Spacetime States: Quantum, Holography & Beyond

• Spacetime states are mathematical constructs that assign state variables to complete spacetime histories, capturing geometry, temporal correlations, and matter configurations.
• They are applied across frameworks like holography, multi-time quantum mechanics, and curved-spacetime QFT to represent entangled bulk geometries and non-Hermitian time operators.
• These constructions yield practical insights into modeling phenomena such as fixed-area states, metric-dependent matter equations, and latent predictive variables in complex systems.

In the surveyed literature, “spacetime states” denotes state-like objects that encode spacetime geometry, temporal correlations, or geometry-dependent matter configurations by assigning a state not merely to a spatial slice but to a doubled boundary Hilbert space, a collection of spacetime events, a history Hilbert space, or a metric-dependent thermodynamic configuration (Cantcheff, 2012, Zhang et al., 2019, Diaz et al., 10 Jun 2026). The expression is therefore used in non-equivalent ways. In holography it often refers to bulk geometries represented by entangled boundary states; in spacetime formulations of quantum mechanics it denotes operators whose correlators reproduce multi-time statistics; in complex-systems work it can mean latent equivalence classes over lightcones; and in relativistic astrophysics it can denote matter states whose equation of state depends explicitly on the ambient metric (Rupe et al., 2020, Hossain et al., 2022).

1. Major meanings of the term

The principal uses of “spacetime states” in the literature considered here can be organized as follows.

Family of usage	Central object	Representative papers
Holographic and quantum-gravitational	Bulk geometry encoded by a state in an enlarged Hilbert space	(Cantcheff, 2012, Botta-Cantcheff et al., 2017, Nomura et al., 2017, Dong et al., 2022, Cirilo-Lombardo et al., 2023)
Quantum states across time/events	Operator on a tensor product over events or time slices	(Zhang et al., 2019, Lie et al., 25 Jul 2025, Diaz et al., 10 Jun 2026)
Curved-spacetime QFT	Geometry-selected or geometry-split state constructions	(Fewster, 2018, Smolyakov, 2023)
Spatiotemporal inference and dynamics	Predictive latent fields or timelike-slice states	(Rupe et al., 2020, Ippoliti et al., 2021)
Matter–spacetime coupling and statistical macrostate	Metric-dependent matter state or stationary causal population state	(Hossain et al., 2020, Hossain et al., 2022, Simchi, 2021)

A common structural feature is that the relevant object is defined by correlational content across space and time rather than by a single-time density operator alone. A major difference is whether the state is meant to be a literal quantum state, a generalized quasi-state, a latent predictive variable, or a statistical macrostate.

2. Holographic and quantum-gravitational spacetime states

One influential holographic proposal identifies asymptotically AdS spacetimes with vectors in a doubled boundary Hilbert space. In this construction, the non-perturbative gravity Hilbert space is

$$\mathcal H_{\rm QG}\equiv \mathcal H=\mathcal H\otimes \widetilde{\mathcal H},$$

and a general spacetime $M$ is represented by

$$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$

Factorized states $|\psi\rangle\otimes|\tilde\psi\rangle$ represent two disconnected asymptotically AdS components, while entangled states represent connected two-boundary geometries; the thermofield-double state

$$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$

is interpreted as the eternal AdS black hole, and black-hole formation is modeled as a unitary Bogoliubov evolution in the enlarged Hilbert space (Cantcheff, 2012). In this sense, geometry is encoded in entanglement structure.

A more selective holographic criterion holds that not every CFT state is dual to a single classical spacetime. States prepared by Euclidean source insertions,

$$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$

are argued to be the relevant semiclassical states, and in the large-$N$ or linearized regime they take the coherent-state form

$$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$

Such states reproduce classical bulk expectation values, whereas generic Hamiltonian eigenstates need not correspond to any single smooth classical geometry (Botta-Cantcheff et al., 2017). This proposal narrows the class of “spacetime states” to coherent semiclassical sectors rather than the full Hilbert space.

