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In the density matrix formalism I have read in numerous places that coherence is identified with the off-diagonal components of the density matrix. The motivation for this that is usually given is that if a state interacts with the environment in such a way that the basis state amplitudes are phase-shifted through this interaction, then the off-diagonal components will average to [itex]0[/itex] if the degree to which each basis amplitude is phase-shifted are uncorrelated.

I think I am missing something critical about this definition. Given the density matrix

[itex]\rho = \sum_{\psi}P(\psi)|\psi\rangle\langle\psi|[/itex]

in the [itex]\{ |m\rangle \}[/itex] representation, [itex]|\psi\rangle = \sum_{m}|m\rangle\langle m|\psi\rangle[/itex], it's easy to think of trivial examples where the coherence is [itex]0[/itex] for pure states, which to me seems absurd. It is also easy to think of examples where the coherence is basis-dependent, which also seems strange to me.

For instance a pure spin-up state as written in the basis [itex]\{ |x\uparrow \rangle , |x \downarrow \rangle \}[/itex] versus the basis [itex]\{ |z \uparrow \rangle , |z \downarrow \rangle \}[/itex] gives, as expected, two different representations of the density matrix: [itex]\rho = \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}[/itex] in the [itex]z[/itex]-basis and [itex]\rho = \begin{pmatrix}1/2 & 1/2\\1/2 & 1/2\end{pmatrix}[/itex] in the [itex]x[/itex]-basis. So here, we have a pure state, which in one basis has a coherence of [itex]0[/itex], and in another has a coherence of [itex]1/2[/itex] - it seems to me that a good definition of coherence would identify the coherence of a pure state as [itex]1[/itex] regardless to basis.

Furthermore, every time I have seen this definition of coherence of a state, the only examples given are for [itex]2\times 2[/itex] density matrices. How does the definition extend to density matrices written for larger bases - do we only define the coherence for one pair of basis amplitudes at a time?

I have a guess to the answers to the above questions, but haven't been able to verify it. The coherence could be a comparative proberty between substates only, not a property of the state as a whole. In this case, a change of basis giving different coherences makes a lot more sense, and the extension of the definition to larger density matrices becomes trivial. Furthermore, the spin-[itex]z[/itex] having a coherence of [itex]0[/itex] above would also make more sense, since if the energy of the spin-up state is nonzero, then the complex amplitude of [itex]| \uparrow \rangle [/itex] will oscillate, while that of [itex]| \downarrow \rangle [/itex] won't, and instead will always remain [itex]0[/itex]. So, the complex amplitudes of the two phases won't be correlated at all.

With this, the value of coherence being zero versus nonzero has an interpretation in terms of complex amplitude correlation between substates. However, it's still hard to see an interpretation of different nonzero values for coherence - e.g., what does a coherence of [itex]1/2[/itex] versus [itex]1/3[/itex] tell you?

Anyway, I think I've said plenty to illustrate my confusion. Any help on the topic would be very much appreciated.

Thanks very much.

-HJ Farnsworth