In quantum mechanics, measuring the joint observable Z₁ ⊗ Z₂ is not strictly equivalent to sequentially measuring Z₁ ⊗ ???? followed by ???? ⊗ Z₂, though their outcomes are mathematically related. Here’s the breakdown:
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Mathematical Relationship:
- The product of the individual operators satisfies: (Z₁ ⊗ ????)(???? ⊗ Z₂) = Z₁ ⊗ Z₂.
- The product of sequential measurement outcomes (e.g., +1 for Z₁ and -1 for Z₂) matches the outcome of directly measuring Z₁ ⊗ Z₂.
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Physical Difference:
- Joint measurement (Z₁ ⊗ Z₂): Projects the state into an eigenstate of Z₁ ⊗ Z₂, preserving entanglement (e.g., Bell states remain intact).
- Sequential measurements: First measuring Z₁ ⊗ ???? collapses the state into a Z₁ eigenstate, then measuring ???? ⊗ Z₂ further collapses it into a product state (e.g., |00⟩ or |11⟩), destroying entanglement.
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Key Implications:
- Outcomes: The product of sequential results matches Z₁ ⊗ Z₂, but the post-measurement states differ.
- Entanglement: Joint measurements preserve it; sequential measurements destroy it.
- Commutativity: Z₁ ⊗ ???? and ???? ⊗ Z₂ commute, but this does not equate to equivalence with Z₁ ⊗ Z₂.
Summary:
- The product of sequential outcomes matches Z₁ ⊗ Z₂, but the physical processes are distinct. Joint measurements preserve entanglement, while sequential measurements collapse the system into a product state. This distinction is critical in quantum protocols relying on entanglement (e.g., teleportation, error correction).
总结概要
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