## Quantum Fidelity, or, how to compare quantum states

Say you are given two quantum states $\rho$ and $\sigma$, and you are asked the following questions:

How similar are they? Can you distinguish them?

In practice, this situation arises in many scenarios. For instance if you have an imperfect experiment (due to noise) that produces a quantum state $\sigma$, and you wish to find out how close it is to the state $\rho$ that you originally wanted to create.

While there are many ways to quantify the distinguishability (i.e., the degree of similarity) between two quantum states, we will focus here on one quantity of particular interest in Quantum Information theory: the Quantum Fidelity (or the so-called Uhlmann-Jozsa Fidelity). The Fidelity is defined as

$F(\rho,\sigma) = \text{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}=\| \sqrt{\rho}\sqrt{\sigma} \|_1,$             (1)

where $\|A \|_1=\text{Tr} \sqrt{A^\dagger A}$ is the 1-norm.

### A brief History

Let us first try to understand why the Fidelity is defined the way it is.

The history of the Quantum Fidelity can be traced back to the late 70’s and early 80’s to the works of Uhlmann [1] and Alberti [2], who were studying generalizations of the quantum mechanical transition probability between two states. In later years the Fidelity was studied by Richard Jozsa (who coined the term “Fidelity“) [3] and then by Benjamin Schumacher [4] in the context of quantum communications.

## My new life as a Postdoc

Hello World!

Whelp… it seems like my last post was published in March 4th 2018, which I now realize means that I haven’t been active for over a year (!). So, let me give you a brief summary of what I’ve been up to:

During most of 2018 I was working in La Plata, Argentina, on my Ph.D Thesis entitled “Factorization and Criticallity in Spin Systems” which sums the research I did during my 3 years and 9 months as a Ph.D Student. (If by any chance you are interested in my thesis AND you happen to speak spanish, you can find it here. Yeah… I know… It’s in spanish… don’t get me started on that one.) Let me just start by saying that I was truly lucky to work under the supervision of Dr. Raul Rossignoli and Dr. Norma Canosa. I will always appreciate and remember their support as mentors and as human being.

Big shout out to Raul and Norma!

I defended my thesis on December the 18th, a day which will live in infamy, and managed to obtain the highest qualification (Approved with a Special Mention). By then  I had several job offers for postdoctoral programs and I decided to move to New Mexico to work at Los Alamos National Laboratory (LANL) under the supervision of Dr. Patrick Coles (here’s his google scholar page).

While many things were taken into consideration to determine where, with whom, and in what field I wanted to work as a Postdoc I chose this particular one for several reasons: First and foremost because I wanted to get my hands dirty and fully transition into the field of quantum computing. Second, I had read several of the papers published by Patrick’s work-group and had thoroughly enjoyed them. And finally because “Have you seen pictures of New Mexico? It’s beautiful!“.

Hence, long story short, here I am at LANL, working as a postdoc and having submitted to the arXiv my first paper on quantum  algorithms. I’ll write a post on it later this weekend. I promise *wink*.

So, with a “See You soon World!” I close this post, which (hopefully) will be the first of many to come.

## Quantum Computation with Light

Optical devices are among the most used and widespread technologies currently employed to perform quantum computations and quantum information processing tasks. They are usually cheaper than other implementations and are natural candidates for realizing communications protocols: Once a photonic state is produced, it can travel long distances while exhibiting low decoherences and it’s state can be manipulated with high precision.

In this article I will discuss a brief introduction to the basics of performing quantum computations with single photons.

## Our latest paper: “Factorization and criticality in finite interacting spin systems under a nonuniform magnetic field”

Our latest paper just got published by Physical Review Letter! Let me briefly explain what our manuscript is about:

## Factorization and criticality in finite interacting spin systems under a nonuniform magnetic field

Exact ground states of strongly interacting spin systems are normally highly entangled and  complex. Yet at particular finite values of the applied magnetic field such systems may exhibit a completely separable exact ground state, i.e. a simple product state where each spin points along a specific direction. Such remarkable phenomenon is known as ground state factorization.

Here we show the existence of an exceptionally degenerate set of completely separable ground states in XXZ systems immersed in a nonuniform field, arising at a remarkable quantum critical point where all magnetization plateaus merge. These points can emerge in arrays of any size, spin and dimension for a wide range of nonuniform field configurations, and exhibit other special critical properties like long range pair entanglement in their immediate vicinity.

The XXZ model is an archetypal quantum spin system which has helped to understand interacting many-body systems and their quantum phase transitions, and which has recently received renewed attention due the possibility of its finite size simulation with state-of-the-art technologies and its potential for implementing quantum information processing tasks. Among other possibilities, present results open the way to design exactly separable ground states in such system through a finite controlled field, which can be used as initial states for quantum protocols.

