Compatible, Incompatible, and Sequential Measurements

Sequential measurements are measurements performed in a specified order, with the state left by earlier measurement operations used as the input for later ones. The order is part of the experiment, not an annotation added after the fact.

The central rule is:

$$\text{probabilities for later outcomes} \quad \text{depend on the earlier instrument,} \quad \text{not only on the earlier POVM}.$$

For ideal projective measurements this becomes the familiar lesson that noncommuting observables can have order-dependent statistics. For realistic detectors, the stronger lesson is that the whole measurement operation matters: conditioning, unread records, coarse graining, detector kicks, and loss channels can all change later probabilities.

The compact projective formulas are reviewed in Sequential Measurements in Core Formalism. This page treats sequential measurement as a measurement-theory object: records, instruments, compatibility, nondisturbance, and temporal correlations.

Ordered Instruments

Let the first measurement have outcome operations $\{\mathcal I_a\}$ and the second measurement have outcome operations $\{\mathcal J_b\}$. The ordered joint probability for first $a$, then $b$, is

$$p(a,b) = \operatorname{Tr} \!\left[ \mathcal J_b(\mathcal I_a(\rho)) \right].$$

If only the second outcome probability is needed after the first outcome $a$ is known, write

$$p(a)=\operatorname{Tr}\mathcal I_a(\rho), \qquad \rho_a = \frac{\mathcal I_a(\rho)}{p(a)}$$

when $p(a)\ne0$. Then

$$p(b|a) = \operatorname{Tr}\mathcal J_b(\rho_a), \qquad p(a,b)=p(a)p(b|a).$$

For an $n$-step measurement record $r_1,\ldots,r_n$, the same idea becomes an ordered composition:

$$p(r_1,\ldots,r_n) = \operatorname{Tr} \!\left[ \mathcal I^{(n)}_{r_n} \circ \cdots \circ \mathcal I^{(1)}_{r_1} (\rho) \right].$$

The rightmost operation acts first. This is the measurement analogue of time-ordered dynamics.

Projective Special Case

Let the first sharp measurement have projectors $\{P_a\}$ and the second have projectors $\{Q_b\}$. In the ideal Lüders model,

$$\mathcal I_a(\rho)=P_a\rho P_a, \qquad \mathcal J_b(\rho)=Q_b\rho Q_b.$$

The ordered joint probability is

$$p(a\ \text{then}\ b) = \operatorname{Tr} \!\left[ Q_bP_a\rho P_aQ_b \right].$$

Using $Q_b^2=Q_b$ and trace cyclicity, this is often written as

$$p(a\ \text{then}\ b) = \operatorname{Tr}(P_aQ_bP_a\rho).$$

The reversed experiment has

$$p(b\ \text{then}\ a) = \operatorname{Tr}(Q_bP_aQ_b\rho).$$

These are generally different operators and therefore different experiments. The difference is not only that two symbols are written in another order; the first measurement changes the state used to predict the second.

Compatible Projective Measurements

For two projective measurements, compatibility is expressed by commuting projectors:

$$P_aQ_b=Q_bP_a \qquad \text{for all }a,b.$$

Then $P_aQ_b$ is itself a projector onto the joint outcome subspace, and

$$p(a\ \text{then}\ b) = \operatorname{Tr}(P_aQ_b\rho) = p(b\ \text{then}\ a).$$

The nonselective first measurement also leaves the later statistics of the second projective measurement unchanged:

$$\operatorname{Tr} \!\left[ Q_b \sum_aP_a\rho P_a \right] = \operatorname{Tr}(Q_b\rho).$$

Compatibility does not mean statistical independence. It means the alternatives can be refined into a common projective measurement with joint outcomes $(a,b)$.

The algebraic background is Compatible Observables and Noncommuting Observables.

Incompatible Measurements and Disturbance

If $P_a$ and $Q_b$ do not commute, an intermediate unread measurement of $\{P_a\}$ can change the later $Q_b$ statistics. Without the intermediate measurement,

$$p_0(b)=\operatorname{Tr}(Q_b\rho).$$

With the intermediate measurement performed but ignored,

$$p_{\mathrm{unread}}(b) = \sum_a \operatorname{Tr}(Q_bP_a\rho P_a).$$

The difference comes from the off-diagonal blocks of $\rho$ in the $P_a$ decomposition:

$$\rho = \sum_{a,c}P_a\rho P_c.$$

The unread measurement removes the terms with $a\ne c$. If $Q_b$ is sensitive to those coherences, the later outcome distribution changes.

This is measurement disturbance in its most operational form: the first measurement interaction changes the probabilities assigned to a later experiment.

Selective and Nonselective Branches

Sequential experiments force one to say whether an intermediate record is retained.

If the first outcome $a$ is known, the later probability is computed from the selective state:

$$p(b|a) = \operatorname{Tr} \!\left[ F_b \frac{\mathcal I_a(\rho)} {\operatorname{Tr}\mathcal I_a(\rho)} \right],$$

where $F_b$ is the later effect if the second measurement is only being used for probabilities.

