Generic aspects of the resource theory of quantum coherence

The class of incoherent operations induces a preorder on the set of quantum pure states, defined by the possibility of converting one state into the other by transformations within the class. We prove that if two $n$-dimensional pure states are chosen independently according to the natural uniform distribution, then the probability that they are comparable vanishes as $n\to \infty$. We also study the maximal success probability of incoherent conversions and find an explicit formula for its large-$n$ asymptotic distribution. Our analysis is based on the observation that the extreme values (largest and smallest components) of a random point uniformly sampled from the unit simplex are distributed asymptotically as certain explicit homogeneous Markov chains.

Figure 1

Middle: The $n$ components of a uniform point $\mu$ in ${\mathrm{\Delta }}_{n-1}$ lie in the unit interval [0,1]. Top: For large $n$, the smallest components ${\mu }_n^{\downarrow },{\mu }_{n-1}^{\downarrow },\cdots ,{\mu }_{n-k+1}^{\downarrow }$ after a proper rescaling behave statistically as the first $k$ points of a Poisson process ${\left(V_j\right)}_{j⩾1}$ with exponential spacings. Bottom: The largest components ${\mu }_1^{\downarrow },{\mu }_2^{\downarrow },\cdots ,{\mu }_k^{\downarrow }$ after a proper rescaling behave statistically as the first $k$ points of a Poisson process ${\left(W_j\right)}_{j⩾1}$ with double-exponential (or Gumbel) spacings.

Figure 2

The probability of incoherent convertibility $P(\psi \prec {\psi }^{\prime })$ for $\psi$ and ${\psi }^{\prime }$ independently chosen from the uniform distribution on pure states of ${\mathcal{S}}_n$. The fit (dotted line) confirms an algebraic decay (24) with exponent $\theta =0.4052$. Here $n=2,4,8,...,1024$.

Figure 3

Probability distribution of $N_n$, the time spent by the process $\left(S_k\right)$ above zero, compared with the arcsine law. Here $n=32$.

Figure 4

Distribution $F_n\left(p\right)$ for various values of $n$ versus the limit distribution $F_{\infty }\left(p\right)$ (distribution functions computed from numerical simulations).