Spacetime Density Matrix: Formalism and Properties

Wu-zhong Guo

REVIEW 4 major objections 6 minor 1 cited by

Spacetime Density Matrix: Formalism and Properties

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper defines the spacetime density matrix: a single operator for each pair of time slices whose trace with two operators reproduces their correlation across times, reducing to the ordinary density matrix as a special case.

desk verdict Sound formalism with true but under-proven moment identities; worth refereeing after the author supplies the missing algebraic proof. read the letter →

arxiv 2508.20397 v1 pith:FDAYAKPX submitted 2025-08-28 hep-th cond-mat.stat-mechgr-qcquant-ph

Wu-zhong Guo This is my paper · ORCID

classification hep-thcond-mat.stat-mechgr-qcquant-ph

keywords spacetimedensitymatrixtransitionoperatorSchwinger-Keldyshpathintegralentanglementintimepseudoentropyreducedshort-timeexpansionperturbationtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that the ordinary density matrix, which encodes all information on a single time slice, can be promoted to a spacetime density matrix living on two or more time slices: one operator for each pair of Cauchy surfaces whose trace with any two operators exactly reproduces their (anti-)time-ordered correlation across those surfaces. The construction is explicit and universal — for any initial state and Hamiltonian the operator is written down in Eq. (9), represented as a Schwinger–Keldysh path integral, and shown to obey a Liouville–von Neumann type equation of motion. For subsystems, partial tracing yields reduced spacetime density matrices whose moments are time-dependent, with a universal short-time behavior and a weak-coupling perturbative method developed to compute them; a key finding is that the coupling between subsystems is what makes these moments nontrivial. A sympathetic reader would care because a single-time density matrix cannot see correlations between different times, and this operator gives a concrete, computable probe of dynamics, subsystem interactions, and the proposed notion of entanglement in time.

What carries the argument

The load-bearing object is the spacetime density matrix $\mathrm{T}_{C_0C_1}$ of Eq. (9), an operator on the doubled Hilbert space whose matrix elements $\langle l|\mathrm{U}\rho_0|i\rangle\langle j|\mathrm{U}^\dagger|k\rangle$ are fixed uniquely by the requirement that its trace with $O_0\otimes O_1$ reproduce the anti-time-ordered correlator $\mathrm{Tr}(\rho_0O_0O_1(t_1))$; it generalizes to $N$ slices as $\mathrm{T}_{C_0C_1\cdots C_{N-1}}$ and reduces to $\rho_0$ or $\rho(t_1)$ when either factor is traced out. The companion mechanism is the super-operator $\mathcal{T}(O_0\otimes O_1)=\mathrm{U}^\dagger|i\rangle\langle j|\mathrm{U}\,O_0\otimes|j\rangle\langle i|O_1$, which generates $\mathrm{T}_{C_0C_1}=\mathcal{T}(\rho_0\otimes I)$, the dual $\tilde{\mathrm{T}}_{C_0C_1}=\mathcal{T}(I\otimes\rho_0)$, and the two-state operator $\mathcal{T}(\rho_0\otimes\sigma_0)$, and which makes the equation of motion $\partial_t\mathcal{T}(\rho_0\otimes I)=i[H\otimes I,\mathcal{T}(\rho_0\otimes I)]+\mathcal{T}(\partial_t\rho_0\otimes I)$ compact. The Schwinger–Keldysh path integral with cuts at each time slice is the same object in a different representation; it supplies the moment identities (66)–(67) by gluing boundary conditions and provides the route toward replica computations of entropy-related quantities.

What would settle it

Build $\mathrm{T}_{C_0C_1}$ by explicit matrix multiplication from Eq. (9) for a small system, say two or three qubits, with a generic Hamiltonian and a non-product initial state, and compute $\mathrm{tr}(\mathrm{T}^2_{C_0C_1})$, $\mathrm{tr}(\mathrm{T}^3_{C_0C_1})$, and $\mathrm{tr}(\mathrm{T}_{C_0C_1}\mathrm{T}^\dagger_{C_0C_1})$ at several times $t_1$; the identities (66)–(67) predict the constant values $(\mathrm{Tr}\rho_0)^2$, $\mathrm{Tr}\rho_0^3$, and $d\,\mathrm{Tr}\rho_0^2$, so any $t_1$-dependence refutes the construction. Separately, expand the exactly computed $\mathrm{tr}(\mathrm{T}^2_{A_0A_1})$ in the two-qubit Heisenberg model to first order in $\delta t$ and check the linear coefficient against $-2i\,\delta t\,\mathrm{Tr}[(\rho_0-\rho_{A_0}\otimes\rho_{\bar{A}_0})H]$.

