Once you're convicted, you're toast. SCOTUS today decided District Attorney’s Office v. Osborne (opinion here) addressing the right of a convicted offender to have DNA testing that might prove innocence. The short answer: Don't be silly. Of course not.
See, before you're convicted, you have a presumption of innocence, and the government can't stop you (well, it can, but there are limits) from trying to show that it hasn't met its burden of proving you guilty. But once you're convicted, you have the burden of proof, and the state can do whatever it wants to stop you from meeting it, including keeping the evidence of innocence from you.
OK, that's not quite what they said. I'm being a bit unfair. But not much. Chief Justice Roberts, writing for the majority, did say this in a footnote:
The dissent asserts that our position “resembles” Justice Harlan’s dissent in Miranda v. Arizona, 384 U. S. 436 (1966). Post, at 15–16, n. 10 (opinion of STEVENS, J.). Miranda devised rules to safeguard a constitutional right the Court had already recognized. Indeed, the underlying requirement at issue in that case that confessions be voluntary had “roots” going back centuries. Dickerson v. United States, 530 U. S. 428, 432–433 (2000). In contrast, the asserted right to access DNA evidence is unrooted in history or tradition, and would thrust the Federal Judiciary into an area previously left to state courts and legislatures.
Got that. Because there isn't a centuries-old right to a scientific procedure that didn't exist even 30 years ago, it can't exist.
DNA testing has an unparalleled ability both to exonerate the wrongly convicted and to identify the guilty. It has the potential to significantly improve both the criminal justice system and police investigative practices. The Federal Government and the States have recognized this, and have developed special approaches to ensure that this evidentiary tool can be effectively incorporated into established criminal procedure—usually but not always through legislation.So, it's valuable. It can "significantly improve" both "the criminal justice system" and "police investigative practices." But it's not important enough to guarantee anyone the right to it - at least, not when some less sure method has convinced a jury of guilt.
In case you've forgotten the lesson of Herrera v. Collins, the Court has not. The Court has never yet, and won't now, recognize
a federal constitutional right to be released upon proof of “actual innocence.”And there you have it.