Quantum Measurement in Different Bases

Hello! Welcome to your fifth lesson in our course on Quantum Computing Foundations.

In the last few lessons, we have focused on the dynamics of quantum systems, exploring how single- and two-qubit gates manipulate quantum states. We culminated this with the powerful result that any arbitrary single-qubit unitary operation can be decomposed into a sequence of simple rotations.

Now, we turn to the crucial question of how to extract classical information from a quantum system after these operations have been performed. This process, known as measurement, is the bridge between the quantum world of superpositions and the classical world of definite outcomes.

The learning outcome for this lesson is to calculate measurement outcomes and probabilities for quantum states in computational and arbitrary bases.

To achieve this, we will:

1. Formalize the concept of projective measurement using observables and projection operators.
2. Calculate measurement probabilities for pure states in both the computational basis and arbitrary bases.
3. Generalize our framework to mixed states using the density matrix formalism.
4. Introduce the concept of a Positive Operator-Valued Measure (POVM) as the most general description of a quantum measurement.

Given your background in physics, the foundational postulates of measurement will likely be familiar. Our goal here is to frame them within the language of quantum information and computation, building towards the more general formalisms used in modern quantum theory.

1. Projective Measurements: The Foundational Postulate

The most common type of measurement in introductory quantum mechanics is the projective measurement. It is defined with respect to a specific physical observable, which is represented by a Hermitian operator $A$.

The core idea is that measuring the observable $A$ on a system in state $|\psi ⟩$ will yield one of the eigenvalues of $A$ as the outcome. The state of the system then "collapses" into the corresponding eigenstate.

For a rigorous statement of this principle, we turn to the foundational lecture notes by John Preskill.

Let's summarize the key points from Preskill's notes:

• Observable: A physical quantity is represented by a Hermitian operator $A$.
• Spectral Decomposition: Any Hermitian operator can be written as $A={\sum }_n{a}_n{E}_n$, where $a_n$ are the eigenvalues (the possible measurement outcomes) and $E_n$ are the projection operators onto the eigenspaces corresponding to each $a_n$.
• Probability (Born Rule): If the system is in state $|\psi ⟩$, the probability of obtaining outcome $a_n$ is given by: $$\text{Prob}(a_n)=⟨\psi |E_n|\psi ⟩=\Vert E_n|\psi ⟩{\Vert }^2$$
• State Collapse: If outcome $a_n$ is observed, the state of the system immediately after the measurement is the normalized projection of $|\psi ⟩$ onto the corresponding eigenspace: $$|{\psi }^{\prime }⟩=\frac{E_n|\psi ⟩}{\Vert E_n|\psi ⟩\Vert }$$
• Completeness: The projectors sum to the identity, ${\sum }_n{E}_n=I$, which ensures that the probabilities sum to 1.

The following video provides a more hands-on, operational view of these rules using bra-ket notation.

2. Measurement in Different Bases

The choice of observable determines the basis in which the measurement is performed.

Computational Basis Measurement

The most common measurement is in the computational basis {$|\text{0}⟩,|\text{1}⟩$}. This corresponds to measuring the Pauli $Z$ operator, ${\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$.

• Eigenvalues: $+1$ (for state $|0⟩$) and $-1$ (for state $|1⟩$).
• Projectors: $E_0=|0⟩⟨0|$ and $E_1=|1⟩⟨1|$.

For a general qubit state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$:

• Prob(0) = $⟨\psi |0⟩⟨0|\psi ⟩=|⟨0|\psi ⟩{|}^2=|\alpha {|}^2$.
• Prob(1) = $⟨\psi |1⟩⟨1|\psi ⟩=|⟨1|\psi ⟩{|}^2=|\beta {|}^2$.

Arbitrary Basis Measurement

Measuring in an "arbitrary basis" simply means measuring an observable other than $Z$. For example, let's consider measuring in the $X$-basis, which corresponds to the Pauli $X$ operator, ${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.

• Eigenstates: $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$ and $|-⟩=\frac{1}{\sqrt{2}}(|0⟩-|1⟩)$.
• Eigenvalues: $+1$ and $-1$, respectively.
• Projectors: $E_{+}=|+⟩⟨+|$ and $E_{-}=|-⟩⟨-|$.

To find the probabilities, we must express our state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ in this new basis. We can do this by inverting the definitions of $|+⟩$ and $|-⟩$:
$|0⟩=\frac{1}{\sqrt{2}}(|+⟩+|-⟩)$
$|1⟩=\frac{1}{\sqrt{2}}(|+⟩-|-⟩)$

Substituting these into $|\psi ⟩$:

$$|\psi ⟩=\alpha \frac{1}{\sqrt{2}}(|+⟩+|-⟩)+\beta \frac{1}{\sqrt{2}}(|+⟩-|-⟩)=\frac{\alpha +\beta }{\sqrt{2}}|+⟩+\frac{\alpha -\beta }{\sqrt{2}}|-⟩$$

Now, applying the Born rule in the X-basis:

• Prob(+) = $|⟨+|\psi ⟩{|}^2={\left|\frac{\alpha +\beta }{\sqrt{2}}\right|}^2$
• Prob(-) = $|⟨-|\psi ⟩{|}^2={\left|\frac{\alpha -\beta }{\sqrt{2}}\right|}^2$

This demonstrates a key principle: the measurement statistics depend entirely on the basis (i.e., the observable) you choose. To fully characterize an unknown quantum state, you need to perform measurements in multiple, non-commuting bases.