Watch

Extended reading notes

Core claim

For a system on two Cauchy surfaces $C_0$ at time $t_0$ and $C_1$ at $t_1$, the paper establishes that a unique operator $\mathrm{T}_{C_0C_1}$ on $\mathcal{H}_0\otimes\mathcal{H}_1$, given explicitly by $\mathrm{T}_{C_0C_1}=\langle l|\mathrm{U}\rho_0|i\rangle\langle j|\mathrm{U}^\dagger|k\rangle\,|j\rangle\langle i|\otimes|l\rangle\langle k|$, satisfies $\mathrm{tr}(\mathrm{T}_{C_0C_1}O_0O_1)=\mathrm{Tr}(\rho_0O_0O_1(t_1))$ for all operators $O_0,O_1$; the adjoint operator handles the time-ordered correlator. The author extends the same construction to $N$ time slices, to a super-operator from which $\mathrm{T}_{C_0C_1}$ and several generalizations descend, and to reduced operators obtained by tracing out complementary subsystems. The central results are that the full transition operator has time-independent moments, $\mathrm{tr}(\mathrm{T}^n_{C_0C_1})$ equal to $\mathrm{Tr}\rho_0^n$ for odd $n$ and $(\mathrm{Tr}\rho_0^{n/2})^2$ for even $n$, with $\mathrm{tr}(\mathrm{T}_{C_0C_1}\mathrm{T}^\dagger_{C_0C_1})^n=d\,\mathrm{Tr}\rho_0^{2n}$; the reduced operators acquire time-dependent moments whose short-time form is universal in $\delta t$ and whose leading dependence on the coupling between subsystems is computed to first order in the interaction.

Load-bearing premise

The entropy interpretation rests on treating powers and logarithms of a reduced transition operator that is generally not Hermitian as genuine Rényi-style and von Neumann-style entropies, with complex values read as pseudoentropy — a step the paper carries over from earlier work and itself flags at the end as still lacking a clear physical interpretation.

Editorial extensions

If this is right

The full transition operator's moments are time-independent — $\mathrm{tr}(\mathrm{T}^n_{C_0C_1})$ is $\mathrm{Tr}\rho_0^n$ for odd $n$ and $(\mathrm{Tr}\rho_0^{n/2})^2$ for even $n$, while $\mathrm{tr}(\mathrm{T}_{C_0C_1}\mathrm{T}^\dagger_{C_0C_1})^n=d\,\mathrm{Tr}\rho_0^{2n}$ — so all time dependence in correlation data is carried by the reduced operators.
Tracing out one time slice returns $\rho_0$ and tracing out the other returns $\rho(t_1)$, so the ordinary density matrices at both ends are special cases and the formalism is a strict generalization.
All spacetime density matrices for different time separations are unitarily equivalent, $T_{C_0C_2}=(I\otimes \mathrm{U}(t_2,t_1))\,T_{C_0C_1}\,(I\otimes \mathrm{U}(t_2,t_1)^\dagger)$, which ties the whole family to one initial state and Hamiltonian.
The second moment of a reduced spacetime density matrix has a universal short-time linear term $-2i\,\delta t\,\mathrm{Tr}[(\rho_0-\rho_{A_0}\otimes\rho_{\bar{A}_0})H]$, which vanishes for product initial states and for zero coupling, making these moments a direct probe of subsystem interactions.
In the absence of coupling between subsystems the reduced moments reduce to time-independent functions of $\rho_0$ alone, so nontrivial time dependence of $\mathrm{tr}(\mathrm{T}_{A_0A_1}^n)$ is itself a signature that an interaction is present.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the moment identities hold for every integer $n$, the spectrum of $\mathrm{T}_{C_0C_1}$ is fixed by the spectrum of $\rho_0$ alone, which would make the full spacetime density matrix an encoding of the initial state rather than a record of the dynamics; writing out that reconstruction explicitly would settle its information content.
The singular-value bound $\sum_i\sigma_i=d$ together with the eigenvalue inequality $\sum_i|\lambda_i|\le d$ suggests that the non-normality of $\mathrm{T}_{C_0C_1}$ is tightly constrained, and measuring how close reduced operators come to saturating these bounds could quantify how far a causally connected pair of subsystems is from Hermiticity.
The sharp sign inversion in the imaginary part of the two-qubit entanglement entropy suggests that the pseudoentropy of reduced transition operators may encode directional information about the evolution; turning this feature into a quantitative diagnostic is a natural next step.
Extending the Dyson-series expansion to higher orders in the coupling, along the lines the paper outlines, should connect these moments to the diagrammatic language of perturbative QFT and to open-system questions such as decoherence rates and approach to equilibrium.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a referee's time. The spacetime density matrix is a genuine object: it is the Choi/Jamiolkowski representation of the two-time correlator, and the paper builds a clean formalism around it—multi-time-slice generalization, superoperator and dual-operator framing, Liouville–von Neumann-type equations, reduced moments, and a perturbative method that is actually usable. The basic partial-trace properties are derived carefully and are correct. The two-qubit and TFD examples are concrete and mostly checkable. I went through the central definition and the low moments by hand; the algebra is sound.

