Closed-Loop Quantum Probability Decomposition

• Closed-loop decomposition of quantum probabilities is a framework that examines how tomographically complete measurement representations can be inverted without losing operational significance.
• The approach analyzes cyclic loop products and adaptive feedback in sequential or continuous measurements to reinterpret interference and non-additivity as inherent properties of quantum dynamics.
• Key insights reveal that while exact reconstruction uniquely determines quantum states, it fails to ensure robustness in unbounded dimensions, highlighting a no-go for robust operational closure.

Searching arXiv for the primary and closely related papers to ground the article in current literature. In the arXiv literature surveyed here, “closed-loop decomposition of quantum probabilities” does not denote a single formalism. It names a cluster of problems about whether quantum probabilities can be organized into a representation from which one can pass to probability data, reason there, and return to the underlying quantum description without losing physically relevant structure; whether transition probabilities can be resolved into closed cyclic products of amplitudes; whether sequential or continuous measurements admit adaptive decompositions into weak steps; and whether probabilities for closed systems can be assigned to histories without external observers. A central recent result is that exact reconstructibility from probabilities is weaker than operational closure: in unbounded-dimensional quantum theory, a tomographically complete probability representation can fail to preserve the physically meaningful notion of approximation, even when it determines the state uniquely (Hausmann et al., 26 Jan 2026).

1. Multiple meanings of “closed loop” in quantum probability

The phrase appears in several technically distinct settings. In one line of work, a quantum state is represented by the tuple of outcome distributions associated with a tomographically complete measurement family, and the “closed-loop” question is whether approximate reasoning in that probability representation can be pulled back to density operators without loss. In another, standard transition probabilities are decomposed into cyclic products of amplitudes associated with closed sequences of states, with interference reinterpreted as contributions from different loop classes. Elsewhere, “closed loop” refers to a feedback-controlled weak-measurement process, to sequential collapse-based constructions of joint probabilities for noncommuting observables, or to closed-system probabilities for cosmological histories in the absence of external measurement (Hausmann et al., 26 Jan 2026, Rave, 1 Jun 2026, Florjanczyk et al., 2015, Morgan, 2021, Craig, 2016).

Taken together, these usages suggest that the unifying issue is not a single algebraic object but a recurring demand for closure under physically relevant inference. Depending on context, closure means robust invertibility of a state-to-probabilities map, path-independent composition of weak measurements, positivity-preserving construction of sequential joint probabilities, or decoherence-based assignability of probabilities to histories of a closed system. This also explains why controversies around the topic are usually controversies about operational meaning rather than about exact formal reconstruction.

2. Probability representations of quantum states and the failure of operational closure

For a separable Hilbert space $\mathcal H$, the state-to-probabilities map studied in “Against probability: A quantum state is more than a list of probability distributions” is

$$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$

where $\mathcal M$ is a tomographically complete family of measurements and $P_M(\rho)$ is the outcome distribution of $M$ on $\rho$. The associated operational metric is

$$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$

to be compared with the trace distance

$$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$

The paper defines topological robustness by the implication

$$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$

for all subsets $\Sigma$ of state space and all sequences $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$0. Since $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$1, the converse implication always holds, so robustness means that the representation induces the same approximation theory as trace distance. Proposition 2.2 states that robustness is equivalent to identity of the $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$2- and $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$3-topologies on $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$4, and also equivalent to completeness of that space with respect to $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$5, where

$$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$6

The central conclusion is sharply negative: injectivity alone does not yield an operationally meaningful closed loop, because the inverse map need not be continuous in the relevant norms (Hausmann et al., 26 Jan 2026).

The paper’s antisymmetric-state example exhibits the failure explicitly. For $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$7-dimensional $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$8 and $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$9, the states

$\mathcal M$0

are compared with the ideal-randomness set

$\mathcal M$1

They satisfy

$\mathcal M$2

so they are not approximately ideal in trace distance. Yet for local/product measurements $\mathcal M$3,

$\mathcal M$4

and

$\mathcal M$5

so

$\mathcal M$6

while

$\mathcal M$7

This is the failure of robustness for subsystem-preserving local statistics.

