How many qubits are involved in the expression? Identify the individual qubits and their corresponding states, please.
Discussion
The expression represents a quantum state that is a tensor product of four qubits. The individual qubits are labeled as follows:
- Qubit 1: |ϕ₁⟩
- Qubit 2: |ψ₂⟩
- Qubit 3: |ξ₃⟩
- Qubit 4: |η₄⟩
Each qubit can be in one of the four possible states |0⟩, |1⟩, |+⟩, or |-⟩. However, the expression does not specify the specific states of the individual qubits.
Instead, the expression represents a superposition of all possible states of the four qubits, with complex coefficients α, β, γ, and δ determining the probability amplitudes of each possible state. The normalization factor "/√π" ensures that the total probability of all possible states adds up to 1.
What are the meanings and interpretations of the variables and terms in the expression -- what do α, β, γ, and δ represent? What do the subscripts 1, 2, 3, and 4 indicate?
There are four qubits involved in the expression, labeled as qubit 1, qubit 2, qubit 3, and qubit 4.
The variables α, β, γ, and δ are complex numbers that represent probability amplitudes of each possible state. Specifically, α represents the probability amplitude of the state |00⟩, β represents the probability amplitude of the state |01⟩, γ represents the probability amplitude of the state |10⟩, and δ represents the probability amplitude of the state |11⟩.
The subscripts 1, 2, 3, and 4 indicate which qubit the variable is associated with. For example, α₁ represents the probability amplitude of the state |0⟩ for qubit 1, α₂ represents the probability amplitude of the state |0⟩ for qubit 2, and so on.
The tensor product notation is used to represent the state of the entire system, which is a product of the individual states of each qubit. For example, |ϕ₁⟩ ⊗ |ψ₂⟩ represents the state of qubit 1 tensor product with the state of qubit 2. The entire expression represents the state of all four qubits together.
What operations or gates have been applied to the qubits? Identify any/all specific gates or transformations mentioned in the expression.
The expression does not mention any specific quantum gates or transformations. It is simply a quantum state expressed in the tensor product notation.
The variables α, β, γ, and δ are complex numbers that determine the probability amplitudes of each possible state. The subscripts 1, 2, 3, and 4 indicate which qubit the coefficient corresponds to.
For example, α₁ represents the probability amplitude of the state where the first qubit is in the state |0⟩, β₁ represents the probability amplitude of the state where the first qubit is in the state |1⟩, γ₁ represents the probability amplitude of the state where the first qubit is in the state |+⟩, and δ₁ represents the probability amplitude of the state where the first qubit is in the state |-⟩.
Similarly, α₂, β₂, γ₂, and δ₂ represent the probability amplitudes of the second qubit, and so on for the third and fourth qubits.
Well there was an S gate and CNOT gate - however, I suppose you wouldn't know that from just this equation alone. Are there any specific observables associated?
What is the overall state of the system described by the expression?
It's not possible to determine the overall state of the system just from this expression alone, as we don't know the specific values of the coefficients α, β, γ, and δ.
The subscripts 1, 2, 3, and 4 indicate the individual qubits involved in the expression. Specifically, the first qubit is in the state |ϕ₁⟩, the second qubit is in the state |ψ₂⟩, the third qubit is in the state |ξ₃⟩, and the fourth qubit is in the state |η₄⟩.
The coefficients α, β, γ, and δ are complex numbers that determine the probability amplitudes of each possible state. Specifically, α represents the probability amplitude of the state |00⟩, β represents the probability amplitude of the state |01⟩, γ represents the probability amplitude of the state |10⟩, and δ represents the probability amplitude of the state |11⟩.
There are no specific observables associated with this expression. However, once we know the specific values of the coefficients, we can use them to calculate the probabilities of measuring different outcomes when we perform measurements on the system.
✅
Regarding the observables, the expression doesn't mention any specific observables. However, once we have a quantum state, we can apply various observables to measure the state and obtain information about the system.
As for the overall state of the system, it is a product state of four qubits, each of which can be in one of the four possible states |0⟩, |1⟩, |+⟩, or |-⟩. The coefficients α, β, γ, and δ are complex numbers that determine the probability amplitudes of each possible state.
