Symmetrizing cost of quantum states

We introduce and analyze a task that we call symmetrization, in which a state of a quantum system, associated with a symmetry group, is transformed by a random unitary operation to a symmetric state. Each element of the unitary ensemble is required to be symmetry preserving, in the sense that it keeps the set of symmetric states invariant. We consider an asymptotic limit of infinitely many copies and vanishingly small error, and analyze the symmetrizing cost, that is the minimum cost of randomness per copy required for symmetrization. We prove that the symmetrizing cost of an arbitrary quantum state is equal to the relative entropy of frameness, thereby providing it with a direct operational meaning.

Figure 1

A schematic diagram of the task of symmetrization is depicted. The symmetrizing cost, defined as the minimum cost of randomness required for symmetrization, quantifies the degree of asymmetry of quantum states operationally. Note that a state $\sigma$ is defined to be symmetric if it satisfies $U_g\sigma U_g^{†}=\sigma$ for all $g\in G$, where $G$ is a symmetry group with a unitary representation ${\left\{U_g\right\}}_{g\in G}$.

Figure 2

A schematic diagram of the task of symmetrization is depicted. $n$ copies of an asymmetric state $\rho$ on system $A$ is transformed by a random application of symmetry preserving unitaries $\{V_1,\cdots ,V_{2^{nR}}\}$ on $A^n$ with the uniform distribution. We require that the state after the transformation is a symmetric state ${\sigma }_n$ up to a small error $\epsilon$. The symmetrizing cost of $\rho$ is defined as the minimum rate $R$ such that $\epsilon \to 0$ can be accomplished in the limit of $n\to \infty$, by properly choosing $\{V_1,\cdots ,V_{2^{nR}}\}$ and ${\sigma }_n$ for each $n$.