Quantum Instruments

A quantum instrument is the operational object that describes a measurement when both the classical outcome and the resulting quantum state matter.

A POVM answers one question:

$$\text{What are the probabilities of the reported outcomes?}$$

An instrument answers two questions:

$$\text{What is the probability of outcome }m, \qquad \text{and what state is left after }m?$$

This distinction is essential for sequential measurements, detector backaction, postselection, continuous monitoring, and realistic readout models. Different instruments can have the same POVM but make different predictions for later measurements.

For the relationship between instrument likelihoods and Bayesian conditioning, see Bayesian Quantum Measurement.

Definition

For a finite outcome set, a quantum instrument is a family of completely positive, trace-nonincreasing maps

$$\{\mathcal I_m\}$$

acting on density operators. The map $\mathcal I_m$ is the unnormalized state transformation associated with outcome $m$.

For an input state $\rho$, the probability of outcome $m$ is

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

If $p(m)\ne0$, the conditional output 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

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

For a complete measurement with no discarded failure branch, the sum

$$\Phi = \sum_m\mathcal I_m$$

is trace preserving. Equivalently,

$$\sum_m \operatorname{Tr}\mathcal I_m(\rho) = \operatorname{Tr}\rho$$

for every trace-class input $\rho$.

Kraus Form

In finite dimensions, each outcome operation can be written using Kraus operators:

$$\mathcal I_m(\rho) = \sum_\alpha K_{m\alpha}\rho K_{m\alpha}^\dagger.$$

Trace nonincrease for each outcome means

$$\sum_\alpha K_{m\alpha}^\dagger K_{m\alpha} \le I.$$

Completeness of the whole instrument means

$$\sum_{m,\alpha} K_{m\alpha}^\dagger K_{m\alpha} =I.$$

The label $\alpha$ represents microscopic alternatives that are not distinguished in the reported outcome $m$. Examples include unresolved detector modes, lost photons, unobserved decay channels, or internal apparatus states that are later ignored.

Associated POVM

Every instrument has an associated POVM. In Kraus form, the effect for outcome $m$ is

$$F_m = \sum_\alpha K_{m\alpha}^\dagger K_{m\alpha}.$$

Then

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

More invariantly, let $\mathcal I_m^*$ be the adjoint map in the trace pairing:

$$\operatorname{Tr} \left[ A\,\mathcal I_m(\rho) \right] = \operatorname{Tr} \left[ \mathcal I_m^*(A)\rho \right].$$

The associated effect is

$$F_m = \mathcal I_m^*(I).$$

This formulation makes the hierarchy clear:

instrument -> POVM
POVM       -> probabilities only

The reverse arrow is not unique. A POVM does not determine the post-measurement state.

Why POVMs Are Not Enough

Consider a sharp two-outcome measurement with projectors

$$P_0=|0\rangle\langle0|, \qquad P_1=|1\rangle\langle1|.$$

The Lüders instrument is

$$\mathcal I_0^{\mathrm L}(\rho)=P_0\rho P_0, \qquad \mathcal I_1^{\mathrm L}(\rho)=P_1\rho P_1.$$

Its associated POVM is $\{P_0,P_1\}$. It is repeatable: immediately measuring the same sharp observable again gives the same outcome with probability one.

Now compare a destructive measure-and-prepare instrument:

$$\mathcal I_0^{\mathrm{prep}}(\rho) = \operatorname{Tr}(P_0\rho)\sigma_0, \qquad \mathcal I_1^{\mathrm{prep}}(\rho) = \operatorname{Tr}(P_1\rho)\sigma_1,$$

where $\sigma_0$ and $\sigma_1$ are fixed output states. This instrument has the same POVM $\{P_0,P_1\}$, because the same probabilities are reported. But the post-measurement states are $\sigma_0$ and $\sigma_1$, not the projected input components.

Thus the phrase “measure $\sigma_z$” is incomplete unless the later state does not matter or the instrument is specified by context.

Measurement in Circuits applies this same-probability/different-future distinction to circuit readout by declaring the classical record, surviving output, selected or unread branch, shot model, and postprocessing. This page retains the general instrument structure.

Nonselective Channel

The nonselective channel

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

is what remains when the measurement interaction happens but the classical record is ignored. It is a completely positive, trace-preserving map for a complete instrument.