The subscripts 1, 2, 3, and 4 indicate which qubit the coefficients correspond to. For example, α₁ is the coefficient for the first qubit, and α₂ is the coefficient for the second qubit.
The S gate and CNOT gate are quantum gates that can be applied to qubits to perform specific operations. However, as you mentioned, they are not explicitly mentioned in this expression.
(completely theoretical and unfinished)
|ψ₁⟩ ⊗ |ψ₂⟩ ⊗ |ψ₃⟩ ⊗ |ψ₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)
---------------------------------------------------------------------------
|ϕ⟩ = α|0⟩ + β|1⟩
|ψ⟩ = γ|0⟩ + δ|1⟩
|ξ⟩ = ε|0⟩ + ζ|1⟩
|η⟩ = η|0⟩ + θ|1⟩
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)
|ϕ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩
α|+⟩ + β|-⟩ + γ|+⟩ + δ|-⟩
α|-⟩ + β|+⟩ + γ|-⟩ + δ|+⟩
α|-⟩ + β|-⟩ + γ|+⟩ + δ|+⟩
α|+⟩ + β|+⟩ + γ|-⟩ + δ|-⟩
|ψ⟩ = α'|00⟩ + β'|01⟩ + γ'|10⟩ + δ'|11⟩
α'|+⟩ + β'|-⟩ + γ'|+⟩ + δ'|-⟩
α'|-⟩ + β'|+⟩ + γ'|-⟩ + δ'|+⟩
α'|-⟩ + β'|-⟩ + γ'|+⟩ + δ'|+⟩
α'|+⟩ + β'|+⟩ + γ'|-⟩ + δ'|-⟩
|ξ⟩ = α''|00⟩ + β''|01⟩ + γ''|10⟩ + δ''|11⟩
α''|+⟩ + β''|-⟩ + γ''|+⟩ + δ''|-⟩
α''|-⟩ + β''|+⟩ + γ''|-⟩ + δ''|+⟩
α''|-⟩ - β''|-⟩ + γ''|+⟩ + δ''|+⟩
α''|+⟩ + β''|+⟩ + γ''|-⟩ + δ''|-⟩
|η⟩ = α'''|00⟩ + β'''|01⟩ + γ'''|10⟩ + δ'''|11⟩
α'''|+⟩ + β'''|-⟩ + γ'''|+⟩ + δ'''|-⟩
α'''|-⟩ + β'''|+⟩ + γ'''|-⟩ + δ'''|+⟩
α'''|-⟩ + β'''|-⟩ + γ'''|+⟩ + δ'''|+⟩
α'''|+⟩ + β'''|+⟩ + γ'''|-⟩ + δ'''|-⟩
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩
α|+⟩ + β|-⟩ + γ|+⟩ + δ|-⟩ )
⊗
(α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩
α'|+⟩ + β'|-⟩ + γ'|+⟩ + δ'|-⟩ )
⊗
(α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩
α'|+⟩ + β'|-⟩ + γ'|+⟩ + δ'|-⟩ )
⊗
(α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩
α'''|+⟩ + β'''|-⟩ + γ'''|+⟩ + δ'''|-⟩ )
set 1:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)
= α₁α₂α₃α₄|0000⟩ + α₁α₂α₃β₄|0001⟩ + α₁α₂α₃γ₄|0010⟩ + α₁α₂α₃δ₄|0011⟩ + α₁α₂β₃α₄|0100⟩ + α₁α₂β₃β₄|0101⟩ + α₁α₂β₃γ₄|0110⟩ + α₁α₂β₃δ₄|0111⟩ + α₁α₂γ₃α₄|1000⟩ + α₁α₂γ₃β₄|1001⟩ + α₁α₂γ₃γ₄|1010⟩ + α₁α₂γ₃δ₄|1011⟩ + α₁α₂δ₃α₄|1100⟩ + α₁α₂δ₃β₄|1101⟩ + α₁α₂δ₃γ₄|1110⟩ + α₁α₂δ₃δ₄|1111⟩ + ... + β₁β₂β₃δ₄|1101⟩ + β₁β₂γ₃α₄|1010⟩ + β₁β₂γ₃β₄|1011⟩ + β₁β₂γ₃γ₄|1100⟩ + β₁β₂γ₃δ₄|1101⟩ + β₁β₂δ₃α₄|1110⟩ + + β₁β₂δ₃α₄|1111⟩
