Projective Measurements

A projective measurement is the ideal sharp measurement model in which mutually exclusive outcomes are represented by mutually orthogonal projectors. It is the cleanest bridge between observables, the Born rule, state update, and the channel obtained when the outcome is ignored.

A projection-valued measure specifies sharp outcome probabilities. To predict the final state, one must also specify an instrument. Here we use the ideal Lüders instrument, which preserves states inside each outcome subspace, and compare it with other implementations. Except in the continuous-variable section, the examples are finite dimensional.

Helpful background. Projectors describe orthogonal subspaces, and the basic projective measurement rule supplies the Born probabilities.

Projection-Valued Measure

For a discrete outcome set, a projective measurement is specified by a family of projectors $\{P_a\}$ satisfying

$$P_a^\dagger=P_a, \qquad P_aP_b=\delta_{ab}P_a, \qquad \sum_a P_a=I.$$

This family is also called a projection-valued measure, or PVM. The label $a$ denotes the reported outcome, not necessarily a unique eigenvector. If the projective measurement is associated with an observable $A$, then the spectral form is

$$A=\sum_a aP_a,$$

where the sum is over distinct eigenvalues. The projectors are the operational measurement data. The numerical eigenvalues $a$ label the readout. The projectors fix the probabilities; they fix the update only after the Lüders implementation is specified.

Two observables can define the same projective measurement if they have the same spectral projectors but different relabelings of the outcomes. Conversely, two devices with the same eigenvalue labels but different projectors are different measurements.

Outcome Probabilities

For a pure input state $|\psi\rangle$, the probability of outcome $a$ is

$$p(a)=\langle\psi|P_a|\psi\rangle = \|P_a|\psi\rangle\|^2.$$

For a density operator $\rho$, the same rule is

$$p(a)=\operatorname{Tr}(P_a\rho) = \operatorname{Tr}(\rho P_a).$$

The equality of the two trace forms follows from cyclicity of the trace. Positivity of $P_a$ makes $p(a)\ge0$, and completeness gives

$$\sum_a p(a) = \operatorname{Tr}\!\left(\rho\sum_aP_a\right) = \operatorname{Tr}\rho =1.$$

A sum of orthogonal projectors is still a projector: grouping exact outcomes into disjoint bins can therefore remain a PVM. Detector noise, false counts, or overlapping response functions generally produce nonprojective POVM effects. Finite resolution alone does not determine which model is appropriate.

Selective Update as an Instrument

If outcome $a$ is obtained and $p(a)\ne0$, the ideal selective Lüders update is

$$\rho \longmapsto \rho_a = \frac{P_a\rho P_a} {\operatorname{Tr}(P_a\rho)}.$$

Before normalization, the outcome-resolved operation is

$$\mathcal I_a(\rho)=P_a\rho P_a.$$

This map is completely positive and trace nonincreasing. Its trace is the outcome probability:

$$\operatorname{Tr}\mathcal I_a(\rho) = \operatorname{Tr}(P_a\rho P_a) = \operatorname{Tr}(P_a\rho) = p(a).$$

Thus the Lüders implementation of a PVM is an instrument: each outcome has both a probability and a conditional output state. In generalized measurement theory, this is the special case with one measurement operator $M_a=P_a$ for each outcome.

Nonselective Measurement

If the measurement is performed but the outcome is not retained, the final state is the outcome average:

$$\mathcal M(\rho) = \sum_a P_a\rho P_a.$$

This is a quantum channel. It is trace preserving because

$$\operatorname{Tr}\mathcal M(\rho) = \sum_a\operatorname{Tr}(P_a\rho P_a) = \operatorname{Tr}\rho.$$

The map removes coherence between different outcome subspaces:

$$\rho = \sum_{a,b}P_a\rho P_b \quad \longmapsto \quad \mathcal M(\rho) = \sum_a P_a\rho P_a.$$

Only the diagonal blocks in the measurement decomposition remain. This is why an ignored projective measurement is a dephasing channel in the measured decomposition.

The channel is idempotent:

$$\mathcal M(\mathcal M(\rho))=\mathcal M(\rho).$$

After the coherences between distinct outcome sectors have been removed once, repeating the same nonselective ideal measurement changes nothing further.

