Quantum Mechanics — Lecture notes for PHYS223 XIX Dynamics for time-dependent Hamiltonians XXI Many particles

XX Dirac notation for composite systems

In preparation for the concluding chapter of this course, we now extend in this the Dirac notation to systems with more than one particle or several degrees of freedom, and also introduce a generalised notion of a state (the density matrix).

XX.1 Many degrees of freedom

Simultaneous observables appear in particular when a particle has many degrees of freedom, which may be continuous (like the three components of the position vector ${\bf r}$ , discrete (like spin), or both (as for a single electron, which possess all these degrees freedoms). In ordinary notation, the state of the electron is described by a position-dependent spinor wavefunction

\psi=\left(\begin{array}[]{c}\psi^{\uparrow}({\bf r})\\ \psi^{\downarrow}({\bf r})\\ \end{array}\right),

(445)

where $|\psi^{\uparrow}({\bf r})|^{2}$ is the probability density to find the electron at position ${\bf r}$ with spin $S_{z}=\hbar/2$ (spin up), while $|\psi^{\downarrow}({\bf r})|^{2}$ is the probability density to find the electron at position ${\bf r}$ with spin $S_{z}=-\hbar/2$ (spin down).

In Dirac notation, we can write these states in terms of a basis $|{\bf r},S_{z}\rangle$ , such that

|\psi\rangle=\int d{\bf r}\,\psi^{\uparrow}({\bf r})|{\bf r},\hbar/2\rangle+% \int d{\bf r}\,\psi^{\downarrow}({\bf r})|{\bf r},-\hbar/2\rangle.

(446)

XX.1.1 Factorisation and entanglement

For the special case

|\psi\rangle=\int d{\bf r}\,\psi({\bf r})(\psi^{\uparrow}|{\bf r},\hbar/2% \rangle+\psi^{\downarrow}|{\bf r},-\hbar/2\rangle),

(447)

where the amplitudes $\psi^{\uparrow}$ and $\psi^{\downarrow}$ do not depend on position, one says that the spinor wavefunction factorises into the orbital wavefunction $\psi({\bf r})$ and the spinor ${\psi^{\uparrow}\choose\psi^{\downarrow}}$ . In the case that the wavefunction does not factorise one says that the spin and the position of the electron are entangled.

As a matter of fact, we encountered entanglement already in the context of the Stern Gerlach experiment. Ignoring the individual degrees of freedom, we can describe a silver atom by a spinor wavefunction (445), where ${\bf r}$ is the centre of mass of the atom and the spin is the total spin of the atom (in its ground state, the total spin of a silver atom is $\hbar/2$ , just as for a single electron). In a Stern-Gerlach experiment, the wavefunction factorises before an atom enters the apparatus, since the path on which the atoms move from the oven towards the apparatus is independent of their spin state. Behind the apparatus, however, spin and position of the atoms have become entangled: The position at which an atom is collected on the screen depends on its spin (two spots are observed, where one corresponds to atoms with $S_{z}=\hbar/2$ and the other corresponds to $S_{z}=-\hbar/2$ ). This entanglement is created inside the apparatus, through the spin- and position-dependent interaction with the magnetic field.

XX.2 Density matrix ${}^{*}$

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 $|\psi_{i}\rangle$ , where $\sum_{i}P_{i}=1$ and $\langle\psi_{i}|\psi_{i}\rangle=1$ . Starting from a pure ensemble with all members in state $|\psi\rangle$ , such a mixed ensemble is obtained, e.g., by measurement of an observable, with $P_{i}$ and $|\psi_{i}\rangle$ obtained as described in the previous section. In the ensemble, expectation values are defined by $\langle A\rangle=\sum_{i}P_{i}\langle\psi_{i}|\hat{A}|\psi_{i}\rangle$ .

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

\hat{\rho}=\sum_{i}P_{i}|\psi_{i}\rangle\langle\psi_{i}|,

(448)

and the expectation values are obtained by

\langle A\rangle_{\rho}={\rm tr}\,(\hat{A}\hat{\rho}).

(449)

Here, ${\rm tr}\,\hat{B}$ denotes the trace of an operator, which in any given orthonormal basis can be calculated as ${\rm tr}\,\hat{B}=\sum_{n}\langle n|\hat{B}n\rangle=\sum_{n}B_{nn}$ .

Normalisation of states carries over to the property ${\rm tr}\,\hat{\rho}=1$ . Moreover, the density matrix is hermitian and positive definite. This entails that in its eigenrepresentation $\hat{\rho}=\sum_{n}p_{n}|n\rangle\langle n|$ , 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 $|\psi_{i}\rangle$ used to define the ensemble are orthogonal to each other.)

