Selective and Nonselective Measurements

Selective and nonselective measurements answer different prediction questions.

A selective measurement is conditioned on a known outcome. A nonselective measurement is the state transformation after the measurement interaction when the outcome is not retained, not read, or deliberately averaged over.

The distinction is not philosophical bookkeeping. It changes the density operator used for later predictions.

The General Instrument Formula

Let a measurement have outcomes $m$ and outcome-resolved operations $\mathcal I_m$. Each $\mathcal I_m$ is completely positive and trace nonincreasing. The probability of outcome $m$ is

$$p(m) = \operatorname{Tr}\mathcal I_m(\rho).$$

If $m$ is observed and $p(m)\ne0$, the selective state is

$$\rho_m = \frac{\mathcal I_m(\rho)} {\operatorname{Tr}\mathcal I_m(\rho)}.$$

If the measurement is performed but the outcome is ignored, the nonselective output is

$$\rho' = \sum_m \mathcal I_m(\rho).$$

The same apparatus can therefore lead to two different state assignments:

$$\text{known record }m \quad\Rightarrow\quad \rho_m, \qquad \text{ignored record} \quad\Rightarrow\quad \sum_m p(m)\rho_m.$$

The nonselective state is an ensemble average of the selective states, weighted by their probabilities.

Projective Special Case

For an ideal projective measurement with projectors $\{P_a\}$, the operation for outcome $a$ is

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

The selective update is

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

The nonselective update is

$$\rho' = \sum_a P_a\rho P_a.$$

These formulas look similar, but they represent different information states. The selective state is used after learning a definite outcome. The nonselective state is used when the outcome information is unavailable or irrelevant.

Record Kept Versus Record Ignored

It is often useful to represent the classical record explicitly. If the outcome is stored in a classical register $R$ with orthonormal labels $\{|m\rangle_R\}$, then after the measurement the joint state of system and record may be written schematically as

$$\rho_{SR}' = \sum_m p(m)\rho_m\otimes |m\rangle_R\langle m|.$$

If the record is read and equals $m$, the conditional system state is $\rho_m$. If the record exists but is ignored, the system state is obtained by tracing out the record:

$$\operatorname{Tr}_R\rho_{SR}' = \sum_m p(m)\rho_m.$$

Thus “the outcome was produced” and “the outcome is known to the agent making predictions” are different statements. The first says a physical correlation exists. The second says which conditional branch should be used for later probabilities.

Qubit Example

Consider an ideal $\sigma_z$ measurement of a qubit with projectors

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

Let the initial state be

$$\rho= \begin{pmatrix} a & c\\ c^* & 1-a \end{pmatrix}, \qquad 0\le a\le1.$$

The outcome probabilities are

$$p(+)=a, \qquad p(-)=1-a.$$

If $+$ is observed, the selective state is

$$\rho_+ = |0\rangle\langle0| = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}.$$

If $-$ is observed, the selective state is

$$\rho_- = |1\rangle\langle1| = \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}.$$

If the outcome is ignored, the nonselective output is

$$\rho' = P_+\rho P_+ + P_-\rho P_- = \begin{pmatrix} a & 0\\ 0 & 1-a \end{pmatrix}.$$

The off-diagonal coherence $c$ is gone. The nonselective measurement has acted as a complete dephasing channel in the $\sigma_z$ basis.

Why the Average Is Not the Original State

A common mistake is to think that if the outcome is ignored, nothing has happened. That is generally false. Ignoring the outcome does not undo the physical system-apparatus correlation.

Before the measurement, the coherence between $|0\rangle$ and $|1\rangle$ appears in $c$. After an ideal unread $\sigma_z$ measurement, that coherence is not available in the system alone. It has been transferred to, or destroyed by coupling with, the record-bearing degrees of freedom.

This is the same logic behind decoherence calculations. When alternatives become correlated with distinguishable records and those records are ignored, local interference terms are suppressed.

Selective Does Not Mean More Physical

The selective state is not more real and the nonselective state is not merely ignorance in every interpretation. They are state assignments for different experimental information.

If an experimenter sorts data by outcome, each sorted subensemble is described by a selective state. If the same experimenter combines all runs without sorting, the ensemble is described by the nonselective state. Both are operationally meaningful and both can be tested by later measurements.

For example, after the qubit measurement above:

• the $+$ subensemble gives $\sigma_z=+1$ with certainty;
• the $-$ subensemble gives $\sigma_z=-1$ with certainty;
• the unsorted ensemble gives expectation value $2a-1$.

Different later predictions are expected because the data sets are different.

