Pseudo-Density Matrices in Quantum Processes

• Pseudo-density matrices are Hermitian, unit-trace operators representing quantum dynamics over time, with negativity encoding causal and temporal correlations.
• They are constructed using multi-time Pauli correlators and bloom maps, unifying descriptions of quantum states, channels, and process matrices.
• Their non-positivity and reconstructive properties enable modeling of indefinite causal order, temporal entanglement, and non-Hermitian generalizations in quantum theory.

Searching arXiv for foundational and papers on pseudo-density matrices and related formalisms. Searching arXiv for foundational and papers on pseudo-density matrices and related formalisms. Pseudo-density matrices are operator-valued representations of quantum processes across time. In the finite-dimensional formulation developed for “states over time,” a pseudo-density matrix is a Hermitian unit-trace operator on a tensor product of Hilbert spaces associated with distinct times, with each single-time marginal a genuine density matrix, but without any requirement of positivity. This departure from ordinary density matrices is the point of the construction: negative eigenvalues encode temporal and causal correlations that cannot be represented as purely spatial correlations (Fullwood, 2023). In the operational qubit-based literature, the same kind of object is reconstructed from multitime Pauli correlators, while a parallel usage appears in work on non-Hermitian reduced transition matrices, where a reduced transition operator may be regarded as a pseudo-density matrix in a bi-orthogonal spectral framework (Liu et al., 2023, Jana et al., 2024).

1. Definition, scope, and basic mathematical structure

In the algebraic formulation, let $A=A_0\otimes\cdots\otimes A_n$ be a tensor product of finite-dimensional multi-matrix algebras. A self-adjoint element $\tau\in A$ is a pseudo-density operator with respect to this factorization if, for every $i=0,\dots,n$,

$\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$

where $\operatorname{tr}_i$ denotes the partial trace over all factors except $A_i$, and $\mathcal{S}(A_i)$ denotes the density matrices on $A_i$. When $A$ is a simple matrix algebra, such an object is called a pseudo-density matrix. Hermiticity and correct one-time marginals are required; positivity is not. Because each marginal has trace $1$, the global operator also has unit trace (Fullwood, 2023).

This definition makes precise the slogan that PDMs are “states over time.” The factors $\tau\in A$0 are not interpreted as distinct spatial subsystems, but as the same physical system at times $\tau\in A$1. A PDM therefore lives on a tensor product over time rather than over space. The same operator-theoretic syntax as multipartite quantum states is retained, but the semantic content changes: the full operator need not be positive because temporal correlations involve intervention, disturbance, and causal order.

In the operational qubit formalism, a PDM over $\tau\in A$2 times for an $\tau\in A$3-qubit system is reconstructed from multitime Pauli correlators: $\tau\in A$4 Tracing out all time slots except one returns the ordinary density matrix at that time. Thus, reduction to the usual single-time state is built into the formalism (Liu et al., 2023).

A standard quantitative indicator of non-positivity is the PDM negativity

$\tau\in A$5

It satisfies $\tau\in A$6, with equality iff $\tau\in A$7 is positive semidefinite, and it is used as a “causal monotone” or a measure of temporal quantum correlations. In this sense, negative eigenvalues are not a defect of the construction but the algebraic signature that the operator represents genuinely temporal structure rather than an ordinary spatial state (Liu et al., 2023).

2. Construction from dynamics and the “state over time” program

A central construction associates a PDM to a quantum state evolving through a finite sequence of channels. For a single step $\tau\in A$8, the key ingredient is a factorization of the channel into a bloom map

$\tau\in A$9

satisfying

$i=0,\dots,n$0

Three concrete bloom maps are defined: $i=0,\dots,n$1

$i=0,\dots,n$2

$i=0,\dots,n$3

where $i=0,\dots,n$4 is the Jamiołkowski channel state. The symmetric bloom is Hermitian, classically reducible, and associative for temporal composition (Fullwood, 2023).

Given a process $i=0,\dots,n$5, the associated state over time is

$i=0,\dots,n$6

Its marginals satisfy

$i=0,\dots,n$7

and the operator is Hermitian and unit trace. It is therefore a PDM on $i=0,\dots,n$8. In this construction, the input state is the “slice” at $i=0,\dots,n$9, the output state is the slice at $\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$0, and the PDM is the local patch connecting them (Fullwood, 2023).

The qubit literature also supplies a closely related closed form in terms of Choi–Jamiołkowski matrices. For two times,

$\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$1

and for $\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$2 times one iterates

$\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$3

This recursive symmetrized-product construction is operationally matched to the Pauli-correlator definition for coarse-grained Pauli projectors (Liu et al., 2023).

The underlying conceptual aim is explicit. General relativity unifies space and time into spacetime, whereas standard quantum theory usually treats states and dynamics separately. PDMs were proposed as the appropriate state-like object that unifies these roles without discarding causal structure. Their non-positivity plays a role analogous to the distinction between Euclidean and Lorentzian signatures: each single-time slice is positive, while the full multi-time object need not be (Fullwood, 2023).

