A solution of the generalised quantum Stein's lemma (2408.06410v2)

Abstract: We solve the generalised quantum Stein's lemma, proving that the Stein exponent associated with entanglement testing, namely, the quantum hypothesis testing task of distinguishing between $n$ copies of an entangled state $\rho_{AB}$ and a generic separable state $\sigma_{A^n:B^n}$, equals the regularised relative entropy of entanglement. Not only does this determine the ultimate performance of entanglement testing, but it also establishes the reversibility of all quantum resource theories under asymptotically resource non-generating operations, with the regularised relative entropy of resource governing the asymptotic transformation rate between any two quantum states. As a by-product, we prove that the same Stein exponent can also be achieved when the null hypothesis is only approximately i.i.d., in the sense that it can be modelled by an "almost power state". To solve the problem we introduce two techniques. The first is a procedure that we call "blurring", which, informally, transforms a permutationally symmetric state by making it more evenly spread across nearby type classes. Blurring alone suffices to prove the generalised Stein's lemma in the fully classical case, but not in the quantum case. Our second technical innovation, therefore, is to perform a second quantisation step to lift the problem to an infinite-dimensional bosonic quantum system; we then solve it there by using techniques from continuous-variable quantum information. Rather remarkably, the second-quantised action of the blurring map corresponds to a pure loss channel. A careful examination of this second quantisation step is the core of our quantum solution.

• The paper presents an explicit solution to the generalised quantum Stein’s lemma, resolving a conjecture on reversibility in quantum resource theories.
• It employs innovative techniques, 'blurring' to smooth state distributions and 'second quantisation' to interpret blurring in infinite dimensions, akin to a pure loss channel.
• A key finding is the equivalence of the Stein exponent for entanglement testing with the regularised relative entropy of entanglement, providing insights into entanglement manipulation and efficiency.

Insights into the Generalised Quantum Stein’s Lemma

The paper "A solution of the generalised quantum Stein’s lemma" by Ludovico Lami offers a well-constructed proof and analysis of the generalised quantum Stein’s lemma, a fundamental result in the field of quantum hypothesis testing and quantum resource theory. At its core, the paper addresses the quantum hypothesis testing problem of distinguishing between multiple copies of an entangled state and a separable state. This work not only provides an explicit solution but also explores its implications on the reversibility of quantum resources under asymptotic limits with specific operations.

Main Contributions and Technical Approaches

The primary contribution of this paper lies in demonstrating that the Stein exponent associated with the entanglement testing problem equals the regularised relative entropy of entanglement. This result resolves a conjecture by Brandão and Plenio related to the reversibility in quantum resource theories. The approach adopted in this paper hinges on two innovative techniques—'blurring' and 'second quantisation.'

1. Blurring Technique: This technique involves smoothing the distribution of input states to enhance their symmetry, thereby spreading the concentration of states across type classes. While blurring suffices for proving the generalised Stein's lemma in classical scenarios, it does not fully extend to the quantum setting without additional considerations.
2. Second Quantisation: This technique lifts the problem to infinite-dimensional physics by interpreting the blurring map's second quantised action as a type of quantum channel, specifically akin to a pure loss channel broadly studied in continuous-variable quantum information theory.

Implications and Results

The equivalence between the Stein exponent and the regularised relative entropy of entanglement showcased in this paper has significant ramifications. It provides essential insights into the transformations and manipulations of entangled quantum states, establishing the operational significance of the regularised relative entropy in quantifying the efficiency of entanglement testing and manipulation. Furthermore, it addresses Plenio's problem of whether there exists a reversible theory of entanglement conversion, hinting at the broader implications of resource efficiency across quantum systems.

Another significant aspect explored is the extension to "almost power states," which accommodate practically relevant scenarios where systems deviate slightly from the idealised i.i.d. structure. Extending the generalised Stein's lemma to such states illustrates that as long as the number of non-i.i.d. or 'defective' systems remains bounded, the result still holds. This consideration reflects the robust nature of the regularised relative entropy of entanglement in capturing entanglement's essence despite imperfections.

Future Perspectives

The paper lays the groundwork for future explorations of non-i.i.d. effects in quantum hypothesis testing. Extending these findings to broader types of resource theories, such as quantum coherence or thermality, might yield intriguing results in quantum technologies. Moreover, investigating efficient computational techniques for practical scenarios of the Stein and Sanov exponents could streamline these concepts' applicability in real-world quantum systems. Finally, understanding how these insights interact with other quantum processes, like error correction and quantum communication, represents a promising avenue for advancing quantum informational sciences.

In conclusion, Lami’s paper on the generalised quantum Stein’s lemma offers a profound theoretical breakthrough with potential practical impact. It rigorously addresses the asymptotic efficiency of entanglement testing and manipulation, reaffirming the foundational role of the relative entropy of entanglement within quantum information theory.