state

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A state is an abstraction of all the quantities necessary to uniquely specify how to prepare system.

1 Introduction
2 Mathematical properties
- 2.1 Norm and dual space
3 Hilbert space
4 Probabilistic interpretation
5 Pure state
6 Mixed state
7 Separable and entangled states
8 References

Introduction

In the case of the Stern-Gerlach experiment, measurement of an observable such as the $z\,$ -component of the intrinsic angular momentum, $S_z\,$ , of the atom may have yielded a value of $\tfrac{\hbar}{2}\,$ . Without performing any other measurements, any additional measurement of this observable will yield $\tfrac{\hbar}{2}\,$ . If the original measurement yielded $-\tfrac{\hbar}{2}\,$ instead, then any further measurements of the same observable would yield $-\tfrac{\hbar}{2}\,$ also. Therefore we can uniquely prepare the system to yield a desired measurement by specifying the state, which we will simply call $\left|\psi\right\rangle\,$ , as being either $\left|\uparrow\right\rangle\,$ , or $\left|\downarrow\right\rangle\,$ , respectively.

On the other hand, suppose we prepare system so that a measurement of the intrinsic angular momentum along a direction at some angle to the $z\,$ -axis yields, say, $\tfrac{\hbar}{2}\,$ , and then subsequently measure $S_z\,$ . If we repeat this experiment a number of times on identically prepared systems, then some fraction $P_\uparrow\,$ of the measurements would yield $\tfrac{\hbar}{2}\,$ , while some fraction $P_\uparrow\,$ would yield $-\tfrac{\hbar}{2}\,$ , with $P_\uparrow + P_\downarrow = 1\,$ . It is an experimentally verifiable fact of quantum mechanics that we can completely specify the prepared state $\left|\psi\right\rangle\,$ using two complex numbers $c_\uparrow\,$ and $c_\downarrow\,$ , with

$\left|c_\uparrow\right|^2 = c_\uparrow^* c_\uparrow = P_\uparrow\,$ ,

$\left|c_\downarrow\right|^2 = c_\downarrow^* c_\downarrow = P_\downarrow\,$ ,

and the statement is more generally known as Bell's theorem. For now we will take it as an observation and use the shorthand

$\left|\psi \right\rangle = c_\uparrow \left|\uparrow\right\rangle + c_\downarrow \left|\downarrow\right\rangle\,$ .

Mathematical properties

In the above notation, we have suggestively multiplied the states $\left|\uparrow\right\rangle\,$ and $\left|\downarrow\right\rangle\,$ by complex numbers and added the results. If we temporarily relax the requirement that $|c_\uparrow|^2 + |c_\downarrow|^2 = 1\,$ , then we see that the set of states form a complex vector space, and we can think about states as corresponding to vectors, known as state vectors, which we can also write as column vectors,

$c_\uparrow \left|\uparrow\right\rangle + c_\downarrow \left|\downarrow\right\rangle \leftrightarrow \begin{pmatrix} c_\uparrow\\ c_\downarrow \end{pmatrix}\,$ .

More generally, we may choose a basis for our complex vector space, denoted $\left\{\left|\phi_i\right\rangle\right\}\,$ , and write

$\left|\Psi\right\rangle = \sum_i c_i \left|\phi_i\right\rangle\,$ .

A state vector written in this way is also called a ket or ket vector.

Norm and dual space

A very useful concept is that of a dual state, which is a linear map from the set of possible states $\mathcal{H}\,$ onto the complex numbers,

$\left\langle\Phi\right| : \mathcal{H} \to \mathbb{C}\,$ ,

and we denote the action of a dual state on a state, known as an inner product, as

$\left\langle\Phi\right| \left( \left|\Psi\right\rangle\right) \equiv \left\langle\Phi | \Psi \right\rangle\,$ .

The dual state vector is also called a bra or bra vector, so that the action of a bra on a ket is called a bracket, a convention known as bra-ket notation.

Furthermore, we can associate to every state $\left|\Psi\right\rangle\,$ one and only one dual state $\left\langle\Psi\right| \equiv (\left|\Psi\right\rangle)^{\dagger}\,$ . We will denote the set of dual states by $\mathcal{H}^*\,$ . In the above example, such a map is given by

$\left\langle\psi \right| = c_\uparrow^* \left\langle\uparrow\right| + c^*_\downarrow \left\langle\downarrow\right|\,$ .