3. Multistep composition, canonicality, and reconstruction of dynamics

For a chain

$\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$4

the symmetric bloom defines a canonical $\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$5-step state-over-time function

$\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$6

The construction is independent of parenthesization, a fact established using associahedral combinatorics: different ways of grouping tensor factors yield the same $\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$7-bloom when the bloom map is associative (Fullwood, 2023).

The corresponding multistep PDM satisfies the expected marginal and composition laws. For two steps,

$\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$8

with marginals

$\operatorname{tr}_i(\tau)\in \mathcal{S}(A_i),$9

and

$\operatorname{tr}_i$0

More generally, for any subset of retained times, tracing out the complementary indices yields the PDM of the reduced process connecting only the retained times. In the classical limit $\operatorname{tr}_i$1 and classical channels $\operatorname{tr}_i$2, the $\operatorname{tr}_i$3-function reduces to the ordinary joint probability distribution

$\operatorname{tr}_i$4

Thus PDMs interpolate between classical stochastic trajectories and quantum dynamics represented as a single Hermitian state-like operator (Fullwood, 2023).

A notable structural result is that, for qubits, this canonical multistep construction exactly recovers the Fitzsimons–Jones–Vedral PDM: $\operatorname{tr}_i$5 The qubit PDM therefore appears not merely as an ad hoc correlator expansion but as the unique symmetric-bloom state over time singled out by the factorization formalism (Fullwood, 2023).

The same work also solves the inverse problem: when can a given PDM be realized by an actual quantum process, and how can the dynamics be recovered explicitly? In the one-step case, given $\operatorname{tr}_i$6 with invertible $\operatorname{tr}_i$7, one solves the Sylvester equation

$\operatorname{tr}_i$8

and sets

$\operatorname{tr}_i$9

Then $A_i$0. For general $A_i$1, the paper defines a subclass $A_i$2 of realizable PDMs and proves a bijection

$A_i$3

with inverse

$A_i$4

Under invertibility assumptions on the intermediate states, a PDM in this subclass corresponds to a unique sequence of channels. This addresses the open problem of reconstructing a quantum process from a given PDM (Fullwood, 2023).

4. Relation to process matrices, multiple-time states, and indefinite causal order

Pseudo-density matrices belong to a broader landscape of spatiotemporal formalisms that includes process matrices, multiple-time states, process tensors, and quantum combs. Their distinctive feature is that they are state-like operators on tensor products of Hilbert spaces indexed by times, rather than higher-order maps from operations to probabilities. This makes them formally simpler than process tensors, but it also means that their operational role depends on the chosen measurement scheme (Fullwood, 2023, Liu et al., 2023).

A major unification result establishes a bridge between PDMs, two-time states, and process matrices. Bipartite process matrices $A_i$5 assign probabilities through the generalized Born rule

$A_i$6

Previous work had shown that such process matrices can be represented by bipartite two-time states. The subsequent step is to show that every operationally realizable two-time state can be represented by a PDM whose probabilities for CP-map outcomes agree exactly with those of the two-time formalism. As a result, process matrices with measurements can be mapped to PDMs (Liu et al., 2023).

The mapping is constructed operationally. One realizes the two-time state by preparing an initial state, applying a CP map between two times, post-selecting at a later time, possibly using ancillas, and then traces out the ancillary system. The associated three-time PDM reproduces the same conditional probabilities as the Aharonov–Bergmann–Lebowitz rule for the corresponding pre- and post-selected process. For ensembles of two-time states, the PDM is the corresponding convex combination of the individual PDMs. This establishes statistical equivalence for the operational tasks considered (Liu et al., 2023).

A direct implication is that PDMs can model indefinite causal order. Since process matrices were introduced precisely to describe correlations incompatible with any fixed causal order, the existence of a mapping from process matrices with POVMs to PDMs implies that PDMs are not restricted to causally ordered dynamics. The same paper emphasizes, however, that the mapping is not a statement that every arbitrary process matrix is literally a single measurement-independent PDM; rather, the PDM construction can bake part of the measurement structure into the representation (Liu et al., 2023).

The quantum-switch example in the appendix of that work illustrates the point. For a switch built from two coherently controlled uses of a constant channel, the reduced PDM $A_i$7 has strictly positive negativity $A_i$8 over a wide parameter range, while $A_i$9. The interpretation given is that there is a causal influence from $\mathcal{S}(A_i)$0 to the joint system $\mathcal{S}(A_i)$1, but not to $\mathcal{S}(A_i)$2 or $\mathcal{S}(A_i)$3 separately. The causal content is therefore encoded in spatiotemporal correlations rather than in individual marginals (Liu et al., 2023).

5. Non-Hermitian extensions and pseudo-entropic constructions

A parallel strand of literature uses “pseudo-density matrix” language for non-Hermitian reduced transition matrices. Given two generally non-orthogonal states $\mathcal{S}(A_i)$4 and $\mathcal{S}(A_i)$5, the transition matrix is

$\mathcal{S}(A_i)$6

and for a bipartition $\mathcal{S}(A_i)$7, the reduced transition matrix is

$\mathcal{S}(A_i)$8

This reduced operator is typically non-Hermitian and may have complex eigenvalues. In that line of work, one may regard $\mathcal{S}(A_i)$9 as a pseudo-density matrix on subsystem $A_i$0 (Jana et al., 2024).