For a pure ensemble, $p_{n}=1$ for one state, while all the other $p_{m}=0$ ( $m\neq n$ ). In this case, $\hat{\rho}=|n\rangle\langle n|=\hat{E}_{n}$ is a projection operator, and therefore fulfills $\hat{\rho}^{2}=\hat{\rho}$ . It follows that for a pure state ${\rm tr}\,\hat{\rho}^{2}=1$ . For a mixed state, however ${\rm tr}\,\hat{\rho}^{2}=\sum_{n}p_{n}^{2}<1$ . The quantity ${\cal P}={\rm tr}\,\hat{\rho}^{2}$ , also known as the purity, therefore easily distinguishes pure from mixed states. The maximally mixed state is described by the density matrix $\hat{\rho}=\frac{1}{{\cal N}}\hat{I}$ (where ${\cal N}$ is the Hilbert space dimension), and has purity ${\cal P}=1/{\cal N}$ .

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

The time evolution of the density matrix follows from the Schrödinger equation, and is given by $\frac{d}{dt}\hat{\rho}=\frac{i}{\hbar}[\hat{\rho},\hat{H}]$ . The general solution can be written as $\hat{\rho}(t)=\hat{U}(t,t_{0})\hat{\rho}(t_{0})\hat{U}^{\dagger}(t,t_{0})$ , where $\hat{U}$ is the unitary time evolution operator.

XX.3 Example: Two-state systems ${}^{*}$

Nature provides us with a wealth of quantum systems for which only two states are important. Besides spin, further examples are atomic transitions between the ground state and a selected excited state, the tunnelling of a particle between the almost-degenerate lowest-energy states of a double well, or two the orthogonal linear polarisation states of a photons. The quantum system can then be described by a spinor wavefunction ${\alpha\choose\beta}$ , where the amplitudes $\alpha$ and $\beta$ are now associated, e.g., with the occupation probability of the ground or excited state in the atom, or describe whether the particle is located in the right or left trough of the double well.

In the Dirac notation, given an orthonormal basis $|0\rangle$ , $|1\rangle$ the state of the system can be written as

|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,

(450)

while the density matrix is of the general form

\hat{\rho}=\rho_{00}|0\rangle\langle 0|+\rho_{01}|0\rangle\langle 1|+\rho_{10}% |1\rangle\langle 0|+\rho_{11}|1\rangle\langle 1|.

(451)

It is again useful to characterize the state (be it mixed or pure) by the polarisation vector

\vec{P}=(\langle X\rangle,\langle Y\rangle,\langle Z\rangle).

(452)

For a normalised pure state,

\vec{P}=(2\,{\rm Re}\,\alpha^{*}\beta,2\,{\rm Im}\alpha^{*}\beta,|\alpha|^{2}-% |\beta|^{2})

(453)

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

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

(454)

For a mixed state, $|\vec{P}|<1$ so that the vector lies within the sphere. In terms of these expectation values, the components of the density matrix can be written as

\rho=\frac{1}{2}\left(\begin{array}[]{cc}1+P_{z}&P_{x}-iP_{y}\\ P_{x}+iP_{y}&1-P_{z}\end{array}\right)=\frac{1}{2}(I+P_{x}X+P_{y}Y+P_{z}Z).

(455)

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

XX.4 Composite systems ${}^{*}$

Another important example where simultaneous observables occur are composite systems (say, a system composed of distinguishable 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 $|n\rangle$ ( $n=1,\ldots,{\cal N}_{1}$ ) for system 1 and $|m\rangle$ ( $m=1,\ldots,{\cal N}_{2}$ ) for system 2, the joint state $|\psi\rangle=\sum_{nm}\psi_{nm}|nm\rangle$ of the composite system can be written by using combined basis states $|nm\rangle$ , sometimes also written as $|n\rangle|m\rangle$ or $|n\rangle\otimes|m\rangle$ . The corresponding dual basis vectors are denoted by $\langle nm|$ . The Hilbert space dimension of the composite system is therefore given by ${\cal N}={\cal N}_{1}{\cal N}_{2}$ . General operators can be written as $\hat{A}=\sum_{nmkl}A_{nk,ml}|nk\rangle\langle ml|$ . Operators acting on subsystem 1 will be denoted by $\hat{A}_{1}$ , and have representation $\hat{A}_{1}=\sum_{nmk}A^{(1)}_{nm}|nk\rangle\langle mk|$ . Operators acting on subsystem 2 will be denoted by $\hat{A}_{2}$ , and have representation $\hat{A}_{2}=\sum_{nkl}A^{(2)}_{kl}|nk\rangle\langle nl|$ . This results in the convenient block matrix form

A_{1}=\left(\begin{array}[]{ccc}A^{(1)}_{11}I&A^{(1)}_{12}I&\cdots\\ A^{(1)}_{21}I&A^{(1)}_{22}I&\cdots\\ \vdots&\vdots&\ddots\\ \end{array}\right),\quad A_{2}=\left(\begin{array}[]{ccc}A^{(2)}&0&\cdots\\ 0&A^{(2)}&\cdots\\ \vdots&\vdots&\ddots\\ \end{array}\right),

(456)

where $I$ is the ${\cal N}_{2}\times{\cal N}_{2}$ -dimensional identity matrix. Here, the basis states are ordered as $|1,1\rangle,|1,2\rangle,\ldots,|1,{\cal N}_{2}\rangle,|2,1\rangle,|2,2\rangle,\ldots$ .

