arxiv: 2604.07471 · v1 · submitted 2026-04-08 · 🪐 quant-ph

Recognition: unknown

On Lorentzian symmetries of quantum information

James Fullwood , Vlatko Vedral , Edgar Guzm\'an-Gonz\'alez

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:18 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum informationLorentz symmetrieslinear entropyqubitpre-spacetimeW-matrixSL(2,C)Minkowski metric

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The pith

Lorentzian symmetries arise on qubit states from preserving linear entropy without assuming spacetime.

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

The authors demonstrate that the restricted Lorentz group SO+(1,3) acts on the internal degrees of freedom of a qubit when one requires that linear entropy is preserved under transformations. This is done in a setting with only qubit degrees of freedom and no external spacetime variables. The result supports the 'It from Qubit' view by deriving relativistic structure from quantum information principles. They further show that linear entropy and related mutual information measures are invariant under larger symmetry groups like SL(2,C) tensor n. The correlation in the singlet state recovers the Minkowski metric.

Core claim

We derive the natural action of the restricted Lorentz group SO+(1,3) on the internal degrees of freedom of a single qubit from the information-theoretic principle of preservation of linear entropy. This holds in a pre-spacetime setting with only internal qubit degrees of freedom. It is shown that the Lorentz invariance of linear entropy is a special case of spectral invariants of the W-matrix being SL(2,C) tensor n invariants. The linear n-partite quantum mutual information is thus SL(2,C) tensor n invariant, and the singlet correlation function yields the Minkowski metric whose symmetry is the full Lorentz group.

What carries the argument

The preservation of linear entropy under transformations of qubit states, which induces the action of the restricted Lorentz group SO+(1,3).

If this is right

The linear entropy is Lorentz invariant for relativistic qubits.
Any spectral invariant of the W-matrix is an SL(2,C) tensor n invariant scalar.
The linear n-partite quantum mutual information is an SL(2,C) tensor n invariant for all n-qubit states.
The correlation function of the singlet state yields the Minkowski metric on the space of qubit observables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework might allow deriving other aspects of spacetime geometry from different quantum information invariants.
Extensions could connect this to quantum field theory by considering continuous degrees of freedom.
Experimental verification in quantum optics setups could test if linear entropy remains constant under boosts on entangled qubits.

Load-bearing premise

Imposing preservation of linear entropy in a pre-spacetime setting with only internal qubit degrees of freedom is sufficient to recover the Lorentz group action without circularly assuming relativistic structure.

What would settle it

A mathematical counterexample of a linear transformation on qubit states that preserves linear entropy but is not an element of SO+(1,3) would show the derivation is incomplete.

read the original abstract

A foundational result in relativistic quantum information theory due to Peres, Scudo, and Terno, is that von Neumann entropy is not Lorentz invariant. Motivated by the "It from Qubit" paradigm, here we show that Lorentzian symmetries of quantum information emerge naturally in a pre-spacetime setting, without any reference to external variables such as position or momentum. In particular, we derive the natural action of the restricted Lorentz group $\text{SO}^+(1,3)$ on the internal degrees of freedom of a single qubit from a simple, information-theoretic principle we refer to as preservation of linear entropy. It is then shown that the Lorentz invariance of the linear entropy of a relativistic qubit is a special case of a much more general phenomenon, namely, that any spectral invariant of an operator we term the '$W$-matrix' is an $\text{SL}(2,\mathbb C)^{\otimes n}$ invariant scalar. Consequently, the linear $n$-partite quantum mutual information is shown to be an $\text{SL}(2,\mathbb C)^{\otimes n}$ invariant for all $n$-qubit states. Finally, we show that the correlation function associated with a pair of qubits in the singlet state yields the Minkowski metric on the space of qubit observables, whose symmetry group is the full Lorentz group $\text{SO}(1,3)$. In accordance with the "It from Qubit" paradigm, our results thus establish the natural emergence of relativistic spacetime structure from intrinsic properties of quantum information.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gets the restricted Lorentz group from linear entropy preservation on single qubits and extends it to SL(2,C) invariants plus a Minkowski metric from singlet correlations, but the W-matrix step looks like it may carry the 4-vector structure in from the start.

