Information Theoretic Representations of Qubit Channels

Tanner Crowder

The Bloch representation of an n-qubit channel provides a way to represent quantum channels as certain affine transformations on [special characters omitted]. In higher dimensions (n > 1), the correspondence between quantum channels and their Bloch representations is not well-understood. Partly motivated by the ability to simplify the calculation of information theoretic quantities of a qubit channel using the Bloch representation, in this thesis we investigate the correspondence between a channel and its Bloch representation with an emphasis on unital n-qubit channels, in which case the Bloch representation is linear. The thesis is divided into three main sections. First we focus our attention on qubit channels. For certain sets of quantum channels, we establish the surprising existence of a special isomorphism into the set of classical channels. We classify the sets of qubit channels with this property and show that information theoretic quantities are preserved by such classical representations. In a natural progression, we prove some well-known facts about SO(3), the proofs of which are either nonexistent or difficult to find in the literature. Some of this work is based on [12, 13]. In the next section, we consider the multi-qubit channels and show that every finite group can be realized as a subgroup of the quantum channels; this approach allows for the construction of a quantum representation for the free affine monoid over any finite group and gives a classical representation for it. We extend some fundamental results from [26, 28] to the multi-qubit case, including that the set of diagonal Bloch matrices is equal to the free affine monoid over the involution group [special characters omitted]. Some of this work appeared in [10]. Lastly, we study the extreme points for the set of n-qubit channels. There are two types of extreme points: invertible and non-invertible; invertible channels are non-singular maps for which the inverse is also a channel. We briefly study the non-invertible extreme points and then parameterize and analyze the invertiblen-qubit Bloch matrices, which form a compact connected Lie group. We calculate the Lie algebra and give an explicit generating set for the invertible Bloch matrices and a maximal torus.

Hidetosi Takahasi

Keye Martin

Bo Sun, Phuc Q. Le, Abdullah M. Iliyasu +4 more

A Multi-Channel Representation for Quantum Image (MCRQI) is proposed to facilitate the further image processing tasks based on the Flexible Representation for Quantum Image (FRQI). Channel Swapping Operation, One Channel Operation, are proposed as basic image processing operations on MCRQI representation. The simulation experiment results on classical computer show that the channel information of R G B (color) and α (transparency) can be carried easily on quantum computer by employing three qubits to represent the color space, and also indicate that this MCRQI is very flexible to realize some classic-like operations. The MCRQI provides a foundation not only express image in RGBα color space, but also to explore theoretical and practical aspects of image processing on quantum computer.

Mark M. Wilde

Bo Sun, Abdullah M. Iliyasu, Fei Yan +2 more

RGB multi channel representation is proposed for images on quantum computers (MCQI) that captures information about colors (RGB channels) and their corresponding positions in an image in a normalized quantum state. The proposed representation makes it possible to store the RGB information about an image simultaneously by using 2n+3 qubits for encoding 2n× 2npixel images, whereas pixel-wise processing is necessary in many other quantum image representations, e.g., qubit lattice, grid qubit, and quantum lattice. Simulation of storage and retrieval of MCQI images using human facial images demonstrated that 15 qubits are required for encoding 64 × 64 colored images, and encoded information is retrieved by measurement. Perspectives of designing quantum image operators are also discussed based onMCQI representation, e.g., channel of interest, channel swapping, and restrict version of color transformation.