A complementary entanglement-based criterion sharpens the relation between state and reconstructable geometry. For holographic states, a directly reconstructable bulk exists only when subregion entanglement is intermediate rather than maximal. The maximally entropic law

$$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$

forces HRRT surfaces either onto a bifurcation surface or onto a non-expanding null hypersurface, so maximally entangled states have no reconstructable spacetime in the relevant sense (Nomura et al., 2017). In this framework, “spacetime states” are exceptional, non-typical, intermediately entangled states rather than generic Hilbert-space vectors.

Fixed-area states provide a different gravitational construction. Starting from a seed state $|\psi\rangle$, one forms

$M$0

where $M$1 restricts the HRT area $M$2 to a narrow band around $M$3. The Euclidean saddles that prepare these states may have conical singularities at $M$4, but the Lorentzian spacetime intrinsic to the state is real at real times and has no conical singularity. With enough symmetry it is smooth; more generally, the leading curvature singularity along null congruences orthogonal to the fixed-area surface behaves as

$M$5

The paper argues that these classical divergences are mild, but quantum fields are more singular, so fixed-area states are expected to be well-defined only when the fixed-area surface is appropriately smeared (Dong et al., 2022).

A more group-theoretic program identifies spacetime itself with coherent states of the complete covering of the $M$6 structure, concretely the metaplectic group $M$7. The complete Hilbert space splits into even and odd sectors,

$M$8

or equivalently $M$9, generated by $$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$0 and $$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$1. In this scheme, large $$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$2 yields the classical continuum manifold, while the Planck scale corresponds to the coherent-state eigenvalue $$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$3 and to the discrete level $$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$4. Coherent and squeezed states, together with their Wigner functions, are then used to describe black holes, de Sitter spacetime, and Euclidean instantons without singularities (Cirilo-Lombardo et al., 2023). This suggests a distinctly algebraic notion of spacetime state in which continuum and discrete geometry are different sectors of one coherent-state formalism.

3. Quantum states over spacetime in quantum mechanics

A separate research line constructs quantum states directly over spacetime events. In continuous-variable form, the basic move is to treat different instances of time as different quantum modes. For an event-labeled collection of quadratures $$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$5, a Gaussian spacetime state is specified by a mean vector $$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$6 and a covariance matrix

$$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$7

where for different times the anticommutator notation refers to the product of measurement results at two events. More generally, the spacetime Wigner function is defined by

$$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$8

and the associated spacetime density matrix by

$$|M\rangle=\sum_{n,\tilde m}G_{n\tilde m}\,|n\rangle\otimes|\tilde m\rangle \in \mathcal H_{\rm QG}.$$9

These objects are Hermitian and unit-trace but, in general, need not be positive, so they function as continuous-variable analogues of pseudo-density matrices (Zhang et al., 2019).

An explicitly operational reformulation defines quantum states over arbitrary spacetime regions through causally agnostic measurements. The central theorem states that a measurement is causally agnostic if and only if it can be implemented by interferometry. In that framework, a spacetime state $|\psi\rangle\otimes|\tilde\psi\rangle$0 is defined by the interferometric kernel

$|\psi\rangle\otimes|\tilde\psi\rangle$1

where $|\psi\rangle\otimes|\tilde\psi\rangle$2 are unitary interventions at spacetime regions $|\psi\rangle\otimes|\tilde\psi\rangle$3. In the purely spatial case this recovers the ordinary density operator. In a one-step temporal process $|\psi\rangle\otimes|\tilde\psi\rangle$4, the same operational rule yields the left-product quantum state over time,

$|\psi\rangle\otimes|\tilde\psi\rangle$5

Under time-reversal symmetry the natural temporal state becomes the Fullwood–Parzygnat product $|\psi\rangle\otimes|\tilde\psi\rangle$6 (Lie et al., 25 Jul 2025). Mixed temporal states are then convex mixtures of QSOTs, and non-factorizable QSOTs become useful probes of non-Markovianity.

