Instruments

A quantum instrument is a countably additive measure whose values are completely positive state transformations. It combines an outcome law with an unnormalized output state, so it can predict subsequent measurements. The canonical Quantum Instruments article gives the operational introduction; this page formulates instruments on trace-class spaces and distinguishes event conditioning from a formal update at a continuous singleton.

Helpful background. Completely positive maps supplies normal operations and their duals; POVMs supplies measurable outcome spaces.

A measure with operation-valued events

Let $(\Omega,\Sigma)$ be a measurable outcome space, with separable input and output Hilbert spaces. A CP instrument assigns each $\Delta\in\Sigma$ a completely positive map

$$\mathcal I(\Delta): \mathcal S_1(\mathcal H_{\mathrm{in}}) \longrightarrow\mathcal S_1(\mathcal H_{\mathrm{out}})$$

such that the total map $\mathcal I(\Omega)$ preserves trace and, for disjoint $\Delta_n$,

$$\mathcal I\!\left(\bigcup_n\Delta_n\right)(T) =\sum_n\mathcal I(\Delta_n)(T)$$

in trace norm for every $T\in\mathcal S_1$. Each event map is trace nonincreasing on positive inputs, because its complement is also positive and the total map preserves trace.

For a positive input, all event outputs are positive and their total traces sum to the input trace. Consequently the trace norm of a tail is exactly its remaining probability. The definition is stronger than requiring scalar additivity for one fixed input state: it describes a single linear apparatus model applicable to every input.

Some older terminology permits merely positive instruments. Here instrument means completely positive, so the apparatus remains valid in the presence of entangled reference systems.

Recovering probabilities and the dual observable

For an input density operator $\rho$,

$$p_\rho(\Delta)=\operatorname{Tr}\mathcal I(\Delta)(\rho).$$

The normal Heisenberg adjoint gives the effect

$$E(\Delta)=\mathcal I(\Delta)^*(I_{\mathrm{out}}), \qquad p_\rho(\Delta)=\operatorname{Tr}(\rho E(\Delta)).$$

The event effects form a normalized POVM. Normality concerns continuity of the adjoint as a function of the bounded output observable; countable additivity concerns its dependence on the outcome event. These are different requirements.

When $p_\rho(\Delta)>0$, the event-conditioned state is

$$\rho_\Delta= \frac{\mathcal I(\Delta)(\rho)}{p_\rho(\Delta)}.$$

The numerator is linear in $\rho$; normalization makes the conditional update generally nonlinear. Keeping the trace-deficient numerator is often preferable when composing operations, because it retains the probability weight automatically.

The same POVM can describe different apparatuses

Fix any POVM $E$ and any output density operator $\tau$. The measure-and-prepare formula

$$\mathcal I_\tau(\Delta)(T) =\operatorname{Tr}(T E(\Delta))\,\tau$$

defines a CP instrument with POVM $E$. Positivity of the measurement functional gives complete positivity; countable additivity follows from the POVM probability law, and the total trace is $\operatorname{Tr}T$.

Changing $\tau$ changes every conditional output while leaving every first measurement probability unchanged. For a discrete PVM, this instrument can be compared with the Lüders instrument $T\mapsto P_kTP_k$. One destroys the input and prepares $\tau$; the other retains within-eigenspace information. A POVM alone cannot choose between them.

More generally, outcome-dependent prepared states give additional instruments when the requisite measurability and integrability are established. The fixed-state construction already proves the essential nonuniqueness without assuming a pointwise operator density.

Coarse graining and sequential measurements

For a countable partition into events $\Delta_k$, forgetting the outcome gives the channel $\sum_k\mathcal I(\Delta_k)=\mathcal I(\Omega)$. Merging a collection of labels adds their unnormalized operations; it does not average their individually normalized states with equal weights.

If a second apparatus has instrument $\mathcal J$ and POVM $F$, then

$$\begin{aligned} p_\rho(\Delta,\Gamma) &=\operatorname{Tr}\mathcal J(\Gamma) \bigl(\mathcal I(\Delta)(\rho)\bigr)\\ &=\operatorname{Tr}\bigl( \rho\,\mathcal I(\Delta)^*(F(\Gamma))\bigr). \end{aligned}$$

This formula makes backaction observable through later statistics. Summing over the first outcome gives the second measurement performed after the first apparatus’s total channel. It need not equal the probability of applying the second apparatus directly to $\rho$.

Continuous records require conditional-measure care

For a continuous outcome law, a singleton can have zero probability. Then $\mathcal I(\{x\})(\rho)=0$ and the normalized event formula divides by zero. A conditional state indexed by $x$, when constructed through a measurable disintegration or a suitable instrument density, is specified only almost everywhere relative to the input outcome distribution.

It is therefore invalid to infer a unique physical posterior at an arbitrary zero-probability point from the event measure alone. Finite detector bins are ordinary nonzero-probability events; infinitesimal-record formulas require their own density and regularity assumptions. Busch and collaborators develop this distinction in the general instrument framework.

Exercises

1. Suppose two disjoint events have probabilities $p_1,p_2>0$ and conditional states $\rho_1,\rho_2$. Find the state conditioned only on their union.

Solution

Add the unnormalized outputs and divide by the total probability: $\rho_{1\cup2}=(p_1\rho_1+p_2\rho_2)/(p_1+p_2)$. Equal weighting would be wrong unless the event probabilities coincide. This is the coarse-graining rule for quantum instruments.

2. For $\mathcal I_\tau$, what is the output when the measurement record is discarded?

Solution

The total channel is $\mathcal I_\tau(\Omega)(\rho)=\operatorname{Tr}(\rho I)\tau=\tau$. Every normalized input is replaced by the same state, although the recorded outcomes can still contain information about the original input.

References

• Busch, Paul, Pekka Lahti, Juha-Pekka Pellonpää, and Kari Ylinen, Quantum Measurement, Springer, 2016 — CP instruments, measurable outcomes, and sequential measurement. Publisher.
• Heinosaari, Teiko, and Mário Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement, Cambridge University Press, 2012, Chapter 5 — instruments and measurement models. Publisher.