Conservation of quantum coherence?

Question

Is coherence conserved under unitary transformation? As we saw in the process of generating entanglement, coherence between states of the subsystem is transferred to quantum correlations (related to quantum coherence?) between the two subsystems. Is there a quantity that represents total coherence which is conserved under this unitary transformation?

This question was inspired by reading several other excellent posts on entanglement: Entanglement and coherence, "FTL" Communication with Quantum Entanglement?, What is coherence in quantum mechanics?.

Note:- Its my understanding/interpretation that the quantum correlations that arise in entangled systems are a result of coherence between the subsystems. And in my view entanglement generation is transfer of coherence and the total coherence is conserved.

Progress:- If there is a quantity which represents quantum coherence in system denoted by $C(\rho)$, shouldn't it have following properties?

1. $C(U^{\dagger}\rho U) = C(\rho)$, where $\rho$ is density matrix and $U$ is any unitary operation.

2. $C(\rho^{a} \otimes \rho^{b}) =C(\rho^{a}) + C(\rho^{b}) $

Answer 1

No, coherence is not conserved by unitary transformations, in general. It's easiest to see this with a simple example. Consider a one-dimensional quantum harmonic oscillator, with Hamiltonian ($\hbar = 1$) $$ H = \omega a^\dagger a,$$ possessing energy eigenstates $H\lvert n\rangle = n\omega \lvert n\rangle$. Now, coherence (in the usual sense of the word) can only be defined with respect to a particular choice of basis. In quantum optics, in the study of nano-mechanical oscillators, and in many other applications of the quantum harmonic oscillator, coherence is usually defined with respect to the energy eigenbasis. That is, a state $\rho$ possesses coherence if its expansion in the energy eigenbasis $$ \rho = \sum_{m,n} \rho_{mn} \lvert m \rangle \langle n\rvert, $$ has at least one term where $\rho_{mn}\neq 0$ for $m\neq n$. Indeed, such a term is technically referred to as a coherence (in the energy eigenbasis).

Thus, the ground state of the system $\lvert 0 \rangle$ does not possess coherence. On the other hand, a coherent state $\lvert \alpha\rangle$, such that $a \lvert \alpha\rangle = \alpha \lvert \alpha \rangle$, does possess lots of coherence (surprise!). However, the two are related by a unitary transformation, the well known unitary displacement operation $\lvert \alpha\rangle = D(\alpha)\lvert 0 \rangle$, where $$ D(\alpha) = \exp \left ( \alpha a^\dagger - \alpha^* a \right ), $$ and clearly $D^\dagger(\alpha) D(\alpha) = 1$. A coherence measure $C(\rho)$ satisfying the OP's condition 1 thus implies $$ C(\lvert \alpha \rangle \langle \alpha \rvert) = C(D(\alpha)\lvert 0 \rangle \langle 0 \rvert D^\dagger(\alpha)) = C(\lvert 0 \rangle \langle 0 \rvert ). $$ Therefore $C(\rho)$ is a rather poor measure, as it assigns the same amount of "coherence" to the vacuum state (normally considered to have no coherence) and to a coherent state (normally considered to have "maximal" coherence).

It is straightforward to generalize this to multipartite systems. One finds that, for example, $C(\rho)$ assigns the same amount of "coherence" to maximally entangled pure states and to separable pure states. Again, this is exactly the opposite of what one would normally call coherence.

Overall, we see that no sensible coherence measure can be invariant under all unitary transformations. In fact, a coherence measure should only be generally invariant under unitaries which are diagonal in the chosen reference basis (i.e. the energy eigenbasis in these examples).

Answer 2

If we understand coherence as "coherent superposition", then yes, coherence is conserved in a certain sense.

A simple superposition of two states evolves unitarily as $$ \alpha\;|a\rangle + \beta\;|b\rangle \;\; \rightarrow\;\; \alpha\;e^{-iHt}|a\rangle + \beta\;e^{-iHt}|b\rangle \equiv \alpha\;|a(t)\rangle + \beta\;|b(t)\rangle $$ so we can say that at any time the relative phase of the evolved components $|a(t)\rangle$ and $|b(t)\rangle$ is the same as the relative phase of initial components $|a\rangle$ and $|b\rangle$.

In fact, the same goes for density matrix elements of mixed states. We have $\rho(t) = e^{-iHt} \rho(0) e^{iHt}$, but also $$ \langle a(t) |\rho(t)|b(t)\rangle = \langle a |\rho(0)|b\rangle $$

Further, in a non-interacting bipartite system evolving under Hamiltonian $H = H_A + H_B$, the relative phase of contributions to a total pure state is preserved in the sense that $$ |\Psi_{AB}(0)\rangle = \alpha\;|\psi_A \otimes \psi_B\rangle + \beta\;|\phi_A \otimes \phi_B\rangle \;\; \rightarrow\;\; |\Psi_{AB}(t)\rangle = \alpha\;|\psi_A(t) \otimes \psi_B(t)\rangle + \beta\;|\phi_A(t) \otimes \phi_B(t)\rangle $$ where $$ \alpha\;|\psi_A(t) \otimes \psi_B(t)\rangle + \beta\;|\phi_A(t) \otimes \phi_B(t)\rangle \equiv \alpha\;e^{-iHt}|\psi_A \otimes \psi_B\rangle + \beta\;e^{-iHt}|\phi_A \otimes \phi_B\rangle $$ for $$ |\psi_A(t)\rangle = e^{-iH_At}|\psi_A\rangle, \;\; |\phi_A(t)\rangle = e^{-iH_At}|\phi_A\rangle\\ |\psi_B(t)\rangle = e^{-iH_Bt}|\psi_A\rangle, \;\; |\phi_B(t)\rangle = e^{-iH_Bt}|\phi_A\rangle $$ And although in this case the states of subsystems A and B are no longer "coherent" pure states, but "incoherent" mixed states, $\rho_{A(B)} = Tr_{B(A)}|\Psi_{AB}\rangle\langle \Psi_{AB}|$, we can say that a certain degree of coherence is still conserved in time even in the mixed states, since their matrix elements still satisfy $$ \langle \psi_A(t) |\rho_A(t)|\psi_A(t)\rangle = \langle \psi_A |\rho_A(0)|\psi_A\rangle,\;\;\;\langle \psi_A(t) |\rho_A(t)|\phi_A(t)\rangle = \langle \psi_A |\rho_A(0)|\phi_A\rangle\;\;, \text{etc} $$ and similarly for B.

Note however that we cannot talk of a single conserved quantity representing coherence. We can only say that a unitary evolution preserves relative phase relationships between unitarily evolved pure states, both in pure state superpositions and in mixed states.