[Paper Review] Noiseless linear amplification via weak measurements

David Menzies, Sarah Croke|ArXiv.org|Mar 24, 2009

Quantum Information and Cryptography3 references5 citations

TL;DR

This paper proposes a weak measurement-based protocol for probabilistic noiseless linear amplification of small-amplitude coherent states, achieving approximate amplification with heralded success. The scheme leverages the weak value imprinting of the system's number operator onto a probe, enabling probabilistic cloning with significantly higher success probability than prior linear optics methods.

ABSTRACT

We discuss the recently introduced concept of non-deterministic noiseless linear amplification, demonstrating that such an operation can only be performed perfectly with vanishing probability of success. We show that a weak measurement, which imprints the weak value of an observable of a pre-selected and post-selected system onto a probe system, can be used to approximate probabilistic noiseless amplification. This result may be applied to various tasks in continuous variable quantum information, including entanglement concentration, probabilistic cloning, and in quantum repeaters. We discuss the application of our scheme to probabilistic cloning of weak coherent states.

Motivation & Objective

• To address the fundamental limitation that deterministic noiseless linear amplification is impossible due to the no-cloning theorem and unitarity constraints.
• To develop a practical, probabilistic alternative to exact noiseless amplification using weak measurements and post-selection.
• To demonstrate how weak values can be harnessed for useful quantum information tasks such as entanglement concentration and probabilistic cloning.
• To compare the performance of the proposed weak measurement protocol with existing linear optical schemes, particularly in terms of success probability and fidelity.

Proposed method

• Utilizes a pre- and post-selected quantum system interacting weakly with a probe via a cross-Kerr Hamiltonian $ H_I = ar{h} ilde{ heta}(t) ilde{A} ilde{P} $, where $ ilde{A} $ is the system's number operator and $ ilde{P} $ the probe's momentum.
• Imprints the weak value $ n_W = rac{raket{p| ilde{n}|eta}}{raket{p|eta}} $ of the system's number operator onto the probe's quadrature, enabling conditional amplification based on measurement outcomes.
• Employs balanced homodyne detection on the probe to measure its quadrature $ p $, with success heralded by outcomes in the range $ -1.6 \≤ p \≤ -\frac{\ln 2}{2\sqrt{2}\alpha\kappa_T} $, ensuring gain $ g \geq \sqrt{2} $.
• Constructs an approximate noiseless amplifier $ \hat{\Gamma}_{\text{approx}}(g) = c_N \sum_{n=0}^{N} g^n |n\rangle\langle n| $, truncated to finite Fock states to satisfy physical constraints.
• Calculates the success probability $ p_S = |c_N|^2 e^{-(g^2-1)|\alpha|^2} $, which decreases exponentially with improved approximation (larger $ N $) or higher gain.
• Applies the scheme to probabilistic cloning by mapping $ |\beta\rangle \to |g\beta\rangle $ with $ g \geq \sqrt{2} $, followed by a 50-50 beam-splitter to produce two copies.

Experimental results

Research questions

• RQ1 Can noiseless linear amplification be implemented non-deterministically despite the no-cloning theorem forbidding deterministic unitary amplification?
• RQ2 To what extent can weak measurements approximate noiseless amplification, and what are the trade-offs between fidelity and success probability?
• RQ3 How does the weak measurement framework compare to existing linear optical schemes in terms of success probability for probabilistic cloning of coherent states?
• RQ4 What is the role of the weak value of the number operator in enabling heralded, approximate amplification via probe measurement?
• RQ5 Can the proposed scheme be optimized for higher success probability while maintaining high fidelity in practical quantum information tasks?

Key findings

• Exact noiseless linear amplification is impossible with non-zero success probability, as the required operator $ \hat{\Gamma}(g) $ violates the trace-non-increasing completely positive map constraint unless $ |c|^2 = 0 $.
• Approximate noiseless amplification can be achieved arbitrarily well by truncating the Fock state space, but success probability decays exponentially with improved approximation fidelity.
• The weak measurement protocol achieves a success probability of 20% at fidelity >0.99 for $ \beta = 0.2 $, significantly outperforming the linear optics scheme (0.5% success probability at similar fidelity).
• For $ \kappa_T = 2 \times 10^{-5} $, the protocol achieves fidelity >0.995 with 4% success probability for $ \beta = 0.5 $, demonstrating scalability to larger coherent states.
• The scheme enables probabilistic cloning of weak coherent states with high fidelity and heralded success, using only a small cross-Kerr nonlinearity, coherent states, and homodyne detection.
• The probability density $ \rho(p) = \frac{1}{\sqrt{\pi}} e^{-p^2 + (g^2 - 1)\beta^2} $ and fidelity $ \mathcal{F}(p) $ are analytically derived, enabling optimization of measurement windows for desired performance.