The soft spots are real but mostly presentation-level. The load-bearing moment formulas (66)–(67) are introduced with "we expect" and supported by path-integral diagrams for thermal states, not by a general algebraic proof. That is a gap because the eigenvalue and SVD claims in Section 3.1 rest on those formulas. The claims themselves are true: (12) makes T_C0C1 unitarily equivalent to the equal-time case, and a direct computation for the equal-time operator gives (66)–(67). So the fix is straightforward—add the proof, or at least state the unitary-equivalence argument.

The n = 1/2 step in (69) is less problematic than it looks. T T^\dagger is positive, so the half-integer power is defined, and the integer-n identities force the multisets of eigenvalues to match. Still, the paper should say this instead of just taking n → 1/2. Right now the reasoning looks like an unjustified analytic continuation even though it can be made rigorous in one line.

Smaller issues: the von Neumann entropy formula in Section 3.6.1 is mistyped (S = -tr log T log T is not what anyone means), and the author explicitly says at the end that the physical interpretation of the complex entropy-like quantities remains unclear. That honesty is fine, but it should be moved into the body so readers do not mistake pseudoentropy for an established interpretation.

Citation practice is fine; the prior work [5,6] is credited, and the self-citations are to independent results, not used to force the main claims. No fitted parameters, no invented entities.

Who gets value: people working on pseudoentropy, real-time QFT, open systems, and holographic probes of dynamics. The paper deserves a serious referee. I would ask the referee to push for the algebraic proof of the moment identities and clarification of the n = 1/2 step, but I would not desk-reject.

Referee Report

4 major / 6 minor

Summary. The paper develops a formalism for a 'spacetime density matrix' TC0C1 acting on H0⊗H1, defined so that tr(TC0C1 O0 O1) reproduces the anti-time-ordered correlator Tr(ρ0 O0 O1(t1)). The main construction is explicit: Eq. (9) gives the operator in a basis, Eq. (10) gives the compact form U†|k⟩⟨l|Uρ0 ⊗ |l⟩⟨k|, and Section 2.1 provides a Schwinger–Keldysh path-integral representation. The paper extends the construction to multiple time slices, introduces reduced transition operators for spacetime subsystems, and proposes Rényi/von Neumann-type entropies for these generally non-Hermitian operators. It derives a Liouville–von Neumann-type equation of motion, studies moments of the full and reduced operators, claims universal time-independent formulas (66)-(67) and universal short-time behavior in Section 3.3, develops a weak-coupling perturbative method, and illustrates the framework on a two-qubit model and on the thermal field double state.

Significance. If the moment formulas were proven, the paper would provide a clean, parameter-free operator formalism with potentially broad diagnostic value: the moments of reduced spacetime density matrices are sensitive to subsystem interactions (Section 3.4), and the path-integral representation in Section 2.1 connects the construction to standard Schwinger–Keldysh techniques. The explicit derivation of TC0C1, the partial-trace consistency relations (13), and the super-operator formulation are genuine strengths, and the two-qubit and TFD examples give concrete, checkable expressions. The quantitative core, however, is not yet at the same standard: the central identities (66)-(67) are presented as expectations supported by low-order path-integral diagrams, and the SVD constraint (69) depends on an unjustified analytic continuation.

major comments (4)

[Section 3.1, Eqs. (66)-(67) and Appendix B] The identities (66) and (67) are load-bearing: they underlie the pseudo-Hermiticity statement, the SVD discussion, and the constraint (69)-(70). In the text they are introduced with 'We expect the following results', and Appendix B supplies path-integral diagrams for a thermal initial state at low n rather than a derivation for arbitrary ρ0 and U. Please provide a direct algebraic proof from the explicit form (10), or alternatively state these results as conjectures and remove the spectral and SVD conclusions that depend on them. The n=2 case in (65) is checked, but that does not establish the general integer-n formula.
[Section 3.1, Eq. (69)] Equation (69) sets n=1/2 in (67), but (67) was stated for integer n. The step from integer moments to the sum of singular values requires an explicit analytic-continuation argument. Since TT† is positive, tr(TT†)^n = Σ σ_i^{2n} is analytic in n on a right half-plane, so a proof of (67) for all positive integers combined with analyticity of the right-hand side would suffice; the manuscript currently gives neither the continuation argument nor a non-integer proof. As written, the constraint Σ σ_i = d is not established.
[Section 2.3, Eqs. (38)-(39); Section 4] The Rényi and von Neumann entropies of the reduced transition operator are defined by log tr T^n/(1-n) and its n→1 limit for a generally non-Hermitian T. This is a definition imported from the pseudoentropy literature, not a theorem, and the paper itself notes at the end that the physical interpretation of these quantities remains unclear. Please state explicitly that these are formal definitions, specify the branch and domain assumptions needed when the spectrum of T is complex or has non-positive real eigenvalues, and separate the validity of the moment computations from the interpretive claim that they define entanglement entropies for causally connected subregions.
[Section 3.3.1, Eqs. (88)-(91)] The short-time expansion results for tr(T^2_A0A1) and tr(T^2_A0\bar A1) are presented after 'With some calculations' without derivation; these formulas support the claimed universal short-time behavior and the coupling-sensitivity analysis in Section 3.4. Please include the calculation, or at least an appendix outline, so the reader can verify the contractions and the basis dependence. The two-qubit example in Section 3.6.1 provides a consistency check, but only for one Hamiltonian.

