In my research, I stumbled on a smooth map: $$\pi_{\rho_0}: B \setminus \{\rho_0\} \to \partial B$$ where $B$ is the open Bloch ball, corresponding to the set of mixed states of a single qubit and $\partial B$ is the Bloch sphere proper, consisting of the set of pured states of a single qubit and $\rho_0 \in B$ is a fixed mixed state of a single qubit. The map is defined as follows.

After fixing $\rho_0 \in B$, you define the image of a mixed state $\rho \in B$, $\rho \neq \rho_0$ as the intersection of the ray starting from $\rho_0$ and passing through $\rho$ with $\partial B$ (the Bloch sphere proper). This can be characterized as follows. It is the unique pure state which can be written as a linear combination

$$a \rho_0 + b \rho$$

where $a, b \in \mathbb{R}$ such that $a + b = 1$ and $b > 1$ (and therefore $a < 0$).

Note that $\pi$ can be viewed as a map

$$ B \times B \setminus \Delta \to \partial B $$

mapping $(\rho_0, \rho)$ to $\pi_{\rho_0}(\rho)$, where $\Delta \subset B \times B$ is the diagonal (i.e. $\Delta$ consists of all pairs $(\rho, \rho)$ such that $\rho \in B$).

Then if $G = \mathrm{SU}(2)$, then $G$ acts on $B \times B$ by $$ g.(\rho_0, \rho) := (g \rho_0 g^{-1}, g \rho g^{-1}). $$ Moreover, $G$ acts on $\partial B$ by $$ g.\nu = g \nu g^{-1}.$$

Then, if I am not mistaken, $\pi$ is $G$-equivariant.

This map $\pi$ seems natural. I know it appears in the literature when $\rho_0 = \frac{1}{2}I$ is at the "origin" of the Bloch ball. But did the more general $\pi_{\rho_0}$, where $\rho_0$ is any fixed mixed state of the qubit, appear in the quantum information theory literature please?

I would like to investigate whether there may be a link between a problem in geometry that I am interested in and quantum information theory.

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    $\begingroup$ Is this mapping equivalent to returning the closest pure state to the target state, as measured by trace distance? It might also be equivalent to purifying the state and then uncomputing the ancilla qubits that were introduced by the purification subroutine. I know that there is effort towards, given some noisy data, try to efficiently infer what the noiseless pure state would be; although you tend to want to exercise control over what states you're collecting e.g. arxiv.org/abs/2011.07064 . $\endgroup$ – Craig Gidney Dec 7 '20 at 18:10
  • $\begingroup$ @CraigGidney, thank you for the reference. Well, the closest pure state is actually $\pi_{\rho_0}(\rho)$ when $\rho_0 = \frac{1}{2}I$ is at the "origin" of the Bloch ball. In general, they are different. Thank you though. $\endgroup$ – Malkoun Dec 7 '20 at 18:48
  • $\begingroup$ @CraigGidney, I think my map is also $\mathrm{SL}(2,\mathbb{C})$-equivariant, when one defines how it acts on the source and target of the map $\pi$ in a suitable way. So it has more symmetry than the usual $\mathrm{SU}(2)$ of Quantum Mechanics. $\endgroup$ – Malkoun Dec 7 '20 at 18:57

I am not aware of a direct application of this projection. However, maybe the following geometrical construction is nevertheless of interest to you.

Similar ray constructions appear in resource theories, namely in the definition of robustness and generalised robustness monotones.

Consider a convex set of states $\mathcal{F}$ which we call the free states in the resource theory. Let us denote by $\mathcal{S}$ the convex set of all states. Then, given a state $\rho$ we define the following functions $$ R(\rho) = \inf\big\{ t\geq 0 \; | \; \rho = (1+t)\sigma_0 - t \sigma_1 \text{ for } \sigma_0,\sigma_1\in\mathcal{F} \big\}, $$ $$ GR(\rho) = \inf\big\{ t\geq 0 \; | \; \rho = (1+t)\sigma_0 - t \sigma_1 \text{ for } \sigma_0\in\mathcal{F}, \sigma_1\in\mathcal{S} \big\}. $$ In the definitions, we optimise over all rays going through $\mathcal{F}$ which include $\rho$.

If $\rho$ is pure, than $\rho$ is the projection of $\sigma_1$ w.r.t to $\sigma_0$ under your map.

The monotones have the following geometrical interpretation:

  • For $R$: $\sigma_0$ is the closest state in $\mathcal{F}$ to $\rho$ measured w.r.t. to the length of the ray segment which lies in $\mathcal{F}$ (the diameter of $\mathcal{F}$ in this direction)
  • For $GR$: similar, but compared to the distance of $\sigma_0$ to the opposite boundary of $\mathcal{S}$.

The generalised robustness is equivalent to what is called the max-relative entropy $$ \mathcal{D}_\mathrm{max}(\rho) := \inf_{\sigma\in\mathcal{F}} D_\mathrm{max}(\rho||\sigma) = \log\left( 1 + GR(\rho) \right) $$ where $D_\mathrm{max}(\rho||\sigma)=\log \inf\{\lambda\geq 0 \, | \, \rho \leq \lambda\sigma\}$.

After your remark, I changed $\min$ to $\inf$. However, $\mathcal{F}$ is usually compact.

Here is a review on resource theories which appeared recently in RMP. The above functions and more appear in Sec. VI.

Chitambar and Goul: "Quantum Resource Theories". https://arxiv.org/abs/1806.06107

  • $\begingroup$ Thank you so much. This is interesting. Could you please add a reference or two? $\endgroup$ – Malkoun Dec 8 '20 at 15:30
  • $\begingroup$ What if, say, the set on the RHS in the definition of $R(\rho)$ is empty, do you define $R(\rho)$ to be $\infty$? $\endgroup$ – Malkoun Dec 8 '20 at 15:36
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    $\begingroup$ @Malkoun Yes indeed, the usual convention would yield $\infty$. Anyway, resource theories usually come with some axioms, for example that $\mathcal{F}$ is convex and full-dimensional. From this, it follows that its affine span is the affine space of unit-trace Hermitian matrices, in particular it contains $\mathcal{S}$. I'll add a reference above. $\endgroup$ – Markus Heinrich Dec 9 '20 at 8:31

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