9. Operators on Hilbert spaces#
Recommended reference: Griffiths [Gri95], in particular Section 3.1.
9.1. Operators#
An operator or linear map on a Hilbert space \(\HH\) is a map
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
and multiplication of an operator \(T\) by a scalar \(c\) is defined by the formula
Like matrices (but unlike vectors), operators can be composed: if \(T\) and \(U\) are two operators, then \(TU\) is the operator defined by
9.2. Matrix entries#
In physics, quantities of the form
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 9.1
Consider a linear map \(A\) on \(\CC^n\), viewed as a matrix. Taking the matrix entries as above, where \(\alpha\) and \(\beta\) range over the standard basis vectors, we obtain the usual matrix entries of \(A\); see Exercise 9.1.
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
9.3. The adjoint of an operator#
Given an operator \(T\) on a Hilbert space \(\HH\), there exists an operator \(T^\dagger\) with the property
The operator \(T^\dagger\) is called the adjoint of \(T\).
Property 9.1 (Properties of the adjoint)
For all operators \(T\), \(U\) and scalars \(c\), we have
\((cT)^\dagger = \bar c T^\dagger\)
\((TU)^\dagger = U^\dagger T^\dagger\)
\((T^\dagger)^\dagger = T\)
Example 9.2
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
This shows that the adjoint of the operator \(A\) corresponds to the conjugate transpose of \(A\) viewed as a matrix.
9.4. Hermitian operators#
Definition 9.1 (Hermitian operator)
An operator \(T\) on a Hilbert space \(\HH\) is called Hermitian if it is equal to its own adjoint, i.e.
It is clear from the definition that all real symmetric matrices are Hermitian.
Example 9.3
The matrix
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 9.1
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
Since \(\braket{\alpha|\alpha}\) is non-zero, we can divide by it and obtain \(\lambda=\bar\lambda\).
Theorem 9.2 (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 \(T\).
9.5. Unitary operators#
Definition 9.2 (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.
Alternatively, \(T\) is unitary whenever \(T\) preserves inner products; see Exercise 9.2 for a precise statement.
Example 9.4
For any angle \(\phi\), the matrix
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.
9.6. Orthogonal projection#
Consider a Hilbert space \(\HH\) and a linear subspace (see Definition 8.8). One can show that every vector \(\ket\alpha\in\HH\) can be decomposed uniquely as
Definition 9.3 (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.
Example 9.5
Let \(W\) be a one-dimensional linear subspace of a Hilbert space \(\HH\). Then for any unit vector \(\ket e\) in \(W\), the operator \(\ketbra{e}{e}\) sending every vector \(\ket\psi\) to \(\ket e \braket{e|\psi}=(\braket{e|\psi})\ket e\) is the orthogonal projection onto \(W\).
9.7. Density operators#
Note
The topic of this section is usually not treated in BSc-level mathematics courses. We include it because density matrices are a fundamental mathematical concept in quantum physics and are important examples of Hermitian operators.
Density operators on Hilbert spaces, or density matrices, play a central role in situations where quantum states are combined with classical uncertainty. They describe mixed states (ensembles of quantum systems) rather than individual (‘pure’) quantum states. Density operators are intensively used in the theory of entanglement and in quantum information theory, for example.
Definition 9.4 (Density operator)
A density operator on a Hilbert space \(\HH\) is an operator \(\rho\colon\HH\to\HH\) with the following properties:
\(\rho\) is Hermitian;
\(\rho\) is positive semi-definite, i.e. for all \(\ket\alpha\in\HH\) we have \(\bra\alpha\rho\ket\alpha\ge0\);
\(\rho\) has trace 1.
A density matrix is the matrix of a density operator relative to some choice of basis of \(\HH\).
When \(\HH\) is finite-dimensional, the trace of \(\rho\) is simply the trace of the matrix of \(\rho\) with respect to any choice of basis for \(\HH\); see Definition 5.5. When \(\HH\) is infinite-dimensional, the definition is somewhat more complicated; see Trace class (Wikipedia)
Example 9.6
The matrix
represents a density operator on the Hilbert space \(\CC^2\) with the standard inner product.
