Error Exponent and Strong Converse for Quantum Soft Covering

Abstract

How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem. Namely, we use a random codebook with codewords independently sampled from a prior distribution and send it through a classical-quantum channel to approximate the target state. When using a random independent and identically distributed codebook with a rate above the quantum mutual information, we show that the expected trace distance between the codebook-induced state and the target state decays with exponent given by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the trace distance converges to one exponentially fast. We obtain similar results when using a random constant composition codebook, whereas the sandwiched Augustin information expresses the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime. That is, we show that even when the rate of the codebook size approaches the quantum mutual information moderately quickly, the trace distance still vanishes asymptotically.

Optimal Second-Order Rates for Quantum Soft Covering and Privacy Amplification

“…We would like to point out that although there are essentially no differences between the three deviation regimes in the one-shot setting, there are at least two different types of operational quantities of interest; their characterizations might be different. As observed in[26] and our previous works[27,38], indeed, the Rényi-type quantities are more favorable in characterizing the optimal error given a fixed size or cardinality such as |Z| and |C| considered in this paper. On the other hand, if one concerns the size or cardinality given a fixed error, the hypothesis-testing-type quantities or the information-spectrum-type quantities might be more direct for characterizations.…”

“…The large deviation analysis [16,[61][62][63][64][65][66][67][68][69][70][71] of privacy amplification against quantum side information and quantum soft covering has been investigated in previous literature [21,26,27,38,72,73], wherein one fixes the rate or the size of |Z| and |C| and studies the optimal errors in terms of the trace distance. Also, some moderate deviation analysis [44,45] were studied for characterizing the minimal trace distance while the rates approach the first-order limits with certain speed [27,38]. In this paper, we took another perspective-what are the optimal rates when the trace distances are upper bounded by a constant ε ∈ (0, 1).…”

“…The second task studied in this work is quantum soft covering [38]. Consider a c-q state ρ XB := x∈X p X (x)|x x| ⊗ ρ x B .…”