Joint Splitting Probabilities

• Joint Splitting Probabilities are methods that decompose a marginal probability or density into contributions indexed by auxiliary labels, channels, or observables across diverse fields.
• They reconcile state-dependent and state-independent elements in frameworks such as quantum weak measurements, random orthonormal states, and constrained transportation problems.
• Applications range from resolving conductance channels to modeling first-passage events and optimal control, highlighting the interplay between hidden structures and observable outcomes.

Joint splitting probabilities denote probability assignments that resolve an observed outcome into contributions indexed by additional labels that are not part of the final marginal event itself. In the cited literature, the phrase appears in several distinct but structurally related settings: complex weak-measurement decompositions for non-commuting observables, joint probability densities for how projection weight is distributed across channels or subspaces, transportation policies that split fixed marginals across pairs of indices, and first-passage objects that retain both exit identity and auxiliary hidden-state information (Hofmann, 2013, Gzyl, 2021, Sezik et al., 10 Aug 2025).

1. Domain-specific meanings and common structure

Across the surveyed papers, the term is attached to several mathematically different objects rather than to a single standardized definition. What recurs is a decomposition of a marginal probability, density, or transport mass into contributions labeled by auxiliary variables, channels, observables, or internal states.

Setting	Joint object	Characteristic relation
Weak-measurement quantum mechanics	$p(a,b)$, $p(m|a,b)$	$p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$
Random orthonormal states	$P_{NK}(t_1,\dots,t_R)$	$t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$
Transportation / joint tables	$T_{ij}$	$\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$
Hidden-state first passage	$\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$	backward equation on $(x_0,y_0)$
Active matter in confinement	$\pi_\lambda(z,w)$	$p(m|a,b)$0
Rare-event control	$p(m|a,b)$1	$p(m|a,b)$2

Observable probabilities are then recovered by summing, integrating, or marginalizing over the auxiliary labels. In quantum weak-measurement formalisms those labels are incompatible observables; in random-matrix theory they are subspace projection weights; in first-passage theory they can be boundary identity, exit state, or number of jumps; in transportation problems they are pairwise allocations of prescribed marginals (Hofmann, 2013, Alonso et al., 2015, Gzyl, 2021, Sezik et al., 10 Aug 2025).

2. Complex joint splitting in quantum weak measurements

In the weak-measurement formulation of Hofmann, the starting point is a complex joint probability for two generally non-commuting observables with eigenstates $p(m|a,b)$3 and $p(m|a,b)$4. For a state described by density operator $p(m|a,b)$5, weak measurement of $p(m|a,b)$6 followed by a precise measurement in the $p(m|a,b)$7-basis defines

$p(m|a,b)$8

This quantity is generally complex, is operationally reconstructed from weak values conditioned on post-selection of $p(m|a,b)$9, and completely characterizes the quantum state through

$p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$0

The probability of a different measurement outcome $p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$1 is then written in the splitting form

$p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$2

Here $p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$3 carries the state dependence, while $p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$4 is state-independent and determined solely by overlaps between eigenstates (Hofmann, 2013).

This construction preserves the algebraic structure of classical probability splitting while abandoning positivity and, in general, reality. The objects $p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$5 and $p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$6 can be complex, and there is no classical sample space of simultaneous sharp values for $p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$7 and $p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$8. Nonetheless, the marginals reproduce standard Born probabilities, and the full state is recoverable from the complex joint distribution. In this sense, joint splitting probabilities become quasi-probabilistic weights attached to incompatible measurement labels rather than frequencies of jointly realized events (Hofmann, 2013).

The same paper emphasizes that the phase of the complex conditional probability encodes an action-like quantity,

$p(m)=\sum_{a,b}p(m|a,b)\,p(a,b)$9

and identifies this phase as the source of “logical tension” among incompatible properties. A probability is therefore “split” into complex contributions whose coherent recombination yields the real positive marginal $P_{NK}(t_1,\dots,t_R)$0 (Hofmann, 2013).

3. Deterministic basis transformations and the exclusion of precise non-contextual quantum joints

A related formulation represents the density operator as a complex joint probability

$P_{NK}(t_1,\dots,t_R)$1

for any pair of observables with non-zero mutual overlap. Transformations to a different joint representation are expressed by complex conditional probabilities,

$P_{NK}(t_1,\dots,t_R)$2

and reversibility is encoded by

$P_{NK}(t_1,\dots,t_R)$3

In this framework, determinism is not a hidden-variable assignment of simultaneous values, but the state-independent reversibility of transformations between complete joint representations. Coarse-graining drives these complex conditional probabilities toward sharply peaked classical relations, so classical reality emerges as an approximation to the transformation laws of quantum determinism (Hofmann, 2011).