In Exercise 9.6, you will classify all the possible density matrices in dimension 2.
Example 9.7
Let \(\ket e\) be a unit vector in \(\HH\). The operator \(\ketbra{e}{e}\) sending every vector \(\ket\psi\) to \(\ket e \braket{e|\psi}=(\braket{e|\psi})\ket e\) is a density operator.
The operator \(\ketbra{e}{e}\) considered above is nothing but the orthogonal projection operator onto the 1-dimensional linear subspace \(\CC\ket e\) of \(\HH\); see Example 9.5.
Definition 9.5
A density operator \(\rho\colon\HH\to\HH\) is pure when \(\rho\) is the orthogonal projection onto a 1-dimensional linear subspace of \(\HH\), and mixed otherwise.
On a finite-dimensional Hilbert space, an operator \(\rho\) is a density operator precisely when there exist unit vectors \(e_1,\ldots,e_n\) and real numbers \(p_1,\ldots,p_n\ge0\) with \(p_1+\cdots+p_n=1\) such that \(\rho\) can be expressed as the convex combination
of the pure density operators \(\ketbra{e_i}{e_i}\). Given a density operator \(\rho\), there are in general multiple ways of expressing \(\rho\) as such a convex combination. By definition, \(\rho\) is pure precisely when it admits a decomposition as above with \(n=1\). Furthermore, it can be shown that \(\rho\) is pure precisely when it satisfies the identity \(\rho^2=\rho\).
9.8. Exponential of an operator#
Recall (see The exponential function) that the exponential function can be characterised by the Taylor series
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 9.6 (Exponential of an operator)
If \(T\) is an operator on a Hilbert space \(\HH\), then the exponential of \(T\) is the operator
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
is given by
Using the operator exponential, we can solve the higher-dimensional variant
Namely, the solution is
9.9. Exercises#
Exercise 9.1
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 9.2
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 9.3
Check that the Pauli matrices
are both Hermitian and unitary.
Exercise 9.4
Verify the description of orthogonal projections onto one-dimensional linear subspaces given in Example 9.5.
Exercise 9.5
Consider the Hilbert space \(\HH=\CC^2\) with the standard inner product. Show that an operator \(p\colon\HH\to\HH\) is the orthogonal projection onto some one-dimensional linear subspace of \(\HH\) precisely when \(p\) is given by a matrix of the form
where \(a,b,c,d\) are real numbers satisfying
Exercise 9.6
Show that the density operators \(\rho\) on the Hilbert space \(\CC^2\) with the standard inner product are precisely the operators given by matrices of the form
\[\begin{split} \rho = \begin{pmatrix} a& b - ci\\ b + ci& d \end{pmatrix} \end{split}\]where \(a,b,c,d\) are real numbers satisfying
\[ a+d=1\quad\text{and}\quad b^2+c^2\le ad. \]Deduce that the density operators \(\rho\) as above are precisely the operators given by matrices of the form
\[ \rho=\frac{1}{2}(I_2+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z) \]where \(\sigma_x\), \(\sigma_y\), \(\sigma_z\) are the Pauli matrices (see Exercise 9.3) and \(r_x\), \(r_y\), \(r_z\) are real numbers satisfying \(r_x^2+r_y^2+r_z^2\le1\).
Show that a density operator \(\rho\) is pure precisely when the numbers \(r_x\), \(r_y\), \(r_z\) in the above expression satisfy \(r_x^2+r_y^2+r_z^2=1\).
This shows that density operators for a 2-dimensional system can be visualised as points of the unit ball in \(\RR^3\). The pure states are those on the unit sphere (the boundary of the unit ball), often called the Bloch sphere in this context; the mixed states are those in the interior.
Exercise 9.7
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 9.8
Consider a square matrix \(D\) and an invertible matrix \(P\). Show that the matrix exponential of \(PDP^{-1}\) satisfies
(This formula is especially useful in the case where \(D\) is diagonal, since together with the previous exercise it allows us to efficiently compute the exponential of a diagonalisable matrix.)