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EPJ Quantum Technology

Quantum image representations based on density matrices in open quantum systems

Yingying Hu, Dayong Lu, Qianqian Zhang, Meiyu Xu

Published Apr 19, 2024• 5 citations

Abstract

Abstract So far, research on quantum image representation has gone through more than 20 years. During this time, the quantum image representation models used have almost all been based on state vectors. However, in practical problems, the environment and the principal quantum system cannot be separated, and isolated quantum systems do not exist in principle. This case is often referred to as an open quantum system. In open quantum systems, many problems involve density matrices, such as the calculation of Von Neumann entropy, the quantization of coherence, and the operator-sum representations of quantum operations. Therefore, the existing quantum image representation models are only suitable for closed quantum systems. To this end, the paper proposes three models that can not only represent quantum images in an open quantum system but also decompose the evolution process of quantum images utilizing operator-sum decomposition. These three models are the representation model of quantum gray-scale images, the tensor product representation model of quantum color images, and the representation model of quantum color images based on mixed states in the Bloch sphere, respectively. All these image representation models have strong correlations among them and are very different from their classical analogues. Between them, the biggest difference is that the paper employs density matrices, inspired by incoherent-coherent states, to represent quantum images rather than classical state vectors. By means of one of the representation models proposed in the paper, we finally demonstrate the evolution process of the quantum image going through the amplitude damping channel.

DOI: 10.1140/epjqt/s40507-024-00241-1

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Abstract So far, research on quantum image representation has gone through more than 20 years. During this time, the quantum image representation models used have almost all been based on state vectors. However, in practical problems, the environment and the principal quantum system cannot be separated, and isolated quantum systems do not exist in principle. This case is often referred to as an open quantum system. In open quantum systems, many problems involve density matrices, such as the calculation of Von Neumann entropy, the quantization of coherence, and the operator-sum representations of quantum operations. Therefore, the existing quantum image representation models are only suitable for closed quantum systems. To this end, the paper proposes three models that can not only represent quantum images in an open quantum system but also decompose the evolution process of quantum images utilizing operator-sum decomposition. These three models are the representation model of quantum gray-scale images, the tensor product representation model of quantum color images, and the representation model of quantum color images based on mixed states in the Bloch sphere, respectively. All these image representation models have strong correlations among them and are very different from their classical analogues. Between them, the biggest difference is that the paper employs density matrices, inspired by incoherent-coherent states, to represent quantum images rather than classical state vectors. By means of one of the representation models proposed in the paper, we finally demonstrate the evolution process of the quantum image going through the amplitude damping channel.

Abstract In this work, the operator-sum representation of a quantum process is extended to the probability representation of quantum mechanics. It is shown that each process admitting the operator-sum representation is assigned a kernel, convolving of which with the initial tomogram set characterizing the system state gives the tomographic state of the transformed system. This kernel, in turn, is broken into the kernels of partial operations, each of them incorporating the symbol of the evolution operator related to the joint evolution of the system and an ancillary environment. Such a kernel decomposition for the projection to a certain basis state and a Gaussian-type projection is demonstrated as well as qubit flipping and amplitude damping processes.

An extended Bloch representation of quantum mechanics was recently derived to offer a possible (hidden-measurements) solution to the measurement problem. In this article we use this representation to investigate the geometry of superposition and entangled states, explaining interference effects and entanglement correlations in terms of the different orientations a state-vector can take within the generalized Bloch sphere. We also introduce a tensorial determination of the generators of SU(N), which we show to be particularly suitable for the description of multipartite systems, from the viewpoint of the sub-entities. We then use it to show that non-product states admit a general description where sub-entities can remain in well-defined states, even when entangled. This means that the completed version of quantum mechanics provided by the extended Bloch representation, where density operators are also considered to be representative of genuine states (providing a complete description), not only offers a plausible solution to the measurement problem but also to the lesser-known entanglement problem. This is because we no longer need to give up the general physical principle saying that a composite entity exists and therefore is in a well-defined state, if and only if its components also exist and therefore are also in well-defined states.

