I Quantum mechanics

The state of a quantum system is described by a vector $|\psi ⟩$. These vectors form a complex linear vector space, which entails, in particular, the following properties: Any state $|\psi ⟩$ can be scaled by any complex number $\alpha$, i.e., we can form new states $|\alpha \psi ⟩=\alpha |\psi ⟩$. Furthermore, any two states $|\psi ⟩$, $|\chi ⟩$ can be combined into new states by a forming a superposition $|\psi +\chi ⟩=|\psi ⟩+|\chi ⟩$.

The vector space is a Hilbert space, i.e., it is equipped with a scalar product that associates a complex number $⟨\psi |\chi ⟩$ to any pair of states $|\psi ⟩$, $|\chi ⟩$. The scalar product is positive definite, $⟨\psi |\psi ⟩>0$ for $|\psi ⟩\ne 0|\psi ⟩$, and fulfills $⟨\psi |\chi ⟩={⟨\chi |\psi ⟩}^{*}$. Furthermore, it is linear in the second argument, but conjugate linear in the first argument, i.e., $⟨\psi |\alpha \chi ⟩=\alpha ⟨\psi |\chi ⟩$, $⟨\alpha \psi |\chi ⟩={\alpha }^{*}⟨\psi |\chi ⟩$, $⟨\psi +\phi |\chi ⟩=⟨\psi |\chi ⟩+⟨\phi |\chi ⟩$, $⟨\psi |\phi +\chi ⟩=⟨\psi |\phi ⟩+⟨\psi |\chi ⟩$.

Formally, the scalar product can be interpreted as a product $⟨\psi |\cdot |\chi ⟩$ between the vectors $|\chi ⟩$ and the entities $⟨\psi |$, which form the dual vector space. They represent the left states in the scalar product and therefore are also conjugate linear: $⟨\alpha \psi +\beta \chi |={\alpha }^{*}⟨\psi |+{\beta }^{*}⟨\chi |$. The particular notation introduced here is the so-called Dirac notation. In this notation, a dual vector is also called a bra, and an ordinary vector is called a ket, alluding to the fact that in the scalar product $⟨\psi |\chi ⟩$ they form a bracket (bra-ket).

We call $||\psi ||=\sqrt{⟨\psi |\psi ⟩}$ the length of the vector $|\psi ⟩$. A vector with $⟨\psi |\psi ⟩=1$ is called normalized. The procedure of passing from a vector $|\psi ⟩$ to the normalized vector $|\psi ⟩/||\psi ||$ is called normalization. Two states $|\psi ⟩$, $|\chi ⟩$ fulfilling $⟨\psi |\chi ⟩=0$ are said to be orthogonal to each other.

A basis is a collection of vectors $|n⟩,n=1,2,3,\mathrm{\dots },𝒩$ such that any vector can be written as a superposition $|\psi ⟩={\sum }_{n=1}^{𝒩}{\psi }_n|n⟩$, where the complex coefficients ${\psi }_n$ are unique. The coefficients ${\psi }_n$ give a representation of the state, and can be written as a column vector

$$\psi =\left(\begin{array}{c}\hfill {\psi }_1\hfill \\ \hfill {\psi }_2\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill {\psi }_{𝒩}\hfill \end{array}\right).$$

The corresponding dual vector is written as a row vector ${\psi }^{†}=({\psi }_1^{*},{\psi }_2^{*},\mathrm{\dots },{\psi }_{𝒩}^{*})$. While there are many possible bases, in which the same vector is represented by different coefficients, the number $𝒩$ of basis states required to obtain all vectors is always the same, and is called the dimension of the vector space ($𝒩$ may be $\mathrm{\infty }$).

An orthogonal basis fulfills $⟨n|m⟩=0$ for any $n\ne m$. If furthermore $⟨n|n⟩=1$ for all $n$ one speaks of an orthonormal basis. In such a basis, the coefficients representing a state are given by ${\psi }_n=⟨n|\psi ⟩$, and the scalar product takes the form $⟨\psi |\chi ⟩={\sum }_n{\psi }_n^{*}{\chi }_n={\psi }^{†}\chi$.