minor comments (6)

[Section 2.2, Eq. (25)] The displayed formula contains a malformed factor '|⟨ l_{N-2}|U(t_{N-1},t_{N-2})†|k_{N-1}⟩|'; the bra-ket notation should be cleaned up.
[Section 3.6.1, equation after (113)] The formula for the von Neumann entropy is misprinted as 'S(TA0A1)=−tr logTA0A1 logTA0A1'; it should read −tr(TA0A1 log TA0A1), and similarly for TA0B1.
[Section 3.6.1 and Appendix C.1] The initial state is defined as cosθ|01⟩+sinθ|10⟩ in the main text but as sinθ|01⟩+cosθ|10⟩ in Appendix C.1; the displayed second moments are insensitive to this swap, but the explicit matrices (135)-(138) are not. Please align the conventions.
[Appendix B, figure cross-references] The sentence 'Fig.19 is an example' in the discussion of tr(TC0C1T†)^2 appears to refer to Fig.22 rather than Fig.19; all figure cross-references should be checked.
[Throughout] There are numerous typographical errors, including 'mutiple timeslices' in the heading of Section 2.2, 'expreesion', 'amptitute', 'evlolution', 'inlcude', and 'convinent'; these should be corrected.
[Section 2, Eq. (13)] The definition of ρ(t1) is garbled in the typesetting and should read ρ(t1) := U(t1,t0) ρ0 U(t1,t0)†.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spacetime density matrix is defined by an explicit trace identity and constructed by coefficient matching, with no fitted parameters and no load-bearing self-citation.

full rationale

The central object TC0C1 is introduced in Eq. (5) as the operator that reproduces Tr(ρ0 O0 O1(t1)) by definition, and Eq. (9) is obtained by explicit coefficient matching between the two sides; there is no fitted parameter and no hidden input. The subsequent moment identities (65)-(73) are independent mathematical or diagrammatic statements about this explicit operator, checked for low values of n and illustrated by path-integral figures. Even if (66)-(67) are only conjectured for general integer n, that would be a gap in proof, not a reduction of a prediction to its own input. The reduced operators are defined by partial traces in Eq. (35), and the short-time and weak-coupling expansions are direct Taylor or Dyson expansions of the same definition, with no parameter fitted to the quantities being 'predicted.' The entropy-like quantities in Section 2.3 are explicitly definitions (38)-(39), with the pseudoentropy interpretation openly imported from [29]; the paper itself states that the physical interpretation remains unclear, so no derived claim is disguised as an input. Self-citations to [30] and [34] are pointers to worked examples and replica calculations, not load-bearing premises used to force the central construction. The analytic continuation n to 1/2 leading to Eq. (69) is under-justified and is a correctness risk, but it is not circular: it extrapolates an identity rather than assuming the conclusion. Overall, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameters are fitted to data; the central quantities are defined from rho0, U, and the Hamiltonian. The main invented mathematical objects are operators built out of already-defined Hilbert-space data, not new physical entities.

assumptions (4)

domain assumption Each Hilbert space Hi on a time slice can be identified with a common Hilbert space via a basis choice.
The explicit formula for TC0C1 uses the same basis |i> at t0 and t1; the paper notes this can be relaxed to unitarily equivalent bases.
domain assumption The system is closed and evolves unitarily under a Hamiltonian H; no environment or measurement is included in the definition of TC0C1.
The construction (10) uses U = e^{-iH(t1-t0)} and assumes unitary evolution.
ad hoc to paper For non-Hermitian reduced transition operators, the Renyi-like entropy formula log tr T^n/(1-n) is a meaningful definition.
The paper introduces this formula as a generalization of entanglement entropy and interprets complex values as pseudoentropy; this is not derived from standard entropy axioms.
domain assumption In QFT, subregion Hilbert spaces factorize enough for partial traces and real-time replica twist operators to define the reduced spacetime density matrix.
The paper notes that subregion algebras in QFT do not factorize and that the perturbative basis-dependent results are formal in QFT; the extension relies on a twist-operator method deferred to other work.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Spacetime Density Matrix: Formalism and Properties." pith.science (2026). https://pith.science/paper/FDAYAKPX

@misc{pith2026250820397,
  author       = {Pith},
  title        = {Pith review of: Spacetime Density Matrix: Formalism and Properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDAYAKPX}},
  note         = {Machine review of arXiv:2508.20397}
}

read the original abstract

In this paper, we develop the general formalism and properties of the spacetime density matrix, which captures correlations among different Cauchy surfaces and can be regarded as a natural generalization of the standard density matrix defined on a single Cauchy surface. We present the construction of the spacetime density matrix in general quantum systems and its representation via the Schwinger Keldysh path integral. We further introduce a super-operator framework, within which the spacetime density matrix appears as a special case, and discuss possible generalizations from this perspective. We also show that the spacetime density matrix satisfies a Liouville von Neumann type equation of motion. When considering subsystems, a reduced spacetime density matrix can be defined by tracing over complementary degrees of freedom. We study the general properties of its moments and, in particular, derive universal short time behavior of the second moment. We find that coupling between subsystems plays a crucial role in obtaining nontrivial results. Assuming weak coupling, we develop a perturbative method to compute the moments systematically.

