Does the trace distance specify a unique state In quantum information, we frequently use the trace distance (see definition) to look at how similar two states are.
If I had a known complete set of states $\{\rho_i\}$ and some unknown state $\sigma$ and knew $D(\rho_i,\sigma)=d_i$ where $D$ denotes the trace distance, can I in principle determine $\sigma$ without additional information?
From a geometric perspective, the answer seems to be yes, however, it is not clear to me how to use the properties of the trace distance to actually prove it (or maybe I'm wrong).
Update: I have found a way around addressing the issue for my research but it is still an interesting question that may be worth pursuing.
 A: For qubits, in the Bloch ball representation:
$$\rho=\frac{1}{2}(I+\mathbf{r}\cdot\mathbf{\sigma})$$
$$\rho'=\frac{1}{2}(I+\mathbf{r}'\cdot\mathbf{\sigma})$$
The difference $\rho-\rho'$ is
$$\rho-\rho'=\frac{1}{2}[(\mathbf{r}-\mathbf{r}')\cdot\mathbf{\sigma}]$$
The eigenvalues of $\rho-\rho'$ are
$$\lambda_{\pm}=\pm\frac{1}{2}\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}=\pm\frac{||\mathbf{r}-\mathbf{r}'||_2}{2}$$. Thus the trace distance in terms of Bloch vectors is
$$D(\rho,\rho')=\frac{||\mathbf{r}-\mathbf{r}'||_2}{2}$$.
In this picture, $D(\rho_i,\sigma)$ is the the radius of a ball containing $\sigma$ as a point and centered in $\rho_i$.
I can think two answers to your question: if you give me a table of values $d_i$ for the six projectors of the Pauli basis I would say that no, $\sigma$ is not specified uniquely, since without further restrictions if I give you $d_i=\epsilon$ for all $i$ there is $\epsilon$ such the intersection between the six balls is empty (or any finite number of projectors). Now if you are asking whether a quantum state $\sigma$ can be expressed uniquely via a table of values $d_i$ I would believe so, since a point in $\mathbb{R}^3$can be expressed uniquely as the intersection between six spheres (maybe even less).
For higher dimensions in principle one could use similar arguments in terms of qudit Bloch vectors - https://arxiv.org/abs/0806.1174 . (Maybe this relates to the construction of Mutually Unbiased Bases somehow?)
