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Quantum Computing

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  • $\begingroup$ Maybe a dumb question in regards to the wikipedia article. It states "he quantum relative entropy is a measure of our ability to distinguish two quantum states where larger values indicate states that are more different. Being orthogonal represents the most different quantum states can be". Wouldn't the QRE between $|0\rangle\langle0|$ and $|+\rangle\langle+|$ also be $+\infty$, as neither is supported by the one-dimensional subspace of the other, right? $\endgroup$ Commented Oct 15, 2020 at 15:33
  • $\begingroup$ Or is $|0\rangle\langle0|$ still in the support of $|+\rangle\langle+|$ given $|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt{2}}$? $\endgroup$ Commented Oct 15, 2020 at 15:36
  • $\begingroup$ @GaussStrife The support of an operator $\rho$, denoted $\mathrm{supp}(\rho)$. is defined in this context to be the orthogonal complement of the kernel $\mathrm{ker}(\rho) = \{ |x \rangle \in \mathcal{H} : \rho |x \rangle = 0 $. Thus $\mathrm{ker}(|+ \rangle \langle + |) = \mathrm{span}\{ |-\rangle\}$ and $\mathrm{supp}(|+\rangle \langle +|) = \mathrm{span}\{|+\rangle\}$. $\endgroup$ Commented Oct 16, 2020 at 8:25
  • $\begingroup$ @GaussStrife Wikipedia also says "However, one should be careful not to conclude that the divergence of the quantum relative entropy S (ρ‖σ) implies that the states ρ and σ are orthogonal or even very different by other measures. Specifically, S (ρ‖σ) can diverge when ρ and σ differ by a vanishingly small amount as measured by some norm." Two rank one states can be arbitrary close (in norm/fidelity), but unless they are exactly equal the relative entropy diverges. So yes, $|0\rangle\!\langle 0|$ is not supported on $|+\rangle\!\langle +|$ and the relative entropy between the two is $+\infty$. $\endgroup$ Commented Oct 16, 2020 at 17:00
  • $\begingroup$ @AngeloLucia See I need to learn to be patient and read the next sentence. Alright this makes perfect sense then. All states that have non-trivial intersection with either the kernel(nullspace) of a given state, or even just cannot be located in said support (regardless of the amount the states differ) diverge in regards to entropy. $\endgroup$ Commented Oct 16, 2020 at 18:13