XX.5 Entanglement in composite systems ${}^{*}$

Sometimes, the state of a composite system can still be written as the product $|\varphi\rangle|\chi\rangle$ of two states, where $|\varphi\rangle$ describes system 1 and $|\chi\rangle$ describes system 2. Such states are called separable. This requires that the coefficients can be written as $\psi_{nm}=\varphi_{n}\chi_{m}$ . States that are not separable are called entangled.

XX.5.1 Reduced density matrix

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, $|\psi\rangle=|\varphi\rangle|\chi\rangle$ , the outcome of such measurements only depends on $|\varphi\rangle$ . 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

\hat{\rho}_{1}=\sum_{nmk}\langle nk|\psi\rangle\langle\psi|mk\rangle|n\rangle% \langle m|,

(457)

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

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

\hat{\rho}_{1}=\sum_{nmk}\langle nk|\hat{\rho}|mk\rangle|n\rangle\langle m|,% \quad\rho_{2}=\sum_{nkl}\langle nk|\hat{\rho}|nl\rangle|k\rangle\langle l|.

(458)

These constructions are also called partial trace, and then written as $\hat{\rho}_{1}={\rm tr}_{2}\,\hat{\rho}$ , $\hat{\rho}_{2}={\rm tr}_{1}\,\hat{\rho}$ . This designation becomes clear when one considers the block form

\rho=\left(\begin{array}[]{ccc}\rho^{(2)}_{11}&\rho^{(2)}_{12}&\cdots\\ \rho^{(2)}_{21}&\rho^{(2)}_{22}&\cdots\\ \vdots&\vdots&\ddots\\ \end{array}\right)

(459)

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

\rho_{1}=\left(\begin{array}[]{ccc}{\rm tr}\,\rho^{(2)}_{11}&{\rm tr}\,\rho^{(% 2)}_{12}&\cdots\\ {\rm tr}\,\rho^{(2)}_{21}&{\rm tr}\,\rho^{(2)}_{22}&\cdots\\ \vdots&\vdots&\ddots\\ \end{array}\right),\quad\rho_{2}=\sum_{n}\rho^{(2)}_{nn}.

(460)

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.

XX.5.2 Bell pairs and entanglement

As an example of a composite system, consider two two-state systems. Pure states can be written as $|\psi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle$ , and are normalised if $|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}+|\delta|^{2}=1$ . An particularly interesting state is given by

|\psi\rangle=\frac{1}{\sqrt{2}}|01\rangle+\frac{1}{\sqrt{2}}|10\rangle.

(461)

This state is entangled: if the state of the first part of the system is determined through a measurement, one knows that the other part is in the opposite state, even though one has not carried out a measurement on this part.

In the general case, the entanglement of a state $|\psi\rangle$ is often characterised by the concurrence

{\cal C}=2|\alpha\delta-\beta\gamma|,

(462)

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

	$\displaystyle\|\beta_{00}\rangle=\sqrt{\frac{1}{2}}(\|00\rangle+\|11\rangle),$		(463)
	$\displaystyle\|\beta_{01}\rangle=\sqrt{\frac{1}{2}}(\|01\rangle+\|10\rangle),$		(464)
	$\displaystyle\|\beta_{10}\rangle=\sqrt{\frac{1}{2}}(\|00\rangle-\|11\rangle),$		(465)
	$\displaystyle\|\beta_{11}\rangle=\sqrt{\frac{1}{2}}(\|01\rangle-\|10\rangle).$		(466)

For the pure state given above, the full density matrix

\rho=\left(\begin{array}[]{c}\alpha\\ \beta\\ \gamma\\ \delta\end{array}\right)(\alpha^{*},\beta^{*},\gamma^{*},\delta^{*})=\left(% \begin{array}[]{cc}A&B\\ C&D\\ \end{array}\right)

(467)

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

\rho_{1}=\left(\begin{array}[]{cc}{\rm tr}\,A&{\rm tr}\,B\\ {\rm tr}\,C&{\rm tr}\,D\\ \end{array}\right)

(468)

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

\rho_{1}=\left(\begin{array}[]{cc}|\alpha|^{2}+|\beta|^{2}&\alpha\gamma^{*}+% \beta\delta^{*}\\ \gamma\alpha^{*}+\delta\beta^{*}&|\gamma|^{2}+|\delta|^{2}\\ \end{array}\right).