A broader unifying formalism, spacetime quantum mechanics, introduces a spacetime Hilbert space

$|\psi\rangle\otimes|\tilde\psi\rangle$7

one copy $|\psi\rangle\otimes|\tilde\psi\rangle$8 per time slice, together with a quantum action

$|\psi\rangle\otimes|\tilde\psi\rangle$9

and a state quantum action $$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$0. The corresponding spacetime state is

$$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$1

It is normalized, $$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$2, but is generally non-Hermitian and not positive semidefinite. Its significance is correlational: fixed-time marginals reproduce ordinary Schrödinger states, while

$$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$3

reproduces time-ordered multi-time correlators (Diaz et al., 10 Jun 2026). In that account, path integrals, QSOTs, pseudo-density matrices, Page–Wootters states, superdensity operators, and timelike-entanglement constructions are different manifestations of one underlying spacetime-state object.

4. Curved-spacetime quantum field theory and geometry-selected states

In algebraic QFT on curved spacetime, the natural ambition would be a preferred state determined by spacetime geometry alone. A locally covariant theory assigns algebras $$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$4 to globally hyperbolic spacetimes $$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$5, and a natural state is a compatible assignment $$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$6 such that

$$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$7

for every admissible embedding $$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$8. A no-go theorem shows that if a nontrivial theory admits such a natural state and satisfies standard Minkowski-space assumptions together with dynamical locality and extended locality, then the theory is trivial: every $$|0(\beta)\rangle=\sum_n \frac{e^{-\beta E_n/2}}{Z^{1/2}}\,|n\rangle\otimes|\tilde n\rangle$$9 consists only of multiples of the identity (Fewster, 2018). This rules out a universal, physically acceptable, locally covariant choice of one preferred state per globally hyperbolic spacetime.

The Sorkin–Johnston construction evades that theorem by being globally defined and nonlocal. It starts from the causal propagator $$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$0, defines $$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$1, takes its positive part

$$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$2

and sets

$$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$3

The resulting state is geometry-determined, but on ultrastatic slabs it is generically not Hadamard; by contrast, softened Brum–Fredenhagen constructions

$$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$4

yield an infinite family of Hadamard states rather than a unique canonical one (Fewster, 2018). Within curved-spacetime QFT, “spacetime state” therefore often means a geometry-informed state assignment, but the literature sharply distinguishes such assignments from a universal preferred vacuum.

The background geometry can also reshape the one-particle spectrum itself. For a free massive scalar field in Schwarzschild spacetime, canonical quantization in the exterior region yields a radial Schrödinger problem with potential $$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$5 at the horizon end and $$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$6 at spatial infinity. For energies $$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$7, the initially doubly degenerate scattering-like modes can be reorganized into orthogonal combinations $$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$8 and $$|\Psi^{\phi_-}\rangle=e^{-\int_{\partial_r{\cal M}_-}\mathcal O\,\phi_-}|0\rangle,$$9: $N$0 behaves asymptotically like a properly normalized plane wave for a distant observer, whereas $N$1 decays at large radius and carries norm predominantly from the near-horizon region. Together with the $N$2 exponentially decaying near-horizon modes, they form a complete basis (Smolyakov, 2023). Here the structure of states is genuinely spacetime-dependent in the sense that the topology and asymptotics of Schwarzschild spacetime split the continuum spectrum into asymptotic and near-horizon sectors.

5. Spatiotemporal states in complex systems, machine learning, and circuit duality

Outside fundamental quantum gravity, “spacetime states” also appears as a predictive-state concept for spatiotemporal data. In computational mechanics, local causal states are equivalence classes of past lightcones $N$3 satisfying

$N$4

Applied pointwise, the encoder

$N$5

produces a latent field $N$6 with the same spacetime geometry as the observed field $N$7. The paper extends this to a spacetime autoencoder by introducing a stochastic decoder $N$8 and a latent Markovian update rule $N$9, interpreted as a stochastic cellular automaton over local causal states (Rupe et al., 2020). In this usage, a spacetime state is a minimal predictive summary of a local lightcone history.

