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Ferromagnetism: A state of matter with spontaneous magnetization. Antiferromagnetism: A state of matter in which the neighboring spin are antiparallel with each other, and there is no net magnetization. Ferrimagnetism: A state in which local moments partially cancel. Altermagnetism: A state with zero net magnetization and spin-split electronic ...
In the special case of pure states the definition simplifies: a pure state is separable if and only if it is a product state. A state is said to be entangled if it is not separable. In general, determining if a state is separable is not straightforward and the problem is classed as NP-hard.
The family of finite-rank operators () on a Hilbert space form a two-sided *-ideal in (), the algebra of bounded operators on . In fact it is the minimal element among such ideals, that is, any two-sided *-ideal I {\displaystyle I} in L ( H ) {\displaystyle L(H)} must contain the finite-rank operators.
The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector , the state vector . If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states , both mixed states and pure states .
However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of H − E (this requires the use of a rigged Hilbert space and a renormalization of the norm). This is related to the quantization of constrained systems and quantization of gauge theories. It is ...
The configuration space of all unordered pairs of points on the circle is the Möbius strip. In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space.
We realize that quantum phases of matter (i.e. the zero-temperature phases of matter) can be divided into two classes: long range entangled states and short range entangled states. [2] Topological order is the notion that describes the long range entangled states: topological order = pattern of long range entanglements.