Uncertainty Relation

Quantum Uncertainty:

In quantum mechanics, the Heisenberg uncertainty principle, dictates the uncertainty product of position and momentum of a quantum mechanical system. The inequality arises from definitions of variance and the Cauchy-Schwarz inequality. The proof can be found on wikipedia.

Now, a quantum mechanical state has a specific position and momentum with uncertainty (model 1). A similar analogy exists in the quadratures of light (model 2).

Position and momentum operators (or in-phase and out-of-phase quadratures of light) can be represented as Hermitian operators \hat{q} and \hat{p} respectively.

These operators are canonically conjugate variables, which means they satisfy the following commutation relation

[\hat{\textbf{q}},\hat{\textbf{p}}] = \hat{\textbf{q}}\hat{\textbf{p}} - \hat{\textbf{p}}\hat{\textbf{q}} = i\hbar

Then the uncertainty relation for a given wavefunction \left|\psi\right\rangle can be expressed as follows.

D(\hat{p})D(\hat{q}) \ge \frac{\hbar^2}{4}

where

D_\psi(\hat{q}) = ||(\hat{\textbf{q}}-{x})\psi||^2 = \left \langle \psi|(\hat{\textbf{q}}-x)(\hat{\textbf{q}}-x)|\psi\right\rangle\\ \label{D2}  D_\psi(\hat{p}) = ||(\hat{\textbf{p}}-{y})\psi||^2 = \left \langle \psi|(\hat{\textbf{p}}-{y})(\hat{\textbf{p}}-{y})|\psi\right\rangle

and

{x} = \left \langle \psi |\hat{\textbf{q}}|\psi\right\rangle\\  {y} = \left \langle \psi |\hat{\textbf{p}}|\psi\right\rangle

q represents position and p represents momentum.  D(q) and D(p) are variances.

q represents position and p represents momentum. D(q) and D(p) are variances. The radius of the circle is related to the uncertainty product. Minimum uncertainty condition is met when the circle has the smallest radius.

Naturally occurring states are the number states

The vacuum state (for light quantization) of the ground state (for quantum harmonic oscillator) and the minimum uncertainty states, centered at the origin. As the number of photons in the system/energy level increases, the associated uncertainty also increases

The ground state (for quantum harmonic oscillator) or the vacuum state (for light quantization) are the minimum uncertainty states, centered at the origin. As the number of energy level/number of photons in the system (n) increases, the associated uncertainty also increases.

The uncertainty dependence on photon number/principle quantum number is

D(\hat{q})D(\hat{p}) = \frac{\hbar}{2\omega}(1+2n)\times\frac{\hbar\omega}{2}(1+2n)

The uncertainty increases with increase in photon/quantum number. Hence the ground/vacuum state is a minimum uncertainty state.

All minimum uncertainty states are called coherent states. A coherent state is hence, a ground state/vacuum state displaced by (x,y). The solution \left\langle q|\psi\right \rangle for the minimum uncertainty condition turns out to be a Gaussian.

The transformation from a ground/vacuum state to a coherent state is dictated by the Weyl operator.

Conceptually, the uncertainty in position(or in-phase quadrature) and momentum(or out-of-phase quadrature) are related in a way that, if one increases, the other must decrease, to conserve the product. So for a particle whose position is accurately known, the uncertainty in momentum of the particle is infinitely high and vice versa.

Fourier Uncertainty:

In signal processing, a signal can be represented in time domain or frequency domain. A signal that is ‘spread out’ and unbounded in the time domain, is a ‘squeezed’ and bounded in the frequency domain and vice versa. A function in time domain, and its Fourier representation are also conjugate variables.

Fractional Fourier Domain by Ozaktus and Aytur offers a good fundamental understanding of the time-frequency representation, and the transform between the various bases that lie in the time-frequency plane.

For a signal \psi and a coordinate multiplication operator X_a defined in domain x_a parametrized by a,

(X_a\psi)(x_a) = x_a\psi(x_a)

$x_0$ is the time domain and $x_1$ is the frequency domain.

x_0 is the time domain and x_1 is the frequency domain. x_a and x_a' are intermediate domains.

