# Given partial states, can one construct the best estimate of the full state?

Given some partial states $$\rho_{AB}$$ and $$\sigma_{AC}$$, is there a general procedure to construct a state $$\delta_{ABC}$$ such that the following sum of trace distances

$$||\text{Tr}_C(\delta_{ABC}) - \rho_{AB}||_1 + ||\text{Tr}_B(\delta_{ABC}) - \sigma_{AC}||_1$$

is minimal? That is we want a joint state that under partial traces is as close as possible to some given targets. In particular, when there is entanglement in both $$AB$$ and $$AC$$, I'm not sure how to proceed.

I would probably set this up as a semi-definite program and throw it at the computer. Basically, your problem is a linear problem in the coefficients of the matrix $$\delta_{ABC}$$ except for the constraint that $$\delta_{ABC}$$ is positive semi-definite. This is exactly what semi-definite programming is designed to do.

Note: the calculation below uses a different 1-norm (my bad). See the comment by Norbert, below.

It's a bit of a pain to set it all up in the right way (which is why I'm not directly stating the full formula!). You need to do things like let $$M1=\text{Tr}_C(\delta_{ABC})-\rho_{AB}$$ and the define variables $$x_{ij}$$ such that $$M1_{ij}\leq x_{ij},\qquad -M1_{ij}\leq x_{ij}$$ with the intent of getting (when we're minimising $$x_{ij}$$) $$x_{ij}=|M1_{ij}|$$. Then, you set $$\sum_{j}x_{ij}\leq x\qquad\forall i.$$ In effect, $$x=\|M_1\|_1$$. If you do something similar with a matrix $$M2$$, variables $$y_{ij}$$ and $$y$$, your final problem is to minimise $$x+y$$.

Don't forget to include the constraint $$\text{Tr}(\rho_{ABC})=1$$. It might be implicit in previous constraints, I'm not sure.

• How does this give the trace norm? It seems you are summing all elements in absolute value. (This is a 1-norm, but a different one.) Though you should be able to express the trace norm with semidefinite constraints (using e.g. the dual characterization through the operator norm, i.e. maximizing $\mathrm{tr}(MX)$ subject to $I \le X \le I$). Mar 9, 2020 at 13:27
• @NorbertSchuch Fair point. I was going for sum of absolute values. I saw the equation and filtered out the words around it. Mar 9, 2020 at 13:40
• Well, density matrices beg for trace norms, don't they? Mar 9, 2020 at 13:47
• Sure. I've just been using the other matrix norm for a lot of calculations recently and was in the zone! Mar 10, 2020 at 11:48
• To be honest, I am not sure this can be done with an SDP. Computing the trace norm alone is already an SDP, but it is a maximization. However, if on top of that you have to minimize the trace norm, this is a $\min\max$, which is no longer an SDP. Mar 10, 2020 at 12:00