Quantum spectral entropy and relative-entropy domains

For a trace-class density $\rho$ on a Hilbert space $H$, define its support subspace in either finite or infinite dimension by $\mathrm{supp}\rho :=(\ker \rho {)}^{\perp },\ker \rho :=\{x\in H:\rho x=0\}\mathrm{.}$ It is closed; write $P_{\rho }$ for its orthogonal projection, whose existence and uniqueness follow from Orthogonal decomposition by a closed subspace.

Fix the positive Boltzmann scale $k_B>0$ (in SI units J/K, as in the approved QSM quantities design and Trace-state cyclicity and spectral probability calculus). For a trace-class density $\rho$ with positive eigenvalues $p_i$, define $S(\rho )=-k_B{\sum }_i{p}_i\log p_i\in [0,\mathrm{\infty }]$, with $0\log 0=0$. The logarithm is natural and acts on dimensionless density eigenvalues.

For finite-dimensional densities $\rho ,\sigma$, define $\mathrm{supp}\rho =(\ker \rho {)}^{\perp }$. If $\mathrm{supp}\rho \subseteq ̸\mathrm{supp}\sigma$, set $D(\rho \mathrm{\parallel }\sigma )=+\mathrm{\infty }$. Otherwise define $D(\rho \mathrm{\parallel }\sigma )=\mathrm{Tr}(\rho \log \rho )-\mathrm{Tr}(\rho \log \sigma ),$ where each logarithm is restricted to the positive support of its operator and zero eigenvalues contribute zero to $\mathrm{Tr}(\rho \log \rho )$.

For trace-class normal densities on a separable Hilbert space, write $\rho ={\sum }_i{p}_i\mathrm{\mid }{u}_i⟩⟨u_i\mathrm{\mid }$ and $\sigma ={\sum }_j{s}_j\mathrm{\mid }{v}_j⟩⟨v_j\mathrm{\mid }$ on their positive supports. Define the nonnegative cross-log moment $C(\rho ,\sigma ):={\sum }_{i,j}{p}_i(-\log s_j)\mathrm{\mid }⟨u_i,v_j⟩{\mathrm{\mid }}^2\in [0,\mathrm{\infty }]\mathrm{.}$ This is the spectral first moment of $-\log \sigma$ in the scalar state law of $\rho$; equivalently it is the nonnegative sum over the spectral projections, so it is independent of bases within eigenspaces. When support inclusion holds and $C(\rho ,\sigma )$ and $S(\rho )$ are both finite, define $D(\rho \mathrm{\parallel }\sigma )=C(\rho ,\sigma )-S(\rho )\mathrm{/}{k}_B$. If $C=+\mathrm{\infty }$ and $S(\rho )<\mathrm{\infty }$, set $D=+\mathrm{\infty }$. No value is assigned by this definition to $+\mathrm{\infty }-(+\mathrm{\infty })$; Trace-class cross-log entropy inequality proves that finite $C$ forces finite $S(\rho )$.

The spectral-moment identification in the preceding cross-log clause is restricted to $\mathrm{supp}\rho \subseteq \mathrm{supp}\sigma$, so that the state is supported on the space where $-\log \sigma$ acts. Outside that domain, $C$ denotes only the displayed nonnegative double sum.