An operator $\widehat{A}$ converts any state $|\psi ⟩$ into another state $|\widehat{A}\psi ⟩=\widehat{A}|\psi ⟩$. Linear operators fulfill $\widehat{A}(\alpha |\psi ⟩+\beta |\chi ⟩)=\alpha \widehat{A}|\psi ⟩+\beta \widehat{A}|\chi ⟩$. Operators can be added according to the rule $(\widehat{A}+\widehat{B})|\psi ⟩=\widehat{A}|\psi ⟩+\widehat{B}|\psi ⟩$, and multiplied according to the rule $(\widehat{B}\widehat{A})|\psi ⟩=\widehat{B}(\widehat{A}|\psi ⟩)$.

In an orthonormal basis, linear operators are represented by $N\times N$-dimensional square matrices

with coefficients $A_{nm}=⟨n|\widehat{A}m⟩\equiv ⟨n|\widehat{A}|m⟩$. They then act on vectors by matrix multiplication, i.e., $|\phi ⟩=\widehat{A}|\psi ⟩$ is represented by coefficients ${\phi }_n={\sum }_m{A}_{nm}{\psi }_m$. In a given representation, the operator addition and multiplication rules translate to the usual prescriptions of matrix addition and multiplication.

In Dirac notation, operators are written as $\widehat{A}={\sum }_{nm}{A}_{nm}|n⟩⟨m|$, and the action of an operator is obtained from the multiplication rule $⟨m|\cdot |\psi ⟩=⟨m|\psi ⟩$.

The action of an operator is particularly simple in its eigenrepresentation, defined by a basis fulfilling $\widehat{A}|n⟩=a_n|n⟩$. The numbers $a_n$ are called eigenvalues, and the vectors $|n⟩$ are called eigenvectors. If the eigenvectors form an orthonormal basis, the matrix $A_{nm}$ is diagonal, $A_{nm}=0$ if $n\ne m$ and $A_{nn}=a_n$. In Dirac notation, the operator can then be written as $\widehat{A}={\sum }_n{a}_n|n⟩⟨n|$.

A particularly simple operator is the identity operator $\widehat{I}$, which leaves all states unchanged, $\widehat{I}|\psi ⟩=|\psi ⟩$. Every state is therefore an eigenstate of $\widehat{I}$, with eigenvalue 1. Consequently, in any orthonormal basis this operator takes the same form $\widehat{I}={\sum }_n|n⟩⟨n|$. Representations are simply obtained by multiplying out the identities $|\psi ⟩=\widehat{I}|\psi ⟩$ and $\widehat{A}=\widehat{I}\widehat{A}\widehat{I}$. For a fixed orthonormal basis, it is useful to decompose the identity $\widehat{I}=\sum {\widehat{E}}_n$ as the sum of projection operators ${\widehat{E}}_n=|n⟩⟨n|$, which fulfill ${\widehat{E}}_n{\widehat{E}}_m$=0 if $n\ne m$, and ${\widehat{E}}_n^2={\widehat{E}}_n$.

For all operators we can define an adjoint operator ${\widehat{A}}^{†}$ by $⟨\psi |{\widehat{A}}^{†}\chi ⟩=⟨\widehat{A}\psi |\chi ⟩$. For many operators, we can also define an inverse operator ${\widehat{A}}^{-1}$ by $\widehat{A}{\widehat{A}}^{-1}=\widehat{I}$

Two important types of operators are hermitian operators $\widehat{H}$ and unitary operators $\widehat{U}$. For any two states $|\psi ⟩$, $|\chi ⟩$, hermitian operator fulfill $⟨\psi |\widehat{H}\chi ⟩=⟨\widehat{H}\psi |\chi ⟩$, while unitary operators fulfill $⟨\widehat{U}\psi |\widehat{U}\chi ⟩=⟨\psi |\chi ⟩$. This entails $\widehat{H}={\widehat{H}}^{†}$ and ${\widehat{U}}^{†}={\widehat{U}}^{-1}$. Both classes of operators have the nice property that their sets of normalized eigenvectors form an orthonormal basis. For hermitian operators, the eigenvalues $a_n$ are real, while for unitary operators they fulfill $|a_n|=1$.

The time evolution of quantum states is governed by the Schrödinger equation

$$i\mathrm{\hslash }\frac{d}{dt}|\psi (t)⟩=\widehat{H}(t)|\psi (t)⟩,$$

where $\widehat{H}$ is a hermitian operator called Hamiltonian. Given an initial state $|\psi (t_0)⟩$, the general solution can be written as $|\psi (t)⟩=\widehat{U}(t,t_0)|\psi (t_0)⟩$, where $\widehat{U}(t,t_0)$ is a unitary operator called the time evolution operator. This operator fulfills the Schrödinger equation $i\mathrm{\hslash }\frac{d}{dt}\widehat{U}(t,t_0)=\widehat{H}(t)\widehat{U}(t,t_0)$ with initial condition $\widehat{U}(t_0,t_0)=\widehat{I}$.

In the particular case $\widehat{H}=\mathrm{const}(t)$ of a time-independent Hamiltonian, the time evolution operator takes the form

$$\widehat{U}(t,t_0)=\exp [-i(t-t_0)\widehat{H}/\mathrm{\hslash }],$$

where the exponential function of an operator is defined as $\exp \widehat{A}={\sum }_{k=0}^{\mathrm{\infty }}{\widehat{A}}^k/k!$ . Using the eigenrepresentation $\widehat{H}={\sum }_n{E}_n|n⟩⟨n|$ of the Hamiltonian we can write $\widehat{U}(t,t_0)={\sum }_n\exp [-i(t-t_0)E_n/\mathrm{\hslash }]|n⟩⟨n|$.

Measurements deliver information about observable properties (observables) of quantum systems. Quantum mechanics associates to each observable $A$ a hermitian operator $\widehat{A}$. The eigenvalues $a_n$ of $\widehat{A}$ are the (only) possible outcomes of the experiment. Each outcome occurs with a probability given by

$$P(A=a_n)=|⟨n|\psi ⟩{|}^2=⟨\psi |n⟩⟨n|\psi ⟩=⟨\psi |{\widehat{E}}_n|\psi ⟩,$$

where $|n⟩$ is the eigenvector associated with $a_n$, and ${\widehat{E}}_n=|n⟩⟨n|$ is the associated projection operator. The average ${⟨A⟩}_{\psi }={\sum }_n{P}_n{a}_n$ of the outcome of many identical experiments, known as the expectation value of $A$, can be calculated directly from the quantum state using ${⟨A⟩}_{\psi }=⟨\psi |\widehat{A}|\psi ⟩$.

In the simplest case, a measurement with outcome $a_n$ transforms the state of the system into the eigenstate $|n⟩$. In general, however, the values of one observable alone do not suffice to uniquely determine the state of a system. Instead, a complete description requires to measure a larger set ${\widehat{A}}^{(l)}$ of simultaneous observables. Such observables commute with each other, $[{\widehat{A}}^{(l)},{\widehat{A}}^{(m)}]=0$, where $[\widehat{A},\widehat{B}]=\widehat{A}\widehat{B}-\widehat{B}\widehat{A}$ is the commutator. This property guarantees that one can find a joint eigenbasis, given by states $|{\psi }_{𝐚}⟩=|a^{(1)},a^{(2)},a^{(3)},\mathrm{\dots }⟩$ fulfilling ${\widehat{A}}^{(l)}|{\psi }_{𝐚}⟩=a^{(l)}|{\psi }_{𝐚}⟩$. These states are only fully specified by knowledge of the eigenvalues $a^{(l)}$ of the full set of simultaneous observables, which we here grouped into a vector $𝐚$ with components ${𝐚}_l=a^{(l)}$.

In this situation, a specific measurement outcome $a_n^{(l)}$ for a single observable ${\widehat{A}}^{(l)}$ delivers only incomplete information about the quantum system. In order to describe the effect of such a measurement, let us introduce the projection operator ${\widehat{E}}_n^{(l)}={\sum }_{{𝐚}_l=a_n^{(l)}}|{\psi }_{𝐚}⟩⟨{\psi }_{𝐚}|$ onto all states that share the given eigenvalue $a_n^{(l)}$. A measurement with outcome $a_n^{(l)}$ then occurs with probability $P_n=⟨\psi |{\widehat{E}}_n|\psi ⟩$, and transforms the quantum state into the (not yet normalized) state $|{\chi }_n⟩={\widehat{E}}_n|\psi ⟩$. The normalized post-measurement state is given by $|{\psi }_n⟩=\sqrt{1/P_n}|{\chi }_n⟩$. Since the other observables remain undetermined, such an incomplete measurement does not force the system into a unique final state.

An ensemble is a large collection of physically identical quantum systems, which however can be described by different states. When all the states are identical the ensemble is said to be pure, otherwise it is mixed. In general, we specify that a fraction $P_i$ of states is in state $|{\psi }_i⟩$, where ${\sum }_i{P}_i=1$ and $⟨{\psi }_i|{\psi }_i⟩=1$. Starting from a pure ensemble with all members in state $|\psi ⟩$, such a mixed ensemble is obtained, e.g., by measurement of an observable, with $P_i$ and $|{\psi }_i⟩$ obtained as described in the previous section. In the ensemble, expectation values are defined by $⟨A⟩={\sum }_i{P}_i⟨{\psi }_i|\widehat{A}|{\psi }_i⟩$.

By construction, a mixed ensemble cannot be described by a single quantum state. However, it is possible to define a statistical operator $\widehat{\rho }$, most commonly known as the density matrix, which allows to calculate all expectation values in a given mixed ensemble. This operator is given by

$$\widehat{\rho }=\sum _i{P}_i|{\psi }_i⟩⟨{\psi }_i|,$$

and the expectation values are obtained by

$${⟨A⟩}_{\rho }=\mathrm{tr}(\widehat{A}\widehat{\rho }).$$

Here, $\mathrm{tr}\widehat{B}$ denotes the trace of an operator, which in any given orthonormal basis can be calculated as $\mathrm{tr}\widehat{B}={\sum }_n⟨n|\widehat{B}n⟩={\sum }_n{B}_{nn}$.

Normalization of states carries over to the property $\mathrm{tr}\widehat{\rho }=1$. Moreover, the density matrix is hermitian and positive definite. This entails that in its eigenrepresentation $\widehat{\rho }={\sum }_n{p}_n|n⟩⟨n|$, all eigenvalues are nonnegative, $p_n>0$; they also sum up to unity, ${\sum }_n{p}_n=1$. (The nonvanishing eigenvalues $p_n$ are only identical to the values $P_i$ if the states $|{\psi }_i⟩$ used to define the ensemble are orthogonal to each other.)

For a pure ensemble, $p_n=1$ for one state, while all the other $p_m=0$ ($m\ne n$). In this case, $\widehat{\rho }=|n⟩⟨n|={\widehat{E}}_n$ is a projection operator, and therefore fulfills ${\widehat{\rho }}^2=\widehat{\rho }$. It follows that for a pure state $\mathrm{tr}{\widehat{\rho }}^2=1$. For a mixed state, however . The quantity $𝒫=\mathrm{tr}{\widehat{\rho }}^2$, also known as the purity, therefore easily distinguishes pure from mixed states. The maximally mixed state is described by the density matrix $\widehat{\rho }=\frac{1}{𝒩}\widehat{I}$ (where $𝒩$ is the Hilbert space dimension), and has purity $𝒫=1/𝒩$.

In a given representation, the density matrix of a pure state $|\psi ⟩$ can be obtained from $\rho =\psi {\psi }^{†}$, which is useful for specific calculations.

The time evolution of the density matrix follows from the Schrödinger equation, and is given by $\frac{d}{dt}\widehat{\rho }=\frac{i}{\mathrm{\hslash }}[\widehat{\rho },\widehat{H}]$. The general solution can be written as $\widehat{\rho }(t)=\widehat{U}(t,t_0)\widehat{\rho }(t_0){\widehat{U}}^{†}(t,t_0)$, where $\widehat{U}$ is the unitary time evolution operator defined in section I.3.

Given an orthonormal basis $|0⟩$, $|1⟩$ of a two-state system, each state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ is represented by a two-component vector $\left(\begin{array}{c}\hfill \alpha \hfill \\ \hfill \beta \hfill \end{array}\right)$. Each hermitian operator $\widehat{H}=a_0\widehat{I}+a_x\widehat{X}+a_y\widehat{Y}+a_z\widehat{Z}$ can be formed from four elementary operators with matrix representation

The matrices $X$, $Y$ and $Z$ are the Pauli matrices, most familiar from the description of the spin of an electron where they are often denoted as ${\sigma }_x$, ${\sigma }_y$ and ${\sigma }_z$, respectively. They fulfill $X^2=Y^2=Z^2=I$, $XY=-YX=iZ$, $YZ=-ZY=iX$, $ZX=-XZ=iY$.

It is useful to characterize the state of a two-state system by the vector of expectation values

$$\overrightarrow{P}=(⟨X⟩,⟨Y⟩,⟨Z⟩),$$

which is known as the polarization vector. For a normalized pure state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$,

$$\overrightarrow{P}=(2\mathrm{Re}{\alpha }^{*}\beta ,2\mathrm{Im}{\alpha }^{*}\beta ,{|\alpha |}^2-{|\beta |}^2)$$

is of unit length, and therefore lies on a sphere called the Bloch sphere. In terms of spherical polar coordinates on this sphere,

$$|\psi ⟩=\cos (\theta /2)|0⟩+e^{i\varphi }\sin (\theta /2)|1⟩.$$

The azimuthal angle $\varphi =\arg {\alpha }^{*}\beta$ of this vector is also known as the phase of the state. For a mixed state, so that the vector lies within the sphere. In terms of these expectation values, the density matrix can be written as

The purity of this density matrix is given by $𝒫=\frac{1}{2}(1+{|\overrightarrow{P}|}^2)$.

An important example where simultaneous observables occur are composite systems (say, a system composed of parts 1 and 2), where incomplete information can be acquired by measuring an observable of a subsystem (say, part 1). Starting from an orthonormal basis $|n⟩$ ($n=1,\mathrm{\dots },{𝒩}_1$) for system 1 and $|m⟩$ ($m=1,\mathrm{\dots },{𝒩}_2$) for system 2, the joint state $|\psi ⟩={\sum }_{nm}{\psi }_{nm}|nm⟩$ of the composite system can be written by using combined basis states $|nm⟩$, sometimes also written as $|n⟩|m⟩$ or $|n⟩\otimes |m⟩$. The corresponding dual basis vectors are denoted by $⟨nm|$. The Hilbert space dimension of the composite system is therefore given by $𝒩={𝒩}_1{𝒩}_2$. General operators can be written as $\widehat{A}={\sum }_{nmkl}{A}_{nk,ml}|nk⟩⟨ml|$. Operators acting on subsystem 1 will be denoted by ${\widehat{A}}_1$, and have representation ${\widehat{A}}_1={\sum }_{nmk}{A}_{nm}^{(1)}|nk⟩⟨mk|$. Operators acting on subsystem 2 will be denoted by ${\widehat{A}}_2$, and have representation ${\widehat{A}}_2={\sum }_{nkl}{A}_{kl}^{(2)}|nk⟩⟨nl|$. This results in the convenient block matrix form

where $I$ is the ${𝒩}_2\times {𝒩}_2$-dimensional identity matrix. Here, the basis states are ordered as $|1,1⟩,|1,2⟩,\mathrm{\dots },|1,{𝒩}_2⟩,|2,1⟩,|2,2⟩,\mathrm{\dots }$.

Sometimes, the state of a composite system can still be written as the product $|\phi ⟩|\chi ⟩$ of two states, where $|\phi ⟩$ describes system 1 and $|\chi ⟩$ describes system 2. Such states are called separable. This requires that the coefficients can be written as ${\psi }_{nm}={\phi }_n{\chi }_m$. States that are not separable are called entangled.

In order to determine whether states are separable or entangled, it is useful to consider measurements of observables of one subsystem, say system 1. When a state is separable, $|\psi ⟩=|\phi ⟩|\chi ⟩$, the outcome of such measurements only depends on $|\phi ⟩$. However, when the system is entangled, measurements on one subsystem cannot be described by a single state of that system. It is then still possible to describe these measurements by a density matrix

$${\widehat{\rho }}_1=\sum _{nmk}⟨nk|\psi ⟩⟨\psi |mk⟩|n⟩⟨m|,$$

known as the reduced density matrix. This means that all expectation values can be computed according to $⟨A⟩=\mathrm{tr}\widehat{A}{\widehat{\rho }}_1$. Analogously, measurements of the second subsystem are described by a reduced density matrix ${\widehat{\rho }}_2={\sum }_{nkl}⟨nk|\psi ⟩⟨\psi |nl⟩|k⟩⟨l|$. If a state $|\psi ⟩$ is separable, the reduced density matrices are pure, i.e., $\mathrm{tr}{\widehat{\rho }}_1^2=\mathrm{tr}{\widehat{\rho }}_2^2=1$. If the state $|\psi ⟩$ is entangled, the reduced density matrices are both mixed, i.e., .

Reduced density matrices can also be defined when the composite system is already in a mixed state, described by a density matrix $\widehat{\rho }$. They are then given by

$${\widehat{\rho }}_1=\sum _{nmk}⟨nk|\widehat{\rho }|mk⟩|n⟩⟨m|,{\rho }_2=\sum _{nkl}⟨nk|\widehat{\rho }|nl⟩|k⟩⟨l|.$$

These constructions are also called partial trace, and then written as ${\widehat{\rho }}_1={\mathrm{tr}}_2\widehat{\rho }$, ${\widehat{\rho }}_2={\mathrm{tr}}_1\widehat{\rho }$. This designation becomes clear when one considers the block form

of the density matrix in the composite basis, where ${\rho }_{nm}^{(2)}$ are ${𝒩}_2\times {𝒩}_2$-dimensional matrices. Then,

In this more general case of a composite system with a mixed density matrix, the purities of both reduced density matrices do not need to be identical, and cannot simply be used to decide whether the system is entangled or not; this is discussed in more detail below.

As an (important) example, consider the composition of two two-state systems. Pure states can be written as $|\psi ⟩=\alpha |00⟩+\beta |01⟩+\gamma |10⟩+\delta |11⟩$, and are normalized if ${|\alpha |}^2+{|\beta |}^2+{|\gamma |}^2+{|\delta |}^2=1$. The entanglement of such a state is often characterized by the concurrence

$$𝒞=2|\alpha \delta -\beta \gamma |,$$

which fulfills $0\le 𝒞\le 1$. For separable states, $𝒞=0$, i.e., the concurrences vanishes. For entangled states, $𝒞>0$. States with $𝒞=1$ are called maximally entangled. Examples of maximally entangled states are the four Bell states

	$|{\beta }_{00}⟩=\sqrt{{\displaystyle \frac{1}{2}}}(|00⟩+|11⟩),$
	$|{\beta }_{01}⟩=\sqrt{{\displaystyle \frac{1}{2}}}(|01⟩+|10⟩),$
	$|{\beta }_{10}⟩=\sqrt{{\displaystyle \frac{1}{2}}}(|00⟩-|11⟩),$
	$|{\beta }_{11}⟩=\sqrt{{\displaystyle \frac{1}{2}}}(|01⟩-|10⟩).$

For the pure state given above, the full density matrix

can be conveniently written in block form, where $A$, $B$, $C$, and $D$ are $2\times 2$-dimensional matrices. The reduced density matrix

can then be obtained by taking traces of the blocks, which here results in

Similarly,

The purity of these reduced density matrices is related to the concurrence,

$$\mathrm{tr}{\widehat{\rho }}_1^2=\mathrm{tr}{\widehat{\rho }}_2^2=1-{𝒞}^2/2.$$

Furthermore, we have the identity $\det {\widehat{\rho }}_1={𝒞}^2/4$.

For composite systems in a pure state, the reduced density matrix also delivers the entanglement of formation $ℰ=-\mathrm{tr}({\widehat{\rho }}_1{\log }_2{\widehat{\rho }}_1)=-\mathrm{tr}({\widehat{\rho }}_2{\log }_2{\widehat{\rho }}_2)$, which is identical to the von Neumann entropy of the mixed states of the subsystems (see section III.2).

