Sign-sensitive measurements that resolve a covariance-axis ambiguity ($\log \Lambda =2mz/s^2$)

Sign-sensitive measurements that resolve a covariance-axis ambiguity add an observable whose distribution changes when the fitted axis direction is reversed. In the global orientation ambiguity in a rank-one covariance fit, $\mathbf{u}$ and $-\mathbf{u}$ give the same covariance. An independently calibrated signed projection can distinguish them.

Consider two candidates with an additional observation $z=\pm m+\eta$, where $m=a{\mathbf{w}}^T\mathbf{u}\ne 0$ is known, $a$ has known positive sign and $\eta \sim N(0,s^2)$. Assume this calibration reading is independent of the covariance data conditional on the candidate. The Gaussian densities giveThe normal density and cumulative distribution function used here are documented in the pinned numerical reference.^[ref] The likelihood-ratio expression follows by expanding the two squared residuals.

For an original example, let $m=0.20\mathrm{V}$, $s=0.050\mathrm{V}$ and observe $z=0.10\mathrm{V}$. The log likelihood ratio is 16, strongly favoring the positive candidate under this stated model. With equal prior probabilities and equal decision costs, the sign-threshold decision has repeated-measurement error probability $\Phi (-|m|/s)=\Phi (-4)\approx 3.17\times 10^{-5}$. This decision-error probability is different from assigning a posterior error probability to that one observed value.

A practical calibration applies a known signed perturbation and records the signed detector-response component. Reversing the applied perturbation checks readout polarity. A calibrated phase-sensitive readout can provide such a projection; measuring only noise covariance does not supply its sign.

The failure of sign recovery from magnitude-only calibration data is important. For symmetric Gaussian noise, $|z|$ has the same distribution under means $+m$ and $-m$. If $\mathbf{w}$ is orthogonal to the axis, then $m=0$ and even the signed reading adds no separation. If the amplitude sign itself is unknown, reversing both amplitude and axis preserves the mean. The additional measurement resolves the ambiguity only when its reference sign and response model are independently established.

Failure of sign recovery from magnitude-only calibration data occurs when opposite signed calibration models induce exactly the same distribution of every recorded magnitude. Under this symmetry, more repetitions can estimate the magnitude distribution accurately without deciding the missing sign. This is the limitation relevant to sign-sensitive measurements that resolve a covariance-axis ambiguity.

Let $Z=sm+ϵ$ with sign $s\in \{-1,+1\}$, known $m>0$ and symmetric noise $ϵ$ having density $f(x)=f(-x)$. If the recorded value is $Y=|Z|$, then for $y>0$Symmetry gives $g(y\mid +1)=f(y-m)+f(y+m)=g(y\mid -1)$. Independent repetitions multiply identical densities, so their likelihood ratio remains one. With equal sign priors and equal decision costs, no decision rule based solely on these magnitudes can have average sign-error probability below1/2.

For normal distribution noise with standard deviation $\sigma$, the magnitude has a folded normal distribution:where $\varphi$ is the standard normal density. The formula depends on $|m|$, not its sign. The pinned numerical reference provides the equivalent standardized folded-normal density and its scale transformation.^[ref]

For the original calibration scale $m=0.20\mathrm{V}$ and $\sigma =0.050\mathrm{V}$, recording a signed value can distinguish the two models strongly, but recording only its magnitude cannot do so under the symmetric model. This is an information loss in the measurement map, rather than a shortage of averaging trials.

The symmetry assumption is essential. As an original counterexample, let noise equal−1 with probability2/3 and+2 with probability1/3, giving zero mean but an asymmetric distribution. With $m=1$, the positive-sign magnitudes are0 or3, while the negative-sign magnitudes are2 or1. Their disjoint supports permit sign recovery if this asymmetric noise model is independently established. The calibrated asymmetry itself supplies sign-reference information absent from the symmetric case.

Similarly, a genuinely signed projection or a known phase reference can break the original ambiguity. An unverified readout polarity or unknown amplitude sign can preserve it. Specify which signed information the apparatus actually records before interpreting a magnitude fit as an oriented calibration.