read the letter

The main claim is that preserving Tr(ρ²) on a qubit's internal state, with no external spacetime variables, forces the natural action of SO+(1,3). They then lift this to a W-matrix whose spectral invariants are SL(2,C)⊗n scalars, so linear mutual information is invariant, and the singlet correlation function reproduces the Minkowski metric whose symmetry is the full Lorentz group. That last part is the cleanest piece: if the calculation really starts from the singlet projector and the observable algebra without extra assumptions, it gives a direct information-theoretic origin for the metric signature and its isometries. The generalization beyond von Neumann entropy non-invariance is also new on the surface; linear entropy is simpler and the W-matrix lets them treat multipartite cases uniformly. Those moves sit squarely in the It-from-Qubit line and could be useful to people already thinking about how relativistic structure might arise from quantum correlations alone. The soft spot is exactly the one the stress-test flags. Standard maps that preserve Tr(ρ²) are just unitary conjugations or orthogonal transformations on the Bloch vector; to reach the full Lorentz group you need an embedding of the qubit observables into a 4-dimensional space whose transformation law already knows about boosts and rotations. The paper introduces the W-matrix to do this, but it is not obvious from the abstract whether that object is derived from the preservation principle or chosen so the desired group emerges. If the latter, the emergence is not as information-theoretic as advertised. The proofs of invariance and the singlet-to-metric step would need to be checked line by line for hidden choices. This is the kind of paper that belongs in a foundations journal rather than a general physics one. Readers already working on relativistic quantum information or quantum gravity approaches that start from qubits will want to see the details; others can skip it. The claims are coherent enough on their own terms to deserve referee time, even if the central derivation turns out to rest on an auxiliary structure. I would send it out for review with a specific request that the referees examine whether the W-matrix definition is independent of the Lorentz representation it is meant to produce.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that, in a pre-spacetime setting with only internal qubit degrees of freedom, the principle of preserving linear entropy (invariance of Tr(ρ²)) is sufficient to derive the natural action of the restricted Lorentz group SO+(1,3) on a single qubit. It introduces an auxiliary W-matrix, proves that any spectral invariant of this W-matrix is an SL(2,C)^⊗n-invariant scalar, shows that the linear n-partite quantum mutual information is therefore SL(2,C)^⊗n invariant, and demonstrates that the correlation function of a pair of qubits in the singlet state yields the Minkowski metric whose symmetry group is the full Lorentz group SO(1,3). The work is motivated by the non-Lorentz-invariance of von Neumann entropy and the 'It from Qubit' paradigm.

Significance. If the central derivation is non-circular, the result would be significant for foundational quantum information and relativistic QI: it would supply an information-theoretic origin for Lorentzian structure without external spacetime variables, extending known results on entropy non-invariance and linking qubit correlations directly to the Minkowski metric. The general statement about spectral invariants of the W-matrix and the invariance of multipartite linear mutual information are potentially useful if rigorously established. The significance is tempered by the need to verify that the W-matrix and its embedding are not chosen to reproduce the known SL(2,C) representation.

major comments (3)

[Single-qubit derivation (around the statement of the main result on SO+(1,3) action)] The preservation of Tr(ρ²) on a single qubit is known to be equivalent to orthogonal transformations on the Bloch vector (SO(3) up to phase). The manuscript must therefore explain, in the section deriving the SO+(1,3) action, precisely how the W-matrix is defined and why its transformation law under the preservation principle selects the full restricted Lorentz group rather than merely its SO(3) subgroup. If the W-matrix is introduced by embedding the three Pauli observables plus the identity into a 4-vector with Minkowski signature, this step risks presupposing the target structure; a concrete test is whether the allowed maps can be derived from entropy preservation alone without that embedding.
[Definition and properties of the W-matrix; proof of spectral invariance] The general claim that 'any spectral invariant of the W-matrix is an SL(2,C)^⊗n invariant scalar' appears load-bearing for the subsequent statements on mutual information. The definition of the W-matrix and the proof of this invariance (likely in the section following the single-qubit case) must be inspected to determine whether the invariance follows from the preservation principle or is built into the construction of the W-matrix. If the latter, the result is tautological and does not establish emergence from information-theoretic postulates.
[Singlet-state correlation and metric derivation] The derivation that the singlet-state correlation function yields the Minkowski metric (and hence the full SO(1,3) symmetry) must be checked for independence from prior relativistic assumptions. The choice of the singlet projector and the explicit form of the two-qubit correlation function should be shown to arise solely from the linear-entropy preservation principle rather than from an a-priori identification of observables with 4-vectors.

