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The Deimginarity Cost of Quantum States

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Pith's one-line read The cost to erase imaginarity from a quantum state equals a regularized relative entropy.

desk verdict Plausible result, but the proof as written has load-bearing errors; reject for now. read the letter →

arxiv 2501.12891 v1 pith:PFVKOH7F submitted 2025-01-22 quant-ph

Xian Shi This is my paper · ORCID
classification quant-ph MSC 81P4581P68 PACS 03.65.Ud03.67.Mn
keywords deimaginarityimaginarityresourcetheoryrelativeentropyofquantumrandomnesscostrealstateserasure
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 addresses a task it calls deimaginarity: using random real orthogonal (covariant-free) unitary operations to transform a quantum state into a real state, and asks for the minimum rate of randomness needed in the limit of infinitely many copies with vanishing error. It proves that this minimum rate, the deimaginarity cost $C_d(\rho)$, equals the regularized relative entropy of imaginarity $I_r^{\infty}(\rho)=\lim_{n\to\infty}\frac{1}{n}I_r(\rho^{\otimes n})$. Because the relative entropy of imaginarity had no known operational meaning, this gives the measure an interpretation: it is the asymptotic randomness cost of erasing imaginarity. The result places imaginarity among the resources whose erasure cost is characterized by a regularized entropy.

What carries the argument

The central object is the relative entropy of imaginarity, $I_r(\rho)=\min_{\sigma\in F} D(\rho\|\sigma)=S((\rho+\rho^T)/2)-S(\rho)$, and its regularized version $I_r^{\infty}(\rho)=\lim_{n\to\infty}\frac{1}{n}I_r(\rho^{\otimes n})$. Covariant-free unitaries are real orthogonal operators $O$ that commute with the real-part map $\Theta(\cdot)=((\cdot)+(\cdot)^T)/2$, so they cannot create imaginarity from real states. The proof machinery consists of $\delta$-typical subspaces of $\rho^{\otimes n}$, the gentle measurement lemma, the Fannes inequality, and the operator Chernoff bound. The load-bearing step in the achievability direction is an unproved assumption: that for each $n$ and $\delta$ there exists an ensemble $\{p(dU),U\}$ of covariant-free real orthogonal unitaries on the typical subspace satisfying $\int p(dU)\,U\theta U^\dagger = I_{n,\delta}/D_{n,\delta}$ for every state $\theta$ on that subspace, and that these unitaries extend to real orthogonal unitaries on the full space $H^{\otimes n}$. This ensemble is what converts the projected state into the maximally mixed state on the typical subspace, from which a real state is reached.

What would settle it

Take the qubit maximally imaginarity state $|\phi\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}$, whose regularized relative entropy of imaginarity is 1 bit per copy. For n=2 and n=3, search explicitly over ensembles of real orthogonal unitaries on $(\mathbb{C}^2)^{\otimes n}$ that satisfy the twirl property on the δ-typical subspace; if for any n such an ensemble provably does not exist, the achievability proof breaks. Alternatively, compute the deimaginarity cost for small n by direct optimization over finite ensembles and check whether the sequence converges to $I_r(\rho)$; finding a rate below 1 bit per copy at any n would contradict the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 2: for every finite-dimensional state $\rho$, the deimaginarity cost equals the regularized relative entropy of imaginarity, $C_d(\rho)=I_r^{\infty}(\rho)$. The converse part shows that any protocol achieving rate $R$ must have $R\geq I_r^{\infty}$, using the subadditivity of von Neumann entropy, the Fannes inequality, and the fact that covariant-free unitaries leave the real-part entropy $S((\rho+\rho^T)/2)$ invariant. The direct part constructs an achievability protocol at rate $I_r^{\infty}$ by projecting onto a $\delta$-typical subspace, applying an assumed ensemble of real orthogonal unitaries that twirls the subspace to the maximally mixed state, and using the operator Chernoff bound to replace the continuous ensemble by a finite one. The paper presents this as the first operational interpretation of the relative entropy of imaginarity.

Load-bearing premise

The proof's achievability part assumes, without proof, that for every state and every number of copies there is a set of real orthogonal unitaries on the typical subspace that sends every state on that subspace to the maximally mixed state on average, and that these unitaries can be extended to the full n-copy space; if this fails, the construction that reaches the claimed rate does not exist.