This interpretation has an immediate conceptual limit. “Excluding joint probabilities from quantum theory” studies two non-commuting observables in a two-dimensional Hilbert space and proves that there is no precise joint probability that applies for any quantum state and is consistent with quantum imprecise probabilities. The argument uses lower and upper probability operators $P_{NK}(t_1,\dots,t_R)$4 and $P_{NK}(t_1,\dots,t_R)$5, with a precise joint probability required to lie between the corresponding lower and upper probabilities for all states. In that sense, a universal, non-contextual, everywhere positive precise joint probability is excluded even for two non-commuting observables in dimension two. Context-dependent joint probabilities remain admissible, for example those associated with sequential measurement schemes, but they are still constrained by the imprecise bounds (Allahverdyan et al., 2018).

Taken together, these papers fix a persistent ambiguity. Complex joint splitting rules can be exact and state-complete as representations, but they are not classical joint probabilities. Conversely, demanding a precise positive non-contextual joint probability is too strong: the imprecise-probability analysis shows that such an object is not available in general (Hofmann, 2011, Allahverdyan et al., 2018).

4. Random orthonormal states, projection weights, and channel splitting

In random-matrix theory, joint splitting probabilities arise as joint probability densities for how projection weight is distributed across a fixed subspace by one or several random orthonormal states. Let $P_{NK}(t_1,\dots,t_R)$6 or $P_{NK}(t_1,\dots,t_R)$7 be Haar distributed, and let

$P_{NK}(t_1,\dots,t_R)$8

be the projection probabilities of $P_{NK}(t_1,\dots,t_R)$9 orthonormal columns onto a $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$0-dimensional subspace. The central object is the joint density

$t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$1

For a single vector, the distribution is Beta: $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$2 in the orthogonal case and

$t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$3

in the unitary case. The mean is $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$4 in both symmetry classes, while the variances differ by the usual orthogonal–unitary prefactor shift (Alonso et al., 2015).

For several orthonormal states, the $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$5 are correlated because the underlying columns are orthogonal. The paper derives determinant-based integral representations for general $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$6, and explicit two-point formulas for specific $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$7. In the unitary case with $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$8 and $t_\xi=\sum_{j=1}^K |w_{j\xi}|^2$9,

$T_{ij}$0

whereas the orthogonal case with the same $T_{ij}$1 involves a complete elliptic integral and develops a square-root singularity as $T_{ij}$2 (Alonso et al., 2015).

The same mathematics describes partial conductances in a two-terminal mesoscopic scattering problem. If $T_{ij}$3 denotes the partial conductance associated with an incoming mode, then $T_{ij}$4 is the Landauer–Büttiker conductance, and the joint density $T_{ij}$5 becomes a joint splitting law for conductance across channels. In this usage, “joint splitting” refers to the statistical allocation of total norm or conductance among correlated channels rather than to first-passage competition or quasi-probabilistic observables (Alonso et al., 2015).

5. Joint tables, transportation policies, and entropic geometry

A distinct classical use of the phrase appears in the study of joint probabilities with fixed marginals. Here the unknown is a joint matrix $T_{ij}$6 satisfying

$T_{ij}$7

and the entries are interpreted either as joint probabilities, transportation policies, or pixel intensities. In the transportation interpretation, $T_{ij}$8 is the fraction of supply at site $T_{ij}$9 shipped to demand site $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$0; “joint splitting probabilities” are therefore feasible ways of splitting prescribed marginal masses across pairs $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$1. Additional size constraints $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$2 and cost constraints $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$3 are incorporated by a linear system $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$4 (Gzyl, 2021).

The paper solves this constrained splitting problem by maximum entropy in the mean. The unknown $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$5 is represented as the expectation of a random vector $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$6 supported on the box $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$7, and the relative entropy with respect to a product reference measure on the box endpoints is maximized under the mean constraint $\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$8. The resulting MEM solution is

$\sum_j T_{ij}=p_i,\ \sum_i T_{ij}=q_j$9

and in the special case $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$0, $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$1,

$\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$2

This gives an entropic family of admissible joint splittings compatible with marginals and linear expectations (Gzyl, 2021).

The same work endows the space of such splittings with a Hessian Riemannian geometry. In the “pixel space” of probabilities, the induced diagonal metric is

$\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$3

which diverges at the boundary of the feasible box. Geodesics and distances defined by this metric provide a geometry of alternative joint splittings, while a stepwise reduction of the cost constraint yields an interior-point-like approach to the minimum-cost transportation problem (Gzyl, 2021).

6. Hidden states, first passage, and active-matter confinement

For two-dimensional Markov processes $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$4, a direct joint analogue of splitting probabilities records both the exit boundary of $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$5 and the value of the hidden variable $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$6 at exit. With $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$7 confined to an interval $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$8 and $\Pi(x_{\rm exit},y_{\rm exit}\mid x_0,y_0)$9 taking values in $(x_0,y_0)$0, the joint splitting probability is

$(x_0,y_0)$1

the probability that $(x_0,y_0)$2, initialized at $(x_0,y_0)$3, first exits $(x_0,y_0)$4 through $(x_0,y_0)$5 and that $(x_0,y_0)$6, initialized at $(x_0,y_0)$7, equals $(x_0,y_0)$8 at exit time. It satisfies the stationary backward equation

$(x_0,y_0)$9

with absorbing or Robin boundary conditions, and conditional splitting probabilities are then defined by Bayes’ theorem,

$\pi_\lambda(z,w)$0

The paper develops generic spectral formulas for decoupled Brownian $\pi_\lambda(z,w)$1 with an autonomous hidden state $\pi_\lambda(z,w)$2, and explicit coupled calculations for run-and-tumble motion, diffusion in an intermittent piecewise-linear potential, and diffusion with stochastic resetting (Sezik et al., 10 Aug 2025).

A complementary first-passage reformulation appears for run-and-tumble particles confined between two walls. There, the process is recast as a jump process with jump kernel $\pi_\lambda(z,w)$3, and the splitting probability

$\pi_\lambda(z,w)$4

satisfies the integral equation

$\pi_\lambda(z,w)$5

Its derivative gives the stationary bulk density,

$\pi_\lambda(z,w)$6

and the boundary values encode adsorption,

$\pi_\lambda(z,w)$7

Iterating the integral equation yields a decomposition

$\pi_\lambda(z,w)$8

where $\pi_\lambda(z,w)$9 is the probability of hitting the opposite wall for the first time in exactly $p(m|a,b)$00 jumps without intermediate return. In one dimension,

$p(m|a,b)$01

with $p(m|a,b)$02 (Frydel, 2024).

The phase-space formulation of confined active matter sharpens this relation. For RTP, ABP, and AOUP dynamics in dimensionless variables $p(m|a,b)$03, the velocity-resolved splitting probability $p(m|a,b)$04 and the stationary phase-space density $p(m|a,b)$05 obey the exact correspondence

$p(m|a,b)$06

After integration over velocities,

$p(m|a,b)$07

Thus the stationary spatial density is literally the derivative of the splitting probability, while the wall-adsorbed fraction equals the wall splitting probability. The correspondence holds in arbitrary spatial dimension for Markovian active dynamics in phase space (Frydel, 27 Jun 2026).

7. Memory effects, optimal control, and monitored quantum competition

For non-Markovian Gaussian first-passage problems, splitting probabilities depend on post-first-passage path statistics rather than only on geometry. In the two-target one-dimensional setting with absorbing points at $p(m|a,b)$08 and $p(m|a,b)$09, the splitting probabilities $p(m|a,b)$10 and $p(m|a,b)$11 satisfy

$p(m|a,b)$12

but the key identity is

$p(m|a,b)$13

where $p(m|a,b)$14 are conditional mean trajectories after first passage to target $p(m|a,b)$15 if motion is continued. For fractional Brownian motion with $p(m|a,b)$16, the small-$p(m|a,b)$17 behavior is

$p(m|a,b)$18

The paper derives self-consistent equations for $p(m|a,b)$19 and $p(m|a,b)$20 and validates the resulting splitting probabilities in viscoelastic-fluid experiments, where marked deviations from the Brownian law $p(m|a,b)$21 are directly traced to out-of-equilibrium post-first-passage trajectories (Dolgushev et al., 2024).

In stochastic control of rare reactive events, the splitting probability reappears as the committor $p(m|a,b)$22, the probability to hit metastable set $p(m|a,b)$23 before $p(m|a,b)$24. A time-dependent analogue $p(m|a,b)$25 solves a generalized bridge problem, and in the bistable spectral approximation

$p(m|a,b)$26

The optimal controller that generates a bridge process statistically identical to the original reactive trajectory ensemble is

$p(m|a,b)$27

This makes the committor gradient not only a reaction-coordinate object but also a force field that guarantees reactivity while preserving the statistics of the transition path ensemble (Singh et al., 2024).

A further quantum extension appears in monitored continuous-time quantum walks with two targets. There the splitting probabilities

$p(m|a,b)$28

describe which target is detected first under repeated monitoring. By introducing symmetric and antisymmetric target superpositions $p(m|a,b)$29, the two-target problem maps onto a pair of single-target detection problems. For large systems, sampling times below

$p(m|a,b)$30

produce a universal regime

$p(m|a,b)$31

independent of bulk initial condition and sampling time, whereas $p(m|a,b)$32 yields a nonuniversal regime with pronounced peaks and dips as functions of $p(m|a,b)$33 and $p(m|a,b)$34. At resonant sampling times, dark states can make $p(m|a,b)$35 (Singh et al., 22 Jan 2026).

A neighboring asymptotic theory for symmetric continuous jump processes shows that even the simplest boundary transmission problem retains microscopic information: the transmission probability from the boundary scales as

$p(m|a,b)$36

which explains why the naive continuous-limit prediction $p(m|a,b)$37 fails for genuine jump dynamics (Klinger et al., 2022). This suggests a broader pattern: whenever splitting observables are refined by hidden labels, incompatible observables, or microscopic jump structure, the resulting “joint” object carries information absent from the marginal splitting probability alone.