Furthermore, the states $\left|\uparrow\right\rangle\,$ and $\left|\downarrow\right\rangle\,$ are mutually exclusive, and form a basis for the allowed states, so it is natural to define a dual basis and require that

$\left\langle \uparrow \| \uparrow \right\rangle\,$	$= 1\,$ ,
$\left\langle \downarrow \| \downarrow \right\rangle\,$	$= 1\,$ ,
$\left\langle \uparrow \| \downarrow \right\rangle\,$	$= 0\,$ ,
$\left\langle \downarrow \| \uparrow \right\rangle\,$	$= 0\,$ .

The action of one dual state on another state is then completely specified by the action of the basis. Let

$\left|\phi \right\rangle = d_\uparrow \left|\uparrow\right\rangle + d_\downarrow \left|\downarrow\right\rangle\,$ .

Then

$\left\langle\psi|\phi\right\rangle = \left( c_\uparrow^* \left\langle\uparrow\right| + c^*_\downarrow \left\langle\downarrow\right| \right) \left( d_\uparrow \left|\uparrow\right\rangle + d_\downarrow \left|\downarrow\right\rangle \right) = c_\uparrow^* d_\uparrow \left\langle \uparrow | \uparrow \right\rangle\ + c^*_\downarrow d_\downarrow \left\langle \downarrow | \downarrow \right\rangle + c_\uparrow^* d_\downarrow \left\langle \uparrow | \downarrow \right\rangle + c_\downarrow^* d_\uparrow \left\langle \downarrow | \uparrow \right\rangle\,$ ,

which simplifies to

$\left\langle\psi|\phi\right\rangle = c_\uparrow^* d_\uparrow + c^*_\downarrow d_\downarrow\,$ .

The matrix notation is suggestive, so if we let

$c^*_\uparrow \left\langle\uparrow\right| + c^*_\downarrow \left\langle\downarrow\right| \leftrightarrow \left( c^*_\uparrow \quad c^*_\downarrow \right)\,$ ,

then

$\left\langle\psi|\phi\right\rangle = \begin{pmatrix} c_\uparrow\\ c_\downarrow \end{pmatrix}^{\dagger}\begin{pmatrix} d_\uparrow\\ d_\downarrow \end{pmatrix} = \left( c^*_\uparrow \quad c^*_\downarrow \right) \begin{pmatrix} d_\uparrow\\ d_\downarrow \end{pmatrix} \,$ .

More generally, given a basis $\left\{\left|\phi_i\right\rangle\right\}\,$ , we can define a dual basis $\left\{\left\langle\phi_i\right|\right\} = \left\{\left|\phi_i\right\rangle^{\dagger}\right\}\,$ that satisfies

$\left\langle \phi_i | \phi_j \right\rangle = \delta_{ij}\,$ .

Such a basis is termed an orthonormal basis. A useful property of such a basis is that

$c_i = \left\langle \phi_i | \Psi \right\rangle\,$ .

The inner product between two states $\left|\Psi\right\rangle = \sum_i c_i \left|\phi_i \right\rangle\,$ and $\left|\Phi\right\rangle = \sum_i d_i \left|\phi_i \right\rangle\,$ is then fully determined:

$\left\langle \Psi | \Phi \right\rangle = \sum_i \sum_j c_i^* d_j \left\langle \phi_i | \phi_j \right\rangle = \sum_i c_i^* d_i\,$ .

Two states are termed orthogonal if their inner product vanishes, i.e., if

$\left\langle\Psi|\Phi\right\rangle = 0\,$ .

Furthermore, we define the norm of a state to be the non-negative real number

$\left\Vert \left|\Psi\right\rangle \right\Vert \equiv \sqrt{\left\langle\Psi|\Psi\right\rangle } \,$ .

We quickly note the following general properties inherited from complex vector spaces:

$\left\langle \Psi | \Phi\right\rangle = (\left\langle \Phi | \Psi\right\rangle)^*\,$

$\left\langle \Psi|\Phi\right\rangle^2 \leq \left\langle\Psi|\Psi\right\rangle \left\langle\Phi|\Phi\right\rangle\,$ (Cauchy–Schwarz inequality)

$\left\Vert \left|\Psi\right\rangle + \left|\Phi\right\rangle\right\Vert \leq \left\Vert \left|\Psi\right\rangle\right\Vert + \left\Vert\left|\Phi\right\rangle\right\Vert\,$ (triangle inequality)

It is important to note that duality is a symmetric property. We may equally well view $\left|\Psi\right\rangle\,$ as the dual of $\left\langle\Psi\right|\,$ , since the inner product defines a map from $\mathcal{H}^*\,$ to $\mathbb{C}\,$ .

Hilbert space

With the above definitions, the set $\mathcal{H}\,$ is a complex vector space with an inner product. If we take as a further (not unreasonable assumption) that the space is complete (roughly speaking, if a sequence converges to some value, then that value is also in the space), then $\mathcal{H}\,$ is also a Hilbert space. This last requirement is automatically satisfied by finite dimensional vector spaces, and so some authors reserve the term Hilbert space for infinite dimensional spaces.

Probabilistic interpretation

In our example we gave the coefficients $c_\uparrow\,$ and $c_\downarrow\,$ the physical interpretation that $P_\uparrow\,$ and $P_\downarrow\,$ give the probability of a measurement yielding either $S_z = \tfrac{\hbar}{2}\,$ or $S_z = -\tfrac{\hbar}{2}\,$ respectively when a large number of measurements on identically prepared systems are made. The coefficients $c_\uparrow\,$ and $c_\downarrow\,$ themselves are termed probability amplitudes. The requirement that $P_\uparrow + P_\downarrow = 1\,$ , i.e., $|c_\uparrow|^2 + |c_\downarrow|^2 = 1\,$ , is really the more general requirement that a physical state be normalized, i.e., $\left\langle \Psi | \Psi \right\rangle = 1\,$ . This reflects the requirement that sum of all probabilities to add up to unity. For a general state $\left|\Psi\right\rangle\,$ , the normalized state that corresponds to it is

$\frac{\left|\Psi\right\rangle}{\sqrt{ \left\langle \Psi|\Psi\right\rangle }}\,$ .

Two states vectors correspond to the same physical state if they are related by an overall factor:

$\left|\Psi\right\rangle \sim A \left|\Psi\right\rangle\,$ , $A \in \mathbb{C}\,$ .

Therefore it is more correct to think of a physical state as being represented by a one-dimensional subspace of $\mathcal{H}\,$ , called a ray.

Returning to the example of the Stern-Gerlach experiment, we recall that $\left|\psi \right\rangle = c_\uparrow \left|\uparrow\right\rangle + c_\downarrow \left|\downarrow\right\rangle\,$ contains all the information necessary to describe our system. Suppose, however, that we prepared our state to always yield a measurement of $\tfrac{\hbar}{2}\,$ when the intrinsic angular momentum is measured along a different axis, (say, the $x\,$ -axis). We could repeat our previous analysis, which would amount to choosing a different basis, which we may call $\left|\uparrow\right\rangle_x\,$ and $\left|\downarrow\right\rangle_x\,$ . We can therefore write our state in two ways

$\left|\psi\right\rangle = c_\uparrow \left|\uparrow\right\rangle_z + c_\downarrow \left|\downarrow\right\rangle_z\,$ ,

$\left|\psi\right\rangle = \left|\uparrow\right\rangle_x\,$ .

We then ask what is the probability that the state thusly prepared will yield a measurement of $S_z = -\tfrac{\hbar}{2}\,$ . This is

$P_\uparrow = |c_\uparrow|^2\,$ ,

where

$c_\uparrow = {}_z\left\langle\uparrow|\psi\right\rangle = {}_z\left\langle\uparrow|\uparrow\right\rangle_x\,$ .

At this point in our analysis, the relationship between the two bases is not known, but in principle it can be determined.

Pure state

A pure state is a state that contains in it all the information possible to predict the outcome of a measurement (at least in theory). In the example above, preparing an individual atom to yield a specific measurement of some observable quantity amounts to preparing in a pure state.

Mixed state

Suppose we have a number of atoms polarized in various different ways. Each atom exists in a pure state (for example $\left|\uparrow\right\rangle\,$ or $\left|\downarrow\right\rangle\,$ or some other state). In statistical mechanics such an arrangement is known as an ensemble. Furthermore, suppose that we are unable to distinguish between individual atoms until they reach our detector, either for technical reasons or because they are fundamentally indistinguishable. Our ignorance of the states of the individual atoms forces us to assign a fraction $p_{\psi_1}\,$ of atoms in the state $\left|\psi_1\right\rangle\,$ , a fraction $p_{\psi_2}\,$ of atoms in the state $\left|\psi_2\right\rangle\,$ , and so forth. While $p_{\psi_i}\,$ gives the probability of any particular atom to be in the state $\left|\psi_i\right\rangle\,$ , it is in general a very different sort of probability than the values $P_\uparrow\,$ and $P_\downarrow\,$ encountered earlier^[1]. We refer to the quantities necessary to describe an ensemble thusly prepared as a mixed state, which is usually expressed in terms of a density matrix. It turns out that there is a lack of uniqueness in preparing a mixed state to reproduce a certain set of experimental outcomes, and that the density matrix contains far less data than suggested here, but it contains just as much information.