State Update Maps

The unnormalized state change associated with a measurement event is a linear completely positive operation. Dividing by its success probability produces a generally nonlinear conditional state, defined only for inputs giving that event positive probability. The canonical State Update Rules article compares physical update models; this page analyzes normalization, continuity, and domain assumptions.

Helpful background. Instruments supplies event operations; trace distance supplies the metric for state errors.

An operation and its conditional map

Let $\Phi:\mathcal S_1(\mathcal H_{\mathrm{in}})\to \mathcal S_1(\mathcal H_{\mathrm{out}})$ be completely positive and trace nonincreasing on positive operators. Its success effect is $E=\Phi^*(I)$, with $0\leq E\leq I$.

For an input state $\rho$, define

$$p_\rho=\operatorname{Tr}\Phi(\rho) =\operatorname{Tr}(\rho E).$$

The trace-deficient operator $\Phi(\rho)$ is meaningful for every input. If $p_\rho>0$, the conditional state is $\widehat\Phi(\rho)=\Phi(\rho)/p_\rho$. If $p_\rho=0$, positivity implies $\Phi(\rho)=0$, and there is no normalized conditional output determined by this formula.

In general $\widehat\Phi$ is nonlinear and is not a quantum channel. In special cases it coincides with one: for $\Phi(\rho)=q\rho$ with fixed $0<q\leq1$, the normalized map is the identity channel. Even then, the selected apparatus branch still has success probability $q$. The normalized output alone does not specify that operation’s frequency of occurrence.

How conditioning changes ensemble weights

For $\rho=t\rho_1+(1-t)\rho_2$, let $p_i$ be the branch probabilities of the two components. Linearity of $\Phi$ gives

$$\widehat\Phi(\rho) =\frac{tp_1\widehat\Phi(\rho_1) +(1-t)p_2\widehat\Phi(\rho_2)} {tp_1+(1-t)p_2},$$

when the indicated conditional states exist. Components with zero probability contribute zero unnormalized output and may be omitted.

The posterior weights depend on the likelihoods $p_i$. Thus conditioning a mixture does not generally equal the same mixture of the conditioned states. This is a quantum operation followed by ordinary probabilistic reweighting; the nonlinearity does not require a new nonlinear dynamical law.

Square-root and single-operator updates

For an effect $E$, the operation

$$\Phi_E(\rho)=\sqrt E\,\rho\,\sqrt E$$

has success probability $\operatorname{Tr}(\rho E)$. It is a valid choice of branch, often called the square-root or Lüders operation for that effect. It is not selected by the POVM alone.

A single-operator realization with the same effect satisfies $M^*M=E$. Its polar decomposition is $M=V\sqrt E$, with $V$ a partial isometry on the support of $E$. The choice of $V$ changes the output while preserving all branch probabilities. General operations may require several Kraus operators and cannot be described by one such $M$.

For a projection $P$, the square-root operation is $P\rho P$, and a successful event leaves the state in $\operatorname{ran}P$. An immediate repetition of the same sharp event then succeeds with probability one. For $E=\eta P$, $0<\eta<1$, the normalized successful output is still supported in $\operatorname{ran}P$, but the next occurrence of that same effect has probability $\eta$. Support alone does not make an unsharp event repeatable.

Continuity deteriorates near rare events

Let $A=\Phi(\rho)$ and $B=\Phi(\sigma)$ have positive traces $a,b$. A direct triangle inequality gives

$$\left\|\frac A a-\frac B b\right\|_1 \leq\frac{\|A-B\|_1+|a-b|}{a} \leq\frac{2\|A-B\|_1}{a}.$$

A positive trace-nonincreasing map contracts the trace norm on self-adjoint inputs: decompose the input into its positive and negative parts and use the triangle inequality. Hence, if both success probabilities are at least $p_*>0$,

$$\|\widehat\Phi(\rho)-\widehat\Phi(\sigma)\|_1 \leq\frac{2}{p_*}\|\rho-\sigma\|_1.$$

This supplies uniform stability away from zero probability. It cannot give a uniform bound near inputs for which the selected event becomes impossible.

For an explicit example, on three orthogonal levels take $\rho_\epsilon=(1-\epsilon)P_0+\epsilon P_1$ and $\sigma_\epsilon=(1-\epsilon)P_0+\epsilon P_2$. Their trace distance is $\epsilon$. Select the event $P=P_1+P_2$. Both probabilities are $\epsilon$, but their conditional outputs are orthogonal pure states, with trace distance one. Rare-event selection can amplify a small input difference into a maximal conditional difference.

Bounded operations do not guarantee finite energy

The operation framework maps trace-class inputs to trace-class outputs. It does not automatically preserve a Hamiltonian’s form domain or a finite-energy constraint. A bounded Kraus operator can send a finite-energy vector to a vector outside that domain, and a measure-and-prepare operation can always choose a state of infinite energy.

Therefore statements about energy, variance, or conservation during an update need extra assumptions on the apparatus and its Kraus operators. Likewise, continuous-record conditioning needs an instrument density or measurable conditional-state construction; dividing an event output by a zero singleton probability is undefined.

Exercises

1. Suppose two candidate inputs have success probabilities $p_1=1$ and $p_2=1/4$, and are initially equally likely. Find their posterior weights after success.

Solution

The unnormalized weights are $1/2$ and $1/8$, so the posterior weights are $4/5$ and $1/5$. This is the reweighting required by selective measurement.

2. Does a zero-probability event allow an arbitrary conditional output to be used in later predictions?

Solution

Its unnormalized branch is zero, so assigning an auxiliary state to that impossible branch does not change any joint probability. Such an assignment is a convention, not a physically predicted posterior. It must not be used to claim behavior after an event that the model says cannot occur.

References

• Busch, Paul, Pekka Lahti, Juha-Pekka Pellonpää, and Kari Ylinen, Quantum Measurement, Springer, 2016 — operations, conditional states, and measurement models. Publisher.
• Heinosaari, Teiko, and Mário Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement, Cambridge University Press, 2012, Chapters 4–5 — state transformations and instruments. Publisher.