Unsharp Measurements

An unsharp measurement is a measurement whose outcomes reveal partial, noisy, or finite-resolution information rather than a sharp projective alternative. In operator language, its effects are positive operators that are not generally projectors.

The basic idea is:

$$\text{sharp measurement} \quad \longrightarrow \quad \text{orthogonal projectors,}$$

whereas

$$\text{unsharp measurement} \quad \longrightarrow \quad \text{positive effects with intermediate eigenvalues.}$$

Unsharpness is a property of the probability question. The associated disturbance is a property of the instrument. The same unsharp POVM can be implemented gently, destructively, or with extra detector kicks.

From Projectors to Effects

For a sharp two-outcome projective measurement, the effects are projectors:

$$E_+=P_+, \qquad E_-=P_-, \qquad P_+ + P_- = I.$$

Each effect has eigenvalues only $0$ or $1$. If the system is in the $P_+$ subspace, outcome $+$ occurs with certainty; if it is in the $P_-$ subspace, outcome $+$ is impossible.

For an unsharp two-outcome measurement, the effects still satisfy

$$E_\pm\ge0, \qquad E_+ + E_- = I,$$

but their eigenvalues can lie between $0$ and $1$. The outcome is informative without being perfectly discriminating.

This is the finite-outcome POVM structure developed in POVMs.

Qubit Unsharp Spin Measurement

The standard qubit example is an unsharp measurement of $\sigma_z$. Let

$$P_+=\frac{1}{2}(I+\sigma_z), \qquad P_-=\frac{1}{2}(I-\sigma_z).$$

An unsharp version has effects

$$E_\pm = \frac{1}{2} \left( I\pm\eta\sigma_z \right), \qquad 0\le\eta\le1.$$

Equivalently,

$$E_+ = \frac{1+\eta}{2}P_+ + \frac{1-\eta}{2}P_-,$$

and

$$E_- = \frac{1-\eta}{2}P_+ + \frac{1+\eta}{2}P_-.$$

The parameter $\eta$ is a measurement-strength or sharpness parameter in this model:

• $\eta=1$ gives the sharp projective measurement of $\sigma_z$;
• $\eta=0$ gives $E_+=E_-=I/2$, so the outcome is a fair label independent of the state;
• intermediate $\eta$ gives partial information about the sign of $\sigma_z$.

Outcome Probabilities

Write a qubit state in Bloch form:

$$\rho = \frac{1}{2} \left( I+\mathbf r\cdot\boldsymbol\sigma \right).$$

Then

$$p(\pm) = \operatorname{Tr}(\rho E_\pm) = \frac{1}{2} \left( 1\pm\eta r_z \right).$$

The probabilities depend only on the $z$ component of the Bloch vector, and the contrast is reduced by $\eta$. The measurement does not distinguish $\sigma_z$ eigenstates perfectly unless $\eta=1$.

For an input eigenstate of $\sigma_z$,

$$p(+|+z)=\frac{1+\eta}{2}, \qquad p(+|-z)=\frac{1-\eta}{2}.$$

Thus $\eta$ controls the single-shot distinguishability of the two eigenstates.

Square-Root Instrument

A common minimally disturbing realization of the unsharp effects uses one measurement operator per outcome:

$$M_\pm=\sqrt{E_\pm}.$$

For the qubit effects above,

$$M_+ = \sqrt{\frac{1+\eta}{2}}\,P_+ + \sqrt{\frac{1-\eta}{2}}\,P_-,$$

and

$$M_- = \sqrt{\frac{1-\eta}{2}}\,P_+ + \sqrt{\frac{1+\eta}{2}}\,P_-.$$

These satisfy

$$M_\pm^\dagger M_\pm=E_\pm, \qquad M_+^\dagger M_+ + M_-^\dagger M_- = I.$$

The selective update is

$$\rho_\pm = \frac{M_\pm\rho M_\pm^\dagger} {\operatorname{Tr}(E_\pm\rho)}.$$

This instrument is diagonal in the $\sigma_z$ basis. It partially filters the state toward the outcome-favored eigenspace without projecting all the way in a single shot.

