Quantum necromancy

In the measurement problem, quantum decoherence and quantum foundations, quantum necromancy is the theorem by Leonard Susskind, Scott Aaronson and Yosi Atia in quantum circuit complexity that it's possible to dynamically detect quantum interference in a quantum superposition if and only if it's possible to dynamically swap the quantum states in the superposition.

The name comes from the Schrödinger's cat thought experiment. We can only physically detect the cat in a superposition if it's also possible to physically transform a dead cat into a live cat.

This theorem has relevance to superselection sectors.

Technical details

If we have orthogonal quantum states $\left|\psi \right\rangle$ and $\left|\chi \right\rangle$, we can physically distinguish ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle \pm \left|\chi \right\rangle \right)$ if and only if we can physically swap $\left|\psi \right\rangle$ with $\left|\chi \right\rangle$.

Let $A$ be the unitary measurement operator which returns an output qubit of $\left|0\right\rangle$ for ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle +\left|\chi \right\rangle \right)$ and an output qubit of $\left|1\right\rangle$ for ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle -\left|\chi \right\rangle \right)$ . Then, by measuring with $A$, and then using the Pauli Z gate on the output qubit, followed by unmeasuring with $A^{-1}$, we transform ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle \pm \left|\chi \right\rangle \right)$ to $\pm {\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle \pm \left|\chi \right\rangle \right)$ . This swaps $\left|\psi \right\rangle$ with $\left|\chi \right\rangle$ .

Conversely, if we can swap $\left|\psi \right\rangle$ with $\left|\chi \right\rangle$ with the time-evolution operator $U$ with the relative phase factor $\theta$, as in$U\left|\psi \right\rangle =\left|\chi \right\rangle ,\,\,U\left|\chi \right\rangle =e^{i\theta }\left|\psi \right\rangle$, then we can use the quantum phase estimation algorithm on the eigenstates ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle \pm e^{-i\theta /2}\left|\chi \right\rangle \right)$ with the eigenvalues $\pm e^{i\theta /2}$ .

The exception mentioned in happens when $\theta =\pi$ . Then, we can't necessarily distinguish ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle +\left|\chi \right\rangle \right)$ from ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle -\left|\chi \right\rangle \right)$, even though we can still distinguish ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle +i\left|\chi \right\rangle \right)$ from ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle -i\left|\chi \right\rangle \right)$ .

There's a more general theorem using the same quantum circuits. Define $a\equiv \left\langle \chi \left|U\right|\psi \right\rangle ,\,b\equiv \left\langle \psi \left|U\right|\chi \right\rangle$ and the bias $\Delta \equiv |a+b|$ . Then, there exists a measurement operator $A$ which returns an output qubit of $\left|0\right\rangle$ with probability $p$ for ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle +\left|\chi \right\rangle \right)$ and probability $p-\Delta$ for ${\frac {1}{\sqrt {2}}}\left(\left|\psi \right\rangle -\left|\chi \right\rangle \right)$ . The converse is also true .

References

This article "Quantum necromancy" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Quantum necromancy. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Quantum rate theory
• DACAPO

---