Hilbert spaces and operators

8. Hilbert spaces and operators#

Recommended reference: Griffiths [Gri95], in particular Section 3.1.

So far, we have been considering vectors in an \(n\)-dimensional real or complex space. In quantum mechanics, one encounters infinite-dimensional vector spaces as well. In particular, the state of a quantum system is represented mathematically by a (unit) vector in a Hilbert space.

8.1. Vector spaces#

As is customary in quantum physics and quantum algorithms, we use bra-ket notation for vectors in Hilbert spaces.

Definition 8.1 (Vector space)

A (real or complex) vector space consists of

  • a set of vectors, denoted by symbols like \(\ket\alpha\)

  • a distinguished vector \(\mathbf{0}\) called the zero vector[1]

  • a way of adding vectors, i.e. given two vectors \(\ket\alpha\) and \(\ket\beta\) we can obtain a third vector \(\ket\alpha+\ket\beta\)

  • a way of multiplying a vector by a scalar: given a scalar \(c\) (a real or complex number, depending on whether we consider a real or complex vector space) and a vector \(\ket\alpha\) we can obtain a vector \(c\ket\alpha\)

such that a number of properties hold:

  • \(0\ket\alpha=\mathbf{0}\)

  • \(1\ket\alpha=\ket\alpha\)

  • \(\mathbf{0}+\ket\alpha=\ket\alpha\)

  • \(\ket\alpha+\ket\beta=\ket\beta+\ket\alpha\)

  • \((\ket\alpha+\ket\beta)+\ket\gamma=\ket\alpha+(\ket\beta+\ket\gamma)\)

  • \(c(\ket\alpha+\ket\beta)=c\ket\alpha+c\ket\beta\)

  • \((c+d)\ket\alpha = c\ket\alpha + d\ket\alpha\)

  • \(c(d\ket\alpha)=(cd)\ket\alpha\)

Definition 8.2 (Basis, dimension)

A basis for a vector space is a collection of vectors \((\ket{e_i})_i\) such that every vector can be written in exactly one way as a linear combination of the \(\ket{e_i}\). The number of vectors in a basis is called the dimension of the space.

As in Basis transformations, any two bases contain the same number of vectors, so the dimension does not depend on the choice of bases.

Example 8.1

The simplest examples are the \(n\)-dimensional vector spaces \(\RR^n\) and \(\CC^n\) with coordinatewise addition and scalar multiplication.

An important special case is the two-dimensional space \(\CC^2\). In bra-ket notation, various symbols are used for the standard basis vectors \(\binom{1}{0}\) and \(\binom{0}{1}\) in this space, such as

  • \(\ket+\) and \(\ket-\);

  • \(\ket0\) and \(\ket1\) (in the context of quantum computing);

  • \(\ket\uparrow\) and \(\ket\downarrow\) (in the context of spins).

8.2. Hermitian inner products and Hilbert spaces#

Definition 8.3 (Hermitian inner product)

A Hermitian inner product assigns to vectors \(\ket\alpha\) and \(\ket\beta\) a complex number \(\braket{\alpha|\beta}\) such that the following properties hold:

  • \(\braket{\alpha|\beta}=\overline{\braket{\beta|\alpha}}\);

  • \(\braket{\alpha|\alpha}\ge0\), and \(\braket{\alpha|\alpha}=0\) holds precisely when \(\ket\alpha=\mathbf{0}\) (note that \(\braket{\alpha|\alpha}\) is a real number because of the first property);

  • \(\bra\alpha(\ket\beta+\ket\gamma) = \braket{\alpha|\beta} + \braket{\alpha|\gamma}\);

  • \(\bra\alpha(c\ket\beta)=c(\braket{\alpha|\beta})\);

Definition 8.4 (Norm of a vector)

The norm or length of a vector \(\ket\alpha\) in a Hilbert space is defined as the real number

\[ \|\alpha\|=\sqrt{\braket{\alpha|\alpha}}. \]

Note

The properties above imply that for any scalar \(c\), the inner product of \(c\ket{\alpha}\) with \(\ket\beta\) equals \(\bar c\braket{\alpha|\beta}\). In other words, while the inner product is (complex) linear in its second argument, it is only conjugate linear in its first argument. This is the usual convention in physics. It differs from the convention in mathematics, where inner products are usually linear in the first argument and conjugate linear in the second argument.