No-Click Is an Outcome

In realistic measurements, an apparatus may have a “no click,” “lost,” or “inconclusive” result. If that result is recorded, it is still an outcome. Conditioning on no-click is selective:

$$\rho_{\varnothing} = \frac{\mathcal I_{\varnothing}(\rho)} {\operatorname{Tr}\mathcal I_{\varnothing}(\rho)}.$$

The nonselective state includes all outcomes, including no-click:

$$\rho' = \sum_{\text{clicks }m}\mathcal I_m(\rho) + \mathcal I_{\varnothing}(\rho).$$

Forgetting to include inconclusive outcomes is a frequent way to produce a trace-decreasing “state” where a normalized unconditional state was intended.

Relation to Channels

The nonselective map

$$\Phi(\rho)=\sum_m\mathcal I_m(\rho)$$

is trace preserving when all outcomes of the measurement are included. It is therefore a quantum channel. The selective maps $\mathcal I_m$ are generally trace nonincreasing because their traces are probabilities.

If each instrument element has Kraus operators $M_{m\alpha}$,

$$\mathcal I_m(\rho) = \sum_\alpha M_{m\alpha}\rho M_{m\alpha}^\dagger,$$

then the complete nonselective channel is

$$\Phi(\rho) = \sum_{m,\alpha} M_{m\alpha}\rho M_{m\alpha}^\dagger, \qquad \sum_{m,\alpha} M_{m\alpha}^\dagger M_{m\alpha} =I.$$

This is the algebraic bridge from measurement theory to quantum channels.

Sequential Predictions

The selective/nonselective distinction becomes unavoidable in sequential experiments. Suppose a first measurement has outcomes $m$ and a later measurement has effects $F_n$. If the first outcome is known to be $m$, then

$$p(n|m) = \operatorname{Tr}(F_n\rho_m).$$

If the first outcome is ignored, the later probability is

$$p(n) = \operatorname{Tr} \left[ F_n \sum_m\mathcal I_m(\rho) \right].$$

These are different questions. The first predicts the later statistics in a conditioned subensemble. The second predicts the later statistics in the unsorted ensemble.

Common Mistakes

• Using a selective state when the outcome was not recorded.
• Averaging over outcomes even though a known record is available.
• Treating an unread measurement as if no interaction occurred.
• Forgetting that no-click or loss can be an outcome with its own operation.
• Renormalizing a single trace-decreasing operation and then calling it the unconditional state.
• Assuming the POVM effects alone determine the selective output states.
• Confusing an ensemble average of conditional states with a coherent superposition of outcomes.

Cross-Links

• Projective Measurements
• State Update Rule
• Generalized Measurements Overview
• Compatible, Incompatible, and Sequential Measurements
• Density Operators
• Decoherence Preview
• Common Misconceptions

References

• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer, 1983.
• P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, Springer, 1996.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.

Exercises

1. A qubit in state

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

is measured projectively in the $\{|0\rangle,|1\rangle\}$ basis. Find the selective states and the nonselective state.

Solution

The outcome probabilities are $p(0)=3/4$ and $p(1)=1/4$. The selective states are $|0\rangle\langle0|$ and $|1\rangle\langle1|$. The nonselective state is

$$\rho' = \frac{3}{4}|0\rangle\langle0| + \frac{1}{4}|1\rangle\langle1|.$$

2. Suppose an instrument has two outcomes with operations $\mathcal I_0$ and $\mathcal I_1$, but a calculation keeps only $\mathcal I_0(\rho)$ and normalizes it. Is that selective or nonselective?

Solution

It is selective, conditioned on outcome $0$. The nonselective state is $\mathcal I_0(\rho)+\mathcal I_1(\rho)$, assuming those are all outcomes. Normalizing only $\mathcal I_0(\rho)$ describes the subensemble in which outcome $0$ occurred.

3. Show that the nonselective output can be written as $\sum_m p(m)\rho_m$ whenever $p(m)\ne0$ for all included outcomes.

Solution

By definition,

$$\rho_m = \frac{\mathcal I_m(\rho)} {p(m)}.$$

Therefore $\mathcal I_m(\rho)=p(m)\rho_m$. Summing over $m$ gives

$$\sum_m\mathcal I_m(\rho) = \sum_m p(m)\rho_m.$$

Outcomes with $p(m)=0$ contribute the zero unnormalized state and may be omitted from the weighted sum.

4. A detector has outcomes click and no-click. The click operation has trace $0.3$ on a given input state, and the no-click operation has trace $0.7$. What trace should the nonselective state have if both outcomes are included?

Solution

It should have trace $1$. The two outcome probabilities sum to $0.3+0.7=1$, so the unconditional operation $\mathcal I_{\text{click}}+\mathcal I_{\varnothing}$ is trace preserving on this input. Keeping only click would describe a postselected subensemble and would have to be normalized selectively.