# Solving Systems of Partial Trace Equations

Say I specify a quantum state - pure or mixed - by its partial traces on various subsystems. To what degree could one recover the original state, and what are the known methods for doing so? For example, consider the following partial traces on the state $\hat{\rho}$ of a system of two qubits, $a$ and $b$

$\rm{Tr}_a(\hat{\rho}) = \frac{\hat{\mathbb{1}}_b}{2}$

$\rm{Tr}_b(\hat{\rho}) = \frac{\hat{\mathbb{1}}_a}{2}$

One solution is the state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$, but any state related to this one by unitaries on the individual qubits is also a solution. Do there exist general methods for either finding or characterizing the uniqueness of solutions to such a system?

As a remark, you have not enough information to recover the original state, but you have not enough information even to recover the structure of the original state, think for instance about the following quantum states :

$\rho_{CLASSIC} = \begin{pmatrix} \frac{1}{4}&0&0&0\\0&\frac{1}{4}&0&0\\0&0&\frac{1}{4}&0\\0&0&0&\frac{1}{4}\end{pmatrix}, \quad \quad \rho_{ENTANGLED} = \begin{pmatrix} \frac{1}{2}&0&0& \frac{1}{2}\\0&0&0&0\\0&0&0&0\\ \frac{1}{2}&0&0& \frac{1}{2}\end{pmatrix}$

It is clear that the partial traces are the same for these 2 states : $\rho_A=\rho_B = \begin{pmatrix} \frac{1}{2}&0\\0&\frac{1}{2}\end{pmatrix}$

But the first state $\rho_{CLASSIC}$ is a standard classic state (classical probabilities), while $\rho_{ENTANGLED}$ corresponds to an entangled state, so they have a completely different structure.

• That's a good point, but I fear it may be specific to the systems for which all of the partial traces are the maximally mixed state. From a physical standpoint, the equations I gave make the statement that, upon tracing out either subsystem, there is no information left about the state. This implies either: all of the information about the state was in its entanglement structure, we had no information about the state to begin with, or some convex combination of the two cases. This clearly rules out some states as solutions, but I wonder what we would need to do to specify the state exactly. – Adrian Dec 9 '13 at 20:09
• @Adrian I think you should be a bit more precise with your question, then. – Norbert Schuch Dec 9 '13 at 22:17
• @Norbert Sorry, could you clarify? Which part of my question do you find vague? – Adrian Dec 9 '13 at 23:19
• @Adrian Well, apparently you're not entirely happy with the answer of Trimok, so you must have sth. more specific in mind. Are you e.g. interested in a pure state? Then, the eigenvalues of the two RDMs must be equal. The general multipartite case is computationally hard. But the way it is phrased now, it seems very difficult to give a clear answer. – Norbert Schuch Dec 10 '13 at 0:15
• @Adrian The general problem (determining whether there is a state compatible with certain reduced states) is known as the "quantum marginal problem", and there has been quite some work on that. However, there the RDMs usually overlap etc., I'm not sure if that's what you're after. – Norbert Schuch Dec 10 '13 at 13:45