Irreversibility of entanglement manipulation. Our main result in Theorem 1 shows that the two-qutrit state ω3 cannot be manipulated reversibly under NE transformations. We can only extract log2(3/2)≈7/12 entanglement bits per copy of ω3 asymptotically, but one complete entanglement bit per copy is needed to generate it. Theorem 1 can be strengthened and extended in various ways. Credit: nature physics (2023). DOI: 10.1038/s41567-022-01873-9
The second law of thermodynamics is often regarded as one of the few laws of physics that is absolutely and undeniably true. The law states that the amount of ‘entropy’ – a physical property – of a closed system can never decrease. It adds an ‘arrow of time’ to everyday events and determines which processes are reversible and which are not. It explains why an ice cube placed on a hot stove will always melt, and why compressed gas will always fly out of the container (and never back) when a valve to the atmosphere is opened.
Only states of equal entropy and energy can be converted reversibly from one to the other. This reversibility condition led to the discovery of thermodynamic processes such as the (idealized) Carnot cycle, which puts an upper limit on how efficiently one can convert heat into work, or vice versa, by passing a closed system through different temperatures and pressures. . Our understanding of this process underpinned the rapid economic development during the Western Industrial Revolution.
The beauty of the second law of thermodynamics is its applicability to any macroscopic system, regardless of the microscopic details. In quantum systems, one of these details can be entanglement: a quantum connection that causes separate components of the system to share properties. Intriguingly, quantum entanglement bears many profound similarities to thermodynamics, even though quantum systems are mostly studied in the microscopic regime.
Scientists have discovered a notion of “entanglement entropy” that exactly mimics the role of thermodynamic entropy, at least for idealized quantum systems perfectly isolated from their environment.
“Quantum entanglement is an important resource that underpins much of the power of future quantum computers. To use it effectively, we need to learn how to manipulate it,” said Ludovico Lami, a quantum information researcher. A fundamental question became whether entanglement can always be reversibly manipulated, in direct analogy to the Carnot cycle. Crucially, this reversibility should hold, at least in theory, even for noisy (‘mixed’) quantum systems that have not been kept perfectly isolated from their environment.
It was believed that a ‘second law of entanglement’ could be established, embodied in a single function that would generalize entanglement entropy and govern all entanglement manipulation protocols. This conjecture appeared in a famous list of outstanding problems in quantum information theory.
No second law of entanglement
To solve this long-standing open question, research conducted by Lami (formerly at the University of Ulm and currently at QuSoft and the University of Amsterdam) and Bartosz Regula (University of Tokyo) shows that manipulation of entanglement is fundamentally irreversible, making each hoping to establish a second law of entanglement.
This new result is based on the construction of a certain quantum state that is very ‘expensive’ to create with pure entanglement. Creating this state will always result in a loss of some of this entanglement, as the invested entanglement cannot be fully recovered. As a result, it is inherently impossible to transform this state into another and back again. The existence of such states was previously unknown.
Because the approach used here does not assume which exact transformation protocols are used, it rules out the reversibility of entanglement in all possible environments. It applies to all protocols, assuming they do not generate new entanglements themselves. Lami explains, “Using entanglement operations would be like running a distillery in which alcohol is secretly added to the drink from elsewhere.”
Says Lami: “We can conclude that no single quantity, such as the entropy of entanglement, can tell us everything there is to know about the permissible transformations of entangled physical systems. The theory of entanglement and thermodynamics are thus governed by fundamentally different and incompatible sets. of laws.”
This may mean that describing quantum entanglement isn’t as easy as scientists had hoped. However, rather than being a disadvantage, the much greater complexity of the theory of entanglement compared to the classical laws of thermodynamics may allow us to use entanglement to achieve feats that would otherwise be completely unthinkable. “What we now know for sure is that entanglement hides an even richer and more complicated structure for which we assumed it,” Lami concludes.
The article will be published in the magazine nature physics.
Ludovico Lami et al. Yet no second law of entanglement manipulation, nature physics (2023). DOI: 10.1038/s41567-022-01873-9
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