Definition 8.5 (Hilbert space)

A Hilbert space is a complex vector space \(\HH\) equipped with a Hermitian inner product \(\braket{\enspace|\enspace}\) and satisfying a condition called completeness.

We will not make the notion of completeness precise in these notes. Roughly speaking, it means that we do not have to worry too much about convergence of sequences or series in our Hilbert space, and there is a notion of infinite linear combinations.

Example 8.2

We take \(\HH=\CC^n\) and define an inner product by

\[ \braket{\alpha|\beta} = \sum_{i=1}^n\bar\alpha_i\beta_i. \]

Check for yourself that this satisfies all the properties of a Hermitian inner product.

Example 8.3

Consider the space \(L^2(\RR)\) of square-integrable functions on the real line, i.e.\ functions \(f\colon\RR\to\CC\) such that the integral \(\int_{-\infty}^\infty|f(x)|^2 dx\) exists and is finite. This is a Hilbert space with inner product

\[ \braket{f|g} = \int_{-\infty}^\infty \bar f(x)g(x)dx. \]

Example 8.4

Consider the space \(\ell^2\) of sequences \(z=(z_0,z_1,\ldots)\) such that

\[ \sum_{n=0}^\infty |z_n|^2<\infty. \]

Then \(\ell^2\) is a Hilbert space with inner product

\[ \braket{w|z}=\sum_{n=0}^\infty \bar w_n z_n. \]

8.3. The dual Hilbert space#

With any Hilbert space \(\HH\), we can associate a dual Hilbert space \(\HH^*\). In the bra-ket formalism used in quantum mechanics, for every “ket vector” \(\ket\alpha\) in \(\HH\) we have a dual “bra vector” \(\bra\alpha\) in \(\HH^*\). This correspondence has the following properties:

vector in \(\HH\)

corresponding vector in \(\HH^*\)

ket vector \(\ket\alpha\)

bra vector \(\bra\alpha\)

\(\ket\alpha+\ket\beta\)

\(\bra\alpha+\bra\beta\)

\(c\ket\alpha\)

\(\bar c\bra\alpha\)

In mathematical language, \(\HH^*\) is the space of continuous linear maps \(\HH\to\CC\). The bra vector \(\bra\alpha\) then represents the linear map that sends \(\ket\beta\) to \(\braket{\alpha|\beta}\).

Example 8.5 (The dual of \(\CC^n\))

In the case \(\HH=\CC^n\) we can identify \(\HH^*\) with \(\CC^n\) as well; a vector \(\ket\alpha\) in \(\CC^n\) is then simply the same vector. However, it can also be convenient to distinguish \(\HH\) from \(\HH^*\) by denoting a ket vector \(\ket\alpha\) as a column vector and the corresponding bra vector \(\bra\alpha\) by the conjugate transpose of \(\ket\alpha\), so \(\bra\alpha\) is the row vector whose entries are the complex conjugates of those of \(\alpha\). In \(\CC^2\), for example, we have

\[\begin{split} \ket\alpha=\begin{pmatrix}1\\ i\end{pmatrix}\Longrightarrow \bra\alpha=\begin{pmatrix}1& -i\end{pmatrix}. \end{split}\]

This allows us to view \(\bra\alpha\) as a linear map sending \(\ket\beta\) to the scalar obtained by multiplying the row vector \(\bra\alpha\) by the column vector \(\ket\beta\).

Note

If you know about continuous linear maps of Hilbert spaces, here is a useful connection between the mathematical definition of \(\HH^*\) and the bra-ket formalism: the Riesz representation theorem states that every continuous linear map \(\HH\to\CC\) is of the form \(\braket{\alpha|\enspace}\) for some \(\alpha\) in \(\HH\).

8.4. Orthonormal bases#

Definition 8.6 (Orthonormal basis)

An orthonormal basis of a Hilbert space \(\HH\) is a collection of vectors \((\ket{e_i})_{i\in I}\), where \(I\) is some index set, such that

  1. the system is orthonormal: \(\braket{e_i|e_j} = \begin{cases} 1&\quad\text{if }i=j,\\ 0&\quad\text{if }i\ne j;\end{cases}\)

  2. the system spans \(\HH\): every vector in \(\HH\) can be written as a (possibly infinite) linear combination of the \(\ket{e_i}\).

