GlobalPhase

 Trying to understand the notion that global phase shifts make no difference to

measurements.

META

Quantum mechanics; what is a global phase

In quantum mechanics, a global phase refers to a complex phase factor that is applied to a wave function or a state vector. It is a mathematical representation of the phase of a quantum system.

Mathematically, a global phase can be represented as:

e^{iα}

where α is a real number and i is the imaginary unit.

Physically, a global phase has no observable effects on a quantum system. It does not change the probabilities of measurement outcomes or the expectation values of observables.

Example: Consider a wave function ψ(x). If we multiply it by a global phase e^{iα}, the resulting wave function e^{iα}ψ(x) represents the same physical state as the original wave function ψ(x).

Importantly, relative phases between different components of a wave function or between different states are physically significant, but a global phase is not.

I hope this helps! Let me know if you have further questions.

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If ς can take any value, how does this not change the reading of other observables

If ς (the global phase) can take any value, it might seem like it should affect the readings of other observables. However, the key point is that the global phase cancels out when calculating expectation values or probabilities.

Here's why:

Expectation values: When calculating the expectation value of an observable A, we use the formula:

⟨A⟩ = ∫ψ*(x)Aψ(x)dx

If we multiply ψ(x) by a global phase e^{iς}, the expectation value becomes:

⟨A⟩ = ∫(e^{-iς}ψ*(x))A(e^{iς}ψ(x))dx = ∫ψ*(x)Aψ(x)dx

The global phase terms e^{iς} and e^{-iς} cancel out, leaving the expectation value unchanged.

Probabilities: When calculating probabilities, we use the square of the absolute value of the wave function:

P(x) = |ψ(x)|²

If we multiply ψ(x) by a global phase e^{iς}, the probability becomes:

P(x) = |e^{iς}ψ(x)|² = |ψ(x)|²

Again, the global phase term cancels out, leaving the probability unchanged.

In summary, the global phase ς can take any value without affecting the readings of other observables because it cancels out when calculating expectation values and probabilities. This is a fundamental property of quantum mechanics, ensuring that global phases do not have physical consequences.

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mathematics; why is the modulus of a complex exponential always equal to 1

The modulus (or magnitude) of a complex exponential e^{ix} is always equal to 1 because of the following reason:

Euler's formula:

e^{ix} = cos(x) + i sin(x)

Modulus definition:

The modulus |z| of a complex number z = a + bi is defined as:

|z| = √(a² + b²)

Applying to e^{ix}:

|e^{ix}| = √(cos²(x) + sin²(x))

Using the Pythagorean identity:

cos²(x) + sin²(x) = 1 (for all real x)

Therefore:

|e^{ix}| = √1 = 1

So, the modulus of a complex exponential e^{ix} is always equal to 1, regardless of the value of x. This is a fundamental property of complex exponentials, which makes them useful for representing rotations and phases in many areas of mathematics and physics.

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why is it called the modulus

The term "modulus" comes from the Latin word "modulus," meaning "measure" or "size." In mathematics, the modulus of a complex number is a measure of its distance from the origin (0) in the complex plane.

Historical context:

In the 18th century, mathematicians like Euler and Gauss used the term "modulus" to describe the absolute value or magnitude of a complex number. They borrowed the term from geometry, where the modulus of a line segment or a vector referred to its length or magnitude.

Analogies:

Length of a vector: The modulus of a complex number can be thought of as the length of a vector in the complex plane.

Magnitude of a quantity: In physics, the modulus of a quantity (like velocity or acceleration) represents its magnitude or size.

Absolute value: The modulus of a real number is simply its absolute value, which measures its distance from 0 on the number line.

In summary, the term "modulus" was chosen because it represents a measure of size or magnitude, which is a fundamental property of complex numbers. This terminology has been adopted across various mathematical and scientific fields to describe the absolute value or magnitude of a quantity.

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So a vector represented on the Bloch sphere remains the same

Yes, that's correct! A vector represented on the Bloch sphere remains the same, regardless of the global phase.

Bloch sphere representation:

A vector on the Bloch sphere represents a quantum state, typically a qubit (a two-level quantum system). The vector's direction and length encode the state's properties.

Global phase invariance:

When a global phase e^{iς} is applied to the quantum state, the vector on the Bloch sphere remains unchanged. This is because the global phase cancels out when calculating the expectation values of observables, as we discussed earlier.

Geometric interpretation:

Think of the Bloch sphere as a representation of the quantum state's "direction" in a high-dimensional space. The global phase is like a rotation around the sphere's axis, which doesn't change the vector's direction or length.

Consequence:

This invariance under global phase transformations has important implications for quantum computing and quantum information processing. It means that global phases can be ignored when manipulating quantum states, simplifying many calculations and operations.

