Quantum Discord and the Power of One Qubit
Abstract
We use quantum discord to characterize the correlations present in the quantum computational model DQC1 (deterministic quantum computation with one quantum bit), introduced by Knill and Laflamme [Phys. Rev. Lett. 81, 5672 (1998)]. The model involves a collection of qubits in the completely mixed state coupled to a single control qubit that has nonzero purity. The initial state, operations, and measurements in the model all point to a natural bipartite split between the control qubit and the mixed ones. Although there is no entanglement between these two parts, we show that the quantum discord across this split is nonzero for typical instances of the DQC1 circuit. Nonzero values of discord indicate the presence of nonclassical correlations. We propose quantum discord as a figure of merit for characterizing the resources present in this computational model.
Significance
This paper introduced quantum discord as a resource for quantum computation, demonstrating that quantum advantage can arise from correlations beyond entanglement. It is one of the most-cited papers in quantum information science, influencing the fields of quantum correlations and mixed-state quantum computing.
Background & Context
The DQC1 Model
Deterministic quantum computation with one quantum bit (DQC1), introduced by Knill and Laflamme in 1998, is a model of mixed-state quantum computation that can efficiently estimate the normalized trace of a unitary matrix — a task believed to be classically intractable. Despite its apparent power, DQC1 circuits generate very little — often zero — entanglement between the control qubit and the mixed-state register.
The Puzzle
If DQC1 achieves an exponential speedup over classical computation yet lacks entanglement, what quantum resource drives this speedup? This paper addresses that question directly by turning to quantum discord — a measure of quantum correlations more general than entanglement.
Quantum Discord
Quantum discord, originally introduced by Ollivier and Zurek (2001) and Henderson and Vedral (2001), captures the amount of quantum correlations in a bipartite state that cannot be attributed to classical correlations alone. Unlike entanglement, discord can be nonzero even for separable (unentangled) states.
Discord is defined as the difference between two classically equivalent expressions for mutual information:
δ(A|B) = I(A:B) − J(A|B)
where I(A:B) is the quantum mutual information and J(A|B) is the classical correlation obtained by optimizing over measurements on B.
Main Results
No Entanglement in DQC1
For a typical instance of the DQC1 circuit acting on an n-qubit mixed state register, the entanglement across the control–register bipartition is zero. This rules out entanglement as the resource driving the computational speedup.
Nonzero Quantum Discord
Despite the absence of entanglement, the quantum discord across the same bipartition is nonzero — and in fact scales as a constant fraction of the maximum possible discord for a system of that size. The discord is generated even by classically incoherent operations.
Discord as a Resource
The authors propose quantum discord as the figure of merit characterizing the resources present in DQC1 computation, suggesting that nonclassical correlations beyond entanglement can be computationally useful — a paradigm shift in quantum information theory.
Impact & Follow-up Work
This paper sparked an entire research field around quantum correlations beyond entanglement. Key follow-up developments include:
- Geometric measures of discord — alternative computable formulations of quantum discord.
- Discord in open quantum systems — study of discord dynamics under decoherence and noise.
- Operational interpretations of discord — connections to quantum state merging, channel capacity, and thermodynamics.
- Experimental observation — NMR experiments demonstrating DQC1 without entanglement (Lanyon et al., 2008; Passante et al., 2009).
- Reviews and surveys — Modi et al. Rev. Mod. Phys. 84, 1655 (2012) consolidating the field.
Cite This Paper
@article{datta08a,
title = {Quantum Discord and the Power of One Qubit},
author = {Datta, Animesh and Shaji, Anil and Caves, Carlton M.},
journal = {Physical Review Letters},
volume = {100},
pages = {050502},
year = {2008},
month = {February},
doi = {10.1103/PhysRevLett.100.050502},
eprint = {0709.0548},
archivePrefix = {arXiv},
primaryClass = {quant-ph}
}Datta, A., Shaji, A., & Caves, C. M. (2008). Quantum discord and the power
of one qubit. Physical Review Letters, 100, 050502.
https://doi.org/10.1103/PhysRevLett.100.050502Datta, Animesh, Anil Shaji, and Carlton M. Caves. "Quantum Discord and the
Power of One Qubit." Physical Review Letters, vol. 100, 2008, p. 050502.
DOI: 10.1103/PhysRevLett.100.050502.