Example 8.6

Consider the space \(L^2([0,1])\) of square-integrable functions on the unit interval. An orthonormal basis for this space consists of the functions \(e_n(x)=\exp(2\pi i n x)\) for all integers \(n\). This reflects the fact that every “nice” function on \([0,1]\) can be expressed as a Fourier series.

8.5. Operators#

An operator or linear map on a Hilbert space \(\HH\) is a map

\[ T\colon\HH\to\HH \]

such that

  • \(T(c\ket\alpha) = c(T\ket\alpha)\)

  • \(T(\ket\alpha+\ket\beta) = T\ket\alpha+T\ket\beta\)

Note

It is also possible to define linear maps between different Hilbert spaces, but we will not go into this.

Note

In physics, operators are often denoted with a hat, so what we call \(T\) would be written as \(\hat T\). One reason for this is to distinguish a physical quantity (like momentum or energy) from the corresponding mathematical operator. Since we do not consider physical quantities in this module, we do not use the hat notation.

Operators form a vector space: the sum of two operators \(T\) and \(U\) is defined by the formula

\[ (T+U)\ket\alpha=T\ket\alpha+U\ket\alpha \]

and multiplication of an operator \(T\) by a scalar \(c\) is defined by the formula

\[ (cT)\ket\alpha=c(T\ket\alpha). \]

Like matrices (but unlike vectors), operators can be composed: if \(T\) and \(U\) are two operators, then \(TU\) is the operator defined by

\[ (TU)\ket\alpha=T(U\ket\alpha). \]

8.6. Matrix entries#

In physics, quantities of the form

\[ \bra\alpha T\ket\beta \]

are often of special interest. These are called matrix entries of \(T\), especially if \(\ket\alpha\) and \(\ket\beta\) are basis vectors in some fixed orthonormal basis of \(\HH\).

Example 8.7

Consider a linear map \(A\) on \(\CC^n\), viewed as a matrix. Taking the matrix entries as above, letting \(\alpha\) and \(\beta\) range over the standard basis vectors, we obtain the usual matrix entries of \(A\); see Exercise 8.2.

Note

A remark on notation: you will often see notation like \(\braket{\alpha|T\beta}\). This is literally meaningless: the operator \(T\) can be applied to vectors, which we denote by \(\ket\beta\), while the symbol \(\beta\) is just a label. However, in practice it is very convenient and completely unambiguous to write \(\braket{\alpha|T\beta}\) instead of the strictly speaking more correct \(\bra\alpha(T\ket\beta)\).

Similarly, we will write \(\braket{T\alpha|\beta}\) to mean the inner product of \(T\ket\alpha\) with \(\ket\beta\). Extending this notation to scalars \(c\), we can also write

\[ \braket{c\alpha|\beta}=\overline{\braket{\beta|c\alpha}} =\bar c\overline{\braket{\beta|\alpha}} =\bar c\braket{\alpha|\beta}. \]

8.7. The adjoint of an operator#

Given an operator \(T\) on a Hilbert space \(\HH\), there exists an operator \(T^\dagger\) with the property

\[ \braket{T^\dagger\alpha|\beta}=\braket{\alpha|T\beta} \quad\text{for all }\ket\alpha,\ket\beta\in\HH. \]

The operator \(T^\dagger\) is called the adjoint of \(T\).

Property 8.1 (Properties of the adjoint)

For all operators \(T\), \(U\) and scalars \(c\), we have

  1. \((cT)^\dagger = \bar c T^\dagger\)

  2. \((TU)^\dagger = U^\dagger T^\dagger\)

  3. \((T^\dagger)^\dagger = T\)

Example 8.8

Take \(\HH=\CC^n\) with the standard inner product. For a matrix \(A\), we write \(A^\dagger\) for the conjugate transpose of \(A\). We extend this to vectors by viewing column vectors as \(n\times 1\)-matrices and row vectors as \(1\times n\)-matrices. For all \(\ket\alpha,\ket\beta\in\HH\) we then have

\[ \braket{\alpha|A|\beta}=\ket{\alpha}^\dagger A\ket\beta =(A^\dagger\ket{\alpha})^\dagger\ket\beta =\braket{A^\dagger\alpha|\beta}. \]

This shows that the adjoint of the operator \(A\) corresponds to the conjugate transpose of \(A\) viewed as a matrix.

8.8. Hermitian operators#

Definition 8.7 (Hermitian operator)

An operator \(T\) on a Hilbert space \(\HH\) is called Hermitian if it is equal to its own adjoint, i.e.

\[ T = T^\dagger. \]

It is clear from the definition that all real symmetric matrices are Hermitian.

Example 8.9

The matrix

\[\begin{split} \begin{pmatrix} 3& 2+i& 4i\\ 2-i& 1& 1-i\\ -4i& 1+i& 3 \end{pmatrix} \end{split}\]

is Hermitian.

The following two results are of fundamental importance in quantum mechanics. Together, they explain why in the mathematical formalism of quantum mechanics, Hermitian operators correspond to physical observables.

Theorem 8.1

The eigenvalues of a Hermitian operator \(T\) are real.

Proof. Suppose \(\lambda\) is an eigenvalue of \(T\) and \(\ket\alpha\) is a corresponding eigenvector. Then we have

\[ \lambda\braket{\alpha|\alpha}=\braket{\alpha|T\alpha} =\braket{T\alpha|\alpha}=\braket{\lambda\alpha|\alpha} =\bar\lambda\braket{\alpha|\alpha}. \]

Since \(\braket{\alpha|\alpha}\) is non-zero, we can divide by it and obtain \(\lambda=\bar\lambda\).

Theorem 8.2 (Spectral theorem)

If \(T\) is a Hermitian operator on a Hilbert space \(\HH\), then there exists an orthonormal basis of \(\HH\) that consists of eigenvectors for \(T\).

Note

Besides the fact that they encode observables, Hermitian operators also appear in quantum physics as density operators, or density matrices, which describe mixed states (ensembles of quantum systems) rather than individual (‘pure’) quantum states. Density operators play a central role in the theory of entanglement and in quantum information theory; we will not discuss them further here.

8.9. Unitary operators#

Definition 8.8 (Unitary operator)

An operator \(T\) on a Hilbert space \(\HH\) is called unitary if the adjoint \(T^\dagger\) is a two-sided inverse of \(T\), i.e.

\[ TT^\dagger = \id\quad\text{and}\quad T^\dagger T=\id. \]

Alternatively, \(T\) is unitary whenever \(T\) preserves inner products; see Exercise 8.3 for a precise statement.

Example 8.10

For any angle \(\phi\), the matrix

\[\begin{split} \begin{pmatrix} \cos\phi& \sin\phi\\ i\sin\phi& -i\cos\phi \end{pmatrix} \end{split}\]

is unitary.

It follows immediately from the definition that the identity operator on a Hilbert space (sending every vector to itself) is unitary, and if \(T\) is a unitary operator, then its inverse \(T^{-1}\) (which equals \(T^\dagger\)) is also unitary. Similarly, if \(T\) and \(U\) are unitary operators, then so is the product \(TU\). This means that the collection of all unitary operators on a Hilbert space \(\HH\) has the mathematical structure of a group.

8.10. Linear subspaces and projection operators#

Definition 8.9 (Linear subspace)

A linear subspace of a vector space \(V\) is a set of vectors \(W\) contained in \(V\) such that the following properties hold:

  1. The zero vector of \(V\) lies in \(W\).

  2. For all vectors \(w\) and \(w'\) in \(W\), the sum \(w+w'\) is also in \(W\).

  3. For every vector \(w\) in \(W\) and every scalar \(\lambda\), the scalar multiple \(\lambda w\) is also in \(W\).

In quantum physics, linear subspaces are important because they encode quantum states that a physical state can collapse to when a measurement is performed. To make this more precise, we need the notions of orthogonal complement of a subspace and orthogonal projection onto a subspace.

Definition 8.10 (Orthogonal complement)

Consider a Hilbert space \(\HH\) and a linear subspace \(W\) of \(\HH\). The orthogonal complement of \(W\) in \(\HH\) is the set of all vectors \(\ket{v}\) in \(\HH\) that are orthogonal to all of \(W\), i.e. such that

\[ \braket{v|w}=0\quad\text{for all }w\in W. \]

It is not hard to verify that the orthogonal complement of a linear subspace of \(\HH\) is again a linear subspace; see Exercise 8.7.

