In quantum physics, **quantum state** refers to the state of a quantum system. A quantum state is given as a vector in a vector space, called the **state vector**. The state vector theoretically contains statistical information about the quantum system. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vector is given by the principal quantum number . For a more complicated case, consider Bohm formulation of EPR experiment, where the state vector involves superposition of joint spin states for 2 different particles.

In a more general usage, a quantum state can be either "pure" or "mixed." The above example is pure. Mathematically, a pure quantum state is represented by a state vector in a vector space, which is a generalization of our more usual three dimensional space. A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. Quantum states, mixed as well as pure, are described by so-called density matrices, although these give probabilities, not densities.

For example, if the spin of an electron is measured in any direction, e.g., with a Stern-Gerlach experiment, there are two possible results, up or down. The vector space for the electron's spin is therefore two-dimensional. A pure state is a two-dimensional complex vector, with a length of one. That is, . A mixed state is a matrix that is Hermitian, positive-definite, and has trace 1.

Before a particular measurement is performed on a quantum system, the theory usually gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. These probability distributions arise for both mixed states and pure states: it is impossible in quantum mechanics (unlike classical mechanics) to prepare a state in which all properties of the system are fixed and certain. This is exemplified by the uncertainty principle, and reflects a core difference between classical and quantum physics. Even in quantum theory, however, for every observable there are states that determine its value exactly.

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