(469)

Similarly,

\rho_{2}=A+D=\left(\begin{array}[]{cc}|\alpha|^{2}+|\gamma|^{2}&\alpha\beta^{*% }+\gamma\delta^{*}\\ \beta\alpha^{*}+\delta\gamma^{*}&|\beta|^{2}+|\delta|^{2}\\ \end{array}\right).

(470)

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

{\rm tr}\,\hat{\rho}_{1}^{2}={\rm tr}\,\hat{\rho}_{2}^{2}=1-{\cal C}^{2}/2.

(471)

Furthermore, we have the identity ${\rm det}\,\hat{\rho}_{1}={\cal C}^{2}/4$ .

For composite systems in a pure state, the reduced density matrix also delivers the entanglement of formation ${\cal E}=-{\rm tr}\,(\hat{\rho}_{1}\log_{2}\hat{\rho}_{1})=-{\rm tr}\,(\hat{% \rho}_{2}\log_{2}\hat{\rho}_{2})$ .

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 ${\cal C}=\rm{max}\,(0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}% -\sqrt{\lambda_{4}})$ . The entanglement of formation ${\cal E}={\rm min}_{\rm dec}\sum_{i}P_{i}{\cal E}(\psi_{i})$ is generalised by minimizing the averaged pure-state entanglement of formation over all possible decompositions $\hat{\rho}=\sum_{i}P_{i}|\psi_{i}\rangle\langle\psi_{i}|$ of the density matrix (where the states $|\psi_{i}\rangle$ do not need to be orthogonal). Remarkably, both entanglement measures are related by the general formula ${\cal E}=h(\frac{1}{2}+\frac{1}{2}\sqrt{1-{\cal C}^{2}})$ , where $h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$ .

XX.5.3 Einstein, Podolsky, and Rosen

Here are some interesting thoughts about Bell pairs, raised by Einstein, Podolsky, and Rosen at a time (1935) when quantum mechanics was not yet totally accepted: When one measures a part of a system in a Bell pair, one gets a random result $0$ or $1$ with probability $1/2$ . However, when one determined the state of part one of the system and communicated the result to the location of part 2, one would be able to predict the outcome of the measurement of the second part with certainty. This influence of the one experiment on the other was termed the ’ghostly action at a distance’.

Maybe the randomness of quantum mechanics is always of this kind, namely encoded in other degrees of freedom (so-called hidden variables) and all randomness would disappear when one would account for the state of these hidden variables? In 1965, John Bell showed that quantum mechanics predicts correlations between the two two-state systems which cannot be explained by a local hidden variable theory (we discuss the details of these considerations below). In the early 1980’s, the existence of these correlations have been tested and confirmed by Alain Aspect and co-workers for the polarisation states of entangled photons.

These experiments are considered to be a proof of Bell’s theorem: No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. Hence, the uncertainties and randomness in the outcome of experiments in quantum mechanics are fundamentally different than mere statistical uncertainties in classical theories (e.g., in classical thermodynamics the statistical uncertainties arise from the uncontrolled microsocopic motion of the particles, which may still be deterministic, i.e., based on Newton’s laws of motion).

XX.5.4 Bell inequalities

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 $|0\rangle=|\!\!\uparrow\rangle$ and $|1\rangle=|\!\!\downarrow\rangle$ . On each spin we carry out two different experiments, described by observables $\hat{A}_{1}$ , $\hat{A}_{1}^{\prime}$ , $\hat{B}_{2}$ , and $\hat{B}_{2}^{\prime}$ , 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

\hat{F}=(\hat{A}_{1}+\hat{A}_{1}^{\prime})\hat{B}_{2}-(\hat{A}_{1}-\hat{A}_{1}% ^{\prime})\hat{B}_{2}^{\prime}.

(472)

Classically, for each combination of outcomes, $F$ is either $2$ or $-2$ , and therefore on average $\langle F\rangle\leq 2$ , which is the CHSH inequality. Quantum-mechanically, the average is obtained by an expectation value. Let us choose $\hat{A}_{1}=Z$ , $\hat{A}_{1}^{\prime}=X$ , $\hat{B}_{2}=-\sqrt{1/2}(X+Z)$ , and $\hat{B}_{2}^{\prime}=\sqrt{1/2}(X-Z)$ , so that $\hat{F}=-\sqrt{2}(X_{1}X_{2}+Z_{1}Z_{2})$ , which is represented by the matrix

F=-\sqrt{2}\left(\begin{array}[]{cccc}1&0&0&1\\ 0&-1&1&0\\ 0&1&-1&0\\ 1&0&0&1\\ \end{array}\right).

(473)

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

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

(474)

We then find $\langle\hat{F}\rangle=\psi^{\dagger}F\psi=2\sqrt{2}$ , 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.