Ideally you need to ascertain whether this is has crossed into a clinical depression (i.e. you have a medical condition as opposed to simply "feeling sad").
I'm not in a position to give you a diagnosis one way or the other, and unfortunately I expect all the pandemic disruption will make getting a clinical evaluation harder than it would have been otherwise. Depending on your however location there may still be services and resources you can access remotely, whether that's over the phone or online.
Even if this isn't something that needs clinical treatment that doesn't mean this isn't a real problem and it doesn't mean that you can't potentially benefit from some counseling at the very least. A great many people are going to be having a very tough time with this - it's a pretty big deal all round.
How should I handle this?
Your boss may or may not be sympathetic to the difficulties - I'd like to think they would be but there's no guarantee of that. In general approaching them with a heads up that your having difficulties is likely to go better than simply doing nothing, but honestly it wouldn't be the first port of call for me. You're too new and don't have an established rapport or reserve of good will with them.
So what to do?
First - start investigating what options you can access regarding mental health support/treatment. Maybe this feeling will resolve itself once the current crisis passes - but that's going to be months away by most predictions. Waiting-and-seeing that long is doing yourself a pretty serious disservice, and I say that irrespective of your job situation. And if this turns out to be something that needed clinical treatment all you've done is suffer unnessecarily and delayed yourself getting better.
Alongside this make sure you're doing as much self-care as you can - eat right, sleep right, exercise where possible, I'm not saying these things are a miracle cure and will take away all your depressive feelings, that's foolish naiviety at best. But ignoring these aspects and neglecting yourself will definitely make such feelings worse.
Second - try not to think about the work as a monolith that you're doing 100% or not doing at all. The one thing you do have in your favor with the whole new job situation is that no-one expects a new starter to be immediately fully productive, so that's going to provide some decent wiggle room for you.
Pretty much all jobs can be broken down into smaller units - pick the smallest one you can find, break it down to the simplest step and do it. Just the first step.
If you get that step done - give yourself a bit of kudos, have a cookie, whatever you want to do for a quick reward. Then, do the next one, and so on.
If you find yourself getting some momentum and start to feel more confident then you can make the steps/tasks bigger - but don't let yourself feel pressured. Remember if your current productivity is zero, anything is better than that, and give yourself credit for how much more than zero you do rather than how much less than "100%" you do.