Non-deterministic quantum noiseless linear amplifiers are a new technology with interest in both fundamental understanding and new applications. With a noiseless linear amplifier it is possible to perform tasks such as improving the performance of quantum key distribution and purifying lossy channels. Previous designs for noiseless linear amplifiers involving linear optics and photon counting are non-optimal because they have a probability of success lower than the theoretical bound given by the theory of generalised quantum measurement. This paper develops a theoretical model which reaches this limit. We calculate the fidelity and probability of success of this new model for coherent states and Einstein-Podolsky-Rosen (EPR) entangled states. A deterministic noiseless, phase insensitive, linear amplifier, as seen in classical systems is unphysical in quantum theory [1]. However it has been demonstrated that an analogous probabilistic amplifier is approximately physically realisable [2-4] and has a wide variety of potential uses in quantum computing and communication technology protocols. These protocols include error correction [5], quantum key distribution [6] and other protocols where distillation of entanglement is desirable [3].In order to translate these systems to useful quantum technologies an investigation into the optimal probabilities of success that can be achieved is important. Low probabilities of success reduce the range of possible experimental and commercial applications of these devices. Ralph and Lund [2] proposed a linear optics implementation of a heralded noiseless linear amplifier which has been theoretically investigated [8][9][10] and experimentally demonstrated with good agreement in visibility and effective gain for small amplitudes α < 0.04 and gains |g| 2 ≤ 5 [3, 7, 11-13]. The probability of success for low amplitude inputs α 1 using this design is P = 1 g 2 +1 . The probability of success of other linear optical designs are similar [4,14]. For higher amplitudes,n, the probability scales as P ≈ 1 (g 2 +1) N where N |α| 2 . The theoretical maximum probability of success for a noiseless linear amplifier in the low photon number regime is P = 1 g 2 [2] and is expected to scale as 1 g 2N .Our aim in this paper is to identify and analyse a physical model for noiseless linear amplification which saturates this maximum probability of success. Our approach is related to the idea that noiseless amplification can be implemented via a weak measurement model [15]. The paper is arranged in the following way. In the first section we will introduce a measurement model for noiseless amplification. In section 2 we will translate this into a physical model for the amplifier and particularly look at the low photon number limit. The following two sections will analyse the performance of the amplifier with respect to coherent state inputs and the distillation and purification of Einstein, Podolsky, Rosen (EPR) entanglement (2-mode squeezing). In the final section we will conclude.

We expand the class of holographic quantum error correcting codes by developing the notion of block perfect tensors, a wider class that includes previously defined perfect tensors. The relaxation of this constraint opens up a range of other holographic codes. We demonstrate this by introducing the self-dual CSS heptagon holographic code, based on the 7-qubit Steane code. Finally we show promising thresholds for the erasure channel by applying a straightforward, optimal erasure decoder to the heptagon code and benchmark it against existing holographic codes.