I hope this helps! Let me know if you have further questions.

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Does the Bloch sphere representation correspond to physical space or abstract space

The Bloch sphere representation corresponds to an abstract space, specifically a mathematical representation of the quantum state's properties.

Abstract space:

The Bloch sphere is a three-dimensional abstract space, where each point on the sphere represents a unique quantum state. This space is not directly related to physical space, but rather provides a convenient and intuitive way to visualize and manipulate quantum states.

Mathematical structure:

The Bloch sphere is a representation of the complex projective space CP^1, which is a mathematical construct used to describe the state space of a qubit. This space has a well-defined geometric and algebraic structure, allowing for efficient calculations and transformations.

Physical interpretation:

While the Bloch sphere itself is an abstract representation, the quantum states it represents correspond to physical properties, such as:

Spin orientation (in spin-1/2 systems)

Polarization (in photon systems)

Energy levels (in atomic systems)

These physical properties can be measured and manipulated in laboratory experiments, but the Bloch sphere representation provides a more abstract and mathematical framework for understanding and working with these properties.

In summary, the Bloch sphere representation is an abstract mathematical tool for visualizing and manipulating quantum states, which correspond to physical properties but don't directly represent physical space.

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Is that true for Hilbert spaces generally

Yes, that's true for Hilbert spaces generally. Hilbert spaces are abstract mathematical structures used to describe quantum states, wave functions, and linear operators. They provide a framework for working with quantum systems, but they don't directly represent physical space.

Hilbert spaces:

Hilbert spaces are complete inner product spaces, meaning they have:

A well-defined notion of distance (norm) and angle (inner product)

A complete set of basis vectors (orthonormal basis)

A way to combine vectors (linear combinations)

Abstract nature:

Hilbert spaces are abstract because they:

Don't rely on a specific coordinate system or basis

Can be infinite-dimensional

Can represent non-physical entities, like wave functions or density operators

Physical interpretation:

While Hilbert spaces themselves are abstract, the objects they represent, like wave functions or quantum states, have physical interpretations. For example:

Wave functions describe probability distributions in physical space

Quantum states correspond to physical properties, like energy or spin

In summary, Hilbert spaces are abstract mathematical frameworks for working with quantum systems, providing a powerful tool for calculations and analysis. The objects they represent, however, can have direct physical interpretations.

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Are spin_up and spin_down physical or abstract

Spin_up (|↑⟩) and spin_down (|↓⟩) are abstract representations of physical properties.

Physical property: spin:

Spin is a fundamental physical property of particles like electrons, protons, and neutrons. It's a measure of their intrinsic angular momentum.

Abstract representation:

Spin_up and spin_down are abstract vectors in a Hilbert space, used to represent the spin states of a particle. They form a basis for the spin state space, allowing us to describe and manipulate spin states mathematically.

Not direct physical entities:

Spin_up and spin_down aren't direct physical entities; they don't represent specific locations or directions in physical space. Instead, they represent the possible states of a particle's spin, which can be measured and manipulated in experiments.

Measurement outcomes:

When measuring spin, the outcome is either "up" or "down", corresponding to the abstract states |↑⟩ and |↓⟩. However, this doesn't mean the particle is physically "up" or "down" in space; it's just a way to describe the spin state.

In summary, spin_up and spin_down are abstract representations of physical spin states, allowing us to work with and understand spin properties mathematically, but they aren't direct physical entities themselves.

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COPILOT:

Are spin_up and spin_down physical or abstract

Spin-up and spin-down are physical properties of particles, specifically related to their intrinsic angular momentum, or spin. Here's a bit more detail:

### Physical Nature of Spin

- **Spin**: This is a fundamental property of particles, much like charge or mass. It describes the intrinsic angular momentum of a particle.

- **Spin-Up and Spin-Down**: These terms refer to the orientation of a particle's spin relative to an external magnetic field. For example, in the case of an electron, "spin-up" means the spin is aligned with the magnetic field, while "spin-down" means it is anti-aligned.

### Measurement and Observation

- **Physical Measurement**: The spin state of a particle can be measured using devices like Stern-Gerlach apparatus, which can physically separate particles based on their spin orientation.

- **Quantum Mechanics**: In quantum mechanics, spin states are represented by abstract mathematical objects (like vectors in a Hilbert space), but these representations correspond to measurable physical properties.

### Summary

While the mathematical description of spin states can be abstract, the properties of being spin-up or spin-down are very much physical, as they can be observed and measured in experiments.

If you have more questions or need further clarification, feel free to ask!

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The takeaway from all this. The Bloch sphere represents one qubit. Below, my two

vectors each on it's own sphere...

                                                                              

                                                              

I can visualize doing an inner oroduct of two vectors on this version of a qubit:

                                

                                                  source: IBM