Editorial extensions

If this is right

  • The relative entropy of imaginarity now has an operational meaning: it is the asymptotic randomness cost of converting a state into a real state using random real orthogonal unitaries.
  • For states where the relative entropy of imaginarity is additive, the deimaginarity cost equals the single-copy relative entropy of imaginarity, so the resource can be erased at that rate per copy.
  • Deimaginarity joins correlation, coherence, and asymmetry as resources whose optimal erasure cost is a regularized entropy, extending the erasure-cost family to imaginarity.
  • The result provides a target for constructive protocols: any protocol that deimaginarizes at a rate below the regularized relative entropy of imaginarity is impossible, and the optimal rate is known even before a protocol is found.

Reading between the lines

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

  • The unproved twirl-ensemble assumption is the point most likely to need repair; a failure of existence for even one state would turn the achievability statement into an open problem rather than a theorem.
  • A natural next step is to check whether the same formula holds when the free operations are restricted to covariant-free channels rather than unitaries, which would generalize the result to a noisy setting.
  • The equality suggests a quantitative duality between imaginarity and randomness: the amount of imaginarity a state carries is exactly the random bits needed to wash it out, a relation that could be tested experimentally by realizing the optimal ensembles for small states.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 5 minor

Summary. The paper introduces a task called deimaginarity, in which n copies of a quantum state ρ are mapped close to a real state by applying a randomly chosen covariant-free (real orthogonal) unitary, and the cost C_d(ρ) is the minimum asymptotic rate of randomness needed. The main result, Theorem 2, states that C_d(ρ) equals the regularized relative entropy of imaginarity I_r^∞(ρ) = lim_{n→∞} (1/n) I_r(ρ^⊗n). The proof is given in Appendix A: a converse part uses entropy inequalities and Fannes' inequality, and a direct part uses typical subspaces, an assumed twirl ensemble, and an operator Chernoff bound.

Significance. If Theorem 2 were established, it would provide a clean operational interpretation of the regularized relative entropy of imaginarity as a randomness cost, analogous to results for coherence and entanglement erasure. The converse part follows a standard template and contains the right entropy estimates modulo a normalization slip. The paper also correctly identifies the free operations and the covariance condition. However, the direct part relies on an unproved and generally unavailable twirl assumption and contains a false assertion that the constructed channel output is real; this invalidates the proof of achievability. The false claim that the relative entropy of imaginarity is additive is an additional factual error, although the theorem itself uses the regularized quantity.

