bibkey: xu2015oblique authors: Jianwei Xu year: 2015 title: “Oblique discord” doi: 10.1142/S0217979216502568 url: https://arxiv.org/abs/1506.00404v1 claim: “The Conjecture of Eq. (22) asserts I(rho^AB) >= I(Phi_A rho^AB) for every normalized oblique operation of Eq. (13) and every bipartite density state.” strata_touched:

• D5/S3/Quantum/Information/ObliqueDiscordMutualInformationRefutation license: citation-only triage: anchor

Oblique discord

Jianwei Xu, Oblique discord, arXiv:1506.00404v1; Int. J. Mod. Phys. B 31, 1650256 (2017). The arXiv text defines a state-dependent normalized operation using a normalized basis and its biorthogonal dual.

The definition on p. 2 preceding Eq. (13) reads:

Suppose $\{|i⟩{\}}_{i=1}^{n_A}$ is a normalized basis of $H^A$ which not necessarily orthogonal to each other. There exists an unique basis $\{|\tilde{i}⟩{\}}_{i=1}^n$ of $H^A$ such that $⟨i|\tilde{j}⟩={\delta }_{ij}$, note that $\{|\tilde{i}⟩{\}}_{i=1}^n$ not necessarily orthogonal and not necessarily normalized. $\{|\tilde{i}⟩{\}}_{i=1}^{n_A}$ is called the dual basis of $\{|i⟩{\}}_{i=1}^{n_A}$. We define the quantum operation ${\Phi }_A=\{|i⟩⟨\tilde{i}|{\}}_{i=1}^{n_A}$ which operates the bipartite state ${\rho }^{AB}$ as

$${\Phi }_A{\rho }^{AB}=\frac{{\sum }_{i=1}^{n_A}|i⟩⟨\tilde{i}|{\rho }^{AB}|\tilde{i}⟩⟨i|}{tr[{\sum }_{i=1}^{n_A}⟨\tilde{i}|{\rho }^{AB}|\tilde{i}⟩]}.$$

The Conjecture on p. 3, Eq. (22), reads:

Conjecture: $I({\rho }^{AB})\ge I({\Phi }_A{\rho }^{AB})$ for any ${\Phi }_A$ and any ${\rho }^{AB}$, where $I({\rho }^{AB})$ is the mutual information, ${\Phi }_A$ is defined in Eq.(13).

The sentence following Eq. (21) on p. 3 reads:

where $I(\rho )=S({\rho }^A)+S({\rho }^B)-S(\rho )$ is the mutual information.

The paper defines $D_O^A(\rho )={\inf }_{{\Phi }_A}[I(\rho )-I({\Phi }_A\rho )]$. The Lean encoding uses finite complex Hilbert spaces, actual bases and their normalization and duality equations, and the frozen density-state subtype. Tensoring each Kraus operator with the identity makes its action on the untouched factor explicit. The denominator is defined literally as tr_B(∑_i ⟨w_i|ρ|w_i⟩), using the existing partialTraceRight on the one-dimensional factor times B. The unit norm of each basis vector proves its equality with the trace of the unnormalized output inside phi; result checks the equality at the witness. Entropy uses natural logarithms; changing to any logarithm base greater than one multiplies mutual information by a positive constant.

Verified locator

• URL: https://arxiv.org/abs/1506.00404v1, Eq. (13) and the Conjecture of Eq. (22).
• DOI: https://doi.org/10.1142/S0217979216502568. The journal full text is unverified; the formal claim concerns the arXiv v1 text.