For the Lüders instrument above,

$$\Phi_{\mathrm L}(\rho) = P_0\rho P_0+P_1\rho P_1.$$

This channel removes coherence between the two eigenspaces.

For the measure-and-prepare instrument,

$$\Phi_{\mathrm{prep}}(\rho) = \operatorname{Tr}(P_0\rho)\sigma_0 + \operatorname{Tr}(P_1\rho)\sigma_1.$$

This channel discards the input state after extracting the classical distribution over outcomes. Both instruments can share the same POVM, but their nonselective channels and future predictions differ.

Sequential Measurements

Instruments are indispensable when one measurement is followed by another. Suppose the first measurement has instrument $\{\mathcal I_m\}$ and the second measurement has POVM effects $\{G_n\}$. The joint probability for first outcome $m$ and second outcome $n$ is

$$p(m,n) = \operatorname{Tr} \left[ G_n\,\mathcal I_m(\rho) \right].$$

The conditional probability for the second outcome, after learning $m$, is

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

The first POVM alone is not sufficient to compute these probabilities. One needs the actual state update, and that information is precisely the instrument.

This is the operational reason instruments belong between state-update rules and open-system dynamics: they tell us how classical records and quantum state transformations fit together.

Mid-Circuit Measurement and Feedforward is the finite adaptive-circuit application: it composes outcome operations with later record-conditioned channels and audits causal histories, reset, and branch merges.

Indirect Measurement Origin

An instrument often arises from an explicit system-apparatus model. Let $S$ be the system, $A$ an apparatus, $\eta_A$ the initial apparatus state, $U$ a joint unitary, and $\{Q_m\}$ a projective pointer measurement on the apparatus. If the apparatus is discarded after outcome $m$, the system operation can be written as

$$\mathcal I_m(\rho_S) = \operatorname{Tr}_A \left[ (I_S\otimes Q_m) U(\rho_S\otimes\eta_A)U^\dagger (I_S\otimes Q_m) \right].$$

The corresponding probability is

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

This construction explains why instruments are completely positive. They are what a closed unitary model becomes after the apparatus degrees of freedom are measured, partly retained as a classical record, and otherwise ignored.

For the ideal correlation model behind projective measurement, see the von Neumann measurement model.

Coarse Graining and Hidden Records

Suppose a detector has fine-grained outcomes $(m,\alpha)$ but reports only $m$. If the fine-grained operations are $\mathcal J_{m\alpha}$, the coarse-grained instrument is

$$\mathcal I_m(\rho) = \sum_\alpha \mathcal J_{m\alpha}(\rho).$$

The associated effect is likewise coarse grained:

$$F_m = \sum_\alpha F_{m\alpha}.$$

This is not merely classical ignorance. Coarse graining can change the output state because distinct microscopic alternatives may leave different quantum states and may destroy coherences that would survive in a more refined description.

Postselection and Failure Outcomes

Many experiments condition on successful runs. The operations for the retained outcomes may sum to a trace-nonincreasing map:

$$\sum_{m\in\mathrm{kept}} \operatorname{Tr}\mathcal I_m(\rho) \le \operatorname{Tr}\rho.$$

To describe the full experiment as a complete instrument, include the discarded outcomes explicitly. A common notation is

$$\mathcal I_{\mathrm{fail}} = \Phi - \sum_{m\in\mathrm{kept}}\mathcal I_m,$$

when the difference is a completely positive operation. The failure branch accounts for no-click events, rejected trials, leakage out of a selected subspace, or data that do not pass an analysis cut.

Postselection is legitimate, but probabilities and normalized states must be reported conditionally. Forgetting the failure branch often leads to apparent violations of trace preservation or exaggerated success probabilities.

Continuous-Measurement Preview

A continuous measurement can be viewed as a limit of instruments applied over short time steps. For a small interval $dt$, the record increment $r$ has an operation $\mathcal I_r^{dt}$. A full measurement record is a sequence

$$r_1,r_2,\ldots,r_N, \qquad Ndt=t.$$

The unnormalized conditional state is obtained by composing the corresponding operations:

$$\tilde\rho_{r_1\cdots r_N} = \mathcal I_{r_N}^{dt} \circ\cdots\circ \mathcal I_{r_1}^{dt}(\rho_0).$$

Its trace is the probability density or probability weight of that record, and the normalized state is the conditional state along the observed trajectory. This is the discrete skeleton behind Continuous Monitoring, quantum jumps, diffusive trajectories, and stochastic master equations.