Nonselective Backaction

If the outcome is ignored, the square-root instrument gives the channel

$$\Phi_\eta(\rho) = M_+\rho M_+^\dagger + M_-\rho M_-^\dagger.$$

For

$$\rho = \begin{pmatrix} \rho_{00} & \rho_{01}\\ \rho_{10} & \rho_{11} \end{pmatrix}$$

in the $\sigma_z$ basis, this channel acts as

$$\Phi_\eta(\rho) = \begin{pmatrix} \rho_{00} & \sqrt{1-\eta^2}\,\rho_{01}\\ \sqrt{1-\eta^2}\,\rho_{10} & \rho_{11} \end{pmatrix}.$$

The populations are preserved, while coherence is reduced by $\sqrt{1-\eta^2}$. The sharp limit $\eta=1$ gives complete dephasing in the measured basis. The no-information limit $\eta=0$ gives the identity channel for this particular instrument.

This illustrates a basic information-disturbance tradeoff for this model: stronger single-shot information about $\sigma_z$ comes with stronger dephasing of superpositions of $\sigma_z$ eigenstates.

Same POVM, Different Instrument

The unsharp effects do not determine the backaction. The same $E_\pm$ can also arise from a sharp projective measurement followed by classical mislabeling.

For example, define

$$\mathcal I_+^{\mathrm{label}}(\rho) = \frac{1+\eta}{2}P_+\rho P_+ + \frac{1-\eta}{2}P_-\rho P_-,$$

and

$$\mathcal I_-^{\mathrm{label}}(\rho) = \frac{1-\eta}{2}P_+\rho P_+ + \frac{1+\eta}{2}P_-\rho P_-.$$

This instrument has the same effects $E_\pm$, because the probabilities are the same. But its nonselective channel is

$$\Phi_{\mathrm{label}}(\rho) = P_+\rho P_+ + P_-\rho P_-,$$

which fully dephases the $\sigma_z$ basis for every value of $\eta$. Even when $\eta$ is small and the reported label is almost random, the system was sharply measured before the label noise was applied.

Therefore “unsharp POVM” does not by itself mean “gentle measurement.” Gentleness is a property of the instrument.

Unsharp Versus Weak

The terms “unsharp” and “weak” overlap but are not identical.

An unsharp measurement has effects that do not perfectly distinguish the sharp alternatives. A weak measurement is usually a measurement regime in which each individual interaction gains little information and causes small disturbance, often as a step toward a repeated or continuous measurement.

The square-root instrument with small $\eta$ is weak in a common operational sense: each outcome only slightly biases the state and the nonselective dephasing is small:

$$\sqrt{1-\eta^2} \approx 1-\frac{\eta^2}{2} \qquad (\eta\ll1).$$

But a noisy-label instrument with the same small $\eta$ can be highly disturbing because it first performs a sharp measurement. Thus weak measurement requires assumptions about the instrument, not only the effects.

Finite-Resolution Pointers

Unsharpness also appears in continuous pointer models. In the von Neumann measurement model, different eigenvalues of $A$ shift a pointer wavepacket by different amounts. If the shifted packets overlap substantially, the pointer readout cannot distinguish the alternatives sharply.

A finite-resolution readout can be represented by effects such as

$$E_y = \int dx\,R(y|x)\,|x\rangle\langle x|,$$

where $R(y|x)$ is a detector response function. If $R(y|x)$ is broad, the measurement reports only a smeared version of the sharp position question.

The full derivation belongs to the apparatus model in Von Neumann Measurement Model. The lesson here is that finite pointer separation and finite detector resolution naturally produce POVMs.

Sequential Consequences

Unsharp measurements are useful in sequential experiments because they can extract partial information while leaving some coherence for later tests. For the square-root qubit instrument, a later $\sigma_x$ measurement after an unread unsharp $\sigma_z$ measurement has its contrast reduced by the factor $\sqrt{1-\eta^2}$ rather than erased completely.

This makes unsharp measurements central in:

• tests that balance information gain against invasiveness;
• continuous monitoring as a limit of many small updates;
• feedback protocols where the record is noisy but useful;
• measurement tomography, where one estimates detector effects and instruments;
• weak-value experiments, where weak coupling is followed by postselection.

The ordered-probability framework is Compatible, Incompatible, and Sequential Measurements.

