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I am trying to understand single qubit operations from Bloch sphere, but I was told that the limitation of Bloch sphere is that it can only visualize or simulate 1 qubit.

What are some instances do I need to know or visualize multi-qubit interactions? How do Quantum computer scientist or students currently use to visualize or understand multi-qubit interactions? And lastly, what are some information does one need to know from a multi-qubit interaction?

I understand the question is a little broad as I am new to this field, just asking this out of curiosity and I'm also interested to explore ideas on how to visualize this as well!

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    $\begingroup$ Welcome to QCSE! Is this helpful? $\endgroup$ Apr 20, 2021 at 2:49
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    $\begingroup$ Generally, you should not combine multiple questions into a single post since this makes it harder to write answers and hence reduces the odds of your question being answered. It also means your question will likely be closed as needing more focus. Instead, feel free to submit multiple posts, ideally with one self-contained question or issue per post. $\endgroup$ Apr 20, 2021 at 2:50
  • $\begingroup$ well, the Bloch representation works in arbitrary dimensions. But if you have e.g. $n$ qubits, then you get a representation in a space of dimension $2^n$, so you can't really "visualise" that $\endgroup$
    – glS
    Apr 20, 2021 at 7:36
  • $\begingroup$ @AdamZalcman Thank you so much! Wow, that is really helpful!!! $\endgroup$
    – Jazz Ang
    Apr 21, 2021 at 3:21

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This is an answer from the Physics point of view. The Bloch sphere is a kind of "phase space" for the qubit. Now let's see what is a phase space. Phase space typically appears in Classical Mechanics in order to study the dynamics of a classical system. You can do lots of things there, for instance study chaotic systems.

How do you do it in Quantum Mechanics? Historically much of the development has been aimed to study for instance light (technically is the harmonic oscillator but it doesn't matter much). You can have your master equation with dissipation and temperature and study the solutions of $\partial_t \rho = \mathcal{L}[\rho]$. Another view is to map this equation onto the phase space (most of the time it is tricky!) using e.g. the Wigner Function. There you can see how your system behaves as a function of time and what does it do because of the temperature, etc.

This is all well because the harmonic oscillator can be described in principle in continous $(x,p)$ variables. On the other hand, qubits are somehow "discrete", so it is not at all trivial to define a mapping $\rho_{\rm qubits}\rightarrow W(\rho_{\rm qubits})$ over the phase space because one has to fulfill several requierements (!)

Many theoretical uses can be found in the literature, as well as recent developments on how to define a reasonable phase space function for qubits. I leave some references here:

[1] W. P. Schleich. Quantum Optics in Phase Space. 1st ed. Wiley-VCH (2001). (Classical treament of Phase Space for Quantum Mechanics.)

[2] A. Perelomov. Generalized Coherent States and Their Applications. 1986th ed. Springer-Verlag (1986). (You can find partial answers to your questions here!)

[3] C. Muñoz, I. Sainz and A. B. Klimov. Macroscopic approach to N-qudit systems. J. Phys. A: Math. Theor. 53 245302(2020). (This is a recent example of how e.g. to see and use the "phase space" for qudits!)

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  • $\begingroup$ what's your definition of a "phase space" here? Because I don't think people would normally refer to the Bloch representation as a phase-space. $\endgroup$
    – glS
    Apr 20, 2021 at 14:28
  • $\begingroup$ I'm not saying that the Bloch sphere is a phase space. That's why I wrote on purpose 'a kind of "phase space"... '. I tried instead to make the argument into "how to visualize a qudit system". As you say in your comment you have to visualize a complicated object (in principle of dimension $2^n$). One solution that I saw in order to answer the OPs question was to rephrase it in terms of phase space, where many things are already known. $\endgroup$ Apr 20, 2021 at 20:55

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