3. Generalization to Mixed States and Density Matrices

The formalism above works perfectly for pure states. However, we often deal with systems that are in a mixed state—either because they are part of a larger entangled system, or because they are prepared with some classical uncertainty. In these cases, the state is described by a density matrix, $\rho$.

The measurement rules are elegantly generalized by replacing the outer product $|\psi ⟩⟨\psi |$ with the density matrix $\rho$.

From this, we arrive at the generalized measurement rules for a system in state $\rho$:

• Probability: The probability of obtaining outcome $a_n$ (associated with projector $E_n$) is:

  $$\text{Prob}(a_n)=\text{Tr}(E_n\rho )$$

  Note that since $\text{Tr}(AB)=\text{Tr}(BA)$, this is equivalent to $\text{Tr}(\rho E_n)$. For a pure state $\rho =|\psi ⟩⟨\psi |$, this reduces to $\text{Tr}(|\psi ⟩⟨\psi |E_n)=⟨\psi |E_n|\psi ⟩$, recovering the original Born rule.

• Post-Measurement State: If outcome $a_n$ is observed, the new density matrix ${\rho }^{\prime }$ is:

  $${\rho }^{\prime }=\frac{E_n\rho E_n}{\text{Tr}(E_n\rho )}$$

4. Generalized Measurements: POVMs

Projective measurements are powerful, but they are not the most general type of measurement allowed by quantum mechanics. The most general framework is the Positive Operator-Valued Measure (POVM).

A POVM is described by a set of measurement operators $\{M_m\}$ that are not necessarily projectors. The only requirement is that they satisfy a completeness relation:

$$\sum _m{M}_m^{†}{M}_m=I$$

The operators $E_m=M_m^{†}{M}_m$ are called the POVM elements. They are positive semi-definite operators that sum to the identity, but they are not required to be orthogonal projectors ($E_m{E}_k\ne {\delta }_{mk}{E}_m$).

The measurement rules are a natural extension of the projective case:

• Probability: The probability of obtaining outcome $m$ is: $$\text{Prob}(m)=\text{Tr}(E_m\rho )=\text{Tr}(M_m^{†}{M}_m\rho )$$
• Post-Measurement State: If outcome $m$ is observed, the new state is: $${\rho }^{\prime }=\frac{M_m\rho M_m^{†}}{\text{Tr}(M_m^{†}{M}_m\rho )}$$

POVMs are essential for many tasks in quantum information, such as optimally distinguishing non-orthogonal states.

The following video provides an excellent and comprehensive introduction to this general formalism.

A key insight, known as Neumark's theorem, is that any generalized measurement (POVM) on a system can be realized as a standard projective measurement on a larger system, which consists of the original system plus an auxiliary system (an "ancilla"). This provides a deep connection between the two formalisms. For a more mathematical treatment of this, you can refer to Section 3.4.2 in the notes "Density Matrices and Quantum Operations" by Chi-Kwang Li et al. (d126b).

Conclusion

In this lesson, we have built a comprehensive picture of quantum measurement, moving from the simple case of projective measurements on pure states to the fully general POVM formalism for mixed states.

Key Takeaways:

• Projective Measurement: Defined by an observable (Hermitian operator). Outcomes are eigenvalues, and the state collapses to the corresponding eigenstate.
• Arbitrary Bases: Measuring in a different basis corresponds to choosing a different observable. The probabilities of outcomes depend on the representation of the state in the chosen measurement basis.
• Density Matrix Formalism: This generalizes measurement to mixed states. The probability of outcome $n$ is given by $\text{Prob}(n)=\text{Tr}(E_n\rho )$, and the post-measurement state is ${\rho }^{\prime }=E_n\rho E_n/\text{Tr}(E_n\rho )$.
• POVMs: The most general description of measurement. They are defined by a set of positive operators $\{E_m\}$ that sum to identity. They are not restricted to being orthogonal projectors and are crucial for tasks like optimal state discrimination.

Preview of the Next Lesson:

We have now seen how the density matrix arises naturally when considering a subsystem of a larger entangled state. Entanglement is perhaps the most profound feature of quantum mechanics and the primary resource that powers many quantum algorithms. In the next lesson, we will focus directly on it. We will construct the Bell states and demonstrate their entangled nature by calculating their Schmidt decomposition and von Neumann entropy.