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As the simplest example, consider a set of two qubits where the reduced density matrix of each qubit is known. If the two qubits are not entangled, the overall state would be given by the tensor product of the one qubit states. More generally, I could write a set of contraints on the elements of a two-qubit density matrix to guarantee the appropriate reduced description.

Is there is a way to do this more elegantly and systematically for arbitrary bi-partite quantum systems? I'm particularly interested in systems where one of the Hilbert spaces is infinite dimensional, such as a spin 1/2 particle in a harmonic oscillator.

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Density matrices often admit an interesting geometric interpretations when you map them to the space of generalized Bloch vectors, see for example the book I. Bengtsson, K. Życzkowski, Geometry of quantum states, 2006. I won't be surprised if it turns out that the result has something to do with the coset space $SU(2N)/[SU(N)\times SU(N)]$, where N is the dimensionality of the Hilbert space of a single qubit.

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