# Can I parameterize the state of a quantum system given reduced density matrices describing its subparts?

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.

• What you are talking about sounds similar, although probably not equivalent, to the quantum marginal problem. This is the question whether a given pair of reduced states are compatible with some global bipartite state. I am not sure if people have considered the more detailed problem of which global state, which sounds like what you want. Some very mathematical work on the quantum marginal problem can be found by Klyachko; see also Schilling for a solution for certain fermion systems. Perhaps those may be useful. – Mark Mitchison May 6 '14 at 19:00
• @MarkMitchison Whenever I see an interesting question I find your comments and answers. If you're ever in Southern California, please do come visit the Google quantum computing lab in Santa Barbara. – DanielSank Sep 12 '15 at 5:44
• @DanielSank Thanks, I will do for sure. Likewise if you are ever in London or Oxford, although I'm afraid I don't have any cool experiments to show you :) – Mark Mitchison Sep 12 '15 at 10:40

## 2 Answers

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.

Given an n-partite system and observables $\hat{T}_i$ with exspectation values $<\hat{T}_i> = t_i$, you can write your state as a maximum entropy state:

$$\rho = \frac{1}{Z}exp\left(\sum_i \theta_i \hat{T}_i\right)$$

see Theorem 2 in http://arxiv.org/abs/quant-ph/0603012