minor comments (2)

Ensure that all notation for the W-matrix (its explicit matrix elements, its relation to the density operator, and its transformation properties) is introduced with a self-contained definition before it is used in the invariance proofs.
The abstract states that linear entropy invariance is 'a special case of a much more general phenomenon'; the main text should make the logical dependence explicit by first proving the general spectral-invariance statement and only then specializing to linear entropy.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising these important points about potential circularity and the need for explicit derivations. We address each major comment in turn and commit to revisions that strengthen the exposition while preserving the core claims.

read point-by-point responses

Referee: [Single-qubit derivation (around the statement of the main result on SO+(1,3) action)] The preservation of Tr(ρ²) on a single qubit is known to be equivalent to orthogonal transformations on the Bloch vector (SO(3) up to phase). The manuscript must therefore explain, in the section deriving the SO+(1,3) action, precisely how the W-matrix is defined and why its transformation law under the preservation principle selects the full restricted Lorentz group rather than merely its SO(3) subgroup. If the W-matrix is introduced by embedding the three Pauli observables plus the identity into a 4-vector with Minkowski signature, this step risks presupposing the target structure; a concrete test is whether the allowed maps can be derived from entropy preservation alone without that embedding.

Authors: We agree that a more explicit, step-by-step derivation is required to dispel any impression of circularity. In the revised manuscript we will first define the W-matrix directly from the qubit density operator and the Pauli basis as the unique Hermitian operator whose quadratic form reproduces Tr(ρ²) when restricted to the three-vector part. We then derive the allowed linear transformations on the four components by imposing that Tr(ρ²) remain invariant for every input state. Because the Pauli matrices satisfy {σ_i, σ_j} = 2δ_{ij} I, the only quadratic form compatible with this invariance under the full set of linear maps is the Minkowski form (with the identity component acquiring the opposite sign). The resulting group is therefore SO+(1,3) rather than SO(3). We will include an auxiliary calculation that recovers the familiar SO(3) subgroup when the identity component is artificially frozen, thereby showing that the extension to the full Lorentz group follows necessarily from the preservation principle applied to the complete observable algebra. revision: yes
Referee: [Definition and properties of the W-matrix; proof of spectral invariance] The general claim that 'any spectral invariant of the W-matrix is an SL(2,C)^⊗n invariant scalar' appears load-bearing for the subsequent statements on mutual information. The definition of the W-matrix and the proof of this invariance (likely in the section following the single-qubit case) must be inspected to determine whether the invariance follows from the preservation principle or is built into the construction of the W-matrix. If the latter, the result is tautological and does not establish emergence from information-theoretic postulates.

Authors: We will expand the relevant section with a self-contained lemma that derives the spectral invariance directly from the local preservation of linear entropy. The argument proceeds by showing that any map preserving Tr(ρ_k²) for each qubit k induces a similarity transformation on the associated local W-matrix; the spectrum is therefore unchanged. Extending to the multipartite W-matrix (defined via the tensor-product structure of the local operators) yields that every spectral function is invariant under the product group action. The proof makes no reference to an a-priori SL(2,C) representation; the group action is obtained as the set of all transformations compatible with the entropy-preservation condition. We will add the full proof and an explicit example for n=2 to confirm that the invariance is a consequence rather than an assumption. revision: yes
Referee: [Singlet-state correlation and metric derivation] The derivation that the singlet-state correlation function yields the Minkowski metric (and hence the full SO(1,3) symmetry) must be checked for independence from prior relativistic assumptions. The choice of the singlet projector and the explicit form of the two-qubit correlation function should be shown to arise solely from the linear-entropy preservation principle rather than from an a-priori identification of observables with 4-vectors.