major comments (6)
  1. [§VI A 2, Eq. (A.16)] The direct part assumes the existence of an ensemble {p(dU), U} of covariant-free unitaries such that for every state θ on the typical subspace H_{n,δ}, ∫ p(dU) U θ U† = I_{n,δ}/D_{n,δ}. This is a strong twirl (2-design) property on a subspace that is generally complex, and no proof or construction is given. The sentence 'Assume ...' is not an argument. Since the operator Chernoff step and the final trace-norm estimate depend on this ensemble, the achievability claim C_d(ρ) ≤ I_r^∞(ρ) is unsupported without a proof of existence or an alternative construction.
  2. [§VI A 2, after Eq. (A.19)] The assertion 'E(W) is real' is false. From Eq. (A.16), E(W) = (μ_{n,δ}/D_{n,δ}) Π_{n,δ}, where Π_{n,δ} is the projector onto the typical subspace. The projector is a real matrix only if H_{n,δ} is invariant under transposition, which is not generally the case. For ρ = |+i⟩⟨+i| and n=1, H_{1,δ}=span{|+i⟩}, D=1, and every real orthogonal U preserving this line acts by a phase on |+i⟩; the averaged output is then ρ itself, not a real state. Hence the final state (1/μ_{n,δ}) E(W) in the construction is not free, and the direct part does not produce the claimed real state.
  3. [§VI A 2, Eq. (A.16)–(A.19)] Even if a twirl ensemble existed, the construction targets the maximally mixed state on H_{n,δ}. For non-real typical subspaces this state is not a real density matrix. The example ρ=|+i⟩⟨+i| shows the obstruction persists for all n, since the typical subspace is spanned by |+i⟩^{⊗n}, and the real-orthogonal stabilizer acts by phases. The proof therefore cannot be repaired by a minor adjustment of parameters; the achievability strategy itself must be replaced.
  4. [§VI A 1, Eq. (A.8)] The displayed inequality S(O_n(ρ⊗n)) ≥ (1/2^{nR}) Σ_k S(Θ(O_k ρ⊗n O_k†)) is not a consequence of concavity. For ρ=|+i⟩⟨+i|, n=1, m=1, O=I, the left side is S(ρ)=0 while the right side is S(Θ(ρ))=1. The intended argument should combine Eq. (A.7) with concavity applied to Θ(O_n(ρ⊗n)); this yields an additional entropy term −nη(2ε)log d. The final bound survives after that correction, but the chain as written is mathematically false.
  5. [§VI A 1, Eq. (A.6)] The isometry T is defined as T = (1/2^{nR}) Σ_{k=1}^{2^{nR}} |k⟩_E ⊗ O_k. With this normalization T†T = 2^{-nR} I, so T is not an isometry and the state |ψ_n⟩ is not normalized. The correct coefficient is 1/√(2^{nR}). This is a local error, but it affects the entropy estimates in Eq. (A.6).
  6. [§II B] The statement 'The REI is additive' is false as written. For ρ=|+i⟩⟨+i|, I_r(ρ)=S(I/2)-S(ρ)=1, while I_r(ρ⊗2)=S(Re(ρ⊗2))-S(ρ⊗2)=1 because Re(ρ⊗2) has eigenvalues 1/2,1/2. Thus I_r(ρ⊗2) ≠ 2I_r(ρ). This false claim is not used in the proof of Theorem 2 (which uses the regularized quantity), but it should be corrected or removed.
minor comments (5)
  1. [Title] The title contains a typo: 'Deimginarity' should be 'Deimaginarity'.
  2. [§III, Definition 1] The text uses 'ortthogonal' and 'O_kΘ(·)O_k†' where the adjoint should be the transpose for real orthogonal operators; these are notational inconsistencies.
  3. [§VI A 2, Eq. (A.18)] The lower bound on λ_{n,δ} should be stated as an inequality for the minimum nonzero eigenvalue; the current text says 'the minimum nonzero eigenvalue ... is bounded by' without specifying direction, and the denominator D should be explicitly the dimension of H_{n,δ}.
  4. [§VI A 2] The notation N = 2^{I_r(ρ⊗n)+3nδ} should be understood as an integer ceiling; as written it is not an integer in general.
  5. [§VI A 1, Eq. (A.11)] The η(2ε) term should be η(2ε)log d multiplied by n, since the Fannes inequality is applied on the n-copy Hilbert space of dimension d^n; the displayed formula is consistent with this, but the text should make the dimension explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: deimaginarity cost is derived from the independently defined regularized relative entropy of imaginarity; the unproved twirl-ensemble assumption is a proof gap, not a circular reduction.

full rationale

The paper's central claim, C_d(ρ) = I_r^∞(ρ), is not obtained by assuming what it proves. The deimaginarity cost is defined directly in Definition 1 as the minimum randomness rate of real-orthogonal covariant-free unitaries needed to bring ρ^⊗n close to a real state, while the relative entropy of imaginarity I_r is defined independently in Eq. (4) as min_{σ∈F} D(ρ∥σ), with the closed form I_r(ρ)=S(Re ρ)-S(ρ) taken from the cited prior work [28]. The converse part lower-bounds an arbitrary achievable rate using the data-processing inequality, Fannes' inequality, and Lemma 6, all applied to quantities derived from the definitions rather than from the target equality. The direct part uses typical subspaces, the operator Chernoff bound, and the assumed twirl ensemble to construct an achievable protocol whose rate approaches the regularized REI; no fitted parameter is renamed as a prediction and no equation is used as both input and conclusion. The only concerning passage is the unproved assertion in Sec. VI A 2, 'Assume {p(dU), U} is an ensemble of the covariant-free unitary operators such that for any state θ acting on H_{n,δ}, ∫ p(dU) U θ U† = I_{n,δ}/D_{n,δ}' — this is an omitted existence proof and therefore a correctness gap in the achievability argument, but it is not circularity, since the assumption is not the same as Theorem 2 and does not implicitly encode the result. There are no load-bearing self-citations, no imported uniqueness claims, and no ansatz smuggled in via citation. The derivation chain is therefore self-contained with respect to circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, or entities. The free parameters list is empty. The axioms are a mix of standard mathematical tools, a true identity about real orthogonal matrices, a false additivity claim, and an unproved twirl ensemble assumption. The false additivity claim is not used in the proof of the theorem but is internally inconsistent with the paper's own protocol.

