Quantum States Over Time

• Quantum States Over Time are operator-valued representations that extend ordinary density matrices to capture temporal and spatiotemporal correlations.
• They unify causal and acausal quantum correlations by reformulating dynamics with constructs such as Choi operators, Jordan products, and pseudo-density matrices.
• Their experimental realizations through interferometry, time-bin tomography, and continuous-variable methods provide insights into non-Markovian effects and quantum transport.

Quantum states over time are operator-valued representations of temporal, and more generally spatiotemporal, correlations that extend the density-operator formalism beyond a single time slice. In the modern formulation, a one-step dynamics $(\rho,E)$, with $\rho_A$ an initial state and $E:A\to B$ a quantum channel, is assigned an operator on ${\cal H}_A\otimes{\cal H}_B$ satisfying the marginality conditions $\Tr_A[E\star\rho]=E(\rho)$ and $\Tr_B[E\star\rho]=\rho$; multipartite versions assign operators on tensor products of Hilbert spaces associated with multiple events or regions in spacetime (Lie et al., 25 Jul 2025). These objects are typically normalized and often Hermitian, but they are not required to be positive, so they generalize rather than duplicate ordinary density matrices (Lie et al., 2024).

1. Conceptual scope and the asymmetry between space and time

In standard quantum theory, a composite system at one time is represented by a joint state on a tensor-product Hilbert space, whereas temporal evolution is represented by a channel. This creates a formal asymmetry: acausal relations are encoded by density operators, while causal relations are encoded by completely positive trace-preserving maps. A central motivation for quantum states over time is to remove this asymmetry, or at least to reformulate it so that temporal and spatial correlations can be treated by a common mathematical language (Lie et al., 2023).

A decisive obstacle was identified in the no-go analysis of Horsman, Heunen, Pusey, Barrett, and Spekkens. They considered five desiderata for a state over time—Hermiticity, preservation of probabilistic mixing, the appropriate classical limit, the appropriate single-time marginals, and associativity—and showed that no construction satisfies all these requirements simultaneously (Horsman et al., 2016). In the same analysis, if Hermiticity is dropped, the construction is fixed uniquely up to an ordering convention, namely ordinary matrix multiplication or its reverse (Horsman et al., 2016). This established that any viable formalism must relax at least one feature that ordinary single-time density operators possess.

A later development replaced the earlier axioms by operationally motivated conditions better adapted to arbitrary spacetime regions. Under completeness, compositionality, classical conditionability, and time-reversal symmetry, the Fullwood–Parzygnat state-over-time function is essentially unique (Lie et al., 2023). This shifted the field from asking whether temporal states can literally resemble ordinary states to asking which generalized state-like object is operationally singled out.

2. Bipartite formalisms: Choi operators, Jordan products, and pseudo-density matrices

For a one-step dynamics $(\rho,E)$, a standard starting point is the Choi or Jamiołkowski operator

$$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$

Three basic products used to define a quantum state over time are

$$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$

and the symmetric Fullwood–Parzygnat product

$$E\star_{FP}\rho := \frac{1}{2}\bigl(E\star_L\rho + E\star_R\rho\bigr).$$

All three satisfy the marginality condition, but $\rho_A$0 and $\rho_A$1 need not be positive or even Hermitian, whereas $\rho_A$2 is the Hermitian part (Lie et al., 25 Jul 2025).

For a single channel $\rho_A$3, the multipartite uniqueness program identifies the bipartite Fullwood–Parzygnat state over time as

$\rho_A$4

with $\rho_A$5 the Jamiolkowski operator and $\rho_A$6 the anti-commutator (Lie et al., 2024). This operator is Hermitian and unit trace, linear in $\rho_A$7, and for qubits it coincides with the pseudo-density matrix (Lie et al., 2024). The same structure appears in the “bloom” formalism, where a canonical state over time for a single CPTP map is defined by

$\rho_A$8

with the bloom map built from canonical broadcasting (Fullwood, 2023).

