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.

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    $\begingroup$ Can you please define state distance? $\endgroup$ – Gabriel Golfetti Jul 15 '18 at 4:10
  • $\begingroup$ See en.wikipedia.org/wiki/Trace_distance $\endgroup$ – A15234B Jul 15 '18 at 4:13
  • $\begingroup$ I don't see why this could possibly be true, you can have many states with the same trace of distance to your given state. Why would the distance alone distinguish them? $\endgroup$ – KF Gauss Jul 15 '18 at 4:21
  • $\begingroup$ @user157879 I believe you misread my question. The key idea is that we know the distance of $\sigma$ to each point in a basis for the space of states. To understand the intuition consider the space Rn with the Euclidean metric. The distance between the vector $\sigma$ and each basis vector $\rho_i$ confines $\sigma$ to lie on a hypersphere. Together, the knowledge of all the hyperspheres allows us to specify $\sigma$ uniquely by their intersection. $\endgroup$ – A15234B Jul 15 '18 at 10:23
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    $\begingroup$ What do you mean by a "complete" set of states? The trace distance is zero only if two states are the same. If I have information about the uncountable set of trace distances between my unknown state and every possible quantum state, then of course I can infer what my state is by looking for the state which has zero distance. I suspect this is not what you mean though. $\endgroup$ – Joel Klassen Jul 15 '18 at 18:49

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