In 1975, Swedish physicist Göran Lindblad developed a theorem that describes the change in entropy that occurs during a quantum measurement. Today, this theorem is a foundational component of quantum information theory, underlying such important concepts as the uncertainty principle, the second law of thermodynamics, and data transmission in quantum communication systems.
Now, 40 years later, physicist Mark M. Wilde, Assistant Professor at Louisiana State University, has improved this theorem in a way that allows for understanding how quantum measurements can be approximately reversed under certain circumstances. The new results allow for understanding how quantum information that has been lost during a measurement can be nearly recovered, which has potential implications for a variety of quantum technologies.
Quantum relative entropy never increases
Most people are familiar with entropy as a measure of disorder and the law that “entropy never decreases”—it either increases or stays the same during a thermodynamic process, according to the second law of thermodynamics. However, here the focus is on “quantum relative entropy,” which in some sense is the negative of entropy, so the reverse is true: quantum relative entropy never increases, but instead only decreases or stays the same.
In fact, this was the entropy inequality theorem that Lindblad proved in 1975: that the quantum relative entropy cannot increase after a measurement. In this context, quantum relative entropy is interpreted as a measure of how well one can distinguish between two quantum states, so it’s this distinguishability that can never increase. (Wilde describes a proof of Lindblad’s result in greater detail in his textbook Quantum Information Theory, published by Cambridge University Press.)
One thing that Lindblad’s proof doesn’t address, however, is whether it makes any difference if the quantum relative entropy decreases by a little or by a lot after a measurement.
In the new paper, Wilde has shown that, if the quantum relative entropy decreases by only a little, then the quantum measurement (or any other type of so-called “quantum physical evolution”) can be approximately reversed.
“When looking at Lindblad’s entropy inequality, a natural question is to wonder what we could say if the quantum relative entropy goes down only by a little when the quantum physical evolution is applied,” Wilde told Phys.org. “It is quite reasonable to suspect that we might be able to approximately reverse the evolution. This was arguably open since the work of Lindblad in 1975, addressed in an important way by Denes Petz in the late 1980s (for the case in which the quantum relative entropy stays the same under the action of the evolution), and finally formulated as a conjecture around 2008 by Andreas Winter. What my work did was to prove this result as a theorem: if the quantum relative entropy goes down only by a little under a quantum physical evolution, then we can approximately reverse its action.”
September 30, 2015 by Lisa Zyga