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Suppose I have a qubit which is entangled with another; let's say they are in $A|00\rangle+B|11\rangle$.

If I have another qubit in the state $a|0\rangle+b|1\rangle$ then the combined state is $Aa|000\rangle+Ba|011\rangle+Ab|100\rangle+Bb|111\rangle$.

If I pass my two qubits through a CNOT gate the state becomes $Aa|000\rangle+Ba|011\rangle+Ab|110\rangle+Bb|101\rangle$.

Why does my operation not change the state of other qubit - it is still same? (last element in every three element representation)

Mark Spinelli
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Tamilaruvi Saravanan
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  • Welcome to QCSE. (1) From where did you get this? Is it a textbook? Please edit the question to provide details, by clicking on the "Edit" button next to the "Share" button. (2) I don't know what you mean by "last element in every three element representation". You are probably applying the CNOT with the leftmost qubit acting as the control and the middle qubit acting as the target, but your explanation of your question doesn't include such details. – Mark Spinelli Jan 09 '24 at 16:12

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You are applying a CNOT gate on the leftmost qubit (as the control) and the middle qubit (as the target) of your system.

As you can see, the CNOT has changed the overall state of your system, rearranging the amplitudes in the computational basis decomposition.

Now, coming to the question, the states of your system before and after applying the CNOT gate are entangled. So, in both cases, the state of your system cannot be written as a separable state. The question to ask, therefore, is the following: "is the reduced density matrix of the third qubit of my system affected by the CNOT gate?". To answer the question, compute the reduced density matrix of the third qubit before and after the CNOT gate.

orangonabbo
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