Figures

Figures reproduced from arXiv: 2508.20397 by the authors.

**Figure 2.** Illustration of the properties (13) of the transition operator TC0C1 . The left panel shows the partial trace over C0, which identifies the boundary conditions i and j. The right panel shows the partial trace over C1, corresponding to the identification of k with l, and the forward and backward contributions cancel each other. We can also represent the operator T † C0C1 . Assuming again a pure initial density matrix… view at source ↗

**Figure 3.** Figure illustrating TC0C1 (left) and T † C0C1 (right) for the thermal initial state ρ0 = e −βH. The red dots indicate the identified points. 2.2 Generalization to mutiple timeslices The above construction can be generalized to multiple Cauchy surfaces. For a Cauchy surface at t = ti with i = 0, 1, . . . , N − 1, we associate a Hilbert space Hi. We are also interested in the general N-point anti-time-ordered correlat… view at source ↗

Figures from the paper (25 more)

**Figure 4.** Figure 4: Schwinger-Keldysh representation of the spacetime density matrix [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: (left) Schwinger–Keldysh representation of the operator [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Schwinger-Keldysh representation of the dual transition operator [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Schwinger-Keldysh representation of the transition operator [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Path integral representation of the transition operator [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Path integral representation of the transition operator [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: Path integral representation trT2 C0C1 . The dash line means the identification. The circle on the right hand side of equation refer to the path integral on imaginary time direction, the circumference is β [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: Path integral representation tr(TC0C1 T † C0C1 ). The dash line means the identification. The circle on the right hand side of equation refer to the path integral on imaginary time direction, the circumference is 2β, while the loop means the path integral on the real …

**Figure 12.** Figure 12: Path integral representation trT2 C0C1C2 . 3.2 Reduced spacetime density matrix and its moments In Section 2.3, we showed how to define the reduced transition operator and associated entropy-related quantities, which capture the entanglement between general spacetime …

**Figure 13.** Figure 13: Plot of trT2 A0A1 . We set J = 1 and θ = π 6 . We plot the entanglement entropy for the two reduced transition operators in [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]

**Figure 14.** Figure 14: Plot of the entanglement entropy for TA0A1 and TA0B1 in the two qubits model. We set J = 1 and θ = π 6 3.6.2 Thermal field double state The thermal field double (TFD) state provides a purification of the canonical thermal state e −βH by introducing an auxiliary copy o…

**Figure 15.** Figure 15: Path representation of the operators F † C0C1,s (left), F † L0L1,s (middle) and F † L0R1,s (right). With this expression it is straightforward to evlauate the moments tr(F † L0L1,s) n , tr(F † L0R1,s) n . (118) 31 [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]

**Figure 16.** Figure 16: Path integral representation of the 2-nd moment [PITH_FULL_IMAGE:figures/full_fig_p032_16.png]

**Figure 17.** Figure 17: Path integral representation of the operator [PITH_FULL_IMAGE:figures/full_fig_p035_17.png]

**Figure 18.** Figure 18: Path integral representation trT3 C0C1 . The circle on the right hand side of equation refer to the path integral on imaginary time direction, the circumference is 3β [PITH_FULL_IMAGE:figures/full_fig_p036_18.png]

**Figure 19.** Figure 19: Path integral representation trT4 C0C1 . It is also straightforward to compute the moment for TC0C1...Ci...CN−1 . Here we only show some examples in Fig.20 and Fig.21. The 2-nd moment for TC0C1...Ci...CN−1 is summarized in Eq.(72). In Section.3.1 we use the digram to …

**Figure 20.** Figure 20: Path integral representation trT2 C0C1C2C3 [PITH_FULL_IMAGE:figures/full_fig_p037_20.png]

**Figure 21.** Figure 21: Path integral representation trT2 C0C1C2C3C4 [PITH_FULL_IMAGE:figures/full_fig_p037_21.png]

**Figure 22.** Figure 22: Path integral representation tr(TC0C1 T † C0C1 ) 2 . 37 [PITH_FULL_IMAGE:figures/full_fig_p037_22.png]

**Figure 23.** Figure 23: Path integral representation trT2 C0C1C2 [PITH_FULL_IMAGE:figures/full_fig_p038_23.png]

**Figure 24.** Figure 24: Path integral representation tr(TC0C1C2C3 T † C0C1C2C3 ) [PITH_FULL_IMAGE:figures/full_fig_p038_24.png]

