# What is known about the size of the spectral gap of unital quantum channels?

I am interested in the spectrum of unital quantum channels $$\Phi$$ (which act on finite dimensional spaces). The literature is extremely vast on such objects so I hope some experts can point me along the right direction or references.

I know that the spectrum (i.e. eigenvalues $$\lambda$$ appearing in the eigenvalue-eigenvector equation $$\Phi(X) = \lambda X$$ ) lies within the unit disk $$|\lambda|\leq 1$$. Because of unitality $$\mathbb{I}$$ is always an eigenvector with eigenvalue $$1$$ because $$\Phi(\mathbb{I}) = \mathbb{I}$$. There may be other eigenvectors that lie on the unit disk $$|\lambda| = 1$$ (called `peripheral eigenvectors/eigenvalues').

My question is what is known about the size of the gap of unital quantum channels, defined as $$1 - |\lambda|$$ such that $$\lambda$$ is the eigenvalue with largest magnitude that is not $$1$$? The physical reason why I am interested in because this sets the rate of convergence to the peripheral eigenvectors.

Perhaps what I am asking is too general, and it depends on the explicit quantum channel in question? In which case my question would then be, is there a way to estimate the magnitude of the gap? Like a variational principle of sorts?

• Technically, spectral gap is ill-defined since the maximum of the set of eigenvalues other than one may not exist. Consider the identity channel. Nov 18, 2023 at 14:21
• It should be possible to estimate the spectral gap of a given channel numerically using eigenvalue algorithms, such as Arnoldi iteration. Nov 18, 2023 at 14:27
• Thanks! Sorry I realized my question was phrased in a slightly confusing manner. Obviously the spectral gap depends on the channel in question and there is no universal value for all unital channels that I might be giving the wrong impression of (I also take your point that the set of non-one eigenvalues may not exist, but in which case I would simply just define the gap as 1). I intended to ask is there an analytic way to bound the spectral gap given a channel. I suppose one can try to employ Arnoldi iteration "by hand" instead of numerically but I don't know how much mileage that will give Nov 18, 2023 at 15:26

TL;DR: Spectral gap depends on the specific channel. Moreover, for any $$g\in[0,1]$$ there is a channel with spectral gap $$g$$.
We can make simple observations about properties of eigenoperators of $$\Phi$$ by hitting both sides of the eigenvalue equation $$\Phi(A)=\lambda A$$ with the trace to get $$\mathrm{tr}(A)=\lambda\,\mathrm{tr}(A)\tag1$$ where we used the fact that $$\Phi$$ is a channel and hence trace-preserving. Clearly, $$(1)$$ can only be true if $$\lambda=1$$ or $$\mathrm{tr}(A)=0$$, so all non-peripheral eigenoperators of $$\Phi$$ are traceless.
This suggests a simple construction of a unital channel with a prescribed spectral gap. Consider depolarizing channel $$\mathcal{D}_p(A)=pA+(1-p)\frac{\mathbb{I}}{d}\mathrm{tr}(A)\tag2$$ where $$d$$ is the dimension of the Hilbert space and $$p\in\big[-\frac{1}{d^2-1}, 1\big]$$. Clearly, every non-zero traceless operator $$A$$ is an eigenoperator of $$\mathcal{D}_p$$ with eigenvalue $$p$$.
Thus, the spectral gap of a unital channel indeed depends on the specific channel. Moreover, for any $$g\in[0,1]$$ there is a unital channel with spectral gap $$g$$.