A related transport-oriented reformulation uses the Jordan product

$\rho_A$9

and defines a “stote” by

$E:A\to B$0

Here $E:A\to B$1 is the Jamiołkowski matrix of a CPTP map, $E:A\to B$2 has marginals $E:A\to B$3 and $E:A\to B$4, and the set of all such $E:A\to B$5 plays the role of a quantum analogue of classical couplings (Hoogsteder-Riera et al., 7 Apr 2025).

The non-positivity of these operators is not accidental. The multipartite QSOT literature states explicitly that such objects are Hermitian and unit trace but not necessarily positive, and that negative eigenvalues witness genuinely temporal, non-spatially embeddable correlations (Lie et al., 2024). A plausible implication is that temporal correlations are being represented by a different kind of operator constraint than the positivity familiar from purely spatial subsystems.

3. Operational realizations: interferometry, time-bin tomography, and continuous variables

An operational definition based on measurement was provided through the notion of a causally agnostic measurement. In that framework, systems $E:A\to B$6 are in a quantum state over spacetime $E:A\to B$7 if the interference term in an interferometric protocol is

$E:A\to B$8

whenever the unitary intervention at region $E:A\to B$9 is ${\cal H}_A\otimes{\cal H}_B$0 (Lie et al., 25 Jul 2025). The central structural result is that a quantum measurement is causally agnostic if and only if it can be implemented by a multi-arm interferometry, so ordinary density operators, temporal QSOTs, and process-matrix-like objects are unified through interference rather than through a direct Born rule on positive states (Lie et al., 25 Jul 2025).

Quantum optics provides a concrete experimental realization of temporal degrees of freedom through time-bin encoding. In a single-photon setting, a qubit can be encoded as

${\cal H}_A\otimes{\cal H}_B$1

where ${\cal H}_A\otimes{\cal H}_B$2 and ${\cal H}_A\otimes{\cal H}_B$3 are Gaussian wave packets separated by a delay ${\cal H}_A\otimes{\cal H}_B$4, and more generally a qudit can be delocalized over ${\cal H}_A\otimes{\cal H}_B$5 temporal modes (Sedziak-Kacprowicz et al., 2020). After propagation through a dispersive fiber, time-resolved single-photon detection defines a continuous POVM ${\cal H}_A\otimes{\cal H}_B$6 on the time-bin subspace, with

${\cal H}_A\otimes{\cal H}_B$7

after including detector jitter (Sedziak-Kacprowicz et al., 2020). Numerical tomography using this POVM reported average fidelities over 9261 qubit states such as LS ${\cal H}_A\otimes{\cal H}_B$8 and MLE ${\cal H}_A\otimes{\cal H}_B$9 for $\Tr_A[E\star\rho]=E(\rho)$0 m and $\Tr_A[E\star\rho]=E(\rho)$1 ps, and for entangled time-bin Bell states at $\Tr_A[E\star\rho]=E(\rho)$2 ps and $\Tr_A[E\star\rho]=E(\rho)$3 km the paper reports LS $\Tr_A[E\star\rho]=E(\rho)$4 with 36 outcomes (Sedziak-Kacprowicz et al., 2020). Time-bin states and temporal modes therefore supply a direct operational instance of “quantum states over time” in quantum optics.

For continuous variables, the proposal is to treat different instances of time as different quantum modes. Six definitions of spacetime states are proposed, based on four measurement processes: quadratures, displaced parity operators, position measurements, and weak measurements (Zhang et al., 2019). In one formulation, the spacetime Wigner function is

$\Tr_A[E\star\rho]=E(\rho)$5

where $\Tr_A[E\star\rho]=E(\rho)$6 is the displaced parity operator, and the corresponding spacetime density matrix is obtained by the inverse transform over $\Tr_A[E\star\rho]=E(\rho)$7 (Zhang et al., 2019). This makes the finite-dimensional pseudo-density-matrix logic available in the continuous-variable multi-mode regime.