**Figure 25.** Figure 25: Path integral representation tr(TC0C1C2C3C4 T † C0C1C2C3C4 ). 38 [PITH_FULL_IMAGE:figures/full_fig_p038_25.png]

**Figure 26.** Figure 26: Path integral representation tr(F † L0L1,s) 3 [PITH_FULL_IMAGE:figures/full_fig_p041_26.png]

**Figure 27.** Figure 27: Path integral representation tr(F † L0L1,s) 4 [PITH_FULL_IMAGE:figures/full_fig_p041_27.png]

**Figure 28.** Figure 28: Path integral representation tr(F † L0R1,s) 3 . 41 [PITH_FULL_IMAGE:figures/full_fig_p041_28.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Selecting Complex Extremal Surfaces with the Kontsevich--Segal--Witten Criterion
hep-th 2026-07 conditional novelty 7.0 of 10

In AdS3, dS3, and AdS4 hyperbolic examples, the Kontsevich-Segal-Witten criterion uniquely selects a three-piece complex contour for timelike extremal surfaces, while timelike strips in AdS4 violate the criterion near...

Reference graph

Works this paper leans on

47 extracted references · 10 canonical work pages · cited by 1 Pith paper

[1]

PCT, spin and statistics, and all that,

R. F. Streater and A. S. Wightman, “PCT, spin and statistics, and all that,”

work page
[2]

Entanglement in many-body systems,

L. Amico, R. Fazio, A. Osterloh and V. Vedral, “Entanglement in many-body systems,” Rev. Mod. Phys. 80 (2008), 517-576 [arXiv:quant-ph/0703044 [quant-ph]]

arXiv 2008
[3]

Entanglement entropy and conformal field theory,

P. Calabrese and J. Cardy, “Entanglement entropy and conformal field theory,” J. Phys. A 42 (2009), 504005 [arXiv:0905.4013 [cond-mat.stat-mech]]

arXiv 2009
[4]

Essay: Emergent Holographic Spacetime from Quantum Informa- tion,

T. Takayanagi, “Essay: Emergent Holographic Spacetime from Quantum Informa- tion,” Phys. Rev. Lett. 134 (2025) no.24, 240001 [arXiv:2506.06595 [hep-th]]

arXiv 2025
[5]

Observable and computable entanglement in time,

A. Milekhin, Z. Adamska and J. Preskill, “Observable and computable entanglement in time,” [arXiv:2502.12240 [quant-ph]]

arXiv
[6]

Superdensity Operators for Spacetime Quantum Mechanics,

J. Cotler, C. M. Jian, X. L. Qi and F. Wilczek, “Superdensity Operators for Spacetime Quantum Mechanics,” JHEP 09, 093 (2018) [arXiv:1711.03119 [quant-ph]]

arXiv 2018
[7]

Entanglement between the future and past in the quantum vacuum,

S. J. Olson and T. C. Ralph, “Entanglement between the future and past in the quantum vacuum,” Phys. Rev. Lett.106 (2011), 110404 [arXiv:1003.0720 [quant-ph]]

arXiv 2011
[8]

Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,

K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki, “Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,” Phys. Rev. Lett. 130, no.3, 031601 (2023) [arXiv:2210.09457 [hep-th]]

arXiv 2023

Show all 47 references

[9]

Holographic derivation of entanglement entropy from AdS/CFT,

S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006) [arXiv:hep-th/0603001]

2006 arXiv
[10]

A Covariant holographic entan- glement entropy proposal,

V. E. Hubeny, M. Rangamani and T. Takayanagi, “A Covariant holographic entan- glement entropy proposal,” JHEP 07, 062 (2007) [arXiv:0705.0016 [hep-th]]

2007 arXiv
[11]

Building up spacetime with quantum entanglement,

M. Van Raamsdonk, “Building up spacetime with quantum entanglement,” Gen. Rel. Grav. 42, 2323 (2010) [Int. J. Mod. Phys. D 19, 2429 (2010)] [arXiv:1005.3035 [hep-th]]

2010 arXiv
[12]

Entanglement Renormalization and Holography,

B. Swingle, “Entanglement Renormalization and Holography,” Phys. Rev. D 86 (2012), 065007 [arXiv:0905.1317 [cond-mat.str-el]]

2012 arXiv
[13]

Bulk Locality and Quantum Error Correction in AdS/CFT,

A. Almheiri, X. Dong and D. Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” JHEP 04 (2015), 163 [arXiv:1411.7041 [hep-th]]

2015 arXiv
[14]

Relative entropy equals bulk relative entropy,

D. L. Jafferis, A. Lewkowycz, J. Maldacena and S. J. Suh, “Relative entropy equals bulk relative entropy,” JHEP 06 (2016), 004 [arXiv:1512.06431 [hep-th]]

2016 arXiv
[15]

Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,

X. Dong, D. Harlow and A. C. Wall, “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117 (2016) no.2, 021601 [arXiv:1601.05416 [hep-th]]. 42