In 2006 we proposed Quantum Fuzzy Sets, observing that states of a quantum register could serve as characteristic functions of fuzzy subsets, embedding Zadeh's unit interval into the Bloch sphere. That paper was deliberately preliminary. In the two decades since, the idea has been taken up by researchers working on quantum annealers, intuitionistic fuzzy connectives, and quantum machine learning, while parallel developments in categorical quantum mechanics have reshaped the theoretical landscape. The present paper revisits that programme and introduces two main extensions. First, we move from pure states to density matrices, so that truth values occupy the entire Bloch ball rather than its surface; this captures the phenomenon of semantic decoherence that pure-state semantics cannot express. Second, we introduce the Q-Matrix, a global density matrix from which individual quantum fuzzy sets emerge as local sections via partial trace. We define a category QFS of quantum fuzzy sets, establish basic structural properties (monoidal structure, fibration over Set), characterize the classical limit as simultaneous diagonalizability, and exhibit an obstruction to a fully internal Frobenius-algebra treatment.

Difficulties in algorithmic simulation of natural thinking point to the inadequacy of information encodings used to this end. The promising approach to this problem represents information by the qubit states of quantum theory, structurally aligned with major theories of cognitive semantics. The paper develops this idea by linking qubit states with color as fundamental carrier of affective meaning. The approach builds on geometric affinity of Hilbert space of qubit states and color solids, used to establish precise one-to-one mapping between them. This is enabled by original decomposition of qubit in three non-orthogonal basis vectors corresponding to red, green, and blue colors. Real-valued coefficients of such decomposition are identical to the tomograms of the qubit state in the corresponding directions, related to ordinary Stokes parameters by rotational transform. Classical compositions of black, white and six main colors (red, green, blue, yellow, magenta and cyan) are then mapped to analogous superposition of the qubit states. Pure and mixed colors intuitively map to pure and mixed qubit states on the surface and in the volume of the Bloch ball, while grayscale is mapped to the diameter of the Bloch sphere. Herewith, the lightness of color corresponds to the probability of the qubit’s basis state «1», while saturation and hue encode coherence and phase of the qubit, respectively. The developed code identifies color as a bridge between quantum-theoretic formalism and qualitative regularities of the natural mind. This opens prospects for deeper integration of quantum informatics in semantic analysis of data, image processing, and the development of nature-like computational architectures.

In the Bloch sphere picture, one finds the coefficients for expanding a single-qubit density operator in terms of the identity and Pauli matrices. A generalization to

n

qubits via tensor products represents a density operator by a real vector of length

{4}^{n}

, conceptually similar to a state vector. Here, we study this approach for the purpose of quantum circuit simulation, including noise processes. The tensor structure leads to computationally efficient algorithms for applying circuit gates and performing few-qubit quantum operations. In view of variational circuit optimization, we study ``backpropagation'' through a quantum circuit and gradient computation based on this representation, and generalize our analysis to the Lindblad equation for modeling the (nonunitary) time evolution of a density operator.

Photon number state representation, especially the single-photon state representation, and the coherent state representation are widely used to specify the state of a radiation field. The concept of density operator or density matrix is introduced in the process of calculating the expectation value of a quantum observable. People then introduce and distinguish the pure state and mixed state. The quantum states of composite system, especially the state of two-photon field, are studied by introducing an 2-D Hilbert space constructed as the direct or tensor product of the Hilbert spaces of two subsystems. A simple model for the creation of single-photon state and multi-photon state is given with details. The concepts of product state, entangled state, and mixed state are introduced and distinguished. Coherent states form a complete set of vector space. Similar to number states, coherent states can be used as a vector basis for characterizing radiation field, except coherent states are nonorthogonal vectors in general.