Coherence in density matrix formalism

[HJ Farnsworth]

Hello,

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 $0$ 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
$\rho = \sum_{\psi}P(\psi)|\psi\rangle\langle\psi|$
in the $\{ |m\rangle \}$ representation, $|\psi\rangle = \sum_{m}|m\rangle\langle m|\psi\rangle$, it's easy to think of trivial examples where the coherence is $0$ 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 $\{ |x\uparrow \rangle , |x \downarrow \rangle \}$ versus the basis $\{ |z \uparrow \rangle , |z \downarrow \rangle \}$ gives, as expected, two different representations of the density matrix: $\rho = \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}$ in the $z$-basis and $\rho = \begin{pmatrix}1/2 & 1/2\\1/2 & 1/2\end{pmatrix}$ in the $x$-basis. So here, we have a pure state, which in one basis has a coherence of $0$, and in another has a coherence of $1/2$ - it seems to me that a good definition of coherence would identify the coherence of a pure state as $1$ regardless to basis.

Furthermore, every time I have seen this definition of coherence of a state, the only examples given are for $2\times 2$ 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-$z$ having a coherence of $0$ above would also make more sense, since if the energy of the spin-up state is nonzero, then the complex amplitude of $| \uparrow \rangle$ will oscillate, while that of $| \downarrow \rangle$ won't, and instead will always remain $0$. 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 $1/2$ versus $1/3$ 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

 

[kith]

Talking about coherence can be a bit confusing because the term has many slightly different meanings and is often used informally.

Mathematically well-defined are the terms "coherences" for the off-diagonal elements of the density matrix and "purity" for the trace of the density matrix squared.

The purity of a density matrix doesn't depend on the basis. It is one for pure states and minimal for a maximally mixed state (which is represented by a density matrix which is proportional to the identity matrix). So if you want to talk about the "coherence" of a (not necessarily pure) state, you may want to use the purity.

As you already suspected, the off-diagonal elements or coherences refer to the coherence between basis states. A non-zero coherence between two states means that you cannot decompose your ensemble into a merely statistical mixture of sub-ensembles such that no sub-ensemble contains both basis states. Sakurai has a number of nice examples on this in his chapter on angular momentum. He probably doesn't use modern terminology though.

 

[HJ Farnsworth]

Thanks kith, that's great.

So to make sure I understand you correctly, a simple interpretation that follows from the definition of coherence in the original post would be something like, "coherence between two basis states is a measure of the degree to which those two basis states must be regarded as part of the same, additively inseparable ensemble".

Other concepts of coherence that I have seen relate coherence between two waves to the relative frequency and phase difference of those waves - i.e., "two waves are coherent iff they have the same frequency and constant phase difference". I think that I can relate the interpretation above to this interpretation.

First, the additive inseparability interpretation motivates this guess: "The statement that the coherence between two basis states $|a \rangle$ and $|b \rangle$ in a density matrix is $0$ is equivalent to the statement that all of the quantum states that were classically weighted in writing the density matrix had either the amplitude for $|a \rangle$ equal to $0$, or that for $|b \rangle$ equal to $0$ - none of them had the amplitudes for both $|a \rangle$ and $|b \rangle$ nonzero." The reason the additive inseparability interpretation motivates this guess is that, if the states composing the ensemble have this property, then it is obvious that the coherence will be $0$ and the density matrix will be additively separable, so it is at least a sufficient condition for incoherence. Testing this guess, let $|\psi_{1}\rangle =a_{1}|a\rangle + b_{1}|b\rangle$, and write
$\rho = \sum{P_{i}|\psi_{i}\rangle \langle \psi_{i} |}=\begin{pmatrix} & a_{1}b_{1}^{*}+\cdots\\a_{1}^{*}b_{1}+\cdots\end{pmatrix}.$
It is immediately obvious that the guess was wrong - e.g., if $|\psi_{2}\rangle =-a_{1}|a\rangle + b_{1}|b\rangle$, then even if both $a_{1}$ and $b_{1}$ are nonzero, we could have offdiagonal matrix elements of $0$. However, note that if this is the case, then the phase difference between the two amplitudes in $\psi_{1}$ is exactly canceled out by that in $\psi_{2}$: $|\psi_{2}\rangle =-a_{1}|a\rangle + b_{1}|b\rangle=e^{i\pi}a_{1}|a\rangle + b_{1}|b\rangle$, vs. $|\psi_{1}\rangle =e^{i0}a_{1}|a\rangle + b_{1}|b\rangle$. The offdiagonal terms from $|\psi_{1} \rangle$ could also be canceled out by multiple other states in the ensemble - the only requirement for this to happen is that the phase differences between $|a\rangle$ and $|b\rangle$ in each state of the ensemble combine so as to cancel each other out (I am ignoring the different relative amplitudes between$|a\rangle$ and $|b\rangle$ among states, different amplitudes will just put a sort of weight on different relative phase differences between the two basis states).

Based on this, I can replace the above guess with, "The statement that the coherence between two basis states $|a \rangle$ and $|b \rangle$ in a density matrix is $0$ is equivalent to the statement that all of the quantum states that were classically weighted in writing the density matrix had relative phases between the two basis states which ultimately canceled each other out." Relating this to the "common" coherence definition two paragraphs above, I could say something like "two basis states are [completely] coherent, in the density matrix sense, if their phase difference is constant throughout the states of the ensemble". (I brushed frequency under the rug since I'm considering everything at a single time, say $t=0\implies \omega t=0$. This is in pretty good analogy to e.g., classical optics temporal coherence between two waves = constant phase difference at a given point in space as the waves propagate: Temporal coherence between two waves throughout time is analogous to density matrix coherence between two basis states among the states in the ensemble.

Any thoughts on this thought process? Decent analysis, complete BS, or am I attempting to go too far in getting an intuition on what is simply a mathematical definition?

A half-follow-up and half-new-question: Regarding the first sentence in kith's response, the other two definitions of coherence that I have frequently come across are spatial/temporal coherence, defined in terms of the autocorrelation function in classical optics, and coherent states, defined as eigenstates of the HO annihilation operator in quantum mechanics. I think I have begun to answer this for myself in the above paragraph, but does anyone know the degree to which these three definitions of coherence/coherent states are related, or of a good source explaining, mathematically and interpretively, the relations among these concepts?

Thanks again.

 

[HJ Farnsworth]

Sorry for bumping this, but I do want to know whether people who have a bit more experience than me with this topic think that my conceptual understanding in the previous post holds water. Any thoughts?

Thanks.