Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum waste management: Utilizing residual states in quantum information processing

Karol Horodecki, Chirag Srivastava, Leonard Sikorski, Siddhartha Das

TL;DR

This work quantitatively shows that after performing a well-known coherent Devetak-Winter protocol one can locally extract private randomness from its residual, thereby enhancing overall resource utility.

Abstract

We propose a framework for quantum residual management, in which states discarded after a resource distillation process are repurposed as inputs for subsequent quantum information tasks. This approach extends conventional quantum resource theories by incorporating secondary resource extraction from residual states, thereby enhancing overall resource utility. As a concrete example, we investigate the distillation of private randomness from the residual states remaining after quantum key distribution (QKD). More specifically, we quantitatively show that after performing a well-known coherent Devetak-Winter protocol one can locally extract private randomness from its residual. We further consider the Gottesman-Lo QKD protocol, and provide the achievable rate of private randomness from the discarded states that are left after its performance. We also provide a formal framework that highlights a general principle for improving quantum resource utilization across sequential information processing tasks.

Quantum waste management: Utilizing residual states in quantum information processing

TL;DR

This work quantitatively shows that after performing a well-known coherent Devetak-Winter protocol one can locally extract private randomness from its residual, thereby enhancing overall resource utility.

Abstract

We propose a framework for quantum residual management, in which states discarded after a resource distillation process are repurposed as inputs for subsequent quantum information tasks. This approach extends conventional quantum resource theories by incorporating secondary resource extraction from residual states, thereby enhancing overall resource utility. As a concrete example, we investigate the distillation of private randomness from the residual states remaining after quantum key distribution (QKD). More specifically, we quantitatively show that after performing a well-known coherent Devetak-Winter protocol one can locally extract private randomness from its residual. We further consider the Gottesman-Lo QKD protocol, and provide the achievable rate of private randomness from the discarded states that are left after its performance. We also provide a formal framework that highlights a general principle for improving quantum resource utilization across sequential information processing tasks.

Paper Structure

This paper contains 6 sections, 4 theorems, 47 equations, 6 figures.

Table of Contents

  1. Private randomness in Gottesmann-Lo key distillation protocol
  2. Definitions
  3. Devetak-Winter $\mathrm{Key}\rightarrow \mathrm{PRand}$ sequential protocol
  4. Proof of inequality \ref{['cricket']}
  5. Proof of \ref{['eq:haire']}
  6. Private randomness after private key distillation for isotropic states

Key Result

Lemma 1

Consider an arbitrary state $\rho_{ABCDE}$. Let us define $\rho^{cccq}:=\sum_{a,b,c}|abc\rangle\langle abc| \langle abc| \operatorname{tr}_{D}\rho_{ABCDE}|abc\rangle$. Consider unitaries, $U_{CD}=\sum_{c}|c\rangle\langle c|\otimes U^c$ and $V_{BD}=\sum_b |b\rangle\langle b|\otimes V^b$. Define state

Figures (6)

  • Figure 1: Pictorial representations of possible sequential resource distillation for the first toy example (on the left) obtained from processing of two copies of a pure entangled state $\ket{\Psi}_{AB}$ and the second toy example and main examples (on right) obtained from processing $n$ copies of some input mixed entangled state $\rho_{AB}$, considered in this work.
  • Figure 2: Extracted private randomness rate and lower bound on the key rate versus the maximally entangled state fraction, $f$, in the isotropic state using Gottesman-Lo (GL) Key$\rightarrow$PRandsequential distillation protocol. The '$r$ step rand' and '$r$ step key' corresponds to the $\mathrm{Rate}_{\mathrm{Key}_{\mathrm{GL}}\rightarrow \mathrm{PRand}}(r)$ and the lower bound on $\mathrm{Key_{GL}}(r)$, respectively.
  • Figure 3: Randomness rate from the residual states of the BBPSSW entanglement distillation protocol versus the maximally entangled state fraction, obtained from the isotropic state given in Eq.(\ref{['eq:iso_state']}). Here '$r$ steps' indicate the curve for the randomness rate after $r^{th}$ step B.
  • Figure 4: The figure depicts the relation between the rate of private randomness and secret key distilled from the isotropic state (depending on parameter $f \in [0,1]$ as in Eq. (\ref{['eq:iso_state']})). The blue, solid curve consists of two distinct parts with a jump at $f \approx 0.8125$, where the rate of key distillation (orange, dashed curve) $-H(A|B)$ becomes positive. When $-H(A|B)$ is positive, randomness is distilled at rate $I(X;E)_{\sigma^f_{XE}}$ as shown in Result 2 and else (if $H(A|B) > 0$) it is distilled at rate $R_{AB}= \log|A| - H(A|B)_+ = \log|A| - H(A|B)$, which in our case of isotropic state is $2-H(AB)$.
  • Figure 5: Depiction of the output of 3 protocols: a) coherent Devetak-Winter key distillation protocol outputs a state with key parts and shielding systems as well as public message $\mathrm{M}_C$ which enabled Alice and Bob become fully correlated. b) Devetak-Winter hashing entanglement distillation protocol is the coherent DW protocol which outputs maximal entanglement on systems $\mathrm{Ent A}$ and $\mathrm{Ent B}$ thanks to generating additionally public message from Alice to Bob $\mathrm{M}_D$. It is obtained via Fourier transform on the shielding system of Alice at the same time decoupling her system $\mathrm{Ent A}$ from Eve. Having learned it, Bob knows how to decouple system $\mathrm{Ent}_B$ from Eve as well. c) Devetak-Winter key $\rightarrow$ randomness sequential protocol distills key on systems $\mathrm{K}_A$ and $\mathrm{K}_B$. The kept private message $\mathrm{M}_D$, which coincides with result of the measurement of the system $\mathrm{Shield}_A$ of the coherent DW protocol is shown to be randomness for Alice, independent from the generated key, and most importantly private from the Eve after she listened to public communication of the key distillation protocol.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof