Recap Moment

 Did a bit of a recap for myself, this morning:

According to Wikipedia, the Pauli matrices were defined in 1927 to describe

the spin of quantum particles.

                                                                   

*     *     I

From Meta AI:

Bloch Sphere:

The Bloch sphere is a graphical representation of a qubit's state as a point on the surface of a sphere. It's a useful tool for visualizing quantum states and transformations.

Z-Gate:

The Z-gate, also known as the phase gate, is a quantum gate that applies a phase shift to a qubit. It's represented by the matrix:

Z = [1 0; 0 -1]

Global Transformation:

A global transformation is a transformation that affects the entire system, as opposed to a local transformation, which only affects a subsystem.

Applying Z-Gate to |1〉:

When you apply the Z-gate to the qubit state |1〉, it results in a phase shift:

Z|1〉 = -|1〉

This transformation is a global transformation because it changes the overall phase of the qubit, which is a global property of the system.

In the Bloch sphere representation, the Z-gate applied to |1〉 would correspond to a rotation around the z-axis, which is a global transformation that changes the orientation of the entire sphere.

So, yes, applying the Z-gate to a |1〉 qubit is an instance of a global transformation!

Global Phase Shift Consequences:

A global phase shift, like the one caused by the Z-gate on the |1〉 state, has some interesting consequences:

No measurable effects: A global phase shift has no measurable effects on the qubit's behavior. The probabilities of measurement outcomes remain unchanged.

Relative phases matter: While the global phase shift itself is unimportant, relative phases between different states are crucial. This is because quantum mechanics relies on interference between states, which depends on relative phases.

Quantum interference: Global phase shifts can cancel out or enhance quantum interference patterns, depending on the context.

Quantum computing: In quantum computing, global phase shifts are often ignored, as they don't affect the computation's outcome. However, relative phases between qubits are essential for quantum algorithms.

Quantum error correction: Global phase shifts can be a challenge for quantum error correction, as they can accumulate and affect the accuracy of quantum computations.

In summary, while a global phase shift may seem like a trivial transformation, it can have significant consequences in the context of quantum interference, computing, and error correction.

Let me know if you'd like me to elaborate on any of these points!

The phase shift caused by the Z-gate on the |1〉 state is a phase shift of 180 degrees or π radians.

To see why, recall that the Z-gate multiplies the |1〉 state by a factor of -1:

Z|1〉 = -|1〉

This can be represented as:

e^{iπ}|1〉 = -|1〉

The factor e^{iπ} represents a phase shift of 180 degrees or π radians.

So, the Z-gate applies a 180-degree phase shift to the |1〉 state!

*     *     *

Neat, qubit(0) has 50% probability, with phase +180 degrees on the  |0>outcome.

The phase shift is on |1> for qubit(1).