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Which of these statements about tensor products is incorrect?

  1. Jun 23, 2013 #1
    I have read the following three simplifications in various places, but together they give a contradiction, so at least one of them must be an oversimplification. Which one?
    (a) Interaction between two systems A and B is described by A[itex]\otimes[/itex]B
    (b) An entangled state C is a pure state, and hence there does not exist A and B such that A[itex]\otimes[/itex]B = C
    (c) By appropriate interactions one can produce entangled states.

    Thanks.
     
  2. jcsd
  3. Jun 23, 2013 #2
    (A) is not correct. The correct statement should be
    If [itex]H_A[/itex] is the hilbertspace of system A and [itex]H_B[/itex] is space of system B, then the hilbertspace of composite system is [itex]H_A[/itex][itex]\otimes[/itex][itex]H_B[/itex]

    (B) is not correct. The correct statement should be
    If [itex]|C\rangle[/itex][itex]\in[/itex][itex]H_A[/itex][itex]\otimes[/itex][itex]H_B[/itex] represents entangled state then it can not be written in the form [itex]|A\rangle[/itex][itex]\otimes[/itex][itex]|B\rangle[/itex] where [itex]|A\rangle[/itex][itex]\in[/itex][itex]H_A[/itex] and [itex]|B\rangle[/itex][itex]\in[/itex][itex]H_B[/itex]

    (C) is correct.
     
  4. Jun 23, 2013 #3
    (a) interaction between two systems whose individual Hilbert spaces are [itex]H_A[/itex] and [itex]H_B[/itex] can be describe by states [itex]|\psi\rangle[/itex] belonging to the direct product of these Hilbert spaces [itex]H=H_A\otimes H_B[/itex].

    (b) an entangled state [itex]|\psi\rangle_C[/itex] belongs to this direct product Hilbert space, however, it cannot be expressed simply as a tensor product of states belonging to [itex]H_A[/itex] and [itex]H_B[/itex]. That is, for [itex]|\psi\rangle_A\in H_A[/itex] and [itex]|\psi\rangle_B\in H_B[/itex], one cannot write [itex]|\psi\rangle_C=|\psi\rangle_A\otimes|\psi\rangle_B[/itex], rather [itex]|\psi\rangle_C=\sum_{i,j}\alpha_{ij}|\psi_i\rangle_A\otimes|\psi_j \rangle_B[/itex] where [itex]\{|\psi_i\rangle_A\}\subset H_A[/itex], [itex]\{|\psi_i\rangle_B\}\subset H_B[/itex] and there MUST be more than one non-vanishing terms in the sum for an entangled state [itex]|\psi\rangle_C[/itex].

    (c) this is correct. (Edit: Actually, we can get an entangled state simply by appropriately partitioning our total system into subsystems A and B.)
     
    Last edited: Jun 23, 2013
  5. Jun 23, 2013 #4
    Thank you very much, Ravi Mohan and conana. That clears up a lot of confusion on my part. Very helpful.
     
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