Common Mistakes

• Treating a POVM effect $F_m$ as if it were the state update.
• Assuming that the same outcome probabilities imply the same backaction.
• Forgetting that an outcome operation is trace nonincreasing before normalization.
• Dropping failure or no-click branches and then calling the remaining operation trace preserving.
• Confusing a nonselective channel $\Phi$ with the full instrument $\{\mathcal I_m\}$.
• Assuming projective measurements are always Lüders measurements; a degenerate or destructive apparatus can implement a different instrument with the same coarse outcome statistics.

References

• E. B. Davies and J. T. Lewis, “An operational approach to quantum probability,” Communications in Mathematical Physics 17, 239-260 (1970).
• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• M. Ozawa, “Quantum measuring processes of continuous observables,” Journal of Mathematical Physics 25, 79-87 (1984).
• A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland (1982); Edizioni della Normale (2011).
• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer (2016).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).

Exercises

1. Instrument to POVM. Let $\mathcal I_m(\rho)=\sum_\alpha K_{m\alpha}\rho K_{m\alpha}^\dagger$ and assume $\sum_{m,\alpha}K_{m\alpha}^\dagger K_{m\alpha}=I$. Prove that $F_m=\sum_\alpha K_{m\alpha}^\dagger K_{m\alpha}$ defines a POVM.

Solution

Each $F_m$ is positive because for every vector $|\psi\rangle$,

$$\langle\psi|F_m|\psi\rangle = \sum_\alpha \|K_{m\alpha}|\psi\rangle\|^2 \ge0.$$

Completeness follows directly:

$$\sum_mF_m = \sum_{m,\alpha} K_{m\alpha}^\dagger K_{m\alpha} =I.$$

Thus $\{F_m\}$ is a collection of positive effects summing to the identity.

2. Same POVM, different future. Let $\rho=|+\rangle\langle+|$, where $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$. Compare the nonselective output of the Lüders instrument for $\{P_0,P_1\}$ with a destructive instrument

$$\mathcal I_0(\rho)=\operatorname{Tr}(P_0\rho)|0\rangle\langle0|, \qquad \mathcal I_1(\rho)=\operatorname{Tr}(P_1\rho)|0\rangle\langle0|.$$

Both instruments have the same first-measurement POVM. What is the probability of obtaining $0$ in a second $\sigma_z$ measurement after the first outcome is ignored?

Solution

For the Lüders instrument,

$$\Phi_{\mathrm L}(|+\rangle\langle+|) = \frac12 |0\rangle\langle0| + \frac12 |1\rangle\langle1|.$$

A second $\sigma_z$ measurement gives outcome $0$ with probability $1/2$.

For the destructive instrument,

$$\Phi_{\mathrm{dest}}(|+\rangle\langle+|) = \frac12 |0\rangle\langle0| + \frac12 |0\rangle\langle0| = |0\rangle\langle0|.$$

The second $\sigma_z$ measurement gives outcome $0$ with probability $1$. The first measurement statistics were the same, but the instruments produced different later states.

3. Sequential probability formula. A first measurement has instrument $\{\mathcal I_m\}$, and a second measurement has effects $\{G_n\}$. Show that $p(m,n)=\operatorname{Tr}[G_n\mathcal I_m(\rho)]$ is normalized if the first instrument is complete and the second POVM is complete.

Solution

Sum over the second outcome first:

$$\sum_n p(m,n) = \sum_n \operatorname{Tr} \left[ G_n\mathcal I_m(\rho) \right] = \operatorname{Tr} \left[ \left(\sum_nG_n\right)\mathcal I_m(\rho) \right].$$

Since $\sum_nG_n=I$,

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

Now sum over $m$:

$$\sum_{m,n}p(m,n) = \operatorname{Tr} \left[ \sum_m\mathcal I_m(\rho) \right] = \operatorname{Tr}\rho =1,$$

where completeness of the first instrument was used in the second equality and $\rho$ was assumed normalized.