[Paper Review] On superactivation of one-shot zero-error quantum capacity and the related property of quantum measurements

M. E. Shirokov, Tatiana Shulman|arXiv (Cornell University)|Dec 12, 2013

Quantum Information and Cryptography18 references3 citations

TL;DR

This paper presents a low-dimensional quantum channel (4-level input, 3-level environment) that demonstrates symmetric superactivation of one-shot zero-error quantum capacity: although the channel itself has zero one-shot zero-error quantum capacity, its tensor product with itself achieves a positive capacity, enabling perfect quantum state transmission. The phenomenon is linked to the emergence of an indistinguishable subspace in the tensor product of two quantum observables, each individually having no such subspaces.

ABSTRACT

We begin with a detailed description of a low dimensional quantum channel ($d_A=4, d_E=3$) demonstrating the symmetric form of superactivation of one-shot zero-error quantum capacity. This means appearance of a noiseless (perfectly reversible) subchannel in the tensor square of a channel having no noiseless subchannels. Then we describe a quantum channel $Φ$ such that $\,\bar{Q}_0(Φ)=0$ and $\,\bar{Q}_0(Φ\otimesΦ)\geq\log n\,$ for any $\,n\leq+\infty$. We also show that the superactivation of one-shot zero-error quantum capacity of a channel can be reformulated in terms of quantum measurements theory as appearance of an indistinguishable subspace for tensor product of two observables having no indistinguishable subspaces.

Motivation & Objective

• To construct an explicit, low-dimensional example of superactivation in one-shot zero-error quantum capacity, resolving a gap in prior high-dimensional, non-explicit constructions.
• To demonstrate symmetric superactivation, where a single channel Φ satisfies Q̄₀(Φ) = 0 but Q̄₀(Φ⊗Φ) > 0, enabling perfect quantum communication in the tensor product.
• To reformulate superactivation in terms of quantum measurement theory, showing that the tensor product of two observables with no indistinguishable subspaces can generate such subspaces.
• To provide a minimal Kraus representation for the constructed channel using its noncommutative graph, enabling explicit numerical realization.
• To show that the superactivation effect can be scaled to arbitrary n, including infinite-dimensional channels, with Q̄₀(Φ⊗Φ) ≥ log n.

Proposed method

• Constructs a quantum channel Φ with input dimension d_A = 4 and environment dimension d_E = 3 using a noncommutative graph that ensures no noiseless subchannels exist in Φ.
• Employs a modified Kraus representation derived from a positive operator basis of the noncommutative graph, enabling explicit numerical construction of the channel.
• Applies Stinespring's theorem to represent the channel via an isometry V, and derives the complementary channel to analyze reversibility on subspaces.
• Reformulates the superactivation condition as the emergence of an indistinguishable subspace in the tensor product of two quantum observables, each lacking such subspaces.
• Uses the theory of quantum measurements to show that the tensor product of two observables with no indistinguishable subspaces can possess a continuous family of n-dimensional indistinguishable subspaces.
• Establishes a general method to construct a channel with a given noncommutative graph via a basis of positive operators summing to identity and a set of unit vectors in a finite-dimensional Hilbert space.

Experimental results

Research questions

• RQ1 Can superactivation of one-shot zero-error quantum capacity be demonstrated in a low-dimensional, symmetric channel with d_A = 4 and d_E = 3?
• RQ2 Does the tensor product of two quantum channels, each with zero one-shot zero-error quantum capacity, ever achieve a positive capacity?
• RQ3 Can the phenomenon of superactivation be reformulated in terms of quantum measurement theory, specifically via the emergence of indistinguishable subspaces?
• RQ4 What is the minimal dimension required to realize superactivation of one-shot zero-error quantum capacity?
• RQ5 Can the superactivation effect be scaled to arbitrary n, including infinite-dimensional channels?

Key findings

• A symmetric superactivation example is constructed with d_A = 4 and d_E = 3, where Q̄₀(Φ) = 0 but Q̄₀(Φ⊗Φ) ≥ log 2 > 0, demonstrating the phenomenon in minimal dimensions.
• The channel Φ has a minimal Kraus representation with explicit numerical coefficients derived from its noncommutative graph, enabling full analytical and numerical characterization.
• The superactivation effect is reformulated in measurement theory: two observables with no indistinguishable subspaces can generate a continuous family of n-dimensional indistinguishable subspaces in their tensor product.
• For any n ∈ ℕ or n = +∞, a channel Φ exists such that Q̄₀(Φ) = 0 but Q̄₀(Φ⊗Φ) ≥ log n, showing scalability of the effect.
• The construction confirms that superactivation is not limited to asymmetric or high-dimensional channels, but can occur in symmetric, low-dimensional settings.
• The result establishes a direct link between superactivation and the existence of entangled indistinguishable subspaces in quantum measurements, offering a new perspective in quantum information theory.