement strength tending to zero. I study how this limit compares to the situation for the undisturbed (no weak measurement) system. Under certain conditions, which I investigate, this limit is discontinuous in the sense that it does not merge smoothly to the Hilbert space description of the undisturbed system. Hence, in these discontinuous cases, the weak value does not represent the undisturbed system. As a result, conclusions drawn from such weak values regarding the properties of the studied system cannot be upheld. Examples are given.

Appendices

I am grateful to E. Cohen for making me aware of the need for a special treatment of this case.

Appendix 1 Comments on the Case $\varvec{S}\vert$ in $\rangle$ = 0

I am grateful to E. Cohen for making me aware of the need for a special treatment of this case.

The criterion $\langle$ in $\vert \mathcal {S}\vert$ in $\rangle$ = 0 is obeyed if $\mathcal {S} \vert$ in $\rangle$ = 0, i.e. if the operator $\mathcal {S}$, whose weak value is under investigation, projects the preselected state $\vert$ in $\rangle$ onto the null Hilbert space. Trivially, in this case the source term in (5) is zero, and so is therefore the weak value. Whether or not, in the terminology introduced above, this should be characterized as a “derailment”—a projection away from the Hilbert space ray $\vert$ in $\rangle$—cannot, I think, be decided on rational grounds. This is so because the null state is contained in any proper Hilbert subspace but is also orthogonal to all states. So whether a projection onto the null state should be said to project the state $\vert$ in $\rangle$ out of its original ray or not is more a matter of judgment. This situation is related to the maybe futile distinction between having a vanishing weak value representing the undisturbed system or not having a “well-behaved” weak value at all.

More to elucidate the question than to try to resolve it, let me here only consider one special case which has some application, e.g., to the example treated in [16].

Consider a Hilbert space of at least three dimensions and suppose that the state $\vert$ in $\rangle$ has the expansion

$$\begin{aligned} \left| {in\rangle } = \right| a\rangle + \vert b\rangle \end{aligned}$$

(21)

in terms of orthogonal (but not necessarily normalized) state vectors $\vert a \rangle$ and $\vert b \rangle$. Further, let $\mathcal {S}$ be the projector

$$\begin{aligned} \mathcal {S}=\Pi _d = \left| {d\rangle \langle d} \right| {/}\langle d\vert d\rangle \end{aligned}$$

(22)

onto a state $\vert d \rangle$ orthogonal to both $\vert a \rangle$ and $\vert b \rangle .$ Then, trivially,

$$\begin{aligned} \mathcal {S}\vert in\rangle =0. \end{aligned}$$

(23)

Although, as I said, the matter is up for discussion, I judge it appropriate, from the way this example is constructed—that the state $\vert d \rangle$ is orthogonal to $\vert$ in $\rangle$ of (A1)—to say that it is here a question of “derailment” out of the ray (A1) of the undisturbed system.

Appendix 2 Convention for Transition in a Beamsplitter

In the notation of Fig.1, the transition in the first (well-balanced, 50–50) beamsplitter is in my convention described by the unitary transition

$$\begin{aligned} \vert N\rangle \rightarrow U(BS1)\vert N \rangle = ( {\left| {L\rangle +i}\right| R}) /\surd 2, \end{aligned}$$

(24)

with analogous expressions for any other similar beamsplitting process considered in this paper.