The general no-go result is Theorem 3.1: if $\mathcal M$8 has asymptotic entropy at most $\mathcal M$9 for a zero-sequence $P_M(\rho)$0, then $P_M(\rho)$1 is not robust. Corollaries then state that $P_M(\rho)$2 is not robust and that efficient families built from polynomially many fiducial effects are not robust. The paper discusses SIC-POVM-type fiducial representations and QBist probabilistic formulations as examples of such efficient constructions, and argues that in unbounded dimension robust probability representations must be extremely large and unstructured. The same work also proves a GPT counterexample: there exists a GPT for which the topologies induced by $P_M(\rho)$3 and $P_M(\rho)$4 are different on state space, despite $P_M(\rho)$5 being stable. A common misconception addressed here is that tomographic completeness is enough; the paper’s point is that exact reconstructibility does not imply robust invertibility.

3. Loop decompositions, interference, and non-additivity

A different use of the topic begins from transition amplitudes

$P_M(\rho)$6

and associates to a cyclic sequence $P_M(\rho)$7 the loop product

$P_M(\rho)$8

For a finite-dimensional Hilbert space, “Closed-loop Structure of Quantum Probabilities from Unitarity” states that

$P_M(\rho)$9

with each summand interpreted as the loop product for $M$0. The paper’s claim is that the decomposition is a direct consequence of unitarity, not an external reinterpretation. The quadratic form of the Born rule is read as the product of forward and reverse amplitudes, interference is re-described as the contribution of non-self-retracing loops, and the phase-invariant quantities associated with loops are Bargmann invariants with loop phase

$M$1

If $M$2 is the reversed loop, then $M$3, so

$M$4

The paper therefore treats constructive and destructive interference as phase-weighted contributions of reversed loop pairs rather than as unexplained cross terms (Rave, 1 Jun 2026).

A separate, lattice-theoretic decomposition treats quantum probabilities as non-additive because the Birkhoff–von Neumann lattice $M$5 is orthomodular and modular, but not distributive. The basic non-additivity operator is

$M$6

with

$M$7

and the paper derives

$M$8

Higher-order Möbius operators,

$M$9

decompose inclusion–exclusion failures and are tied to nested commutators. The deviation from the law of total probability is quantified by

$\rho$0

with the decomposition

$\rho$1

This framework is not cast as a literal closed-loop theory, but it resolves non-additivity into structured overlap corrections that the paper presents as observables (Vourdas, 2015).

4. Sequential and feedback decompositions of measurement probabilities

In “Continuous decomposition of quantum measurements via Hamiltonian feedback,” a two-outcome generalized measurement is realized as the long-time limit of weak interactions with a stream of probe qubits. Each probe is prepared in $\rho$2, interacts with the target through

$\rho$3

is then measured in the $\rho$4 basis, and the result updates the pointer variable $\rho$5. The weak Kraus operators are

$\rho$6

Path independence is enforced by the reversibility condition

$\rho$7

which yields

$\rho$8

The paper proves that any solution must lie in a subspace closed under anti-commutation, hence in a finite-dimensional Jordan algebra. The total walk operator satisfies

$\rho$9

and the final measurement operators are

$$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$0

with

$$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$1

The main theorem characterizes the continuously realizable measurements as block-diagonal forms determined by the Jordan decomposition of the control algebra. In this setting, a closed-loop decomposition means an adaptive random walk with immediate classical feedback, and the Born-rule probabilities are the hitting probabilities of the absorbing endpoints (Florjanczyk et al., 2015).

A different sequential construction appears in “The collapse of a quantum state as a joint probability construction.” There the issue is how to construct positive normalized joint probabilities for timelike-separated measurements of noncommuting observables. For an observable $$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$2, the Lüders update is

$$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$3

The two-step joint density for $$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$4 followed by $$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$5 is

$$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$6

Morgan formalizes this via the sequential product

$$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$7

and the collapse product

$$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$8

The resulting product is noncommutative, nonassociative, and nonlinear in its left operand. The paper’s central conceptual claim is that collapse is not a mysterious dynamical jump but a mathematical rule for constructing joint probabilities; it then shows an equivalent no-collapse picture in which Lüders transformers act on later measurement operators, and further a commuting representation with the same joint statistics. This rules out a common misconception that collapse and no-collapse descriptions necessarily define different empirical models for sequential measurements.