assumptions (5)
  • domain assumption The relative entropy of imaginarity satisfies I_r(rho) = S((rho+rho^T)/2) - S(rho), from Xue et al. [28].
    Used in Lemma 6 and in the proof of Theorem 2. Without this identity, the quantity in the theorem is not connected to the real part of the state.
  • domain assumption The relative entropy of imaginarity is additive over tensor products, stated in Sec. II.B.
    This claim is false. For the maximally imaginarity qubit rho = |+i><+i|, I_r(rho) = ln2 but I_r(rho^{tensor 2}) = ln2, not 2ln2. If it were true, the paper's own two-unitary protocol would contradict the main theorem.
  • domain assumption There exists an ensemble {p(dU), U} of covariant-free unitaries such that the average of U theta U-dagger equals I/D on the delta-typical subspace, stated in the direct part of the appendix.
    The direct part depends on this twirl property. The paper does not prove the ensemble exists or specify how the unitaries act on the orthogonal complement of the typical subspace.
  • standard math Standard tools: typical subspace bounds, Fannes inequality, Gentle Measurement Lemma, and Operator Chernoff bound.
    These standard results are invoked in the appendix without proof and are needed for the achievability and converse arguments.
  • standard math For real orthogonal O, Theta(O rho O^T) = O Theta(rho) O^T, as stated in Eq. (2).
    This identity is true and implies the covariance condition in Definition 1 is automatically satisfied by all real orthogonal unitaries.

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Pith. "Pith review of The Deimginarity Cost of Quantum States." pith.science (2026). https://pith.science/paper/PFVKOH7F

@misc{pith2026250112891,
  author       = {Pith},
  title        = {Pith review of: The Deimginarity Cost of Quantum States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFVKOH7F}},
  note         = {Machine review of arXiv:2501.12891}
}
abstract

Here we address a task denoted as deimaginarity, which is to transform a state into a real state with the aid of random covariant-free unitary operations. We consider the minimum cost of randomness required for deimaginarity in the scenario of infinite copies and suiciently small error, and we prove that the deimaginarity cost of a state $\rho$ is equal to its regularized relative entropy of imaginarity, which can be seen as an operational interpretation of the relative entropy of imaginarity.

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Figures reproduced from arXiv: 2501.12891 by the authors.

Figure 1
Figure 1. FIG. 1: The task of deimginarity. Here [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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    Direct Part In this subsection, we will present the proof of the fol- lowing inequality, Cd(ρ) ≤ lim n→∞ 1 n Ir(ρ⊗n). The method of this proof is similar to the one shown in [29, 32]. The preliminary knowledge needed here are placed in Sec. VI B. Assume Hn,δ is the δ-typical s...

  36. [44]

    Assume X is a discrete random variable with finite alphabet X and px = P (X = x) when x ∈ X

    Typical Sequences and Subspaces Next we recall the definitions and properties of typical subspaces [39]. Assume X is a discrete random variable with finite alphabet X and px = P (X = x) when x ∈ X. A sequence ⃗ x= ( x1, x2, · · ·, xn) is denoted as δ-typical with respect to {p...

  37. [45]

    Necessary Lemmas Lemma 3 (Gentle Measurement Lemma) [40] As- sume ρ is a density matrix with trρ ≤ 1 acting on a finite dimensional system, then for any ϵ ≥ 0 such that 0 ≤ X ≤ I and tr(ρX) ≥ 1 − ϵ, we have ∥ρ − √ Xρ √ X∥ ≤2 √ 2ϵ. Lemma 4 (F annes Inequality) [38] Assume ρ and...

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Reviewed August 10, 2026 · model on record in the stance chip above.