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Let's say that I have a 4-level quantum state, which is described by a linear combination of the following four eigenbases:

$$|\text{red}⟩ = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} , |\text{blue}⟩ = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} , |\text{green}⟩ = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} , |\text{yellow}⟩ = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}\,.$$

Hence; $|\psi⟩ = \alpha|\text{red}⟩ + \beta|\text{blue}⟩+ \gamma|\text{green}⟩ + \delta|\text{yellow}⟩$ represents an arbitrary quantum state.

What would the Dirac notation of a maximally entangled state of 2 such 4-level quantum states look like? Also, what operations would need to be performed in order to prepare such a state? Lastly, how can one check if an arbitrary state in such an eigenbasis is an entangled state or not?

FDGod
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Ritwik Garg
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  • These are exactly the computational basis we generally use. ${ |0\rangle, |1\rangle, |2\rangle, |3\rangle } \equiv { |00\rangle, |01\rangle, |10\rangle, |11\rangle } \equiv { |\text{red}\rangle, |\text{blue}\rangle, |\text{green}\rangle, |\text{orange}\rangle } $. – FDGod Oct 28 '23 at 22:00
  • Right, but this only describes one quantum system right? Now if I have another quantum system with the same basis, how can I create a maximally entangled state between both of them? – Ritwik Garg Oct 28 '23 at 22:06
  • These basis are sufficient to describe two qubits. I am not sure exactly what your question is. One of the maximally entangled state is just $$ |\psi \rangle = \frac{1}{\sqrt{2}} \bigg( | \text{red} \rangle \otimes | \text{red} \rangle + |\text{yellow} \rangle \otimes | \text{yellow} \rangle \bigg) $$ – FDGod Oct 28 '23 at 22:09
  • Are you saying that you want a maximally entangled state between two systems of size $\mathbb{C}^4$ each? i.e., maximally entangled state you are looking for $\in \mathbb{C}^{16} $? – FDGod Oct 28 '23 at 22:13
  • Yes that would be correct – Ritwik Garg Oct 28 '23 at 22:14
  • Also what initial states and operations would I have to use in order to achieve a state like: |ψ⟩=12–√(|red⟩⊗|red⟩+|yellow⟩⊗|yellow⟩)

    I am asking for a circuit that would be analogous to applying a H gate on qubit 1 and CNOT gate on qubit 2, to prepare the phi + bell state given that the initial qubits are both ket 0

     – Ritwik Garg Oct 28 '23 at 22:22

1 Answers1

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Too long for a comment so writing it here.

Say, for simplicity, $$ \begin{align} |0\rangle &:= |\text{red}\rangle \,, \\ |1\rangle &:= |\text{blue}\rangle \,, \\ |2\rangle &:= |\text{green}\rangle \,, \\ |3\rangle &:= |\text{yellow}\rangle \,. \\ \end{align} $$

The Dirac notation for one of the maximally entangled state between two 4-level systems, $A$ and $B$, that you are looking for, as discussed in the comments, up to a global phase, would be:

$$ |\Phi\rangle_{AB} = \frac{1}{2}\bigg(|0\rangle_A |0\rangle_B + |1\rangle_A |1\rangle_B + |2\rangle_A |2\rangle_B + |3\rangle_A |3\rangle _B \bigg)\,. $$

You can obtain the rest of the maximally entangled states between two 4-level systems by doing the unitary operation on $|\Phi\rangle_{AB}$ as follows:

$$ |\Phi^{x,z}\rangle_{AB} = \bigg(X_A(x)Z_A(z) \otimes I_B \bigg)|\Phi\rangle_{AB} $$

where $x,z \in \{0,1,2,3\}$.

Here, $X(x)$ and $Z(z)$ are generalized Pauli operators for $d=4$, i.e., they are $4 \times 4$ matrices, which are also known as cyclic shift operators and phase operators, respectively.

In conclusion, there will be total $16$ maximally entangled states for two 4-level systems which are precisely $\{|\Phi^{x,z}\rangle_{AB} \}$, where $x,z \in \{0,1,2,3\}$.

As for checking if a given joint state of systems $A$ and $B$ is entangled or not, can be done by calculating the coherent information, $I(A\rangle B)$, of the given state. If it is positive, you can conclude that the given state is entangled (It is not an iff relation).

$$I(A\rangle B) = S(B) - S(AB)\,,$$

where $S(\cdot)$ is the Von Neumann entropy.

As for the circuit, I will need more time. If no one answers in the meantime, I will edit this answer later when I have free time.

FDGod
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  • Kindly check the answer I have posted as a reply to this. Thank you! – Ritwik Garg Oct 28 '23 at 23:36
  • There was a mistake in my previous answer. Please read it again and then repost your reply. Sorry about that. – FDGod Oct 28 '23 at 23:52
  • Excuse me for my ignorance but could you give an example of what the generalised Pauli matrices ‘X(x)’ and ‘Z(z)’ for the system in question are, since I am only familiar with the Pauli matrices for a single qubit system. – Ritwik Garg Oct 28 '23 at 23:53
  • You can get all $X(x)$ by just doing cyclic rotations of coloums of the $4 \times 4$ identity matrix. The operator $Z(z)$ works as follows on our basis states- $$Z(z)|j\rangle = e^{\frac{i 2 \pi z j}{4}}|j\rangle,,$$ and also $$X(x)|j\rangle = |j \oplus x \rangle,.$$ – FDGod Oct 29 '23 at 00:26
  • Clear image, in case it is not visible clearly on your screen. – FDGod Oct 29 '23 at 00:27
  • I am still unsure of what the matrix representation of $X(x)$ and $Z(z)$ is. Could you please give an example of what the matrix representation of $X(0)$ and $X(1)$ might look like, as well as that of $Z(0)$ and $Z(1)$.

    Also what would the initial states need to be inorder to construct the state:

    $$ |\Phi\rangle_{AB} = \frac{1}{2}\bigg(|0\rangle_A |0\rangle_B + |1\rangle_A |1\rangle_B + |2\rangle_A |2\rangle_B + |3\rangle_A |3\rangle _B \bigg),. $$

     – Ritwik Garg Oct 29 '23 at 13:33
  • You can look up some online sources for generalized Pauli matrices. As for the construction of the state, it doesn't matter what your initial state is. You apply operation accordingly so that we get the desired state. Generally, we start with the ground state, $|0\rangle |0\rangle$. Then we apply the unitary operation $U$ so that $$|00\rangle \xrightarrow[]{U} |\Phi \rangle_{AB},. $$ – FDGod Oct 29 '23 at 22:33