If the first measurement happens but the outcome is ignored, the later probability is

$$p(b) = \operatorname{Tr} \!\left[ F_b \sum_a\mathcal I_a(\rho) \right].$$

These are predictions for different ensembles. The selective state describes the subensemble with a known record. The nonselective state describes the unsorted ensemble after the physical measurement interaction. See Selective and Nonselective Measurements for the general distinction.

Same POVM, Different Future Statistics

Sequential measurements are the quickest way to see why a POVM is not a full measurement model.

Suppose one outcome of the first measurement is represented by a measurement operator $M_a$. The associated effect is

$$E_a=M_a^\dagger M_a.$$

If $U_a$ is a unitary, then

$$N_a=U_aM_a$$

has the same effect:

$$N_a^\dagger N_a=E_a.$$

The first-outcome probability is therefore the same for $M_a$ and $N_a$. But a later measurement with effect $F_b$ gives, after conditioning on outcome $a$,

$$p_M(b|a) = \frac{ \operatorname{Tr}(F_bM_a\rho M_a^\dagger) }{ \operatorname{Tr}(E_a\rho) },$$

whereas

$$p_N(b|a) = \frac{ \operatorname{Tr}(F_bU_aM_a\rho M_a^\dagger U_a^\dagger) }{ \operatorname{Tr}(E_a\rho) }.$$

These need not be equal. The two devices have the same first-outcome statistics but different backaction, so they make different predictions for subsequent measurements. The instrument is the canonical object; see Quantum Instruments.

Nondisturbance for a Later Test

For a general first measurement, let the nonselective channel be

$$\Phi(\rho)=\sum_a\mathcal I_a(\rho).$$

The first measurement is nondisturbing for a later effect $F_b$ if

$$\operatorname{Tr}[F_b\Phi(\rho)] = \operatorname{Tr}(F_b\rho)$$

for all input states $\rho$. In the Heisenberg-picture adjoint channel, this is

$$\Phi^\dagger(F_b)=F_b.$$

For an ideal projective measurement of $\{P_a\}$ followed by another projective question $\{Q_b\}$, this nondisturbance condition is guaranteed when all $P_a$ commute with all $Q_b$. For generalized measurements, nondisturbance can be subtler: an unsharp measurement may disturb less than a sharp one, and a noisy instrument may disturb more than its POVM alone suggests.

Spin One-Half Sequence

For a spin-$1/2$ particle, the projector for outcome $s=\pm1$ along unit vector $\boldsymbol n$ is

$$P_{\boldsymbol n,s} = \frac{1}{2} \left( I+s\,\boldsymbol n\cdot\boldsymbol\sigma \right).$$

If the system is first projected into outcome $s$ along $\boldsymbol n$, a subsequent ideal measurement along $\boldsymbol m$ gives outcome $t=\pm1$ with probability

$$p(t|s) = \operatorname{Tr} \!\left( P_{\boldsymbol m,t} P_{\boldsymbol n,s} \right) = \frac{1}{2} \left( 1+st\,\boldsymbol n\cdot\boldsymbol m \right).$$

For perpendicular axes, $\boldsymbol n\cdot\boldsymbol m=0$, so the second outcome is unbiased. Thus a $z$-definite beam sent through an $x$ analyzer and then a $z$ analyzer loses the original certainty of the final $z$ outcome. The intermediate incompatible analyzer prepared an $S_x$ eigenstate.

If the same axis is repeated, $\boldsymbol n\cdot\boldsymbol m=1$. Then $p(t=s|s)=1$ and $p(t=-s|s)=0$, the ideal repeatability result.

The Stern–Gerlach implementation and analyzer language are developed in Stern–Gerlach Revisited.

Degeneracy and Refinement in Sequences

Degenerate measurements require extra care in sequential experiments because later compatible measurements can reveal whether the first apparatus preserved or destroyed coherence inside an eigenspace.

Let $P_a$ project onto a degenerate eigenspace and let $\{P_{a\lambda}\}$ be a refined resolution inside that space. A Lüders measurement with outcome $a$ leaves

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

A refined measurement that records $\lambda$ microscopically but reports only $a$ leaves

$$\rho_a^{\mathrm{ref}} = \frac{ \sum_\lambda P_{a\lambda}\rho P_{a\lambda} }{ \operatorname{Tr}(P_a\rho) }.$$

Both can be repeatable for the coarse outcome $a$. They can differ in later measurements that probe superpositions inside the $a$ subspace. This is why degeneracy is not a cosmetic detail in measurement sequences.

Temporal Correlations

Sequential measurement data often enter through temporal correlation functions. If outcomes $q_i=\pm1$ are recorded at times $t_i$, one can define

$$C_{ij} = \sum_{q_i,q_j=\pm1} q_iq_j\,p(q_i,q_j).$$

The probabilities in this expression are sequential probabilities for a specified measurement protocol. They are not automatically the same as expectation values of operator products computed without measurement backaction.

