א"ר כרוספדאי א"ר יוחנן שלשה ספרים נפתחין בר"ה אחד של רשעים גמורין ואחד של צדיקים גמורין ואחד של בינוניים צדיקים גמורין נכתבין ונחתמין לאלתר לחיים רשעים גמורין נכתבין ונחתמין לאלתר למיתה בינוניים תלויין ועומדין מר"ה ועד יוה"כ זכו נכתבין לחיים לא זכו נכתבין למיתה
(Gemara Rosh HaShanah 16b)
"Rabbi Kruspedai said that Rabbi Yoḥanan said: Three books are opened on Rosh HaShana (before the Holy One, Blessed be He:) One of the completely wicked people, and one of the completely righteous people, and one of middling people (whose good and bad deeds are equally balanced). Wholly righteous people are immediately written and sealed for life; wholly wicked people are immediately written and sealed for death; and middling people are left with their judgment suspended from Rosh HaShana until Yom Kippur, (their fate remaining undecided). If they merit, (through the good deeds and mitzvot that they perform during this period,) they are written for life; if they do not so merit, they are written for death."
(from Sefaria's translation)
"Schrödinger's cat", is (on erev R"HaShannah, hidden from sight) living in both the state of life and death. Therefore, it cannot be considered "completely" or "wholly" righteous or wicked, since it is neither completely or wholly one thing or the other at this point.
Therefore, on R"H, it cannot be inscribed in the Book of Life or Death. However, since it is considered as having aspects of both life and death at the same time, then it is a "Beinoni" by definition (middle group).
So the cat has its judgment suspended until Yom Kippur.
But, since no matter how estranged or how non-religious a Jew may unfortunately be.... it is well known that EVERY JEW is OBSERVANT on Yom Kippur!
Therefore, by Yom Kippur, the cat will be observed. At that point it will be inscribed into its proper book.
The cat (before being observed) is both in a potential state of life and death at the same time. This is a Halachic "Safek" (doubt) #1.
Opening the box on Shabbos, will either determine that the cat is now dead or alive. But we do not know which will be. This is Halachic "Safek" (doubt) #2.
Since before opening the box, the cat may be considered possibly dead, and now opening the box merely causes potential taking of life... then the whole act of opening the box is merely a S'fek S'fekah and is permitted on Shabbos.