In the present article, we consistently develop the main issues of the Bloch vectors formalism for an arbitrary finite-dimensional quantum system. In the frame of this formalism, qudit states and their evolution in time, qudit observables and their expectations, entanglement and nonlocality, etc are expressed in terms of the Bloch vectors—the vectors in the Euclidean space , arising under decompositions of observables and states in different operator bases. Within this formalism, we specify for all d ⩾ 2 the set of Bloch vectors of traceless qudit observables and describe its properties; also, find for the sets of the Bloch vectors of qudit states, pure and mixed, the new compact expressions in terms of the operator norms that explicitly reveal the general properties of these sets and have the unified form for all d ⩾ 2. For the sets of the Bloch vectors of qudit states under the generalized Gell‐Mann representation, these general properties cannot be analytically extracted from the known equivalent specifications of these sets via the system of algebraic equations. We derive the general equations describing the time evolution of the Bloch vector of a qudit state if a qudit system is isolated and if it is open and find for both cases the main properties of the Bloch vector evolution in time. For a pure bipartite state of a dimension d ₁ d ₂, we quantify its entanglement via the characteristics of the Bloch vectors for its reduced states. The introduced general formalism is important both for the theoretical analysis of quantum system properties and for quantum applications, in particular, for optimal quantum control, since, for systems where states are described by vectors in the Euclidean space, the methods of optimal control, analytical and numerical, are well developed.

在本文中，我们一直在开发任意有限维量子系统的 Bloch 向量形式主义的主要问题。在这种形式主义的框架中，qudit 状态及其随时间的演化、qudit 可观察量及其期望、纠缠和非局域性等都用 Bloch 向量表示——欧几里得空间中的向量，在不同的运营商基础。在这种形式主义中，我们指定所有d⩾ 2 无迹qudit observables 的Bloch 向量集并描述其属性；此外，对于纯的和混合的 qudit 状态的布洛赫向量集合，根据算子范数找到新的紧凑表达式，这些表达式明确地揭示了这些集合的一般性质，并具有所有d的统一形式⩾ 2. 对于广义 Gell-Mann 表示下的 qudit 状态的 Bloch 向量集，无法通过代数方程组从这些集的已知等效规范中解析地提取这些一般属性。我们推导出描述 qudit 状态的 Bloch 向量时间演化的一般方程，如果一个 qudit 系统是孤立的，如果它是开放的，并且在这两种情况下找到 Bloch 向量随时间演化的主要属性。对于维度为d ₁ d ₂的纯二分状态，我们通过 Bloch 向量的特征对其减少状态进行量化。引入的一般形式主义对于量子系统性质的理论分析和量子应用都很重要，特别是对于最优量子控制，因为对于状态由欧几里德空间中的向量描述的系统，最优控制的方法，分析和数字，发展得很好。