Quantum Queuing (and Pancha-Prashna Q-Upanishad)

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As a child growing up in India, through Amar Chitra Katha comic series, like many other kids, I was introduced to:

The Panchatantra ( पञ्चतन्त्र, “Five Treatises”) is an ancient Indian collection of interrelated animal fables in Sanskrit verse and prose, arranged within a frame story. The surviving work is dated to roughly 200 BCE – 300 CE, based on older oral tradition. 

This was an attempt, I understand now, to make us street-smart and be successful in the material, practical world. What I found quite appealing, additionally, was its structureit was story within a story, like a Russian dollmise en abyme.

This lower knowledge was complemented with higher knowledge, book-smarts, for spiritual success:

The Upanishads (उपनिषद् Upaniṣad ) are late Vedic Sanskrit texts of Hindu philosophy which form the foundations of Hinduism.  They played an important role in the development of spiritual ideas in ancient India, marking a transition from Vedic ritualism to new ideas and institutions. 

To be clear:

You don’t read Upanishads to become a tech-entrepreneur.

You become a tech-entrepreneur to read Upanishads.

There are many Upanishads; I mentioned Mundaka earlier in Weston Brahmin. For this post, I decided this one is more appropriate (also from Atharva Veda):

The Prashnopanishad (प्रश्नोपनिषद्, Praśnopaniṣad) is a Mukhya (primary) Upanishad, and is listed as number 4 in the canon of 108 Upanishads of Hinduism. It contains six Prashna (questions), and each is a chapter with a discussion of answers.

Beyond the structure, what is notable about this Upanishad is that it most directly discusses the primal energy (Prana) and form-giving substance (Rayi), which, in my view was a valiant attempt to understand Life-Physics (a phrase that I just coined).

Best I stick to inanimate objects, in my view, that is, Theoretical Physics.

Beyond energy-matter discussions (which is quite old news, clearly), I felt it is time (for OR/OM folks in particular, but in general for folks with curiosity about Quantum) to become familiar with something that is considered even more fundamental. Which is? Channeling Sean Connery (see my post The Artist of Bond):

Information. Quantum Information.

It has been nearly two millenia since the last Upanishad was written, in Sanskrit, many in the form of shlokas. That was the lingua franca of that age among the priestly class. Today, in the Universities around the world, including India, it is Mathematics, in the form of formulations. The frame I am using is Neo-Quantum Organon, and I chose Latin because that was the lingua franca for scientific communication in Europe, post-Enlightenment, notably (a valiant effort at understanding inanimate Universe):

Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) by Isaac Newton, often referred to as simply the Principia (/prɪnˈsɪpiə, prɪnˈkɪpiə/), is a work in three books written in Latin, first published 5 July 1687. After annotating and correcting his personal copy of the first edition, Newton published two further editions, during 1713 with errors of the 1687 corrected, and an improved version of 1726.

In my post, as COVID19 began – What would Newton do? – I summarized:

Social Distancing. Mathematical Research.

I actually did that! A prerequisite to be able to follow

Neo-Quantum Organon, Book Three: Quantum Queuing (QuQu)

is a working knowledge of both Quantum Information Theory (which most OR/OM folks do not have) and Queuing Theory (which most OR/OM folks should have).

The Five Questions posed introduce Quantum Information Theory to OR/OM folks through Semi-Definite Program (SDP) formulations, as a bridge from OR/OM to Theoretical Physics, cheekily referring to it here as:

Pancha-Prashna Q-Upanishad.

Beyond simple examples with closed form solutions via algebraic manipulations, most interesting settings require numerical computation, and so we have created companion Jupyter notebooks.

The first question on Quantum State Discrimination, I have discussed earlier as a Śrīdhara Brāhmaṇa

The second question is on Quantum State Fidelity. Given two mixed states:

How close are they to each other?

A measure of closeness is fidelity. Two identical states have fidelity of one; orthogonal states have zero fidelity. There is a semi-definite characterization, and fidelity is the optimum value of certain primal and dual semi-definite programs.

With respect to Quantum Channel Discrimination, our third prashna, we want to know what is the maximum probability of distinguishing without using entanglement, and what additional sharpening is possible with entanglement. This is where the real fun begins in quantum information theory. Again, this admits a SDP formulation.

Question 4 simply wonders if a given state is separable or entangled. It turns out that this is not that easy to determine! SDP to the rescue, again. Question 5 brings us to the very core of Quantum Shannon Theory:

What is the maximum achievable transmission rate through a noisy quantum channel?     

QuQu focuses on this fundamental practical question in Quantum Communication, building on Quantum Shannon Theory, merging it with Queuing Models:

Since quantum information decoheres with time, and sending more information causes queuing delays, due to buffering, what is the maximum information that can be sent?

Seriously, isn’t this intellectual fun? 🤷🏽‍♂️

Right now, applying a maximally inverse tactic to my own weekend pastimes, I am off to see F9: The Fast Saga on Cinemark XD, for ample non-intellectual entertainment. 😏

F9 (also known as F9: The Fast Saga and Fast & Furious 9) is an action film, with stars Vin Diesel, Michelle Rodriguez, Tyrese Gibson, Ludacris, John Cena, Jordana Brewster, Nathalie Emmanuel, Sung Kang, Michael Rooker, Helen Mirren, Kurt Russell, and Charlize Theron. In a limited release, it has already grossed over $300 million, on a budget of $200 million.

Recalling my post Ford v Ferrari where I described my first driving experiences with these exotics:

Mukhya Prashna: Should I drive my Ferrari or my McLaren? 😏

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