4. Multipartite extension, covariance, and Markovian structure

The multipartite problem is not a trivial iteration of the bipartite one. The 2024 uniqueness result shows that two simple assumptions—linearity in the initial state and a quantum analog of conditionability for multipartite probability distributions—uniquely single out a multipartite extension of bipartite quantum states over time (Lie et al., 2024). The resulting multipartite QSOT for an $\Tr_A[E\star\rho]=E(\rho)$8-chain $\Tr_A[E\star\rho]=E(\rho)$9 is obtained by the iterative formula

$\Tr_B[E\star\rho]=\rho$0

where each step uses the same one-step product (Lie et al., 2024).

This iterative structure yields a natural characterization of quantum Markovianity. For an $\Tr_B[E\star\rho]=\rho$1-step process on a $\Tr_B[E\star\rho]=\rho$2-level system, a Markovian dynamics has

$\Tr_B[E\star\rho]=\rho$3

while failure of such a factorization indicates non-Markovianity (Lie et al., 25 Jul 2025). The same paper emphasizes that mixed and non-factorizable QSOTs are crucial in the temporal setting, and that QSOTs can serve as low-order witnesses of non-Markovianity with $\Tr_B[E\star\rho]=\rho$4 parameters, compared with $\Tr_B[E\star\rho]=\rho$5 for process tensors (Lie et al., 25 Jul 2025).

A distinct but related program formulates general covariance for states over time in the setting of finite-dimensional $\Tr_B[E\star\rho]=\rho$6-algebras. There, an $\Tr_B[E\star\rho]=\rho$7-step process is assigned a canonical state over time

$\Tr_B[E\star\rho]=\rho$8

built from canonical broadcasting and bloom maps, with single-time marginals $\Tr_B[E\star\rho]=\rho$9 (Fullwood, 2023). Under $(\rho,E)$0-isomorphisms $(\rho,E)$1, the canonical state satisfies

$(\rho,E)$2

which is their notion of general covariance for states over time (Fullwood, 2023). This suggests that the spacetime analogy can be made structural rather than merely heuristic.

5. Time-symmetric inference and time-asymmetric dynamics

A separate line of work uses “state over time” language for retrodiction and smoothing. The past quantum state formalism assigns to time $(\rho,E)$3 the pair

$(\rho,E)$4

where $(\rho,E)$5 is the usual forward-conditioned density matrix and $(\rho,E)$6 is an effect matrix propagated backward from later measurements (Gammelmark et al., 2013). For a generalized measurement with Kraus operators $(\rho,E)$7, the probability of the outcome $(\rho,E)$8 at time $(\rho,E)$9, conditioned on the full record up to a later time $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$0, is

$$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$1

This formalism generalizes pre- and post-selected quantum states to arbitrary continuous measurement scenarios and Markovian dissipation (Gammelmark et al., 2013).

The same time-symmetric logic extends to hybrid quantum–classical systems undergoing time-irreversible Lindblad dynamics. In that setting, the filtered state $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$2 conditions on past records only, while the smoothed hybrid state $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$3 uses both past and future records through a forward object $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$4 and a backward effect vector $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$5 (Budini, 2017). The construction is explicitly unbiased: $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$6 so averaging over future and past records recovers the unconditional hybrid Lindblad evolution (Budini, 2017).

By contrast, time-asymmetric quantum mechanics develops a mathematically different notion of states over time based on Hardy rigged Hilbert spaces. There, prepared in-states $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$7 and out-observables $$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$8 satisfy semigroup evolution

$$J[E] := \sum_{i,j} \ket{i}\!\bra{j}_A \otimes E(\ket{j}\!\bra{i})_B.$$9

rather than a unitary group on all $$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$0 (Bohm et al., 2011). In this framework, Gamow vectors associated with resonance poles $$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$1 yield exact exponential decay

$$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$2

for $$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$3 (Bohm et al., 2011). This is a distinct usage of temporal state concepts: not an operator on multiple time-labelled Hilbert spaces, but a boundary-condition-based formulation of intrinsic irreversibility.

