There is clear evidence from government websites, the New York Times, and other authoritative publications, that state secrets is written without an apostrophe expressing possession.
The state secrets privilege is a common law evidentiary rule that
allows the government to withhold information from discovery when
disclosure would be inimical to national security."
Center for Constitutional Rights:
FAQs: What Are State Secrets
The New York Times:
State Secrets Privilege Invoked to Block Testimony in C.I.A. Torture
Also in Japan, source: The Guardian:
Japan whistleblowers face crackdown under proposed state secrets law
In conclusion the evidence is clear, there is no apostrophe in state secrets.
Why is this? Because state is acting like an adjective here, to describe the type of secrets, the state doesn't actually possess the secrets.
The same concept is applied to farmers markets, in the following explanation from grammar girl at quick and dirty tips:
Does ‘Farmers Market’ Have an Apostrophe?
Here’s an even trickier one: farmers market. The market is used by the
farmers, populated by the farmers, but generally not owned by the
farmers. So it seems reasonable to conclude that you don't use an
apostrophe because the word farmers is there to identify the type of
market. It's acting like an adjective...