Spatiotemporal Quantum State

• Spatiotemporal quantum state is an operator-valued representation encoding quantum correlations across distinct time slices, analogous to density operators for spatial systems.
• It uniquely merges process matrix and pseudo-density matrix approaches to represent timelike correlations, addressing measurement disturbance and nonpositivity.
• This framework underpins applications in quantum optics, computation, and reference-frame theory by jointly modeling spatial and temporal degrees of freedom.

“Spatiotemporal quantum state” denotes, in its foundational usage, an operator-valued representation of quantum correlations across different times, intended to play for timelike or mixed spacetime scenarios the role that an ordinary density operator plays for spacelike-separated systems. In this literature the object is typically defined on a tensor product of Hilbert spaces associated with distinct time slices, has unit trace, is Hermitian, and is not generally positive semidefinite. Closely related constructions appear under the names “state over time,” “canonical state over time,” “pseudo-density matrix,” and “quantum state over spacetime.” The same phrase also appears in quantum optics, continuous-variable computation, and quantum reference-frame theory for states whose degrees of freedom or operational description are jointly distributed over space and time rather than over space alone (Fullwood et al., 22 Jul 2025, Fullwood et al., 2024, Lie et al., 2023).

1. Conceptual motivation and basic setting

The central motivation is the asymmetry of standard quantum theory between space and time. Spatial correlations are represented by multipartite density operators, whereas causal evolution in time is represented by channels. Several papers seek a “causally neutral” or “state-like” formulation in which a process from an earlier region $A$ to a later region $B$ is encoded by a single operator on $A\otimes B$. In one formulation, a state-over-time function is a map

$$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$

with marginal conditions

$$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$

This turns dynamical evolution into a static bipartite object with the same ordinary marginals as a spatial state, while not requiring positivity in the usual probabilistic sense (Lie et al., 2023).

A broader operational formulation is supplied by the process-matrix framework, in which spacelike, timelike, and hybrid spatiotemporal resources are treated uniformly. Each local laboratory $A$ is assigned input and output spaces $A_I$ and $A_O$, local actions are instruments $\{\mathcal M_{a|x}\}_a$, and correlations are generated by the generalized Born rule

$P(a,b,\dots|x,y,\dots)=\tr\!\left[\left(M^{A_IA_O}_{a|x}\otimes M^{B_IB_O}_{b|y}\otimes\dots\right)W^{A_IA_OB_IB_O\dots}\right].$

Within this framework, every temporal process can be normalized into a state, $B$0, but not every positive density operator corresponds to a valid temporal process, because process matrices satisfy additional causal and normalization constraints (Costa et al., 2017).

2. Canonical state over time and its axiomatic status

A prominent construction is the Fullwood–Parzygnat, or Jordan-product, state over time,

$B$1

equivalently written with the Choi–Jamiołkowski operator $B$2 as

$B$3

This operator is Hermitian, bilinear, compatible with classical limits, and compatible with composition. It is also the operator identified in the canonical-state-over-time literature as the unique encoding of a broad class of temporal correlations, although it need not be positive (Lie et al., 2023, Fullwood et al., 2024).

The uniqueness question is nontrivial. Earlier axioms were shown not to single out a unique state-over-time function; explicit one-parameter families satisfy weakened versions of the old conditions. In response, a new operational axiomatics was proposed, based on completeness, compositionality, classical conditionability, the Jamiołkowski condition, quantum conditionability, and time-reversal symmetry. Under these operational axioms, the Fullwood–Parzygnat construction is established as the essentially unique state-over-time function. A central structural result is that completeness implies state-linearity, while completeness plus compositionality determine the entire map from its time-expansion on the identity channel (Lie et al., 2023).

In the correlation-representation literature, the same operator appears as the “canonical state over time.” For finite-dimensional algebras $B$4 and a process $B$5, it is the unique element $B$6 such that

$B$7

for the admissible observable class discussed below. This places the Jordan-product operator at the center of several otherwise distinct spacetime-state programs (Fullwood et al., 2024).

3. Sequential measurements, quasiprobability, and the spatiotemporal Born rule

For two-point sequential measurements, the basic scenario is the 4-tuple

$B$8

where $B$9 is an initial state on $A\otimes B$0, $A\otimes B$1 is a projective measurement on $A\otimes B$2, $A\otimes B$3 is a quantum channel, and $A\otimes B$4 is a projective measurement on $A\otimes B$5. The standard Lüders-von Neumann sequential probability is

$A\otimes B$6

Unlike the spatial case, this need not be representable as $A\otimes B$7 for a fixed positive density operator $A\otimes B$8. The stated obstruction is measurement disturbance (Fullwood et al., 22 Jul 2025).

