How to understand the Haar measure from a quantum information perspective?

Question

I found it a little difficult to understand it using Wikipedia and some mathematical documents. How to understand the Haar measure from a quantum information theory perspective? Are there any materials that explain it?

Answer 1

Computations in quantum information processing are implemented by means of unitary operations. Sometimes, we need to think not about a specific unitary operation required to execute a specific computation, but about the whole space of unitary transformations. (Examples will be given below). For a single $n$-dimensional qudit, (which can also be a tensor product of a number of qubits), this space is the unitary group $U(n)$.

$U(n)$ is a compact Lie group; all the matrix element magnitudes of its elements are less or equal to one, thus it can be mapped into a hypersurface occupying finite region in some Euclidean space, given by the orthonormality constraints of its raw vectors. In particular, this implies that its Euclidean volume is finite.

Spaces of finite volume can be naturally turned into probability spaces, basically by normalizing their volume to $1$. This is the starting point where they become useful in probabilistic analysis, in our case in quantum probability. Probability spaces are equipped with a measure assigning a number between $0$ and $1$ to subsets, which in the continuous cases, can be taken as the normalized volume of the subset.

For the group $U(n)$, the normalized Euclidean volume element mentioned above is such a measure. This measure is not invariant, in general, under the group action. The normalized volume of a subset changes if we rotate the subset by means of a group element $u$. However, if we define a new measure by averaging the volumes obtained by rotating the subset by all group elements, the resulting averaged measure will be invariant under the group action. This is the Haar measure.

The method described above for the construction of the Haar measure is not practical, since we need to average over an infinite number of elements. More practical methods will be described below. A non-rigorous but reasoned account of the Haar measure of the unitary group U(n), in view of quantum information theory applications, will be given. The sketchy reasoning presented here can be used as a starting point for a more rigorous treatment. The explanations will refer specifically to the unitary group. Their quite straightforward generalized to other classical compact groups, will not be treated.

Properties of the Haar measure

The (volume element of the) Haar measure will be denoted by $d\mu_H(u)$, $u\in U(n)$. The Haar measure has the following properties:

1. It has a finite volume

$$\int_{U(n)} d\mu_H(u) < \infty$$ 2. It is (right and left) invariant under the group action: Given an integrable function $f(u)$, then $$ \int_{U(n)} f(vu) d\mu_H(u) = \int_{U(n)} f(u) d\mu_H(u) , \quad v\in U(n) $$ $$ \int_{U(n)} f(uv) d\mu_H(u) = \int_{U(n)} f(u) d\mu_H(u) , \quad v\in U(n) $$ 3. It is unique up to a multiplication by a scalar: Any another invariant measure $d\mu(u)$ satisfies: $$ d\mu(u) = \mathrm{const.} d\mu_H(u) $$

For $U(n)$, these properties can be proved by the explicit constructions which will be described below; however, it is worthwhile to show an explicit method to modify any non-invariant measure into a Haar measure, for example the Euclidean measure mentioned above:

Since $U(n)$ is a manifold, it can be covered by coordinate patches in bijection with open subsets in $\mathbb{R}^n$: $$U(n) \supset U\ni u \mapsto x(u) \in \mathbb{R}^n$$ In order to change the Euclidean volume element $d^nx$ into a Haar measure, we need to evaluate the action of a general group element $u$ on an element $u_{\epsilon} = 1 + \epsilon$, $|\epsilon|<<1$ close to the identity; we get, $$x(u u_{\epsilon } )= B(x(u)) + A(x(u))x(u_{\epsilon}) + O(|\epsilon|^2), B(u) \in \mathbb{R}^n , A(u )\in \text{Mat}(n)$$ Then the measure: $$ d\mu(u(x)) = |\det A(x)|^{-1} d^nx$$

Is a Haar measure. The proof is not a very difficult exercise, please see the following post in mathematics stackexchange for the details.

A similar construction is based on the fact that the Haar measure can be induced by a metric - The Cartan-Killing metric. If we take the other side of the bijection above:

$$\mathbb{R}^n \ni x \mapsto u(x) \in U \subset U(n) $$

The metric: $$G_{ij}(x) = \mathrm{Tr}\left (\frac{\partial u(x) ^{-1}}{\partial x^i} \frac{\partial u(x) }{\partial x^j} \right)$$

Induces a Haar measure, through:

$$ d\mu_H(u(x)) = \sqrt{\det G(x)} d^nx$$

The existence of this metric allows to define a Dirac delta function by means of the solutions of the time independent Schrödinger equation on U(n). $$\Delta \psi = E \psi$$ Where $\Delta$ is the Laplacian on $U(n)$: $$\Delta = \frac{1}{\sqrt{\det G(x)}} \sum_{i, j} \frac{\partial}{\partial x^i}\left( \sqrt{\det G(x)} G^{ij} \frac{\partial}{\partial x^j}\right)$$