Authors: We will revise the final section to derive the singlet projector from the entropy-preservation principle itself. The singlet is the unique two-qubit state that remains invariant under all joint transformations that preserve the linear entropy of each reduced density operator; this characterization follows from the fact that any other state would allow a local entropy-non-preserving map. With the singlet fixed in this manner, the two-qubit correlation function is computed directly as the expectation value of the product of local observables expressed through the W-matrix. The resulting bilinear form on the space of single-qubit observables is shown to be the Minkowski inner product by explicit matrix calculation, without external relativistic input. We will include the intermediate steps that connect the entropy-invariance condition to the metric signature. revision: partial

Circularity Check

1 steps flagged

W-matrix and SL(2,C) invariants introduced to match derived Lorentz action, reducing central claim to definitional choice

specific steps

self definitional [Abstract (paragraph 2)]
"It is then shown that the Lorentz invariance of the linear entropy of a relativistic qubit is a special case of a much more general phenomenon, namely, that any spectral invariant of an operator we term the 'W-matrix' is an SL(2,ℂ)⊗n invariant scalar. Consequently, the linear n-partite quantum mutual information is shown to be an SL(2,ℂ)⊗n invariant for all n-qubit states."

The W-matrix is introduced and termed by the authors precisely so that its spectral invariants coincide with SL(2,C) invariants; the preservation of linear entropy is then applied within this space, making the claimed derivation of the Lorentz action equivalent to the choice of embedding the qubit observables into the SL(2,C) representation rather than an independent consequence of the entropy principle alone.

full rationale

The paper claims to derive the SO+(1,3) action solely from preservation of linear entropy (Tr(ρ²) invariance) in a pre-spacetime qubit setting. However, the load-bearing step introduces the W-matrix whose spectral invariants are defined to be SL(2,C) invariants, with SL(2,C) being the known double cover of the Lorentz group. This makes the subsequent invariance statements and the emergence of the Minkowski metric from singlet correlations follow by construction from the auxiliary embedding rather than emerging independently from the entropy principle. The derivation chain therefore contains a self-definitional reduction at the point where the information-theoretic principle is applied to the pre-structured W-matrix space.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that linear entropy preservation is the correct principle to impose and on the introduction of the W-matrix as the object whose spectral invariants yield the desired symmetries. No free parameters are mentioned in the abstract.

axioms (1)

domain assumption Preservation of linear entropy is the fundamental information-theoretic principle in a pre-spacetime setting
Invoked directly to derive the natural action of SO+(1,3) on qubit degrees of freedom.

invented entities (1)

W-matrix no independent evidence
purpose: Operator whose spectral invariants are SL(2,C) tensor n invariant scalars
Introduced to generalize the invariance result from single qubits to n-partite states and linear mutual information.

pith-pipeline@v0.9.0 · 5572 in / 1497 out tokens · 59900 ms · 2026-05-10T17:18:00.066558+00:00 · methodology

discussion (0)

Reference graph

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Applying this result to each factor in (A2) then yields Tr(ρρ⋆) = nY i=1 Tr(A1 i ) Tr(A1 j)−Tr(A 1 i A1 j)

= Tr(ρ1) Tr(ρ2)−Tr(ρ 1ρ2). Applying this result to each factor in (A2) then yields Tr(ρρ⋆) = nY i=1 Tr(A1 i ) Tr(A1 j)−Tr(A 1 i A1 j) . When expanding the previous product, each factor contributes either a term of the form Tr(A m i ) Tr(Am j ) or −Tr(A m i Am j ). Consider a choice of indicesα 1 <· · ·< α q for which the first type is selected, andβ 1 <· ...