In their form discussed above, these measures of entanglement only apply to pure states of a composite system. Entanglement measures for multi-component systems with a mixed density matrix are an active field of research. Well understood is only the case of two composite two-level systems, for which entanglement measures can be computed efficiently from the $4\times 4$ dimensional density matrix $\rho$ of the composite system in the standard basis. In order to obtain the concurrence, one needs to compute the four eigenvalues ${\lambda }_i$ of the matrix $\rho (Y_1{Y}_2){\rho }^{*}(Y_1{Y}_2)$, where $Y_1$ and $Y_2$ are the $Y$ Pauli matrix acting on subsystem 1 and 2, respectively. When the eigenvalues are ordered such that ${\lambda }_1>{\lambda }_2>{\lambda }_3>{\lambda }_4$, the concurrence is given as $𝒞=\max (0,\sqrt{{\lambda }_1}-\sqrt{{\lambda }_2}-\sqrt{{\lambda }_3}-\sqrt{{\lambda }_4})$. The entanglement of formation $ℰ={\min }_{\mathrm{dec}}{\sum }_i{P}_iℰ({\psi }_i)$ is generalized by minimizing the averaged pure-state entanglement of formation over all possible decompositions $\widehat{\rho }={\sum }_i{P}_i|{\psi }_i⟩⟨{\psi }_i|$ of the density matrix (where the states $|{\psi }_i⟩$ do not need to be orthogonal). Remarkably, both entanglement measures are related by the general formula $ℰ=h(\frac{1}{2}+\frac{1}{2}\sqrt{1-{𝒞}^2})$, where $h(x)=-x{\log }_{2}x-(1-x){\log }_2(1-x)$.

Entanglement is physically significant because it results in correlations that cannot be described by classical probabilities. These correlations can be uncovered by statistical tests, known as Bell inequalities. The most transparent inequality is the CHSH inequality due to Clauser, Horn, Shimony, and Holt. Consider the composition of two two-state systems; to be explicit, think of the spins of two electrons with basis states $|0⟩=|\uparrow ⟩$ and $|1⟩=|\downarrow ⟩$. On each spin we carry out two different experiments, described by observables ${\widehat{A}}_1$, ${\widehat{A}}_1^{\prime }$, ${\widehat{B}}_2$, and ${\widehat{B}}_2^{\prime }$, which measure whether the spin points into a particular direction. To the outcome of each experiment we designate the value $1$ or $-1$, depending on whether the spin is found to be aligned parallel or antiparallel to the measurement direction, respectively. Now consider the expectation value of

$$\widehat{F}=({\widehat{A}}_1+{\widehat{A}}_1^{\prime }){\widehat{B}}_2-({\widehat{A}}_1-{\widehat{A}}_1^{\prime }){\widehat{B}}_2^{\prime }.$$

Classically, for each combination of outcomes, $F$ is either $2$ or $-2$, and therefore on average $⟨F⟩\le 2$, which is the CHSH inequality. Quantum-mechanically, the average is obtained by an expectation value. Let us choose ${\widehat{A}}_1=Z$, ${\widehat{A}}_1^{\prime }=X$, ${\widehat{B}}_2=-\sqrt{1/2}(X+Z)$, and ${\widehat{B}}_2^{\prime }=\sqrt{1/2}(Z-X)$, so that $\widehat{F}=-\sqrt{2}(X_1{X}_2+Z_1{Z}_2)$, which is represented by the matrix

Furthermore, assume that the system is in the Bell state $|{\beta }_{11}⟩=\sqrt{1/2}(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)$, represented by the vector

$$\psi =\sqrt{1/2}\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \\ \hfill 0\hfill \end{array}\right).$$

We then find $⟨\widehat{F}⟩={\psi }^{†}F\psi =2\sqrt{2}$, which violates the CHSH inequality. The reason are quantum-mechanical correlations that arise as a consequence of the entanglement of the Bell state. Quantum computation taps into this resource to achieve tasks that are classically impossible.