How to check if a qubit in a circuit is equal to another qubit?

Question

If we have a set of qubits q1 ,q2 and q3 for example,

$$ \vert{q_1}\rangle = a \vert{0}\rangle +b \vert{1}\rangle\\ \vert{q_2}\rangle=\vert{0}\rangle\\ \vert{q_3}\rangle=\vert{0}\rangle $$

the state of the system is obtained by the tensor product of the states of the all qubits respectively, $$ \vert{\psi}\rangle=\vert{q_1}\rangle\otimes\vert{0}\rangle\otimes\vert{0}\rangle\\ = a \vert{000}\rangle +b \vert{100}\rangle $$

and then we perform a set of unitary operations on the global state, at the end i wanted to check either the qubit q1 after the operation is the same (equality) as before the operation or not, say we get after the operation :

$$ U\vert{\psi}\rangle= a \vert{001}\rangle +b \vert{101}\rangle\\ =(a \vert{0}\rangle +b \vert{1}\rangle)\otimes\vert{01}\rangle $$

in this case the subsystem q1 is equal to itself before the operation (doesn't mean that the q1 untouched by the $U$)

So how I can separate the subsystem from the rest of the system so I can compare it

EDIT :

I'm working on simulating the circuits of the quantum error correction codes, in which we encode one logical qubit 1 in three or more physical qubits {1,2,3} and I want to check if the code is succeed to correct the error that happens to the qubit and successfully restored the original value of the qubit 1 or not all that is just to calculate the success rate of the code. I'm not talking about the physical implementation in which you already cannot observe the state of the qubits in the first place, but I'm just the simulation i know the state of the qubit which just a vector and also i have the original copy of the qubit

So I need to define a mathematical function (or in python for example) that takes two state vectors and which qubit to compare, and returns True if :(the desired qubit is : separable from the rest of the system (not entangled) And this qubit is equal to the first) and False otherwise

Answer 1

If trying to use the wave function description, one cannot always uniquely determine the wave function of the sub-system qubit 1. This roots from the famous EPR (Einstein-Podolsky-Rosen, 1935) paradox.

To understand this, considering the following entangled state between qubit 1 and qubits 2-3

$$ \begin{align} |\psi \rangle_{1, 23} & = \frac{1}{\sqrt{2}}(|0\rangle|00\rangle + |1\rangle|11\rangle) \\ & = \frac{1}{\sqrt{2}}(|+\rangle|++\rangle + |-\rangle|--\rangle) \end{align} $$ where $$ |+\rangle \equiv \frac{|0\rangle + |1\rangle}{\sqrt{2}} \\ |-\rangle \equiv \frac{|0\rangle - |1\rangle}{\sqrt{2}} \\ |++\rangle \equiv \frac{|00\rangle + |11\rangle}{\sqrt{2}} \\ |--\rangle \equiv \frac{|00\rangle - |11\rangle}{\sqrt{2}} $$

If one chooses to measure the qubits 2-3 in {$|00\rangle, |11\rangle$}, the qubit 1 would be produced in {$|0\rangle, |1\rangle$}. Alternatively, if one chooses to measure the qubits 2-3 in {$|++\rangle, |--\rangle$}, the qubit 1 would be produced in {$|+\rangle, |-\rangle$}. But, {$|0\rangle, |1\rangle$} and {$|+\rangle, |-\rangle$} are two different wave functional basis. This paradox shows that the wave function for the sub-system qubit 1 cannot be uniquely written out. It is dependent on the measurement choice to qubits 2-3.

