What is the relationship between the spread in the standard basis and the spread in the sign basis? How does the uncertainty principle for spreads in these bases relate to the bit value and sign value of a qubit?

/ / Published in Quantum Information, EITC/QI/QIF Quantum Information Fundamentals, Introduction to Quantum Information, Uncertainty principle, Examination review

The relationship between the spread in the standard basis and the spread in the sign basis is a fundamental concept in quantum information theory. To understand this relationship, we must first define what we mean by "spread" in these bases.

In quantum mechanics, the state of a qubit can be represented as a superposition of two basis states, commonly referred to as the standard basis states |0⟩ and |1⟩. The spread in the standard basis refers to the uncertainty or variability in measuring the qubit's state in terms of these basis states. Mathematically, this spread can be quantified using the standard deviation of the probabilities associated with each basis state.

Similarly, the state of a qubit can also be represented in the sign basis, which consists of the basis states |+⟩ and |−⟩. The spread in the sign basis refers to the uncertainty or variability in measuring the qubit's state in terms of these basis states. Again, this spread can be quantified using the standard deviation of the probabilities associated with each basis state.

Now, the uncertainty principle for spreads in these bases relates to the bit value and sign value of a qubit. The bit value of a qubit refers to the probability of measuring the qubit in the standard basis state |1⟩, while the sign value refers to the probability of measuring the qubit in the sign basis state |−⟩. The uncertainty principle states that the product of the spreads in the standard and sign bases is bounded by a minimum value.

Mathematically, let σ_std and σ_sign represent the spreads in the standard and sign bases, respectively. The uncertainty principle can be expressed as:

σ_std * σ_sign ≥ 1/2

This inequality implies that the more precisely we know the bit value of a qubit (i.e., the smaller the spread in the standard basis), the less precisely we can know its sign value (i.e., the larger the spread in the sign basis), and vice versa.

To illustrate this relationship, consider a qubit in the state |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes. If we measure this qubit in the standard basis, the probability of obtaining the outcome |0⟩ is |α|^2, and the probability of obtaining the outcome |1⟩ is |β|^2. The spread in the standard basis can then be calculated as:

σ_std = sqrt(|α|^2 * (1 – |α|^2) + |β|^2 * (1 – |β|^2))

Similarly, if we measure the qubit in the sign basis, the probability of obtaining the outcome |+⟩ is |α|^2 + |β|^2, and the probability of obtaining the outcome |−⟩ is |α|^2 – |β|^2. The spread in the sign basis can be calculated as:

σ_sign = sqrt((|α|^2 + |β|^2) * (1 – |α|^2 – |β|^2) + (|α|^2 – |β|^2) * (1 – |α|^2 + |β|^2))

By applying the uncertainty principle inequality, we can see that as one spread decreases, the other spread must increase to satisfy the inequality.

The relationship between the spread in the standard basis and the spread in the sign basis is governed by the uncertainty principle. The more precisely we know the bit value of a qubit, the less precisely we can know its sign value, and vice versa. This relationship is quantified by the product of the spreads in these bases, which is bounded by a minimum value according to the uncertainty principle.

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