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What is a unitary operator that makes all the amplitudes all negative on the arbitrary state of $n$ qubits? For example suppose, $n=2$, the arbitrary state is: $a_1|00\rangle+a_2|01\rangle-a_3|10\rangle+a_4|11\rangle$ then the unitary operator will give the result $-a_1|00\rangle-a_2|01\rangle-a_3|10\rangle-a_4|11\rangle$ on the above state (where $a_i$ are real positive numbers that are the amplitudes).

In other words the amplitudes are not complex numbers and the negative signs are randomly distributed regarding the $a_i$ for $n=2$; a similar statement is true for any $n$. Also we do not know for which $a_i$is negative or positive without measuring the state (which will destroy the state and we do not want to destroy the state).

An informal description of what the question asks is, is there a unitary operator that gives the version of an arbitrary state which has negated absolute values of all the original amplitudes in the resulting state generated by the unitary operator.

1Are you looking for $-I$ where $I$ is the identity matrix? – Rammus – 2020-11-16T11:26:09.473

@Rammus That would not work since it would give -a1|00>-a2|01>+a3|10>-a4|11> and thus a3 is positive-this does not answer my question since the question asks for a unitary operator to make all the amplitudes negative – Z.E. – 2020-11-16T11:28:12.973

I am not sure exactly what you are asking:

@Z.E. Ok I think I didn't understand your question correctly then. – Rammus – 2020-11-16T12:48:59.293