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Metadata 
bibkey: watrous2018entropicidentities
authors: John Watrous
year: 2018
title: The Theory of Quantum Information — spectral calculus, reductions and entropy
doi: null
url: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf
claim: Spectral functional calculus, partial trace, isometric entropy invariance and the definition of mutual information give the coherent-copy marginals, Gibbs logarithm, uniform-reference entropy identity and unbiased-pinching account by explicit standard substitutions.
strata_touched:
  - D5/S3/Quantum/Information/CoherentCopyCorrelationTax
  - D5/S3/Quantum/Information/PartialTraceMutualInformation
  - D5/S3/Quantum/Information/CorrelatedGibbsEnergyIdentity
  - D5/S3/Quantum/Divergence/GibbsVariationalIdentity
  - D5/S3/Quantum/Divergence/DualAccountFull
license: citation-only
triage: anchor

Spectral calculus, reductions and entropy

Verified locator

DOI: null

The exact upstream locator is: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf

The author-hosted pre-publication PDF (title page: copyright 2018 John Watrous, “draft, pre-publication copy”) was retrieved and the following interior passages read. Page numbers below are printed pages, followed by one-based PDF pages.

  • Section 1.1.2, p. 22 (PDF 30), equations (1.120)–(1.121): partial trace on a tensor product is Tr(X)Y or X Tr(Y).
  • Section 1.1.3, p. 27 (PDF 35), equation (1.145): spectral functional calculus is f(A) = sum_k f(lambda_k) Pi_k.
  • Section 2.1, p. 68 (PDF 76), equations (2.30)–(2.34): a subsystem state is the actual partial trace; (2.34) gives its entries.
  • Definition 4.4, p. 203 (PDF 211), equations (4.7)–(4.8): pinching is Phi(A) = sum_j Pi_j A Pi_j, with sum_j Pi_j = I.
  • Definition 5.17, pp. 265–266 (PDF 273–274), equations (5.82)–(5.84): entropy is the Shannon entropy of the eigenvalues, or -Tr(P log P), with the value of t log t at zero set to zero.
  • Definition 5.18, p. 266 (PDF 274), equation (5.85): relative entropy is Tr(P log P) - Tr(P log Q) when im(P) is in im(Q).
  • Section 5.2.1, p. 267 (PDF 275), equation (5.91): I(X:Y) = H(X) + H(Y) - H(X,Y).
  • Proposition 5.19, p. 268 (PDF 276), equation (5.93): H(V P V*) = H(P) for an isometry, including singular positive P.
  • Proposition 5.22, p. 269 (PDF 277): relative entropy of density operators is nonnegative (Klein’s inequality).

The book uses base-two logarithms for entropy. Multiplication of its entropy and relative entropy by ln(2) gives the repository’s natural-log convention. All entropy identities and inequalities below use nats. The exponential in the Gibbs construction is the ordinary natural exponential.

Coherent-copy entries, marginals and entropy

Set V|i> = |i,i>. Orthonormality gives V*V = I. Expansion in matrix units gives

R = V rho V* = sum_{i,j} rho_ij |i><j| tensor |i><j|.

Consequently R_(i,i),(j,j) = rho_ij. This is the exact specialization behind coherentCopyState_correlated_entry, not an assertion that Watrous names that declaration. Applying (1.120)–(1.121), with Tr(|i><j|) = delta_ij, gives

Tr_record R = sum_i rho_ii |i><i| = Delta(rho),
Tr_system R = sum_i rho_ii |i><i| = Delta(rho).

These are marginalRight_coherentCopyState and marginalLeft_coherentCopyState. The order of the retained subsystem is explicit: marginalRight retains the first factor; marginalLeft the second. Proposition 5.19 with this V gives vonNeumannEntropy_coherentCopyState. Equivalently, V sends an eigenbasis of rho to orthonormal eigenvectors of R; the orthogonal complement contributes only zero eigenvalues and zero entropy.

Actual marginal mutual information and unitary invariance

PartialTraceMutualInformation.quantumMutualInformation is precisely (5.91), with the subsystem states computed by (2.30) and (2.34). It is a definition of a function of one joint state, not three independent entropy parameters.

CorrelatedGibbsEnergyIdentity.von_neumann_entropy_unitary is Proposition 5.19 specialized to a square unitary U. No full-rank assumption is needed. The repository’s proof obtains this equality by subtracting two previously proved pinching identities; that implementation choice does not change the standard mathematical statement or attribute that proof route to the book.

Gibbs logarithm and the uniform reference

Let H = sum_k h_k Pi_k be Hermitian. By (1.145), exp(H) has eigenvalues exp(h_k) > 0. Thus Z = Tr(exp H) > 0 and G = exp(H)/Z has eigenvalues exp(h_k)/Z. Apply the scalar natural logarithm in the same spectral basis:

log G = sum_k (h_k - ln Z) Pi_k = H - ln Z I.

This is log_gibbs_state, a direct spectral-calculus consequence, not a separately located named Gibbs theorem. With Tr(rho) = 1, (5.83) and (5.85) then give D(rho||G) = -S(rho) - Tr(rho H) + ln Z. No commutation between rho and H is used. The reference G is full rank, so the support condition in (5.85) always holds, even for singular rho.

For H = 0, exp(H) = I, Z = d and G = I/d. The same formula gives

S(rho) + D(rho||I/d) = ln d.

This is entropy_uniform_identity. The existing citation attached to the general gibbs_variational_identity is a separate entry and is not replaced.

Unbiased pinching and the entropy/freedom segment

For rank-one projectors Z_j and X_k resolving I, write a Z-fixed density as rho = sum_j p_j Z_j, p_j = Tr(Z_j rho), p_j >= 0 and sum_j p_j = 1. The rank-one identity X_k A X_k = Tr(X_k A) X_k follows immediately by writing X_k = |x_k><x_k|. Under the theorem’s supplied overlap hypothesis Tr(X_k Z_j) = 1/d, Definition 4.4 therefore gives

Phi_X(rho) = sum_{j,k} p_j Tr(X_k Z_j) X_k
           = (sum_j p_j) (sum_k X_k)/d = I/d.

This calculation also covers d = 1. Put sigma = Phi_X(rho) and omega = I/d. The uniform-reference identity gives

sigma = omega,
D(rho||sigma) = D(rho||omega) = ln d - S(rho),
S(sigma) - S(rho) = D(rho||sigma),
Phi_X(rho) = rho iff rho = omega.

These are exactly the conjuncts of dual_account_full. The last equivalence is under the existing Z-fixed and unbiased hypotheses. No general assertion that every measurement depolarizes a state is made.

For entropy_freedom_segment, eigenvalues p_i of a density obey 0 <= p_i <= 1 and sum_i p_i = 1, hence -sum_i p_i ln p_i >= 0. Proposition 5.22 gives D(rho||omega) >= 0; the displayed uniform-reference identity supplies S + D = ln d. This is a pointwise account for each state, without a dynamical equation or a prescribed trajectory.

What this note does and does not attest

Attested by this repository’s own retrieval: the PDF and every interior locator and formula listed above. The substitutions and finite-dimensional algebra connecting them to the ten bound declarations are written out here and checked against the current Lean statements. The PDF receipt is in the stage-3 implementation report; no search-engine snippet is used as evidence.

Not attested here: a page-by-page comparison with the publisher’s printed edition, the book’s historical source references, or earliest priority for any consequence. Those historical attributions, if inferred, are ASSUMED-UNVERIFIED. The book is cited for the stated definitions and propositions; the copy construction and unbiased/Gibbs substitutions above are explicit standard deductions, not quotations of same-named theorems.