6. Applications, controversies, and current frontiers

A persistent misconception is that a state over time should be positive whenever it is called a state. The modern literature rejects this. The multipartite QSOT program states directly that such operators are Hermitian and unit trace but not necessarily positive (Lie et al., 2024). The interferometric account sharpens the point: by tying QSOTs to interference terms rather than direct Born-rule probabilities, the puzzle of non-positive eigenvalues is resolved because the interference term can be any complex number as long as total probabilities remain nonnegative (Lie et al., 25 Jul 2025). The 2016 no-go theorem explains why this departure from positivity is structural rather than optional (Horsman et al., 2016).

Operational indistinguishability is another nonclassical feature. Mixed QSOTs can represent ensembles of dynamics, and different ensembles can yield the same QSOT. An explicit qubit example shows that two different ensembles of $$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$4-rotation channels and pure inputs have the same mixed QSOT, so they are indistinguishable under causally agnostic interferometric schemes (Lie et al., 25 Jul 2025). The same paper identifies a new spatiotemporal correlation called synchronization, defined by

$$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$5

and interprets it as a resource for building a temporal reference frame in a time-reversal-symmetric setting (Lie et al., 25 Jul 2025).

States-over-time methods have also been used to formulate quantum transport. In that setting, the stote

$$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$6

plays the role of a quantum coupling, and the transport cost is postulated to be linear in $$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$7 (Hoogsteder-Riera et al., 7 Apr 2025). For a cost matrix $$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$8, the optimal cost between $$E\star_L \rho := (\rho_A\otimes\mathbb{I}_B)J[E],\qquad E\star_R \rho := J[E](\rho_A\otimes\mathbb{I}_B),$$9 and $$E\star_{FP}\rho := \frac{1}{2}\bigl(E\star_L\rho + E\star_R\rho\bigr).$$0 is

$$E\star_{FP}\rho := \frac{1}{2}\bigl(E\star_L\rho + E\star_R\rho\bigr).$$1

For the unitary-invariant cost $$E\star_{FP}\rho := \frac{1}{2}\bigl(E\star_L\rho + E\star_R\rho\bigr).$$2, the paper finds that for pure qubits the unitary cost equals $$E\star_{FP}\rho := \frac{1}{2}\bigl(E\star_L\rho + E\star_R\rho\bigr).$$3, which suggests that the natural quantum cost behaves like the square of a distance and is qualitatively different from Monge’s classical transport (Hoogsteder-Riera et al., 7 Apr 2025).

Open-system scattering theory offers a different physical frontier. In an open double quantum-dot system with interdot Coulomb interaction, exact time-evolving many-electron states were constructed for arbitrary initial conditions (Nishino et al., 2024). For initial states of localized electrons on the dots, the authors found exact time-evolving resonant states of the form

$$E\star_{FP}\rho := \frac{1}{2}\bigl(E\star_L\rho + E\star_R\rho\bigr).$$4

which are normalizable because their wave functions are restricted to a finite space interval due to causality (Nishino et al., 2024). The survival probability of electrons on the dots decays exponentially in time on one side of an exponential point of resonance energies and oscillates during the decay on the other side because of interference between two resonance energies (Nishino et al., 2024). This suggests that temporal-state concepts also capture transient resonance structure in open, interacting systems.

Across these diverse formulations, the common theme is the replacement of “state at one time plus dynamics” by a single object that encodes correlations across events. The resulting objects differ in ontology, positivity, and operational role, but the accumulated results support a stable core claim: temporal quantum correlations can be represented by operator-valued state-like structures, provided one relinquishes the expectation that every such structure must behave like an ordinary density matrix at a single time.