Qubit Example: Measuring σz

Let the measurement be the ideal measurement of $\sigma_z$ with outcomes $+$ and $-$:

$$P_+=|0\rangle\langle0|, \qquad P_-=|1\rangle\langle1|.$$

For

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle, \qquad |\alpha|^2+|\beta|^2=1,$$

the outcome probabilities are

$$p(+)=|\alpha|^2, \qquad p(-)=|\beta|^2.$$

If $+$ occurs, the selective state is $|0\rangle$. If $-$ occurs, it is $|1\rangle$.

For a general qubit density matrix in the same basis,

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

the nonselective measurement gives

$$\mathcal M(\rho) = \begin{pmatrix} \rho_{00} & 0\\ 0 & \rho_{11} \end{pmatrix}.$$

The system alone has lost its off-diagonal terms because different alternatives have become correlated with distinguishable records. The phase can still be present in the joint system–apparatus state.

This is dephasing in the $z$ basis. Whether it changes the mean energy depends on the Hamiltonian: it preserves energy for every input if $[H,P_\pm]=0$. For example, it preserves $\langle H\rangle$ when $H=\hbar\omega\sigma_z/2$. With $H=\hbar\omega\sigma_x/2$, however, the input $|+\rangle$ has energy $\hbar\omega/2$ and its unread output $I/2$ has energy zero. The measuring interaction then exchanges energy with the system; dephasing in an arbitrary basis is not automatically energy conserving.

Energy Measurements

For a Hamiltonian with a nondegenerate discrete spectrum,

$$H=\sum_n E_n|n\rangle\langle n|,$$

an ideal energy measurement has projectors

$$P_n=|n\rangle\langle n|.$$

For an input state

$$|\psi\rangle=\sum_n c_n|n\rangle,$$

the probability of observing $E_n$ is $|c_n|^2$, and the selective post-measurement state is $|n\rangle$.

This textbook statement hides a physical assumption: the apparatus must be able to resolve the energy eigenvalues sharply enough and must implement an approximately projective coupling to energy. Finite-time, finite-resolution, or destructive energy measurements may not be well described by this ideal PVM.

Degeneracy and Refinement

If several orthogonal states share the same outcome label, the projector has rank greater than one:

$$P_a = \sum_{\lambda=1}^{g_a} |a,\lambda\rangle\langle a,\lambda|.$$

For the Lüders instrument, the conditional output lies in the eigenspace of $A$ with eigenvalue $a$. That subspace can contain many pure states and mixtures; the outcome does not select a particular basis vector.

For a Lüders measurement, the selective update is still

$$\rho_a = \frac{P_a\rho P_a} {\operatorname{Tr}(P_a\rho)}.$$

This preserves coherences inside the degenerate subspace. A different device might secretly measure a refined basis $\{|a,\lambda\rangle\}$ and then report only $a$. That refined-and-forgotten operation gives

$$\rho \longmapsto \sum_{\lambda=1}^{g_a} |a,\lambda\rangle\langle a,\lambda| \rho |a,\lambda\rangle\langle a,\lambda|$$

conditional on the coarse outcome, after normalization. This can erase coherences inside the eigenspace that the Lüders update would preserve.

Therefore “measure the observable $A$” is incomplete unless the intended degeneracy handling is clear.

Repeatability

An ideal projective measurement is repeatable for the same coarse projective decomposition. If outcome $a$ occurs and the Lüders update is applied, then a second immediate measurement of the same PVM gives $a$ with probability one:

$$\operatorname{Tr}(P_a\rho_a)=1.$$

This repeatability is a property of the ideal measurement model, not of arbitrary detectors. A weak measurement, a noisy measurement, an inefficient measurement, or a destructive measurement need not leave the system in a state that guarantees the same repeated result.

Repeatability also does not mean the state is unchanged. If the input state contains components in several outcome subspaces, the selective update changes the state by projecting onto the observed subspace. If the outcome is ignored, the nonselective update changes the state by removing inter-outcome coherence.

Continuous Variables

For continuous spectra, the projective idealization is expressed using spectral projectors. For position on a line, an ideal question of the form “is the particle in region $\Delta$?” is represented by

$$P_\Delta = \int_\Delta dx\, |x\rangle\langle x|.$$

The probability is

$$p(\Delta) = \operatorname{Tr}(\rho P_\Delta).$$

Exact position eigenstates are not normalizable physical states, and real position detectors have finite resolution. A realistic detector is often better represented by a smeared POVM. The PVM remains the mathematically sharp limiting model and the correct ideal reference point.