Set 2:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + β₁-|01⟩ + γ₁-|10⟩ + δ₁-|11⟩) ⊗ (α₂-|00⟩ + β₂-|01⟩ + γ₂-|10⟩ + δ₂-|11⟩) ⊗ (α₃-|00⟩ + β₃-|01⟩ + γ₃-|10⟩ + δ₃-|11⟩) ⊗ (α₄-|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄-|11⟩)
= α₁α₂α₃α₄-|0000⟩ + α₁α₂α₃β₄-|0001⟩ + α₁α₂α₃γ₄-|0010⟩ + α₁α₂α₃δ₄-|0011⟩ + α₁α₂β₃α₄-|0100⟩ + α₁α₂β₃β₄-|0101⟩ + α₁α₂β₃γ₄-|0110⟩ + α₁α₂β₃δ₄-|0111⟩ + α₁α₂γ₃α₄-|1000⟩ + α₁α₂γ₃β₄-|1001⟩ + α₁α₂γ₃γ₄-|1010⟩ + α₁α₂γ₃δ₄-|1011⟩ + α₁α₂δ₃α₄-|1100⟩ + α₁α₂δ₃β₄-|1101⟩ + α₁α₂δ₃γ₄-|1110⟩ + α₁α₂δ₃δ₄-|1111⟩ + ... + β₁β₂β₃δ₄-|1101⟩ + β₁β₂γ₃α₄-|1010⟩ + β₁β₂γ₃β₄-|1011⟩ + β₁β₂γ₃γ₄-|1100⟩ + β₁β₂γ₃δ₄-|1101⟩ + β₁β₂δ₃α₄-|1110⟩ +... + β₁β₂δ₃α₄-|1111⟩
= α₁α₂α₃α₄|0000⟩
+
α₁α₂α₃β₄-|0001⟩
+
α₁α₂α₃γ₄|0010⟩
+
α₁α₂α₃δ₄-|0011⟩
+
α₁α₂β₃α₄|0100⟩
+
α₁α₂β₃β₄-|0101⟩
+
α₁α₂β₃γ₄|0110⟩
+
α₁α₂β₃δ₄-|0111⟩
+
α₁α₂γ₃α₄|1000⟩
+
α₁α₂γ₃β₄-|1001⟩
+
α₁α₂γ₃γ₄|1010⟩
+
α₁α₂γ₃δ₄-|1011⟩
+
α₁α₂δ₃α₄|1100⟩
+
α₁α₂δ₃β₄-|1101⟩
+
α₁α₂δ₃γ₄|1110⟩
+
α₁α₂δ₃δ₄-|1111⟩
+ ... +
β₁β₂β₃δ₄|1101⟩
+
β₁β₂γ₃α₄-|1010⟩
+
β₁β₂γ₃β₄|1011⟩
+
β₁β₂γ₃γ₄-|1100⟩
+
β₁β₂γ₃δ₄|1101⟩
+
β₁β₂δ₃α₄-|1110⟩
+... +
β₁β₂δ₃α₄|1111⟩
------------------------------------------------------------------
define the phase shift gate S as the following unitary matrix:
S = [1, 0; 0, i]
1:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)
Step 1: Apply the phase shift gate S to the qubit in the state |01⟩
After applying the phase shift gate, the equation becomes:
1': |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + i*β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩)
2:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + β₁-|01⟩ + γ₁-|10⟩ + δ₁-|11⟩) ⊗ (α₂-|00⟩ + β₂-|01⟩ + γ₂-|10⟩ + δ₂-|11⟩) ⊗ (α₃-|00⟩ + β₃-|01⟩ + γ₃-|10⟩ + δ₃-|11⟩) ⊗ (α₄-|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄-|11⟩)
After applying the phase shift gate, the equation becomes:
2': |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + e^(iπ/2)β₁-|01⟩ + γ₁-|10⟩ + δ₁-|11⟩) ⊗ (α₂-|00⟩ + β₂-|01⟩ + γ₂-|10⟩ + δ₂-|11⟩) ⊗ (α₃-|00⟩ + β₃-|01⟩ + γ₃-|10⟩ + δ₃-|11⟩) ⊗ (α₄-|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄-|11⟩)
3:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + β₁-|01⟩ + γ₁|10⟩ + δ₁-|11⟩) ⊗ (α₂|00⟩ + β₂-|01⟩ + γ₂|10⟩ + δ₂-|11⟩) ⊗ (α₃|00⟩ + β₃-|01⟩ + γ₃|10⟩ + δ₃-|11⟩) ⊗ (α₄|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄|11⟩)