5. Closed systems, consistent histories, and effective autonomy

In loop quantum cosmology, the probability-assignment problem is explicitly a closed-system problem: there is no external observer, measuring device, or external time parameter in the usual sense. The consistent-histories framework therefore assigns probabilities not to measurement outcomes but to alternative coarse-grained histories, provided the corresponding branch wave functions decohere. For class operators $$d_{\mathcal M}(\rho,\sigma)\coloneq \frac12 \sup_{M\in\mathcal M}\|P_M(\rho)-P_M(\sigma)\|_1,$$9, branch wave functions $$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$0, and decoherence functional

$$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$1

a family decoheres when

$$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$2

Applied to solvable loop quantum cosmology with scalar field $$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$3 as internal time, the volume observable at $$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$4 is represented by

$$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$5

For the asymptotic two-time coarse graining “bounce” versus “singular,” the papers show that in sLQC the singular branch wave function vanishes and the bounce branch equals the full state, so

$$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$6

By contrast, in Wheeler–DeWitt quantization the corresponding result is

$$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$7

A common misconception addressed in both treatments is that single-time probabilities are enough to settle the bounce question; the point made explicitly is that bounce versus singularity is a statement about a history, not about isolated one-time alternatives (Craig et al., 2013, Craig, 2016).

A related but distinct notion of closure appears in the study of effectively isolated subsystems. For a composite system $$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$8, the paper “On the probabilistic nature of quantum mechanics and the notion of closed systems” defines $$\delta(\rho,\sigma)=\frac12\|\rho-\sigma\|_1 =\frac12\sup_M\|P_M(\rho)-P_M(\sigma)\|_1.$$9 as closed when there exists a unitary group $$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$0 such that

$$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$1

In Model 1, with $$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$2 a particle moving away from $$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$3, Lemma 1.1 gives

$$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$4

for sufficiently large separation $$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$5. In Model 2, the effective local subsystem is the dressed atom together with nearby photons, and Theorem 2 gives an explicit error-controlled approximation by autonomous dressed-atom dynamics. The paper’s conclusion is that entanglement does not by itself prevent effective closedness: what matters is approximate dynamical decoupling of local observables.

6. Other closure criteria, structural analogies, and unresolved tensions

Several further formalisms extend the vocabulary of closure and decomposition. “Irreducible decompositions and stationary states of quantum channels” develops an infinite-dimensional analogue of the decomposition of a Markov chain into transient states and closed positive recurrent communicating classes. For a quantum channel $$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$6, the fast recurrent space is

$$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$7

and the Hilbert space decomposes as

$$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$8

Minimal enclosures inside $$\lim_{n\to\infty} d_{\mathcal M}(\rho^{(n)},\Sigma)=0 \quad\Longrightarrow\quad \lim_{n\to\infty}\delta(\rho^{(n)},\Sigma)=0,$$9 are the supports of extremal invariant states, while grouped sectors $\Sigma$0 permit stationary off-diagonal coherences. This is a decomposition of long-run quantum probability flow into persistent recurrent sectors and non-fast-recurrent remainder (Carbone et al., 2015).

“Characterizing common cause closedness of quantum probability theories” uses closure in a causal-explanatory sense. For a quantum probability space $\Sigma$1, common cause closedness means that every positive correlation between compatible projections has a nontrivial Reichenbachian common cause within the same lattice. The paper proves that, for projection lattices of von Neumann algebras with faithful normal state and with at least one correlation present,

$\Sigma$2

Purely nonatomic spaces, including type $\Sigma$3 and type $\Sigma$4 factors with faithful states, are therefore common cause closed, while Hilbert-lattice models of finite systems are not (Kitajima et al., 2015).

“Binary Matroids and Quantum Probability Distributions” supplies a combinatorial decomposition for IQP/X-programs. With binary matrix $\Sigma$5, code $\Sigma$6, and matroid $\Sigma$7, the key quantity is

$\Sigma$8

The paper proves

$\Sigma$9

where $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$00 and $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$01 is the affinification of $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$02. At $$\rho \longmapsto \mathbf P_{\mathcal M}(\rho), \qquad \mathbf P_{\mathcal M}(\rho)\coloneq \bigl(P_M(\rho)\bigr)_{M\in\mathcal M},$$03, Theorem 3 states that the output distribution is uniform on an efficiently computable affine space. This is not formulated as a loop theory, but it does decompose amplitudes and correlations into code and matroid invariants (Shepherd, 2010).

Across these programs, one recurring tension is exactness versus operational significance. Exact tomography, exact loop expansions, exact sequential probability constructions, exact recurrent decompositions, and exact common-cause representations do not automatically yield stable approximation theory, efficient inference, or preserved subsystem structure. The sharpest statement of that tension is the one already identified for state representations: exact reconstructibility from probabilities is available under tomographic completeness, but a robust, subsystem-preserving, efficient, operationally closed probability description is ruled out in the unbounded-dimensional quantum settings of greatest foundational interest (Hausmann et al., 26 Jan 2026). A plausible implication is that future work will continue to search for representation frameworks that are simultaneously robust, structurally meaningful, and suitable for generalization beyond standard quantum theory.