Protective Measurement

Protective measurement couples a meter weakly and slowly to a system while suppressing transitions out of a protected state. In an ideal adiabatic limit, the meter responds to an expectation value in that state. The canonical Protective Measurements page develops the measurement dynamics; the question here is how much the protocol establishes about the physical meaning of a quantum state.

Protection supplies a physical resource

Let $\lvert n\rangle$ be a nondegenerate eigenstate of $H_0$, separated by a nonzero gap. Couple an observable $A$ to a pointer momentum $P$ through

$$H_{\rm int}(t)=g_T(t)A\otimes P,\qquad \int_0^T g_T(t)\,dt=1.$$

For a sufficiently smooth interaction with $g_T$ of order $1/T$, a bounded relevant pointer-momentum range, and suitable adiabatic conditions, the leading effect is

$$\lvert n\rangle\lvert\varphi\rangle \longmapsto e^{-iE_nT/\hbar}\lvert n\rangle e^{-i\langle A\rangle_n P/\hbar}\lvert\varphi\rangle.$$

The pointer translation encodes $\langle A\rangle_n$, with corrections from finite duration, transitions, and meter dynamics. The gap, state preparation, protecting Hamiltonian, coupling, and meter resolution are all resources. A finite noisy pointer reading is not an exact real number.

A protected spin example

Take $H_0=-(\Delta/2)\,\boldsymbol n\cdot\boldsymbol\sigma$, with $\Delta>0$, and prepare its ground state. Protective coupling to $Z$ shifts the pointer by $n_z$, although a strong projective $Z$ measurement has outcomes only $\pm1$. Repeating appropriate protected measurements of $X,Y,Z$ could determine the Bloch vector to finite accuracy if the assumptions continue to hold.

This differs from applying one fixed apparatus to an arbitrary unknown qubit with no promise about its preparation. The protecting Hamiltonian already singles out a direction $\boldsymbol n$. Changing the state generally requires changing the physical protection or knowing that the state belongs to a promised protected family.

Why distinguishability is not violated

Suppose one universal unitary could perfectly preserve any unknown state while writing its distinguishing information into a meter:

$$U\lvert\psi\rangle\lvert M\rangle=\lvert\psi\rangle\lvert M_\psi\rangle.$$

Inner-product preservation for two inputs implies

$$\langle\psi|\phi\rangle =\langle\psi|\phi\rangle\langle M_\psi|M_\phi\rangle.$$

For nonorthogonal inputs, the meter states must have overlap one. Thus a fixed nondisturbing universal device cannot perfectly distinguish all unknown states. Protective protocols avoid that premise through the protection promise, apparatus dependence, approximate limits, or restricted state set.

The proposal by Aharonov and Vaidman motivated arguments for wavefunction reality. The operational result is narrower: expectation values can be encoded by a particular protected interaction. Whether that requires an ontic wavefunction depends on a complete account of the apparatus and protection, and is distinct from the assumptions in the PBR theorem.

Exercise

For the protected spin above, set $\boldsymbol n=(\sqrt3/2,0,1/2)$. Compare the protective $Z$ shift with projective $Z$ statistics.

Solution

The protective shift is $1/2$ in the chosen coupling units. A projective measurement returns $+1$ with probability $3/4$ and $-1$ with probability $1/4$, giving the same mean. The equality of means does not make the procedures equivalent: their disturbances, resources, and single-trial pointer distributions differ.

References

• Aharonov, Y., Anandan, J., and Vaidman, L., “Meaning of the wave function,” Physical Review A 47, 4616–4626 (1993), doi:10.1103/PhysRevA.47.4616 — protocol and interpretive argument.
• Aharonov, Y., and Vaidman, L., “Measurement of the Schrödinger wave of a single particle,” Physics Letters A 178, 38–42 (1993), arXiv:hep-th/9304147 — original protective-measurement proposal.