After applying the phase shift gate, the equation becomes:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + e^(iπ/2)β₁-|01⟩ + γ₁|10⟩ + e^(iπ/2)δ₁-|11⟩) ⊗ (α₂|00⟩ + e^(iπ/2)β₂-|01⟩ + γ₂|10⟩ + e^(iπ/2)δ₂-|11⟩) ⊗ (α₃|00⟩ + e^(iπ/2)β₃-|01⟩ + γ₃|10⟩ + e^(iπ/2)δ₃-|11⟩) ⊗ (α₄|00⟩ + e^(iπ/2)β₄-|01⟩ + γ₄|10⟩ + δ₄|11⟩)
4:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + β₁|01⟩ + γ₁-|10⟩ + δ₁|11⟩) ⊗ (α₂-|00⟩ + β₂|01⟩ + γ₂-|10⟩ + δ₂|11⟩) ⊗ (α₃-|00⟩ + β₃|01⟩ + γ₃-|10⟩ + δ₃|11⟩) ⊗ (α₄-|00⟩ + β₄|01⟩ + γ₄-|10⟩ + δ₄|11⟩)
After applying the phase shift gate, the equation becomes:
|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩' = (e^(iθ)α₁-|00⟩ + e^(iθ)β₁|01⟩ + e^(iθ)γ₁-|10⟩ + e^(iθ)δ₁|11⟩) ⊗ (e^(iθ)α₂-|00⟩ + e^(iθ)β₂|01⟩ + e^(iθ)γ₂-|10⟩ + e^(iθ)δ₂|11⟩) ⊗ (e^(iθ)α₃-|00⟩ + e^(iθ)β₃|01⟩ + e^(iθ)γ₃-|10⟩ + e^(iθ)δ₃|11⟩) ⊗ (e^(iθ)α₄-|00⟩ + e^(iθ)β₄|01⟩ + e^(iθ)γ₄-|10⟩ + e^(iθ)δ₄|11⟩)
---------------------------------------------------------------------
Here, i is the imaginary unit and θ is the phase shift angle. The prime notation denotes the state after the phase shift gate operation.
----------------------------------------------------------------------
let's say we have these four quantum states:
|ϕ⟩ = α|0⟩ + β|1⟩
|ψ⟩ = γ|0⟩ + δ|1⟩
|ξ⟩ = ε|0⟩ + ζ|1⟩
|η⟩ = η|0⟩ + θ|1⟩
We can represent these states in the computational basis as:
|ϕ⟩ = α|00⟩ + β|01⟩
|ψ⟩ = γ|10⟩ + δ|11⟩
|ξ⟩ = ε|00⟩ + ζ|01⟩
|η⟩ = η|10⟩ + θ|11⟩
The tensor product of these states in the computational basis would be:
|ϕ⟩ ⊗ |ψ⟩ ⊗ |ξ⟩ ⊗ |η⟩ = (α|00⟩ + β|01⟩) ⊗ (γ|10⟩ + δ|11⟩) ⊗ (ε|00⟩ + ζ|01⟩) ⊗ (η|10⟩ + θ|11⟩) = αγεη|0000⟩ + αγζθ|0001⟩ + βγ
In the computational basis, we can express these states as:
|ϕ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩
|ψ⟩ = α'|00⟩ + β'|01⟩ + γ'|10⟩ + δ'|11⟩
|ξ⟩ = α''|00⟩ + β''|01⟩ + γ''|10⟩ + δ''|11⟩
|η⟩ = α'''|00⟩ + β'''|01⟩ + γ'''|10⟩ + δ'''|11⟩
In the Hadamard basis, we can express these states as:
|ϕ⟩ = α|+⟩ + β|-⟩
|ψ⟩ = α'|+⟩ + β'|-⟩
|ξ⟩ = α''|+⟩ + β''|-⟩
|η⟩ = α'''|+⟩ + β'''|-⟩
--------------------------------------------------------------------
