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I am reading this paper: Quantum Generative Training Using Rényi Divergences. In it, the authors mention the following multiple times:

"...an unbounded loss function can circumvent the existing no-go results"

"...unbounded loss functions, such as the quantum relative entropy..."

"...unbounded loss function such as maximal Rényi divergence of order two..."

"...other unbounded loss functions such as relative entropy..."

"...entanglement between the hidden and visible layers can still destroy the ability to train the network with respect to the bounded loss function..."

I have no clue what they mean by "unbounded/bounded loss function" here. At one point, they also say:

"Existing algorithms almost exclusively utilize a linear bounded operator as a loss function. This condition is quite reasonable because the loss function is typically estimated by measuring the expectation values of Hermitian operators"

I am not sure if this usage of "linear bounded operator" is related; if not, what does it mean?

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In this context, I think that the authors simply refer to a bounded function

A linear operator is called bounded when it it has finite operator norm. This is equivalent to saying that the linear operator is continuous as a map between normed vector spaces. Note that any linear operator on a finite-dimensional vector space is always bounded, so this is only a non-redundant concept if one is dealing with infinite-dimensional vector spaces.

Finally, if you define a function on states as $f(\rho) = \mathrm{tr}(A\rho)$ and $A$ is a bounded operator (on a Hilbert space), then this function is automatically bounded since we have by Hölder's inequality $$ |f(\rho)| \leq \|A\|_\infty \|\rho\|_1 = \|A\|_\infty < \infty, $$ where $\|\cdot\|_\infty$ is the operator/spectral norm and $\|\cdot\|_1$ is the trace norm.

PS: You could have answered this yourself by mere googling.

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