The appropriate spectral framework is bi-orthogonal rather than Hermitian. One writes

$A_i$1

with complex eigenvalues and left/right eigenvectors satisfying

$A_i$2

On this basis one defines the pseudo-entropy

$A_i$3

and the pseudo-capacity

$A_i$4

Both are generically complex (Jana et al., 2024).

This non-Hermitian framework supports a corresponding pseudo-modular Hamiltonian

$A_i$5

which is itself non-Hermitian. Right and left modular evolutions are generated by $A_i$6 and $A_i$7, and their Krylov-space dynamics are described באמצעות a bi-Lanczos algorithm that produces complex Lanczos coefficients $A_i$8. The first two coefficients retain the same formal role as in the Hermitian case: $A_i$9 The early modular-time growth of right and left pseudo-modular spread complexities is

$A$0

In this sense, pseudo-capacity is the early-time modular-complexity measure for non-Hermitian pseudo-density matrices (Jana et al., 2024).

The examples studied in that framework show that the non-Hermitian generalization is not merely formal. In two-level systems, pseudo-modular complexity can saturate to a nonzero value rather than remain purely oscillatory. In a four-qubit example built from GHZ-like states, right and left pseudo-complexities differ when $A$1, while pseudo-capacity highlights transitions more sharply than pseudo-entropy. In the transverse-field Ising model, pseudo-density matrices built from two ground states in different phases encode cross-phase information, and the associated pseudo-modular complexity acquires an enhanced oscillatory profile near the phase boundary. In $A$2 Chern–Simons theory with Wilson loops, pseudo-capacity distinguishes topological linkings and survives in cases interpreted as entanglement swapping between different Wilson-line pairings (Jana et al., 2024).

6. Temporal nonclassicality, macroscopic realism, and temporal entanglement

The PDM formalism can also be organized around a spatiotemporal Born rule. For two times, given projectors $A$3 and $A$4, one defines the quasiprobability

$A$5

For a channel $A$6 between the two times, this equals the Margenau–Hill time-ordered quasiprobability

$A$7

The usual Lüders–von Neumann sequential probability is

$A$8

The difference between the two is a disturbance term,

$A$9

and the paper proves that $1$0 for all $1$1 iff the two-time non-signaling-in-time condition holds. For three times, an analogous decomposition

$1$2

is obtained, and the vanishing of the total disturbance term is equivalent to the full set of three-time NSIT conditions. Since Arrow of Time is automatic for CPTP dynamics, equality of PDM-Born quasiprobabilities and sequential probabilities is equivalent to macroscopic realism for the corresponding measurement scheme (Comar et al., 23 Mar 2026).

This yields a sharp distinction between positivity of the PDM and particular measurement-induced witnesses. If $1$3, then macroscopic realism holds for any choice of two sequential projective measurements; likewise, if $1$4, then MR holds for any choice of three sequential projective measurements. Positivity therefore implies a measurement-independent form of classical spatiotemporal behavior. Negativity is necessary for MR violation, but not sufficient: there are processes with negative PDMs for which particular measurement choices still satisfy NSIT and MR (Comar et al., 23 Mar 2026).

The same work defines temporal entanglement directly from the structure of a PDM. A multitime PDM $1$5 is time separable if it can be written as

$1$6

If no such decomposition exists, the PDM is time entangled. A central theorem shows that if $1$7, then the PDM is not time separable. Thus PDM negativity implies temporal entanglement (Comar et al., 23 Mar 2026).

Temporal entanglement is also necessary for temporal Bell nonclassicality. For a two-time CHSH expression

$1$8

with correlators computed as $1$9, violation is possible only if $\tau\in A$00 is time entangled. The inclusion is strict: there are negative, temporally entangled PDMs that do not violate temporal CHSH. Likewise, Leggett–Garg inequality violation implies negativity of the three-time PDM, but there are explicitly analyzed cases with strong PDM negativity and no LGI violation. The resulting hierarchy distinguishes positivity, negativity, temporal entanglement, NSIT/MR violation, LGI violation, and temporal Bell inequality violation as related but non-equivalent notions of temporal nonclassicality (Comar et al., 23 Mar 2026).

Taken together, these developments establish pseudo-density matrices as a unifying operator language for quantum dynamics across time. In one direction, they provide a canonical Hermitian, unit-trace, generally non-positive state-over-time object with exact marginal, composition, and reconstruction properties. In another, they connect to process matrices and indefinite causal order. In a further extension, they support non-Hermitian pseudo-entropic and pseudo-complexity constructions. Across these variants, the recurring theme is that a state-like description of time is possible only by relaxing positivity, and that the resulting algebraic nonclassicality encodes precisely those temporal, causal, and post-selected structures that ordinary density matrices cannot represent (Fullwood, 2023, Liu et al., 2023, Jana et al., 2024, Comar et al., 23 Mar 2026).