Physical Reading

A projective measurement can be viewed as the ideal limit of a system-apparatus interaction that perfectly correlates distinct system subspaces with distinguishable pointer states:

$$\left(\sum_a P_a|\psi\rangle\right)|A_0\rangle \longrightarrow \sum_a P_a|\psi\rangle |A_a\rangle.$$

The pointer states $|A_a\rangle$ are mutually orthogonal in this ideal model. Reading and conditioning on $a$ gives the selective update; tracing out the pointer gives the nonselective channel, whether or not anyone reads it. If the apparatus or environment is included explicitly, the combined system can still be described by unitary dynamics before conditioning or tracing.

This is the operational bridge from projective measurement to open-system language.

Common pitfalls

Sharp probabilities do not fix disturbance. The same PVM can be implemented with an extra conditional rotation or a finer internal measurement. The formulas above assume the Lüders instrument.

Discarding a record does not undo a measurement. Use the nonselective channel when the detector acted without a retained outcome; use the selective state only for a known outcome with nonzero probability.

A continuous PVM does not prepare exact position eigenstates. Region projectors are well-defined bounded operators. Exact point outcomes have zero probability for an ordinary position density and do not yield normalizable position eigenvectors.

Exercises

1. Let a qubit state be

$$\rho= \begin{pmatrix} 1/3 & i/4\\ -i/4 & 2/3 \end{pmatrix}.$$

Compute the selective states and the nonselective state for an ideal $\sigma_z$ measurement.

Solution

The projectors are $P_+=|0\rangle\langle0|$ and $P_-=|1\rangle\langle1|$. The outcome probabilities are $p(+)=1/3$ and $p(-)=2/3$. Conditional on $+$, the state is $|0\rangle\langle0|$. Conditional on $-$, the state is $|1\rangle\langle1|$. If the outcome is ignored,

$$\mathcal M(\rho) = \begin{pmatrix} 1/3 & 0\\ 0 & 2/3 \end{pmatrix}.$$

2. Show that the nonselective projective measurement channel $\mathcal M(\rho)=\sum_aP_a\rho P_a$ is idempotent.

Solution

Apply the channel twice:

$$\mathcal M(\mathcal M(\rho)) = \sum_{a,b}P_aP_b\rho P_bP_a.$$

Using $P_aP_b=\delta_{ab}P_a$, only terms with $a=b$ remain:

$$\mathcal M(\mathcal M(\rho)) = \sum_a P_a\rho P_a = \mathcal M(\rho).$$

3. Let $A=0(|0\rangle\langle0|+|1\rangle\langle1|)+2|2\rangle\langle2|$ and

$$|\psi\rangle = \frac{|0\rangle+|1\rangle+|2\rangle}{\sqrt3}.$$

Find the probability of outcome $0$ and the Lüders-updated state after that outcome.

Solution

The projector for outcome $0$ is $P_0=|0\rangle\langle0|+|1\rangle\langle1|$. Hence

$$p(0)=\langle\psi|P_0|\psi\rangle=\frac{2}{3}.$$

The updated state is

$$|\psi_0\rangle = \frac{P_0|\psi\rangle}{\sqrt{p(0)}} = \frac{|0\rangle+|1\rangle}{\sqrt2}.$$

The coherence between $|0\rangle$ and $|1\rangle$ is preserved because the coarse Lüders measurement did not resolve the degeneracy.

4. A detector reports whether a particle lies in a small interval $\Delta x$ but has finite resolution and occasional false counts. Should it be modeled automatically by the sharp projector $P_{\Delta x}$?

Solution

No. The sharp projector is an ideal reference model for the exact question “is the particle in this region?” Finite resolution and false counts generally require a POVM whose effects are smeared and may include inefficiency or background counts. The PVM may be a good approximation only if those imperfections are negligible for the prediction being made.

References

• P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, 2nd ed., Springer (1996). Publisher.
• G. Lüders, “Concerning the state-change due to the measurement process,” Annalen der Physik 15, 663–670 (2006), English translation of the 1951 article. DOI.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018), chapter 2. Author’s text.