[Paper Review] Universal Asymmetric Quantum Cloning Revisited

Anna-Lena K. Hashagen|arXiv (Cornell University)|Jul 13, 2016

Quantum Information and Cryptography33 references3 citations

TL;DR

This paper revisits the universal asymmetric $1\to 2$ quantum cloning problem by leveraging symmetry properties and the bipolar theorem from convex analysis to fully characterize the set of achievable single-clone fidelities and other figures of merit. It derives the optimal cloning map and quantifies the trade-off between clone qualities, providing an analytical characterization of the attainable region for fidelity, trace norm, Hilbert-Schmidt, operator norm, and diamond norm measures.

ABSTRACT

This paper revisits the universal asymmetric $1 o 2$ quantum cloning problem. We identify the symmetry properties of this optimisation problem, giving us access to the optimal quantum cloning map. Furthermore, we use the bipolar theorem, a famous method from convex analysis, to completely characterise the set of achievable single quantum clone qualities using the fidelity as our figure of merit; from this it is easier to give the optimal cloning map and to quantify the quality tradeoff in universal asymmetric quantum cloning. Additionally, it allows us to analytically specify the set of achievable single quantum clone qualities using a range of different figures of merit.

Motivation & Objective

• To re-derive the optimal universal asymmetric $1\to 2$ quantum cloning channel using symmetry and convex analysis.
• To completely characterize the set of all achievable single-clone qualities for various figures of merit, including fidelity, trace norm, and operator norm.
• To quantify the trade-off between the qualities of two asymmetric clones in a state-independent manner.
• To provide an alternative to group representation methods by using the bipolar theorem and duality in convex analysis.
• To establish a unified analytical framework for the attainable region of clone fidelities across different distance measures.

Proposed method

• The paper applies the bipolar theorem from convex analysis to characterize the set of achievable clone qualities in the universal asymmetric $1\to 2$ cloning scenario.
• It exploits the symmetry of the optimization problem to derive the optimal cloning map, which is shown to be unique under the given constraints.
• The method uses the one-sided polar construction to map the set of achievable clone qualities into a convex set, enabling full characterization.
• The authors define a generalized distance function $d^k(T_i, \text{id})$ for different figures of merit $k \in \{F,1,2,\infty,\diamond\}$, including fidelity and Schatten norms.
• The optimal cloning channel is derived as a solution to a semidefinite program, consistent with prior results but re-derived via symmetry and duality.
• The characterization includes explicit expressions for the boundary of the attainable region via the function $g(x_1,x_2)$, which defines the admissible fidelity trade-off.

Experimental results

Research questions

• RQ1 What is the complete set of achievable qualities for two asymmetric clones in universal $1\to 2$ quantum cloning?
• RQ2 How can the optimal cloning map be derived using symmetry and convex duality?
• RQ3 What is the analytical form of the attainable region for different figures of merit, including fidelity and Schatten norms?
• RQ4 How does the trade-off between clone qualities depend on the choice of distance measure?
• RQ5 Can the bipolar theorem be used to re-derive and fully characterize the optimal cloning channel in a way that generalizes beyond fidelity?

Key findings

• The optimal universal asymmetric $1\to 2$ cloning channel is uniquely determined by its symmetry properties and the constraints of the problem.
• The set of all achievable single-clone qualities is fully characterized as a convex set, with the boundary defined by the function $g(x_1,x_2)$ in Equation (6.5b).
• For the fidelity figure of merit, the attainable region is analytically described by the convex hull of the origin, the maximum fidelity points, and the set where $g(f^F(x_1^{(F)}), f^F(x_2^{(F)})) = 0$.
• The method yields explicit analytical expressions for the attainable region under five different figures of merit: fidelity, trace norm, Hilbert-Schmidt norm, operator norm, and diamond norm.
• The characterization via the bipolar theorem provides a rigorous, duality-based alternative to the group representation approach used in prior work, particularly for the fidelity case.
• The paper confirms that the optimal cloning map is consistent with previous results but derives it through a novel, more general framework based on convex analysis.