This distinction matters in Leggett–Garg tests, which compare temporal correlations against assumptions such as macroscopic realism and noninvasive measurability. One common Leggett–Garg form for dichotomic outcomes is

$$K=C_{12}+C_{23}-C_{13}\le1,$$

under its macrorealist assumptions. Quantum protocols can violate such inequalities, but interpreting the violation requires careful control of measurement invasiveness, detector inefficiency, and state preparation. This page only gives the measurement-theory preview; a foundations page should own the full discussion.

Common Mistakes

Multiplying marginal probabilities

For a sequence, $p(a,b)$ is not generally $p(a)p(b)$ using the original state twice. The state after the first outcome enters the second probability.

Ignoring unread measurements

If an intermediate measurement occurred, discarding its outcome does not undo its nonselective channel.

Treating order dependence as notation

$A$ then $B$ and $B$ then $A$ are generally different physical procedures when the measurements are incompatible.

Using only POVMs for sequential predictions

POVM effects determine the probabilities of their own outcomes, but the future depends on the instrument.

Assuming compatibility means independence

Compatible projective measurements can have a joint distribution, but the joint probabilities can still be correlated.

Treating Leggett–Garg inequalities as ordinary two-time expectation algebra

Temporal-correlation tests depend on how the measurements are implemented. The invasiveness assumptions are part of the physics.

Exercises

Ordered Projective Probability

For projectors $P_a$ followed by projectors $Q_b$, derive

$$p(a\ \text{then}\ b) = \operatorname{Tr}(Q_bP_a\rho P_aQ_b).$$

Solution

The first operation is $\mathcal I_a(\rho)=P_a\rho P_a$. The second operation is $\mathcal J_b(\rho)=Q_b\rho Q_b$. The ordered joint probability is the trace after both trace-nonincreasing operations:

$$p(a,b) = \operatorname{Tr} \!\left[ Q_bP_a\rho P_aQ_b \right].$$

This is the desired expression. Since $Q_b^2=Q_b$, cyclicity of the trace also gives $\operatorname{Tr}(P_aQ_bP_a\rho)$.

Unread Intermediate Spin Measurement

A qubit starts in $\lvert+z\rangle$. An unread ideal $\sigma_x$ measurement is performed, followed by a $\sigma_z$ measurement. What is the probability of obtaining $+z$ at the end?

Solution

The unread $\sigma_x$ measurement maps the initial pure state to

$$\rho' = \frac{1}{2} \lvert+x\rangle\langle+x\rvert + \frac{1}{2} \lvert-x\rangle\langle-x\rvert = \frac{I}{2}.$$

Therefore

$$p(+z) = \operatorname{Tr} \!\left[ \lvert+z\rangle\langle+z\rvert \frac{I}{2} \right] = \frac{1}{2}.$$

The intermediate unread measurement erased the coherence needed to guarantee the later $+z$ outcome.

Commuting Projectors

Assume $P_aQ_b=Q_bP_a$ for all $a,b$. Show that the unread $P$ measurement does not change the later probability of outcome $b$ for the $Q$ measurement.

Solution

The unread state is $\sum_aP_a\rho P_a$. The later probability is

$$\sum_a \operatorname{Tr}(Q_bP_a\rho P_a).$$

Using commutation and cyclicity,

$$\operatorname{Tr}(Q_bP_a\rho P_a) = \operatorname{Tr}(P_aQ_bP_a\rho) = \operatorname{Tr}(Q_bP_a\rho).$$

Summing over $a$ gives

$$\sum_a\operatorname{Tr}(Q_bP_a\rho) = \operatorname{Tr} \!\left[ Q_b \sum_aP_a \rho \right] = \operatorname{Tr}(Q_b\rho).$$

Same Effect, Different Sequence

Let $N_a=U_aM_a$ with $U_a$ unitary. Show that $M_a$ and $N_a$ have the same effect but can give different later probabilities for an effect $F_b$.

Solution

The effects agree:

$$N_a^\dagger N_a = M_a^\dagger U_a^\dagger U_aM_a = M_a^\dagger M_a.$$

Thus the probability of outcome $a$ is the same. But the conditional output states are related by

$$\rho_a^N = U_a\rho_a^M U_a^\dagger.$$

The later probability for effect $F_b$ is $\operatorname{Tr}(F_b\rho_a^N)$ instead of $\operatorname{Tr}(F_b\rho_a^M)$. These are equal for all inputs only under additional conditions, such as $F_b$ being invariant under the relevant unitary action on the conditional states.

Cross-Links

• Measurement as an Operation
• Projective Measurements
• Selective and Nonselective Measurements
• State Update Rules
• Measurement Backaction
• Repeatability and QND Measurement
• Quantum Instruments
• Sequential Measurements
• Compatible Observables
• Noncommuting Observables
• Stern–Gerlach Revisited

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press (1955).
• G. Lüders, “Über die Zustandsänderung durch den Messprozeß,” Annalen der Physik 8, 322–328 (1951).
• 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).
• 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).
• A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?”, Physical Review Letters 54, 857–860 (1985).
• C. Emary, N. Lambert, and F. Nori, “Leggett–Garg inequalities,” Reports on Progress in Physics 77, 016001 (2014).