Within a Hilbert space \(\HH\), we will only consider linear subspaces \(W\) that satisfy the completeness condition and are therefore themselves Hilbert spaces. In this case, one can show that every vector \(\ket\alpha\in\HH\) can be decomposed uniquely as

\[ \ket\alpha = \ket{\alpha}_W + \ket{\alpha}_{W^\perp} \quad\text{with}\quad\ket{\alpha}_W\text{ in }W\quad\text{and}\quad \ket{\alpha}_{W^\perp}\text{ in }W^\perp. \]

Definition 8.11 (Orthogonal projection)

Given a Hilbert space \(\HH\) and a linear subspace \(W\) of \(\HH\), the orthogonal projection of \(\HH\) onto \(W\) is the map that sends every vector \(\ket\alpha\) to \(\ket\alpha_W\).

Orthogonal projections onto linear subspaces turn out to be important examples of linear maps.

8.11. Exponential of an operator#

Recall (see The exponential function) that the exponential function can be characterised by the Taylor series

\[ \exp(x) = \sum_{n=0}^\infty\frac{1}{n!}x^n. \]

In quantum physics, it is frequently useful to apply this formula not just to (real or complex) numbers, but to operators as well. One reason is that there is an important application to differential equations.

Definition 8.12 (Exponential of an operator)

If \(T\) is an operator on a Hilbert space \(\HH\), then the exponential of \(T\) is the operator

\[ \exp(T) = \sum_{n=0}^\infty\frac{1}{n!}T^n. \]

In the case where \(\HH=\CC^n\), an operator \(T\) corresponds to an \(n\times n\)-matrix \(A\). The same formula as above then defines the matrix exponential \(\exp(A)\) of \(A\).

Warning

Since operators do not commute in general, the exponential does not have the property \(\exp(T+U)=\exp(T)\exp(U)\) as one might expect from the case of ordinary exponentials.

We have seen in Example 4.3 that the solution of the differential equation

\[ \dot x(t) = \lambda x(t),\quad x(0)=x_0 \]

is given by

\[ x(t) = x_0 \exp(\lambda t) \]

Using the operator exponential, we can solve the higher-dimensional variant

\[ \frac{d}{dt}\ket{x(t)} = T\ket{x(t)},\quad \ket{x(0)}=\ket{x_0}. \]

Namely, the solution is

\[ \ket{x(t)} = \exp(Tt)\ket{x_0}. \]

8.12. Exercises#

Exercise 8.1

Check that the inner product defined in Example 8.2 is indeed a Hermitian inner product on \(\CC^n\).

Exercise 8.2

Consider a matrix \(A=(A_{i,j})_{i,j=1}^n\) and the standard basis \((\ket{e_i})_{i=1}^n\) of \(\CC^n\). Check that the matrix entry \(\braket{e_i|A|e_j}\) is simply the actual matrix entry \(A_{i,j}\).

Exercise 8.3

Show that an operator \(T\) is unitary precisely when it satisfies \(\braket{T\alpha|T\beta}=\braket{\alpha|\beta}\) for all vectors \(\ket\alpha\), \(\ket\beta\).

Exercise 8.4

The Cauchy–Schwarz inequality states that for vectors \(\ket\alpha,\ket\beta\) in a Hilbert space we have

\[ \bigl|\braket{\alpha|\beta}\bigr|^2\le \braket{\alpha|\alpha}\braket{\beta|\beta}. \]

Show this using the basic properties of the inner product.

Exercise 8.5

The triangle inequality states that for vectors \(\ket\alpha,\ket\beta\) in a Hilbert space we have

\[ \|\alpha+\beta\|\le\|\alpha\|+\|\beta\|. \]

Derive the triangle inequality from the Cauchy–Schwarz inequality.

Exercise 8.6

Check that the Pauli matrices

\[\begin{split} \sigma_x = \begin{pmatrix}0& 1\\1& 0\end{pmatrix},\quad \sigma_y = \begin{pmatrix}0& -i\\i& 0\end{pmatrix},\quad \sigma_z = \begin{pmatrix}1& 0\\0& -1\end{pmatrix} \end{split}\]

are both Hermitian and unitary.

Exercise 8.7

Consider a linear subspace \(W\) of a Hilbert space \(\HH\). Show that the orthogonal complement \(W^\perp\) is again a linear subspace.

Exercise 8.8

Suppose \(D\) is a diagonal \(n\times n\)-matrix, viewed as an operator on the space \(\CC^n\). Find an expression for the matrix exponential \(\exp(D)\).

Exercise 8.9

Consider a diagonal matrix \(D\) and an invertible matrix \(P\). Show that that the matrix exponential of \(PDP^{-1}\) satisfies

\[ \exp(PDP^{-1}) = P\exp(D)P^{-1}. \]
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