Measurements in Quantum Systems and Fundamental Theory
In my previous article, I briefly discussed Hermitian products and their importance in Quantum Information Science (QIS). However, the story has just begun. At the heart of quantum systems is a postulate that I will roughly interpret as “the mere idea of measuring a quantity changes it”.
The mere idea of measuring a quantity changes it.
As you may recall, elementary particles such as electrons can have spins: clockwise and counter-clockwise. However, we won’t know which electron is spinning clockwise and which one is spinning counter-clockwise with cent-percent certainty. We are mortal beings and we can only talk about the probability of getting either clockwise spin or counter-clockwise. That’s the gist of the uncertainty principle informally.
In a system like an electron, the description of its condition may be described as a spin — speaking in an oversimplistic manner. Such a description may be referred to as a state.
Recall the Hermitian matrix from my previous article. Any state can be described by a Hermitian matrix called a density matrix (or sometimes just density). We use ρ to denote a density matrix. The trace of the density matrix is 1.
As we are mortal beings, devoid of god-like powers, we can only measure things rather than direct observations of microscopic systems. Hence, if a state of the system, 𝓗 is specified by the density matrix ρ (which is the ground truth in the way that only a divine being knows), and we make some measurement M, that returns a value ω. The measurement M itself is a Hermitian matrix. We do several such measurements and receive multiple ω. In such case, the following condition holds:
A set of M (M_\omega in Equation 1) is called positive-operator valued measurement or POVM in short. For continuous values, the sum in Equation 1 can be replaced by the integration. I in Equation 1 is the identity matrix.
Now, let’s say, for some reason, we receive god-like power and magically know what ρ is, then the probability of obtaining a value ω for the measurement M is given by
You can check that sum of Equation 2 is indeed 1 using Equation 1 and the fact that the trace of ρ is 1.
If a system 𝓗, (say electron) has two states: ρ1 and ρ2 corresponding to clockwise and counter-clockwise spin with a probability of λ and (1-λ). Now, we want to know if we can get some value ω by making a measurement M. What’s the probability that we can get ω? This can be given by
Thus effective, a state of the system is given by a probabilistic mixture λρ1 + (1-λ)ρ2.
For a vector u, if the norm is 1, then it is a pure state in quantum mechanics. Thus, u is a state of the system measured as |u⟩⟨u|. If the state is not a pure state, it is a mixed state.
A pure state cannot be written as a probabilistic mixture of other states.
For many pure states u_i, we can have |x⟩ = \sum_i x_i |u_i⟩ as a new vector, and a quantum-mechanical superposition, not to be confused with the probabilistic mixture.
Hermitian Matrices are Observables or Physical Quantities
If we have a Hermitian matrix X, and its eigenvalues as x_i, then the projection matrix corresponding to the eigenspace is Ex_i and X = sum_i x_i Ex_i. x_iEx_i is nothing but the spectral decomposition of X (not much different from what you already know from Linear Algebra or Machine Learning classes).
We can have more than one eigenvector corresponding to a single eigenvalue, and diagonalization is X = sum_d x_i |u_i⟩⟨u_i| is not unique, however, spectral decomposition will be unique. The expectation and variance of Ex (the set of all Ex_i) are given by TrρX and TrρX² — (TrρX)² respectively.
I hope this article was helpful in understanding what the measurement and states are in quantum systems. This is helpful for understanding qubits and other fundamental concepts of QIS.
References
- Quantum Information: an introduction by Masahito Hayashi
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