The Quantum Superposition Principle and Bloch Sphere Representation

Yuguo Su, Fei Yao, Hongbin Liang +3 more

By describing the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, Majorana’s stellar representation provides an intuitive geometric perspective to comprehend the quantum system with high-dimensional Hilbert space. However, the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, a practical method is presented to resolve the problem for the mixed-spin ( s , 1/2) system and describe the entanglement of the system. The system can be decomposed into two spins: spin-( s + 1/2) and spin-( s – 1/2) at the coupling bases, which can be regarded as independent spins. Besides, any pure state may be written as a superposition of two orthonormal states with one spin-( s + 1/2) state and the other spin-( s – 1/2) state. Thus, the whole initial state can be regarded as a state of a pseudo spin-1/2. In this way, the mixed spin decomposes into three spins. Therefore, the state can be represented by (2 s + 1) + (2 s – 1) + 1 = 4 s + 1 sets of stars on a Bloch sphere. Finally, some examples are given to show symmetric patterns on the Bloch sphere and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on the Bloch sphere.

Michael D. Mazurek, Matthew F. Pusey, Robert W. Spekkens +1 more

Diederik Aerts, Massimiliano Sassoli de Bianchi

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.

Alejandro Corichi

Mark P. Silverman

Bo Li, Xiao-Bin Liang, Shao-Ming Fei

The superposition of states is one of the most fundamental issues in the quantum world. Generally there do not exist physical operations to superpose two unknown random states with nonzero probability. We investigate the superposition problem of unknown qubit states with respect to a known qubit state. It is shown that under trace-nonincreasing completely positive operations the superposable state sets are located in some circles on the Bloch sphere. Meanwhile, we prove that the quantum states in a circle on the Bloch sphere are superposable with respect to a known state. Finally, for the high-dimensional case, we illustrate that any superposition transformation protocols will violate the no-cloning principle for almost all the states. Our results also promote the understanding and applications of the superposition principle in view of quantum no-go theorems.