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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.

glS
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Z.E.
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3 Answers3

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If I understand your question correcly, you are asking for a unitary that, in effect, looks at amplitudes in the computational basis, which are assumed to all be real, and if they are positive, make them negative.

This, quite simply, is impossible for a unitary. To see this, note that you would have (in the 1-qubit case, although you can do exactly the same for any number of qubits) $$ U(|0\rangle-|1\rangle)=-|0\rangle-|1\rangle,\qquad U(|0\rangle+|1\rangle)=-|0\rangle-|1\rangle. $$ In other words, there are two distinct inputs that produce the same output. This is not a reversible procedure, and therefore cannot be unitary (because all unitaries are reversible).

(Technically, I should allow for the introduction of ancilla qubits as well. This will not change the conclusion).

DaftWullie
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First, apply $X$ gate on second qubit and then controlled $Z$ gate. The resulting state would be $a_1|00\rangle + a_2|01\rangle + a_3|10\rangle + a_4|11\rangle$. Now, you can apply operator $-I$ which is a global phase $\pi$.

In fact, you do not have to apply global phase operator as states which differ in global phase only as indistinguishable.

Martin Vesely
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You should know that amplitudes are not OBSERVABLES (they cannot be measured). In general, the minus sign is just a phase factor ($ -1 = e^{i\pi}$), which is irrelevant to the statistical outcome, since $ |a|^2 = |-a|^2 $.

Consequently, we cannot find an unitary operation that distinguishes between a minus signed amplitude and a positive one (or either the amplitude of a bigger magnitude from others).

Appo
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