Absence of quantum Darwinism as a resource in secure quantum communication and computation

TL;DR

This work argues that a key ingredient of an efficient classical simulation algorithm, a knowledge of the local basis in which the multi-party state is diagonal, is made available by quantum Darwinism and gives a protocol for secure quantum communication that exploits this insight.

Abstract

The emergence of classical world from underlying quantum mechanics is characterized by not only vanishing quantum correlations but also an unfolding of objectivity also known as quantum Darwinism. We show that the absence of this objectivity has a quantum advantage in cryptography and also provides the crucial missing link in efficient classical simulation of quantum circuits with zero discord. For this purpose, we consider a model of mixed state quantum computation where one is promised concordant states at all stages of the quantum circuit. A concordant quantum state has zero discord with respect to any part and there exists a basis made up of a tensor product of orthonormal local subsystem basis in which the density matrix is diagonal. Efficient classical simulation of concordant computation has surprisingly been an outstanding question in quantum information theory. We argue that a key ingredient of an efficient classical simulation algorithm, a knowledge of the local basis in which the multi-party state is diagonal, is made available by quantum Darwinism. Concordant states in the absence of quantum Darwinism cannot be efficiently simulated by existing methods and give a cryptographic advantage in communication. We show this by giving a protocol for secure quantum communication that exploits this insight. Our work also has implications for the quantum-classical border and we discuss how objectivity emerging out of Darwinism demarcates this border in three ways - empirical based on our observations and experience of objectivity, information theoretic due to the absence of any quantum correlations and lastly computational in the sense discussed above. Lastly, we show that the quantum-classical boundary as drawn by quantum Darwinism as well by what can be simulated efficiently in a mixed state quantum computation aligns with the boundary given by Hardy

Figures (3)

• Figure 1: Information of the system $S$ stored in the environment $E$. The global system-environment is in a pure state, and $I(S:F_f)$ represents the mutual information between the system and the fraction of the environment ($f = F_f/E$).
• Figure 2: Graphical representation of an example of the protocol we proposed; Let's say Alice wants to communicate a discrete probability distribution $\mathcal{M}$. She encodes her message in a degenerate concordant state as shown inside the box. Then applied the sequence of gates $\{G_i\}$'s and publicly announced $\{G_i'\}$'s. Here $G_i = U_i P_i U_{i-1}^{\dagger}$ and $G_i^{'} = U_i P_i B_i U_{i-1}^{\dagger}$ such that LBF fails for this gate set. They previously decided the set of haar random local unitaries $\{U_i\}$'s. Therefore Bob can get back the actual distribution by simulating the inverse computation in his classical computer.
• Figure 3: Hierarchy of computational models illustrating the transition from fully quantum regimes to classical computation. Quantum computations that exhibit an advantage over classical models occupy the topmost regime, enabled by entanglement and discord as genuine quantum resources. As these resources diminish, one approaches intermediate regimes between quantum and classical descriptions. In particular, the quasi-classical regime arises as an intermediate abstraction, analogous to Generalized Local Theories (GLTs) in the framework of Generalized Probabilistic Theories (GPTs) PhysRevA.75.032304. This regime should be regarded as a purely mathematical construct without physical realization. The progressive restrictions are marked by vanishing entanglement ($\text{Entanglement} \to 0$), vanishing discord ($\text{Discord} \to 0$), and the discretization of basis transformations (continuous $\to$ discrete). The final step, corresponding to full knowledge of the diagonal basis, represents the onset of Darwinism, where classical objective reality emerges.

Theorems & Definitions (5)

• Remark
• Lemma 1
• proof : Heuristic Argument
• Corollary 1.1
• proof