The alternative proposed in “The spatiotemporal Born rule is quasiprobabilistic” is the Margenau–Hill quasiprobability

$A\otimes B$9

which obeys correct marginals and coarse-graining additivity but can be negative. The key relation is

$$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$0

with disturbance correction

$$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$1

Several sufficient conditions make $$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$2: $$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$3 maximally mixed, $$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$4 discard-and-prepare, $$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$5, or $$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$6. The central theorem states that there is a unique bipartite operator

$$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$7

such that

$$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$8

for all projective measurements. This is the spatiotemporal Born rule. The operator is Hermitian, unit trace, depends only on $$\star:\mathcal{C}(A,B)\times \mathcal{S}(A)\to \mathcal{B}(AB),$$9 and $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$0, has reduced states $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$1 and $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$2, and is not generally positive. An ordinary Born-rule representation of the actual sequential probabilities exists if and only if $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$3 for all scenarios; equivalently, the disturbance term vanishes identically. The same paper also develops a quasiprobabilistic Bayes’ rule based on Bayesian inversion of channels and shows that, when a Bayesian inverse exists, reversed quasiprobabilities satisfy a swap symmetry (Fullwood et al., 22 Jul 2025).

A closely related limitation is formulated at the level of observables. For general timelike-separated observables $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$4 and $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$5, the sequential expectation

$$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$6

is not bilinear in $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$7, so no universal operator $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$8 can reproduce all temporal correlations in the way a density operator does for spacelike correlations. This no-go result motivates restricting attention either to quasiprobabilities or to a special observable class (Fullwood et al., 2024).

4. Observable classes, pseudo-density matrices, and unification of formalisms

The maximal observable class presently known to admit a universal operator representation is the class of light-touch observables. An observable is light-touch if its spectrum is either $$\operatorname{Tr}_A\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\mathcal{E}_{B|A}(\rho_A), \qquad \operatorname{Tr}_B\!\left[\mathcal{E}_{B|A}\star \rho_A\right]=\rho_A.$$9 or $A$0 for some $A$1; equivalently, a nonzero light-touch observable is a scalar multiple of a unitary Hermitian operator. Pauli observables are canonical examples. Light-touch observables span the whole real vector space of observables in any finite-dimensional algebra, and they are exactly the largest class for which

$A$2

holds for all processes, all algebras $A$3, and all observables $A$4. In the qutrit case, a SIC-POVM $A$5 yields an orthonormal light-touch basis $A$6 (Fullwood et al., 2024).

In the multi-qubit literature, the corresponding object is the pseudo-density matrix,

$A$7

which is Hermitian and unit trace but may have negative eigenvalues. The generalized construction above recovers pseudo-density matrices as the special case obtained by choosing Pauli observables as the light-touch basis. For qutrit identity evolution, an explicit $A$8 example has eigenvalues $A$9, illustrating nonpositivity directly (Fullwood et al., 2024).

A further unification is obtained by mapping between process matrices, multiple-time states, and PDMs. Operational scenarios underlying two-time states can be represented as PDMs, and the known equivalence between bipartite process matrices and bipartite two-time states then yields a mapping from process matrices with POVMs to PDMs. One consequence is that PDMs can model processes with indefinite causal order rather than only ordinary time-ordered dynamics. This does not erase the differences between the formalisms: process matrices encode local input-output structure without fixed global causal order, multiple-time states rely on pre- and post-selection, and PDMs are reconstructed from multitime correlators; nevertheless, the mappings show that these frameworks are not isolated constructions (Liu et al., 2023).

A recurrent misconception is that a spacetime state should simply be a positive density operator with time labels. The cited literature argues against this in two ways. First, not all temporal processes admit such a representation, even in finite dimensions. Second, several of the best-developed formalisms treat nonpositivity not as a defect but as the signature that timelike correlations are not reducible to ordinary joint probabilities. In this sense, nonpositivity, rather than being ancillary, is often the defining feature of the subject (Fullwood et al., 22 Jul 2025, Fullwood et al., 2024).

5. Operational meaning: interferometry, tomography, and temporal orientation

An explicitly operational account is given by interferometric measurement. A multipartite collection of systems $A_I$0, with no assumed causal relation between the regions, is said to be in a quantum state over spacetime $A_I$1 if interferometry yields an interference term

$A_I$2

The corresponding measurement probabilities have the form

$A_I$3

A central theorem states that a quantum measurement is causally agnostic if and only if it can be implemented by interferometry. In this framework, ordinary density operators, quantum states over time, and the first-order sector of the process-matrix formalism are described by the same interference rule. Mixed QSOTs appear as convex mixtures $A_I$4, are stated to be crucial for modeling quantum non-Markovianity, and can be nonfactorizable across multiple times (Lie et al., 25 Jul 2025).