A distinct many-body usage comes from spacetime duality in quantum circuits. A local unitary gate $$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$0 can be reinterpreted sideways as a generally non-unitary dual gate $$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$1. The resulting dual circuit defines pure states on timelike slices and a non-unitary evolution in the transverse direction. Their entanglement is exactly mapped to a unitary problem with boundary decoherence: $$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$2 This yields logarithmic, volume-law, and fractal steady states; in particular, fractally entangled spacetime-dual states satisfy

$$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$3

The paper’s “spacetime states” are therefore pure states on timelike cuts of a circuit tensor network, experimentally preparable through a teleportation-based protocol with only boundary-scale postselection (Ippoliti et al., 2021).

These two lines share a formal move: both assign state variables to spacetime points or timelike cuts rather than to one-time slices only. The resulting states are not equivalent objects, but both are explicitly geometry-preserving constructions in which local organization across space and time is the primary datum.

6. Matter–spacetime couplings, statistical macrostates, and functional reinterpretations

In relativistic astrophysics, a concrete matter state–spacetime state relation arises when the equation of state of degenerate fermions is derived directly in the curved spacetime of a star rather than imported from Minkowski spacetime. For a static spherical star with

$$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$4

the local box metric retains the redshift factor $$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$5, and the reduced Dirac action becomes

$$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$6

The single-particle energy is therefore

$$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$7

so the curved-space equation of state takes the form

$$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$8

rather than depending on density alone (Hossain et al., 2020). In a slowly rotating star, frame dragging further splits the effective chemical potentials,

$$|\Psi^{\phi_-}\rangle\propto e^{\sum_k \lambda_k a_k^\dagger}|0\rangle.$$9

leading to

$$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$0

For the ideal neutron-star model discussed in the paper, including gravitational time dilation raises the maximum mass from about $$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$1 to about $$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$2, while the direct frame-dragging correction to the EOS is extremely small; for white dwarfs the limit rises from $$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$3 to $$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$4 (Hossain et al., 2022). In this literature, the thermodynamic state of matter is explicitly part of the spacetime state because the state variables already depend on the metric before the stellar structure equations are solved.

A still more unconventional use appears in a causal-set-inspired statistical model of spacetime as a growing population of events. There the causal set $$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$5 is treated as an age-structured population with Leslie evolution matrix

$$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$6

probability elements

$$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$7

and stationary distribution

$$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$8

The associated entropy is

$$S_A=\frac{1}{4l_{\rm P}^{d-1}}\min\{\|A\|,\|\bar A\|\}$$9

Here the stationary population is the spacetime macrostate, while individual causal-event birth configurations are the microstates (Simchi, 2021). This usage is explicitly statistical rather than geometric or Hilbert-space based.

At a more interpretive level, spacetime functionalism recasts the issue by shifting attention from intrinsic spacetime substance to spacetime role realization. The functional-reduction schema requires that higher-level spacetime features be functionalized and then shown to be instantiated by lower-level structures. In loop quantum gravity, weave states satisfy

$|\psi\rangle$0

while in causal set theory manifoldlike behavior is defined through faithful embedding into a relativistic spacetime (Lam et al., 2018). On this reading, a “spacetime state” is whatever lower-level configuration realizes the localization, metric, causal, and topological functions relevant to relativistic spacetime.

Taken together, these usages show that “spacetime states” is not a single technical term but a family of constructions answering a common question: how should statehood be assigned when spacetime itself, temporal succession, or geometry-dependent matter is part of what is being represented? The answers range from entangled AdS boundary states and non-Hermitian history operators to latent predictive fields, metric-dependent equations of state, stationary causal-population macrostates, and functionally defined spacetime realizers.