The proper way to express the state of the subsystem qubit 1 is to adopt the density operator formulation. Essentially the density operator was developed to describe the mixed, uncertain, quantum wave functions. For the pure 3-qubit quantum state, it is simply written as

$$ \hat{\rho}_{1, 2, 3} \equiv |\psi(1, 2, 3)\rangle \langle \psi(1, 2, 3)|$$

The uncertain state for the sub-system qubit 1 is obtained by tracing over the sub-system basis of qubits 2 and 3.

$$ \hat{\rho}_{1} = Tr_{2,3} \left[ |\psi(1, 2, 3)\rangle \langle \psi(1, 2, 3)| \right] $$

More explicitly $$ \hat{\rho}_{1} = \sum_{i_2, i_3 \in \{0, 1\}} {\langle i_2, i_3| \left[ |\psi(1, 2, 3)\rangle \langle \psi(1, 2, 3)| \right] |i_2, i_3\rangle } \space \space (1) $$

Applying equation (1) to your example, $ U|\psi\rangle \equiv (a \vert{0}\rangle +b \vert{1}\rangle)\otimes\vert{01}\rangle$, the density operator for qubit 1 would be $$ \hat{\rho}_{1}(U|\psi\rangle) = (b |1\rangle +a |0\rangle)(a \langle 0| +b \langle 1|) $$ So your example simply demonstrates a pure wave function state for qubit 1, because qubits 2-3 isn't really entangled with qubit 1.

Similarly, applying equation (1) to the entangled example mentioned above, $|\psi \rangle_{1, 23} \equiv \frac{1}{\sqrt{2}}(|0\rangle|00\rangle + |1\rangle|11\rangle)$, the density operator for qubit 1 would be $$ \begin{align} \hat{\rho}_{1}(|\psi \rangle_{1, 23}) &= \frac{1}{2}|0\rangle \langle 0| + \frac{1}{2}|1\rangle \langle 1| \\ &= \frac{1}{2}|+\rangle \langle +| + \frac{1}{2}|-\rangle \langle -| \end{align} $$ This example demonstrates a mixed, uncertain quantum state of qubit 1, as described above.

[Answer to EDIT]

As to the recipe, or an algorithm, to resolve the qubit-1 state and code the comparison function, I came up with a heuristic one. The idea is simple.

1. The disentangled state for qubit-1 and qubit-2-3 can be written as $$ \begin{align} |\psi_1\rangle |\psi_{23}\rangle &\equiv (a'|0\rangle + b'|1\rangle) (c|00\rangle +d|01\rangle +e|10\rangle +f|11\rangle) \\ &= a'|0\rangle (c|00\rangle +d|01\rangle +e|10\rangle +f|11\rangle) \\ &+ b'|1\rangle (c|00\rangle +d|01\rangle +e|10\rangle +f|11\rangle) \end{align} $$ which is simply spanned over the 8-basis of the 3 qubits. The first 4 terms and the last 4 terms contains the reduced two wave functions of qubit-2-3. The coefficients of these vectors are linearly dependent (i.e., another way to say they are the same wave functions). So we can encode any 3-qubit state in an 8-complex-component vector $$ [c_0, c_1, c_2, c_3, \space\space c_4, c_5, c_6, c_7] $$ The state is disentangled iff the first 4 terms has some common factor ($\frac{a'}{b'}$) to the last 4 terms $$ \frac{a'}{b'} \equiv c' = \frac{c_0}{c_4} = \frac{c_1}{c_5} = \frac{c_2}{c_6} = \frac{c_3}{c_7} $$ Just make sure the first and the last 4 terms of the encoded vector correspond to qubit-1's $|0\rangle$ and $|1\rangle$ states. In your python code, you need to handle the edge cases when any of denominators equal to zero.

2. If step 1 returns disentanglement true and the common complex factor c', the qubit-1's state is immediately retrieved by $$ |\psi_1\rangle \propto c'|0\rangle + |1\rangle $$ Therefore the comparison function for qubit 1's state returns true if $$ c' == \frac{a}{b} $$ assuming the original copy of qubit 1 is given by $a|0\rangle + b|1\rangle$. Again, you need to handle the case when the denominator equals to zero in the python code.

Answer 2

If you have many "copies" of your qubits, you can measure both qubits in different bases and do quantum tomography to "scan" state of the qubits. Finally you can compare whether the qubits are same. Note that they can differ in global phase.

See more on quantum tomography for example here.