($G^{ij}$ is the reciprocal matrix of $G_{ij}$)

Let $\{f_E(u)\}$, be an orthonormal set (with respect to the Haar measure) of solutions of the Schrödinger equation for all possible energies (I am skipping the issue of the existence of solutions), then due to the completeness property, the function: $$\delta_H(u, v) = \sum_E \bar{f_E}(u) f_E(v)$$ is an invariant Dirac delta function over the square integrable functions, satisfying the properties: $$\delta_H(u, v) = \delta_H(kg, kh)$$ $$ \int_{U(n)} \delta_H(u, v) d\mu_H(u) = 1$$ and $$ \int_{U(n)} f(u) \delta_H(u, v) d\mu_H(u) = f(v) $$

The delta function can be used to sketch the proof of the Haar measure uniqueness up to a factor. Suppose that there exists a second invariant measure $d\mu(u)$ on $U(n)$. Denote by: $$\langle f \rangle_{\mu} = \int_{U(n)} f(u) d\mu_H(u) $$ and $$\langle f \rangle = \int_{U(n)} f(u) d\mu(u) $$ Then: $$ \int_{U(n)} \int_{U(n)} f(u) \delta_H(u, v) d\mu_H(u) d\mu(v) = <f> $$ Changing the integration order (Fubini's theorem): $$ \int_{U(n)} f(u) \left (\int_{U(n)} \delta_H(u, v) d\mu(v) \right) d\mu_H(u) = <f> $$ Using the invariance property of the delta function $$ \int_{U(n)} f(u) \left (\int_{U(n)} \delta_H(e, u^{-1}v) d\mu(v) \right) d\mu_H(u) = <f> $$ Using the invariance property of the measure:

$$ \int_{U(n)} f(u) \left (\int_{U(n)} \delta_H(e, v) d\mu(v) \right) d\mu_H(u) = <f> $$ Thus $$ (\int_{U(n)} \delta_H(e, v) d\mu(v) ) \int_{U(n)} f(u) d\mu_H(u) = <f> $$ Therefore $$ \mathrm{const.} <f>_{\mu} = <f> $$ (To be precise, in the above, we assumed that $f$ is both $L^1$ and $L^2$)

Some explicit constructions

The explicit construction of the Haar measure in coordinates requires the parametrization of U(n), one possibility is by means of Euler angles, please see for example the following article by Tilma and Sudarshan.

The normalization of the Haar measure to a unit volume, requires the computation of its volume for the purpose of normalization, please see the review Zhang and Boya, Sudarshan and Tilma for computation techniques which do not require explicit parametrizations.

Just to give a taste of how these computations are performed, I'll describe how the volume of $U(n)$ can be computed a product of volumes of spheres: The first row in the matrix representing a group element $u$ is a normalized $n$ dimensional complex vector, thus it spans a sphere $S^{2n-1}$ as $u$ spans the whole unitary group. The second row is also a normalized $n$ dimensional complex vector, but it must be orthogonal to the first one, thus it lives in an $n-1$ dimensional space, thus it spans a sphere $S^{2n-3}$, the third should be orthogonal tom the first two thus it spans a sphere $S^{2n-5}$, etc. Thus, the Haar volume of $U(n)$ is given by: $$\text{Vol}(U(n)) = \text{Vol}(S^{2n-1}) \text{Vol}(S^{2n-3})…\text{Vol}(S^1) $$ Using the volume formula of the n-sphere: $$\text{Vol}(S^{2n-1}) = \frac{2 \pi^n}{\Gamma(n)},$$ the $U(n)$ volume is obtained: $$\text{Vol}(U(n)) = \frac{(2 \pi)^{n^2+n}}{\prod_{m=1}^{n-1} m!}$$

These constructions have already applications to quantum information theory: Tilma and Sudarshan apply their results to the computation of the volumes of unitary orbits of density matrices, while Zhang reviews the computation of the Hilbert-Schmidt volume of the space of density matrices. It should be mentioned that the integration of class functions i.e. functions which are invariant under the conjugation map: $$f(u) = f(v^{-1}uv),$$ over the unitary group $U(n)$ can be simplified to an integration over its maximal torus: $$ \int_{U(n)} f(u) d\mu_H(u) = \frac{1}{n!}\int_{T_n} f(t) V(t)^2 dt$$ where $t = [t_1, t_2, …, t_n]^T$ are the eigenvalues of $u$ and the integration is over the maximal torus $U(1)^n$ and $V(t)$ is the Vandermonde determinant (the determinant of the Vandermonde matrix). This is the Weyl integration formula, please see the following lecture note by: Nico Sprong.

