Is entanglement a property of a state rather than of the spaces, and how should "irreducible to its parts" be stated precisely?

Quantum Computing is part of Stack Overflow’s open communities: specialist spaces where curiosity is welcome, knowledge is shared freely, and the best answers rise to the top. Stack Overflow for Teams is now called Stack Internal. Bring the best of human thought and AI automation together at your work. Bring the best of human thought and AI automation together at your work. Learn more Bring the best of human thought and AI automation together at your work. I'm not a mathematician and I'm trying to make sure I understand this correctly. My understanding is: (1) Hilbert spaces with ⊗ form a symmetric monoidal category, and the product is non-cartesian; (2) entanglement is a property of a state in H_A ⊗ H_B, namely one not in the image of the map from pairs of states; (3) decoherence disperses correlations into the environment, so the pair alone looks mixed while the joint state stays pure. Is this right? And is there a standard way to say a construction is "irreversible" in this setting? Thanks for contributing an answer to Quantum Computing Stack Exchange! Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This comment attacks a person or group. Learn more in our Abusive behavior policy. This comment is rude or condescending. Learn more in our Code of Conduct. A problem not listed above. Try to be as specific as possible. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Site design / logo © 2026 Stack Exchange Inc; user contributions licensed under CC BY-SA . rev 2026.9.25.