I Calculating qubit purity/entanglement in a quantum computer

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Bob Walance
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I had written a simulator for a quantum computer and wanted to be able to calculate the values of the "purity of reduced states".

"Purity of reduced states" is a quantity that IBM provides on their simulator for each qubit. They used to call it "entanglement", and I believe that the two terms are synonymous since all of the qubits in a quantum computer start out in a pure state.

After struggling to find examples for how to calculate these purity/entanglement values for each qubit, I did finally figure it out. Attached is a document describing how to do this. The values calculated with this method jibe with IBM's simulator for various quantum circuits including simple one-qubit gates, CNOT gates, Toffoli gates, and various combinations of those.

Perhaps someone will find this useful.
 

Attachments

Thread 'Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
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Thread 'Is It Known For Sure Infinites In QFT Are Caused Using a Continuum?'

I am reading WHAT IS A QUANTUM FIELD THEORY?" A First Introduction for Mathematicians. The author states (2.4 Finite versus Continuous Models) that the use of continuity causes the infinities in QFT: 'Mathematicians are trained to think of physical space as R3. But our continuous model of physical space as R3 is of course an idealization, both at the scale of the very large and at the scale of the very small. This idealization has proved to be very powerful, but in the case of Quantum...
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Thread 'Lesser Green's function'

Thread 'Lesser Green's function'

The lesser Green's function is defined as: $$G^{<}(t,t')=i\langle C_{\nu}^{\dagger}(t')C_{\nu}(t)\rangle=i\bra{n}C_{\nu}^{\dagger}(t')C_{\nu}(t)\ket{n}$$ where ##\ket{n}## is the many particle ground state. $$G^{<}(t,t')=i\bra{n}e^{iHt'}C_{\nu}^{\dagger}(0)e^{-iHt'}e^{iHt}C_{\nu}(0)e^{-iHt}\ket{n}$$ First consider the case t <t' Define, $$\ket{\alpha}=e^{-iH(t'-t)}C_{\nu}(0)e^{-iHt}\ket{n}$$ $$\ket{\beta}=C_{\nu}(0)e^{-iHt'}\ket{n}$$ $$G^{<}(t,t')=i\bra{\beta}\ket{\alpha}$$ ##\ket{\alpha}##...
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