2016 arXiv
[16]

Quantum Extremal Surfaces: Holographic Entangle- ment Entropy beyond the Classical Regime,

N. Engelhardt and A. C. Wall, “Quantum Extremal Surfaces: Holographic Entangle- ment Entropy beyond the Classical Regime,” JHEP 01 (2015), 073 [arXiv:1408.3203 [hep-th]]

2015 arXiv
[17]

Entanglement Wedge Reconstruction and the Information Paradox,

G. Penington, “Entanglement Wedge Reconstruction and the Information Paradox,” JHEP 09 (2020), 002 [arXiv:1905.08255 [hep-th]]

2020 arXiv
[18]

The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,

A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” JHEP 12, 063 (2019) [arXiv:1905.08762 [hep-th]]

2019 arXiv
[19]

Replica wormholes and the black hole interior,

G. Penington, S. H. Shenker, D. Stanford and Z. Yang, “Replica wormholes and the black hole interior,” JHEP 03 (2022), 205 [arXiv:1911.11977 [hep-th]]

2022 arXiv
[20]

Replica Wormholes and the Entropy of Hawking Radiation,

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” JHEP 05 (2020), 013 [arXiv:1911.12333 [hep-th]]

2020 arXiv
[21]

The entropy of Hawking radiation,

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “The entropy of Hawking radiation,” Rev. Mod. Phys. 93 (2021) no.3, 035002 [arXiv:2006.06872 [hep-th]]

2021 arXiv
[22]

Brownian motion of a quantum oscillator,

J. S. Schwinger, “Brownian motion of a quantum oscillator,” J. Math. Phys. 2 (1961), 407-432

1961
[23]

The Theory of a general quantum system interacting with a linear dissipative system,

R. P. Feynman and F. L. Vernon, Jr., “The Theory of a general quantum system interacting with a linear dissipative system,” Annals Phys. 24 (1963), 118-173

1963
[24]

Diagram Technique for Nonequilibrium Processes,

L. V. Keldysh, “Diagram Technique for Nonequilibrium Processes,” Sov. Phys. JETP 20 (1965), 1018-1026

1965
[25]

Deriving covariant holographic entan- glement,

X. Dong, A. Lewkowycz and M. Rangamani, “Deriving covariant holographic entan- glement,” JHEP 11 (2016), 028 [arXiv:1607.07506 [hep-th]]

2016 arXiv
[26]

Records from the S-Matrix Marathon: Schwinger- Keldysh Formalism,

F. M. Haehl and M. Rangamani, “Records from the S-Matrix Marathon: Schwinger- Keldysh Formalism,” [arXiv:2410.10602 [hep-th]]

arXiv
[27]

Real-time gravitational replicas: Formalism and a variational principle,

S. Colin-Ellerin, X. Dong, D. Marolf, M. Rangamani and Z. Wang, “Real-time gravitational replicas: Formalism and a variational principle,” JHEP 05 (2021), 117 [arXiv:2012.00828 [hep-th]]

2021 arXiv
[28]

Real-time gravi- tational replicas: low dimensional examples,

S. Colin-Ellerin, X. Dong, D. Marolf, M. Rangamani and Z. Wang, “Real-time gravi- tational replicas: low dimensional examples,” JHEP 08 (2021), 171 [arXiv:2105.07002 [hep-th]]

2021 arXiv
[29]

New holographic generalization of entanglement entropy,

Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka and Z. Wei, “New holographic generalization of entanglement entropy,” Phys. Rev. D 103 (2021) no.2, 026005 [arXiv:2005.13801 [hep-th]]. 43

2021 arXiv
[30]

Entanglement measures for causally connected subregions and holography,

X. Gong, W. z. Guo and J. Xu, “Entanglement measures for causally connected subregions and holography,” [arXiv:2508.05158 [hep-th]]

arXiv
[31]

How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,

Y. Aharonov, D. Z. Albert and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60 (1988), 1351-1354

1988
[32]

Colloquium: Understanding Quantum Weak Values: Basics and Applications,

J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, R. W. Boyd, “Colloquium: Understanding Quantum Weak Values: Basics and Applications,” Rev. Mod. Phys. 86 (2014) 307 [arXiv:1305.7154 [quant-ph]]

2014 arXiv
[33]

PseudoHermiticity versus PT symmetry. The necessary condition for the reality of the spectrum,

A. Mostafazadeh, “PseudoHermiticity versus PT symmetry. The necessary condition for the reality of the spectrum,” J. Math. Phys. 43, 205-214 (2002) [arXiv:math- ph/0107001 [math-ph]]

2002
[34]

Constructible reality condition of pseudo entropy via pseudo-Hermiticity,

W. z. Guo, S. He and Y. X. Zhang, “Constructible reality condition of pseudo entropy via pseudo-Hermiticity,” JHEP 05 (2023), 021 [arXiv:2209.07308 [hep-th]]

2023 arXiv
[35]