Common Mistakes

Equating unsharp with harmless

An unsharp POVM can be implemented by a highly disturbing instrument. The effects alone do not determine the backaction.

Treating the sharpness parameter as universal

The symbol $\eta$ in the qubit example is a convenient model parameter. Other detectors use different calibration conventions, and the same number may not compare measurement strength across platforms.

Confusing readout noise with weak interaction

Noisy labels can make a POVM unsharp even if the system was sharply measured before the label was corrupted.

Assuming no information means no disturbance

For the square-root instrument at $\eta=0$, the channel is the identity. But another instrument with effects $I/2$ and $I/2$ could still disturb the system while reporting random labels.

Applying projective repeatability

Unsharp measurements are generally not repeatable in the ideal projective sense. Repeating them accumulates information stochastically rather than producing a guaranteed same outcome after one shot.

Exercises

Positivity and Completeness

Show that

$$E_\pm = \frac{1}{2} \left( I\pm\eta\sigma_z \right)$$

defines a POVM for $0\le\eta\le1$.

Solution

The effects sum to

$$E_+ + E_- = \frac{1}{2}(I+\eta\sigma_z) + \frac{1}{2}(I-\eta\sigma_z) =I.$$

In the $\sigma_z$ basis, the eigenvalues of $E_+$ are $(1+\eta)/2$ and $(1-\eta)/2$, and the eigenvalues of $E_-$ are the same pair in the opposite order. These are nonnegative for $0\le\eta\le1$. Therefore the effects are positive and complete.

Bloch-Vector Probabilities

For

$$\rho = \frac{1}{2} \left( I+\mathbf r\cdot\boldsymbol\sigma \right),$$

derive $p(\pm)=(1\pm\eta r_z)/2$.

Solution

Use $\operatorname{Tr}\rho=1$, $\operatorname{Tr}(\rho\sigma_z)=r_z$, and

$$E_\pm = \frac{1}{2} \left( I\pm\eta\sigma_z \right).$$

Then

$$p(\pm) = \operatorname{Tr}(\rho E_\pm) = \frac{1}{2} \left[ \operatorname{Tr}\rho \pm \eta\operatorname{Tr}(\rho\sigma_z) \right] = \frac{1}{2}(1\pm\eta r_z).$$

Nonselective Dephasing Factor

Using the square-root operators $M_\pm=\sqrt{E_\pm}$, show that the off-diagonal matrix element $\rho_{01}$ is multiplied by $\sqrt{1-\eta^2}$.

Solution

Write

$$M_+ = aP_+ + bP_-, \qquad M_- = bP_+ + aP_-,$$

where

$$a=\sqrt{\frac{1+\eta}{2}}, \qquad b=\sqrt{\frac{1-\eta}{2}}.$$

The off-diagonal element in $M_+\rho M_+^\dagger$ is $ab\rho_{01}$, and the off-diagonal element in $M_-\rho M_-^\dagger$ is also $ab\rho_{01}$. Therefore the total factor is

$$2ab = 2\sqrt{ \frac{1+\eta}{2} \frac{1-\eta}{2} } = \sqrt{1-\eta^2}.$$

Same Effects, Different Disturbance

Compare the square-root instrument with the noisy-label instrument for the same effects $E_\pm$. What happens to $\rho_{01}$ when the outcome is ignored?

Solution

For the square-root instrument, the unread channel maps

$$\rho_{01} \longmapsto \sqrt{1-\eta^2}\,\rho_{01}.$$

For the noisy-label instrument, the unread channel is

$$\rho \longmapsto P_+\rho P_+ + P_-\rho P_-,$$

so

$$\rho_{01}\longmapsto0.$$

The effects, and therefore the outcome probabilities, are the same. The disturbance differs because the instruments are different.

Cross-Links

• Why Generalized Measurements Are Needed
• POVMs
• Kraus Operators
• Quantum Instruments
• Weak Measurements
• Measurement Backaction
• Von Neumann Measurement Model
• Compatible, Incompatible, and Sequential Measurements
• Stochastic Master Equations
• Diffusive Trajectories
• Formula Sheet

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• P. Busch, P. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, 2nd ed., Springer (1996).
• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer (2016).
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic (1995).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).