Accurate quantum state estimation via "Keeping the experimentalist honest"

Question

Bob has a black-box, with the label "V-Wade", which he has been promised prepares a qubit which he would like to know the state of. He asks Alice, who happens also to be an experimental physicist, to determine the state of his qubit. Alice reports $\sigma$ but Bob would like to know her honest opinion $\rho$ for the state of the qubit. To ensure her honesty, Bob performs a measurement $\{E_i\}$ and will pay Alice $R(q_i)$ if he obtains outcome $E_i$, where $q_i={\rm Tr}(\sigma E_i)$. Denote the honest probabilities of Alice by $p_i={\rm Tr}(\rho E_i)$. Then her honesty is ensured if $$ \sum_i p_i R(p_i)\leq\sum_i p_i R(q_i). $$ The Aczel-Pfanzagl theorem holds and thus $R(p)=C\log p + B$. Thus, Alice's expected loss is (up to a constant $C$) $$ \sum_i p_i(\log{p_i}-\log{q_i})=\sum_i {\rm Tr}(\rho E_i)\log\left[\frac{{\rm Tr}(\rho E_i)}{{\rm Tr}(\sigma E_i)}\right]. $$ Blume-Kohout and Hayden showed that if Bob performs a measurement in the diagonal basis of Alice's reported state, then her expected loss is the quantum relative entropy $$ D(\rho\|\sigma)={\rm Tr}(\rho\log\rho)-{\rm Tr}(\rho\log\sigma). $$ Clearly in this example Alice is constrained to be honest since the minimum of $D(\rho\|\sigma)$ is uniquely obtained at $\sigma$. This is not true for any measurement Bob can make (take the trivial measurement for example). So, we naturally have the question: which measurements can Bob make to ensure Alice's honesty? That is, which measurement schemes are characterized by $$ \sum_i {\rm Tr}(\rho E_i)\log\left[\frac{{\rm Tr}(\rho E_i)}{{\rm Tr}(\sigma E_i)}\right]=0\Leftrightarrow \sigma=\rho? $$

Note that $\{E_i\}$ can depend on $\sigma$ (what Alice reports) but not $\rho$ (her true beliefs).

Partial answer: Projective measurement in the eigenbasis of $\sigma$ $\Rightarrow$ yes, Blume-Kohout/Hayden showed that this is the unique scheme constraining Alice to be honest for a projective measurement.

Informationally complete $\Rightarrow$ yes, clearly this constrains Alice to be honest since the measurement will uniquely specify the state (moreover, the measurement can be chosen independent of $\sigma$).

Trivial measurement $\Rightarrow$ no, Alice can say whatever she wants without impunity.

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Answer 1

Trivially for any set of measurements $\{E_i\}$ where $\rho$ and $\sigma$ have equal expectation value for each $E_i$, $$\sum_i\mbox{Tr}(\rho E_i) \log \left[ \frac{\mbox{Tr}(\rho E_i)}{\mbox{Tr}(\sigma E_i)}\right] = \sum_i\mbox{Tr}(\rho E_i) \times 0 = 0.$$

Note that the log-sum inequality theorem says that $$\sum_i a_i \log\left(\frac{a_i}{b_i}\right) \leq \big(\sum_i a_i\big)\times \log \left(\big(\sum_i a_i\big)/\big(\sum_i b_i\big)\right),$$ with equality only if $\frac{a_i}{b_i}$ is constant for all $i$. This implies that $$\sum_i\mbox{Tr}(\rho E_i) \log \left[ \frac{\mbox{Tr}(\rho E_i)}{\mbox{Tr}(\sigma E_i)}\right] = 0$$ if and only if $\frac{\mbox{Tr}(\rho E_i)}{\mbox{Tr}(\sigma E_i)}$ is constant. Since the sum of the traces over $i$ is 1 for both density matrices this implies that you must have $\frac{\mbox{Tr}(\rho E_i)}{\mbox{Tr}(\sigma E_i)} = 1$.

If, however, the projections onto each subspace selected by $E_i$ aren't equal, then there exists some $i$ such that $\frac{\mbox{Tr}(\rho E_i)}{\mbox{Tr}(\sigma E_i)} \neq 1$.

Hence for Alice to have zero expected loss, then $\mbox{Tr}(\rho E_i) = \mbox{Tr}(\sigma E_i) ~~~\forall i$. If expectation value for each $E_i$ uniquely identifies a density matrix, then necessarily $\rho = \sigma$ if Alice has zero expected loss. A sufficient condition for this is when $E_i$ project onto $(\dim \mathcal{H})^2 -1$ linearly independent density matrices, where $\mathcal{H}$ is the Hilbert space of the system.

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Comments

This argument is for the case where the set of measurements is fixed and independent of $\sigma$. If the scheme need only work for certain classes of $\sigma$ then this imposes correlations between entries in the density matrix which reduces the number of linearly independent measurement operators required to uniquely identify it. An example of this is where $\sigma = |+\rangle\langle +|$, where purity implies that the state has expectation value 0 for $Z$ and $Y$ measurements, and if $\mbox{Tr}(\rho X) = \mbox{Tr}(\sigma X)$ then $\mbox{Tr}(\rho Y) = \mbox{Tr}(\sigma Y)$, etc.

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And hence you need only measure in the $X$ basis, even though this does not have sufficiently many linearly independent outcomes to uniquely identify an arbitrary density matrix.

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I made a few changes to clarify which measurement schemes are allowed: they are independent on $\rho$ but not necessarily fixed as they can depend on what state $\sigma$ Alice reports. What is surprising is that if Bob is going to perform a PVM, he must do it in the eigenbasis of $\sigma$. The proof used in the paper is not intuitive to me. Your line of reasoning is more appealing but I'd have to think a bit harder to continue it to reach the same conclusion about PVMs.

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@ChrisFerrie: Sorry, I meant if the measurements can depend on $\sigma$ not $\rho$. I've edited my above comments to reflect this.

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Oh good, thanks. This is a very important point and I think you should add it to your answer.

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Joe, your sufficient condition is that $\{E_i\}$ is informationally complete -- but we already knew that was sufficient and not necessary by the Blume-Kohout/Hayden result. So is there something more interesting that can be said about the condition Tr($\rho E_i$) = Tr($\sigma E_i$), for all $i$?

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@ChrisFerrie: No, that is not the same as the sufficient condition I gave since I only required linear independence and not orthogonality and only for a sufficiently large subset of $\{E_i\}$, so it includes a bunch of weak measurements. However that was only an example. The necessary and sufficient condition is that all expectation values are the same. This includes a huge range of measurements, such as measuring a randomly chosen Pauli operator, which yields only a single bit of information.

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