Superposition, Entropy and Schmidt Decomposition of States

Etang Chen

Dafa Li, Maggie X. Cheng, Xiangrong Li +1 more

Schmidt decomposition (SD) has been proven to be an important tool in quantum information and computation. Ac{\' i}n et al. proposed the SD for three qubits, but in the meanwhile, they indicated that it is possible to have two different SDs for the same state. The more challenging quest of finding the sufficient and necessary condition for the uniqueness of SD has never been undertaken. In this paper, we propose a necessary and sufficient condition for the uniqueness of SD for three qubits. By examining the condition, one can tell what state has one SD and what state has two SDs without actually performing the Schmidt decomposition. We investigate the relation between the uniqueness of SD and the von Neumann entanglement entropy (vNEE). To this end, we prove that any state having the maximal vNEE $S(\rho _{x})=\ln 2$, $ x=A, B,$ or $C$ must have a unique SD. %Therefore, we should choose a state having a unique SD for its maximal vNEE for quantum information theory. This means if a state has two SDs, then the state does not have the maximal vNEE. Therefore, we should not choose a state having two SDs for its maximal vNEE for quantum information theory. In this paper, we also give all the SD states that have the maximal vNEE and a unique SD, as well as all the SD states that have a unique SD.

Vladimir S. Mashkevich

A theory of quantum state reduction is advanced. It is based on two principles: (1) Gauge decomposition; (2) Maximum entropy. To wit: (1) The reduction decomposition of a state vector is the Schmidt decomposition with respect to the states of a set of (dressed) gauge boson modes; (2) The reduction instant is that of the maximum entropy of a resulting mixed state. The theory determines states undergoing the reduction, its instant, resulting pure states and their probabilities. Applications: (Polarized) photon absorption and transmission, emission, particle detection, reduction of a superposition of states, nonintegral photon states, photon and matter-photon entanglement, processes with weak bosons, and the role of gluons.

Antonio Zecca A

Mikhail Vladimirovich Fedorov

In a general form, three-photon polarization states are superpositions of contributions from four three-photon polarization modes with four arbitrary complex coefficients. Entanglement of such states is defined as entanglement between their one-photon and two-photon parts. The degree of entanglement is found explicitly in terms of such entanglement quantifiers as the generalized concurrence, entropy of reduced states and the Schmidt parameter. Polarization Stokes vectors, as well as the Schmidt decompositions of three-photon states are found and discussed. Similarity and differences with respect to biphoton polarization states are discussed too.

Bob Coecke