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This is a common question and there isn't an absolutely clear answer. The problem is that this involves questions of marital/community property laws and probate laws.
Most judges will say that if the credit card was applied for by one spouse only and without the permission of the other spouse, the other spouse didn't use the card, and never agreed to be responsible for it then it is a debt of the applying spouse only and it would be a debt of their estate but it wouldn't pass to the "innocent spouse".
Some judges, instead, rule it is community/marital property debt and thus at least half of it is owed by the living spouse.
I have never seen a credit card company actually pursue the innocent spouse in a case like this other than making some phone calls to them and trying to harass them into paying.
In addition, when there is an estate opened on the deceased spouse's estate the credit card company should be notified as a creditor. Normally they don't file a claim and then that can also be used as evidence that they can't collect from the living spouse.
Please ask any follow up questions in this thread.