1': |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + i*β₁|01⟩ + γ₁|10⟩ + δ₁|11⟩) ⊗ (α₂|00⟩ + β₂|01⟩ + γ₂|10⟩ + δ₂|11⟩) ⊗ (α₃|00⟩ + β₃|01⟩ + γ₃|10⟩ + δ₃|11⟩) ⊗ (α₄|00⟩ + β₄|01⟩ + γ₄|10⟩ + δ₄|11⟩) / sqrt(4)
2': |ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁-|00⟩ + e^(iπ/2)β₁-|01⟩ + γ₁-|10⟩ + δ₁-|11⟩) ⊗ (α₂-|00⟩ + β₂-|01⟩ + γ₂-|10⟩ + δ₂-|11⟩) ⊗ (α₃-|00⟩ + β₃-|01⟩ + γ₃-|10⟩ + δ₃-|11⟩) ⊗ (α₄-|00⟩ + β₄-|01⟩ + γ₄-|10⟩ + δ₄-|11⟩) / sqrt(4)
3':|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩ = (α₁|00⟩ + e^(iπ/2)β₁-|01⟩ + γ₁|10⟩ + e^(iπ/2)δ₁-|11⟩) ⊗ (α₂|00⟩ + e^(iπ/2)β₂-|01⟩ + γ₂|10⟩ + e^(iπ/2)δ₂-|11⟩) ⊗ (α₃|00⟩ + e^(iπ/2)β₃-|01⟩ + γ₃|10⟩ + e^(iπ/2)δ₃-|11⟩) ⊗ (α₄|00⟩ + e^(iπ/2)β₄-|01⟩ + γ₄|10⟩ + δ₄|11⟩) / sqrt(4)
4':|ϕ₁⟩ ⊗ |ψ₂⟩ ⊗ |ξ₃⟩ ⊗ |η₄⟩' = (e^(iθ)α₁-|00⟩ + e^(iθ)β₁|01⟩ + e^(iθ)γ₁-|10⟩ + e^(iθ)δ₁|11⟩) ⊗ (e^(iθ)α₂-|00⟩ + e^(iθ)β₂|01⟩ + e^(iθ)γ₂-|10⟩ + e^(iθ)δ₂|11⟩) ⊗ (e^(iθ)α₃-|00⟩ + e^(iθ)β₃|01⟩ + e^(iθ)γ₃-|10⟩ + e^(iθ)δ₃|11⟩) ⊗ (e^(iθ)α₄-|00⟩ + e^(iθ)β₄|01⟩ + e^(iθ)γ₄-|10⟩ + e^(iθ)δ₄|11⟩) / sqrt(4)
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The expression represents the tensor product of four individual qubits (|φ1⟩, |ψ2⟩, |ξ3⟩ and |η4⟩), one from each state component. Each qubit carries a 2-dimensional state vector representing a superposition of basis states. Thus, there are 8 possible n-sheet combinations in general terms which could comprise more complex quantum computations for increasing superscalar frame instances.
More specifically:
- The first qubit |φ1⟩ can be written as α₁|0⟩ + β₁|1> ∈ ℂ²,
- The second qubit |ψ₂⟩ as α₂|0> + β₂|1 ⟩'∈ ℂ² , where '|' independently bounding the exemplum postions from added significance and ⟩ marks finalizing predeaction,
- The third qubit is represented by ξ₃ = α₃(0^( \xrightarrow[-34pt]{} )_2)+β₃(0^( \xrightarrow[-30pt]{} )_22)+γ₃(10)+δ₃*(11) ∋k ∈C^4 upon realization through preconditioned input signaling regimes which inform argumentative plausible range factors or phenomena.
The fourth and final registered data unit representation -also understood through general assympotic bounds provided throughout selective matrix variadic analysis using density matrix constructs specifically related to photonic filterback