It can be proved that X_a and X_a' are related by a similar commutation and uncertainty equation

[X_a,X_a'] = \frac{i}{2\pi}sin(\phi'-\phi)

D(x_a)D(x_a') \ge \frac{1}{16\pi^2}\sin^2(\phi'-\phi)

The transformation from one domain to the other is performed using Fractional Fourier Transform.

Chirp and Wavelet transforms are special forms of Fractional Fourier Transforms (ref).

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Quantum Mechanical Position, Momentum and Fourier Transform

For people familiar with quantum physics, it’s a known fact that a ‘waveform’ \left|\Psi\right\rangle can be essentially ‘measured’ in either ‘position’ x or ‘momentum’ p basis. We define the inner products using the following notations:

\left\langle x | \Psi \right \rangle = \Psi_p(x)

and

\left\langle p | \Psi \right \rangle = \Psi_x(p)

The point to be noted is that momentum is the time derivative of position. In such a case,

\Psi_p(x) = \mathfrak{F}\{\Psi_x(p)\} \\    \Psi_x(p) = \mathfrak{F}\{\Psi_p(x)\}

This can be proved as follows. A transformation from the x basis to the p basis can be made as follows.

The \hat{p} operator is

\hat{p} = -i\hbar\frac{\partial}{\partial \hat{x}}.

To find eigenstates of \hat{p}, we can call \langle x|p\rangle = f_p(x)

-i\hbar\frac{\partial}{\partial x}f_p(x)=pf_p(x)

which yields to f_p(x) = e^{ipx/\hbar}.

Now, to pass from a basis to the other we can write

\langle p|\psi\rangle= \int \langle p|x\rangle\langle x|\psi\rangle dx

or

\Psi_x(p) = \int e^{-ipx/\hbar}\Psi_p(x)dx

which is a Fourier transform.

Thus \Psi_p(x) and \Psi_x(p) form a Fourier pair. Simply speaking, a waveform that is measured in the time derivative of a certain basis will be the Fourier transform of the waveform being measured in that basis.

Now, coming to the signal processing scenario in which the representation of a ‘signal’ f is effectively a ‘measurement’ in either ‘time’ t or ‘frequency’ \omega basis.

The correspondence between these two models:

x \mapsto t

\frac{p}{\hbar} \mapsto \omega

Classically, time and frequency are inverses of each other.

\omega = \frac{2\pi}{T}
where T is the time period (measured in time domain).

Similarly, going by de Broglie wave equation,

\frac{p}{\hbar} = \frac{2\pi}{\lambda}
where \lambda is the wavelength (measured in position domain).

In the next post, I will elaborate on the uncertainty relation in both signal processing and quantum physics. The whole idea behind creating an analogy between the two is to see if certain tools that have already been developed in quantum physics, can be adapted to enhance signal processing techniques.

Compressed Sensing

  1. Formulating the problem statement:

    There are essentially two ways to ‘compress’ data. Analog data is generally sampled and then quantized. The quantization process is called encoding. Conventionally, there are several encoding techniques in place which give optimal data compression, bounded by Shannon’s compression limit. More recently, however, people have concentrated on optimum techniques of sampling the data. It can be postulated that if the data is ‘sparse’ in a certain basis, then very few measurements are required to sample the data (typically much lesser than the Nyquist rate), which can later be reconstructed with very low margin of error.

  2. Measurement:

    Given a discrete signal or image with n points, there exists a k such that a minimum of m measurements can be performed that map the n point signal to a m point transform (m is much smaller than n). The discrete signal that is being measured can be complex in nature. In such a case,

    b_i = |\left\langle A_i,x \right\rangle|

    where x is the input vector of length n and A_i is the i'th row of the measurement operator A_{m \times n}, i = 1....m. These measurements are in most cases, random in nature. As an example, one may consider the rectangular matrix A to be m permuted rows of a square Walsh-Hadamard matrix \Omega_{n \times n}.

  3. Reconstruction of measured signal:

    The reverse problem is usually a convex optimization problem. Suppose the signal in consideration x is sparse a certain basis, for instance, wavelet basis. The wavelet transformed signal must hence be sparse. The problem hence becomes

    min ||Wx'||_1

    such that x' is the reconstructed signal and W is an n point wavelet transform, subject to the condition

    b_i = |\left\langle A_i,x' \right\rangle|