SVD entanglement entropy,

A. J. Parzygnat, T. Takayanagi, Y. Taki and Z. Wei, “SVD entanglement entropy,” JHEP 12 (2023), 123 [arXiv:2307.06531 [hep-th]]

2023 arXiv
[36]

Spectral projections for density matrices in quantum field theories,

W. z. Guo, “Spectral projections for density matrices in quantum field theories,” JHEP 04 (2025), 033 [arXiv:2408.08031 [hep-th]]

2025 arXiv
[37]

Eternal black holes in anti-de Sitter,

J. M. Maldacena, “Eternal black holes in anti-de Sitter,” JHEP 04 (2003), 021 [arXiv:hep-th/0106112 [hep-th]]

2003 arXiv
[38]

Cool horizons for entangled black holes,

J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” Fortsch. Phys. 61 (2013), 781-811 [arXiv:1306.0533 [hep-th]]

2013 arXiv
[39]

Time Evolution of Entanglement Entropy from Black Hole Interiors,

T. Hartman and J. Maldacena, “Time Evolution of Entanglement Entropy from Black Hole Interiors,” JHEP 05 (2013), 014 [arXiv:1303.1080 [hep-th]]

2013 arXiv
[40]

Complexity and Shock Wave Geometries,

D. Stanford and L. Susskind, “Complexity and Shock Wave Geometries,” Phys. Rev. D 90, no.12, 126007 (2014) [arXiv:1406.2678 [hep-th]]

2014 arXiv
[41]

Holographic Complexity Equals Bulk Action?,

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, “Holographic Complexity Equals Bulk Action?,” Phys. Rev. Lett. 116, no.19, 191301 (2016) [arXiv:1509.07876 [hep-th]]

2016 arXiv
[42]

Traversable Wormholes via a Double Trace Deformation,

P. Gao, D. L. Jafferis and A. C. Wall, “Traversable Wormholes via a Double Trace Deformation,” JHEP 12 (2017), 151 [arXiv:1608.05687 [hep-th]]

2017 arXiv
[43]

W. z. Guo, work in progress
[44]

Quantum quenches in 1 + 1 dimensional conformal field theories,

P. Calabrese and J. Cardy, “Quantum quenches in 1 + 1 dimensional conformal field theories,” J. Stat. Mech. 1606 (2016) no.6, 064003 [arXiv:1603.02889 [cond- mat.stat-mech]]. 44

2016 arXiv
[45]

Breuer, Heinz-Peter, and Francesco Petruccione, The Theory of Open Quantum Systems (Oxford, 2007; online edn, Oxford Academic, 1 Feb. 2010),

2007
[46]

W. H. Zurek, Pointer Basis of Quantum Apparatus: Into What Mixture Does the Wave Packet Collapse?, Phys. Rev. D 24, 1516–1525 (1981)

1981
[47]

W. H. Zurek, Environment induced superselection rules, Phys. Rev. D 26, 1862–1880 (1982). 45

1982

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

PCT, spin and statistics, and all that,

R. F. Streater and A. S. Wightman, “PCT, spin and statistics, and all that,”

work page

[2] [2]

Entanglement in many-body systems,

L. Amico, R. Fazio, A. Osterloh and V. Vedral, “Entanglement in many-body systems,” Rev. Mod. Phys. 80 (2008), 517-576 [arXiv:quant-ph/0703044 [quant-ph]]

arXiv 2008

[3] [3]

Entanglement entropy and conformal field theory,

P. Calabrese and J. Cardy, “Entanglement entropy and conformal field theory,” J. Phys. A 42 (2009), 504005 [arXiv:0905.4013 [cond-mat.stat-mech]]

arXiv 2009

[4] [4]

Essay: Emergent Holographic Spacetime from Quantum Informa- tion,

T. Takayanagi, “Essay: Emergent Holographic Spacetime from Quantum Informa- tion,” Phys. Rev. Lett. 134 (2025) no.24, 240001 [arXiv:2506.06595 [hep-th]]

arXiv 2025

[5] [5]

Observable and computable entanglement in time,

A. Milekhin, Z. Adamska and J. Preskill, “Observable and computable entanglement in time,” [arXiv:2502.12240 [quant-ph]]

arXiv

[6] [6]

Superdensity Operators for Spacetime Quantum Mechanics,

J. Cotler, C. M. Jian, X. L. Qi and F. Wilczek, “Superdensity Operators for Spacetime Quantum Mechanics,” JHEP 09, 093 (2018) [arXiv:1711.03119 [quant-ph]]

arXiv 2018

[7] [7]

Entanglement between the future and past in the quantum vacuum,

S. J. Olson and T. C. Ralph, “Entanglement between the future and past in the quantum vacuum,” Phys. Rev. Lett.106 (2011), 110404 [arXiv:1003.0720 [quant-ph]]

arXiv 2011

[8] [8]

Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,

K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki, “Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,” Phys. Rev. Lett. 130, no.3, 031601 (2023) [arXiv:2210.09457 [hep-th]]

arXiv 2023