Temporal quasiprobability and tomography provide another operational layer. A generalized temporal Kirkwood–Dirac framework defines left, right, and doubled temporal KD quasiprobabilities for arbitrary multitime processes, with their real parts identified as temporal Margenau–Hill quasiprobabilities. These quantities are experimentally accessible through interferometric measurement schemes. Temporal Bloch tomography reconstructs temporal states from KD or MH correlators, thereby relating pseudo-density-operator-like objects, doubled density operators, superdensity-operator-like constructions, and MH variants within one operational framework. The doubled KD object is information complete, its diagonal reduces to the Lüders–von Neumann distribution, and its marginals give the left and right temporal KD distributions (Jia et al., 8 Jan 2026).

Temporal orientation can also be inferred from spacetime-state data. In one approach, a two-time $A_I$5-qubit PDM is reconstructed from Pauli correlators, and a forward process is associated with a CPTP map $A_I$6, while the reverse process is constructed through a recovery map obtained by inverting a unitary dilation. When the initial and final states are full rank, the Choi matrix extracted from the forward interpretation can be CP-compatible while the backward one is not, thereby selecting the arrow of time. The same work stresses that negative eigenvalues of the PDM do not, by themselves, determine temporal orientation (Liu et al., 2023).

6. Broader usages in photonics, computation, and quantum reference frames

Outside the operator-state-over-time literature, “spatiotemporal quantum state” is also used for structured photonic and bosonic states whose degrees of freedom are jointly organized in space and time. In one such line of work, spatiotemporal Laguerre–Gaussian wavepackets are ultrafast 3D structured pulses labeled by radial and azimuthal quantum numbers $A_I$7. They exhibit phase singularities, cylinder-shaped edge dislocations, multi-ring topology, and, when $A_I$8 and $A_I$9, a composite transverse orbital angular momentum made of two directionally opposite components. Strong spatiotemporal astigmatism converts these STLG wavepackets to spatiotemporal Hermite–Gaussian modes, enabling mode recognition through the resulting decoupled space-time intensity pattern (Liu et al., 2024).

A distinct photonic usage treats a single photon as occupying a composite spatiotemporal Hilbert space $A_O$0. A programmable quantum parametric mode sorter based on mode-selective quantum frequency up-conversion was demonstrated for 5 spatial Laguerre–Gaussian modes and 3 temporal Hermite–Gaussian modes, giving 15 composite signal states and 225 pump-signal pairs. In the reported implementation, temporal pump optimization achieved more than 12 dB extinction for mutually unbiased temporal mode sets, and single-mode-fiber collection further improved selectivity (Garikapati et al., 2022).

In continuous-variable measurement-based quantum computation, the relevant object is the 2D spatiotemporal cluster state: a bilayer square-lattice resource generated by multiplexing both temporal and spatial modes. The corresponding architecture combines deterministic cluster-state generation, gate teleportation, and square-lattice GKP error correction. The reported full-inseparability condition requires more than 4.5 dB squeezing for the cluster-state resource, while the complete biased-GKP-plus-repetition-code fault-tolerant architecture yields a squeezing threshold of 12.3 dB (Du et al., 2023).

Spatiotemporal state language also appears in quantum reference-frame theory. A nonrelativistic spatiotemporal quantum reference frame assigns to each subsystem both an external spatial degree of freedom $A_O$1 and an internal clock degree of freedom $A_O$2, with physical states defined on a constrained Hilbert space by total momentum and Wheeler–DeWitt constraints. In that setting, reduced states are perspective-dependent, standard Schrödinger evolution emerges relationally from the static constrained state, and relative clock uncertainty modifies reciprocal spatial spreads even in noninteracting scenarios (Suleymanov et al., 2023).

A plausible implication of this diversity is that “spatiotemporal quantum state” is not a single universally standardized term. In foundational work it usually means a Hermitian unit-trace operator encoding timelike or mixed spacetime correlations; in photonics it often means a jointly space-time structured mode or wavepacket; in continuous-variable computation it denotes a deterministic resource state distributed over temporal and spatial multiplexing; and in reference-frame theory it denotes a relational state on a constrained spacetime Hilbert space. What unifies these usages is the refusal to treat space and time as sharply separate representational domains.