Measures induced from the Haar measure

Homogeneous spaces of the unitary group possess measures induced from the Haar measure. One of the most important of such spaces in quantum information theory is the complex projective space: $\mathbb{C}P^n = \frac{U(n+1)}{U(n)}$ which is the space of pure states in $n+1$ dimensions. The induced measure on the space of pure states is called the Fubini-Study measure, please see Bengtsson and Życzkowski section 4.7. Sometimes the induced measure is also called a Haar measure.

Any $n+1$ dimensional state vector can be obtained by the application of a unitary matrix on some fixed vector. Thus, in order to construct a $\mathbb{C}P^n$-random pure state vector, we can apply a random $U(n)$ matrix drawn from a Haar distributed ensemble.

For mixed states, unitary orbits of density matrices are generalized flag manifolds. These spaces are also homogeneous spaces, therefore equipped with measures induced by the Haar measure. The five-dimensional case is explicitly elaborated also in Boya, Sudarshan and Tilma in section 7.

Some applications in quantum information

1. Identification of unknown quantum channels

As discussed above, the generation of a Haar-random state vector can be obtained by acting on a fixed vector by a Haar distributed unitary matrix. Please see the following article by Francesco Mezzadri describing the random matrix generation algorithm. Randomly generated state vectors can be used to identify or at least gain information about unknown quantum channel.

To give a motivation of this approach, let's consider first a classical channel modeled as a damped harmonic oscillator: $$m \ddot{x} + c \dot{x} + kx = F(t)$$ And we want to estimate the channel parameters $m$, $c$ and $k$. If we knew in advance that our channel is a damped harmonic oscillator we could subject it to two sinusoidal inputs with different frequencies and extract the coefficients from its frequency response. But if we no a-priori idea what the channel dynamics is (except that it is linear), we can use a white noise random signal to identify its spectral density function: $$\langle |x(\omega)|^2 \rangle = \frac{\frac{1}{m^2}}{(\omega^2-\frac{k}{m})^2 + (\frac{c \omega}{m})^2}$$ The analysis of the spectral density response can be used to identify all the system degree and all of its model coefficients.

In the quantum case, by analogy we place random state vectors at the input and measure the outer state. We can gain information about the channel. Please see the following thesis by Easwar Magesan. For example, given a channel $\mathcal{E}$, approximating a unitary channel $\mathcal{U}$, then the fidelity of $\mathcal{E}$, with respect to $\mathcal{U}$, is given by (Equation 2.12 in the thesis). $$F(\mathcal{E} , \mathcal{U} ) = \int_{\psi} d\psi \mathrm{tr} \left( \mathcal{U} (|\psi\rangle\langle\psi|)\mathcal{E} (|\psi\rangle\langle\psi|)\right)$$ Where the integration is over the Fubini-Study measure. The quantum case is much more difficult than the classical one: Experimentally, on needs to input the channel with randomly generated vectors as described above; each time, perform a full tomography of the output state a perform the averaging with the ideal channel output. Both the generation of the random inputs and the full tomography are hard. One can reduce the complexity by cleverly selecting a specific set of random vectors called a design capable of estimating well behaved classes of channels; and averaging the channel over a group of unitarities called twirling which allows extracting partial information in a scalable way.

2. Quasi-probability functions on the group manifold.

In the above applications, the group manifold is utilized as a classical probability space. But in fact, we can encode the full quantum information in functions over the $U(n)$ group manifold. This can be done by transforming the density matrix $\rho$ into a function $\chi$ of positive type over the group manifold (which is a generalization of the Wigner function on the phase space): $$\chi(u) = \mathrm{Tr}(\rho u), \quad u \in U(n)$$ Please see Anillo.

This function has the following properties stemming from the density matrix properties:

1. Normalization $$\chi(e) =1$$
2. Symmetry $$\bar{\chi}(u) =\chi(u^{-1})$$
3. Positive definiteness (Bochner's theorem} $$\sum_{j,k } \chi(g_j^{-1}g_k) \ge 0$$

For any set of group elements ${g_1, g_2, \dots, g_n}$.

The function $\chi(u)$ can be regarded as a generalized Fourier transform of the density matrix or a characteristic function. Several types of quantum channels can be implemented on the group manifold, for example if we multiply two different functions $\chi(u)$ of positive type, we get a function of positive type because the Hadamard product of two positive definite matrices is also a function of positive type, thus the multiplication implements a quantum channel.

A second use of this representation is to find the probability density function of a set of commuting observables (since a set of commuting observables can be characterized by a classical probability density function), which can be computed as an inverse Fourier transform: Defining $$v = e^{i\sum_i T_i v_i}, $$ where $\{T_i\}$ is a set of commuting observables $$f_{\{T_i\}}({t_i}) = \prod_n \left(\int_0^{2\pi} \right) d^nv e^{-i\sum_i t_i v_i} \chi(v)$$

I have given an explicit example of this computation in my answer on this question.