You can find the paper here:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.220605

And on the arXiv:

https://arxiv.org/abs/1708.00565

## I think it’s time we have “The Talk”

If you don’t know Zach Weinersmith’s Saturday Morning Breakfast Cereal, you should go check it out now here!

## Why we should be talking about the ethics of quantum computation

Gyges was a shepherd in the service of the king of Lydia; there was a great storm, and an earthquake made an opening in the earth at the place where he was feeding his flock. Amazed at the sight, he descended into the opening, where, among other marvels, he beheld a hollow brazen horse, having doors, at which he stooping and looking in saw a dead body of stature, as appeared to him, more than human, and having nothing on but a gold ring; this he took from the finger of the dead and reascended.

Now the shepherds met together, according to custom, that they might send their monthly report about the flocks to the king; into their assembly he came having the ring on his finger, and as he was sitting among them he chanced to turn the collect of the ring inside his hand, when instantly he became invisible to the rest of the company and they began to speak of him as if he were no longer present. He was astonished at this, and again touching the ring he turned the collet outwards and reappeared; he made several trials of the ring, and always with the same result-when he turned the collet inwards he became invisible, when outwards he reappeared.

Whereupon he contrived to be chosen one of the messengers who were sent to the court; where as soon as he arrived he seduced the queen, and with her help conspired against the king and slew him, and took the kingdom.

– The Ring of Gyges, Plato [1]

What would you do in the place of Gyges upon finding a ring that would turn you invisible? Would you use it for a greater good? Or for your own personal benefit? What can we expect from humans to do with power over others?

These are some of the questions about human nature arising from Plato’s thought experiment in the Republic, and they remains as valid today as they were 2,400 years ago. The ring of Gyges has been used as a paradigmatic example when discussing ethics and it can be the perfect starting point towards a discussion about the ethics of quantum computation.

## Steven Weinberg on density matrix formalism

Just a friendly reminder as to why the density matrix formalism is so important and why it should be thought along side the state vector formalism.

Nothing that is done in one system can instantaneously affect the density matrix of another isolated system, although it can affect the state vector.

Nobel Laureate Steven Weinberg, [1]

[1] “Quantum mechanics without state vectors”, Steven Weinberg, Phys. Rev. A 90, 042102 (2014).

## The More You Know: Quantum Simulator

A Quantum Simulator is a controllable quantum mechanical system than can be used to study more complex quantum system that cannot be directly studied nor simulated by any classical computer.

One of the most important consequences of the existence of entanglement in quantum mechanics is related to the quantity of information necessary to describe the state of a quantum system. In a classical system, the number of parameters needed to characterize its state is equal to the sum of the degrees of freedom of each of its components. However, for a quantum system, the number of parameters needed to describe its state grows exponentially with the number of constituents. And this makes the task of simulating a quantum system a very difficult one even for systems with few constituents, the computational resources needed for the simulation quickly become unreachable using computational techniques based on classical computers.

In 1982 Nobel laureate Richard Feynman conjectured that these limitations could be overcome using computers based on quantum systems. Based on this conjecture, physicists started exploiting the potential of quantum mechanics to develop new quantum computers (universal quantum simulators) capable of simulating quantum systems.

Nowadays there are several quantum systems than can be used as hardware for quantum simulators: trapped ions, cold atoms in optical lattices, liquid and solid-state NMR, photons, quantum dots, superconducting circuits, and NV centres. [1]

## Tools for Drawing Quantum Circuits

Last night I was writing an article for the series “The more You Know” to add to my Dictionary of Quantum Information and Quantum Computation and I ran across the need to draw a Quantum Circuit.

Although my first thought was to use a generic drawing software the results were not at all satisfying. Thus I proceeded to take the next logical step: an online search for a Quantum Circuit drawing tool. In this article I review two particularly interesting tools I found.

## The More You Know: Quantum Circuit

Quantum Circuits are used to visualize a prescription of quantum gates acting on a quantum state. They describe the changes of the quantum data as it undergoes successive discrete operations.

The data in a Quantum Circuit is usually encoded in a $n$-qubit state (other Quantum Circuit models can use $m$-level quantum systems instead of these $2$-level systems) which is set to a known vale, i.e. $|000...\rangle$. Thus, for every qubit in the circuit a horizontal rail, a wire, is drawn to represent its evolution, time flows from left to right.

Wires in a Quantum Circuit initialized to the $n$-qubit quantum state $|000...\rangle$.

Additionally, a Quantum Circuit is built from elementary quantum gates (unitary operations) acting on a fixed number of qubits. Since Quantum Circuits are read from left to right, leftmost gates act first.

Single-qubit and Two-qubit quantum gates in a Quantum Circuit.

Then, a description of the Quantum System consists of

1. Input: A controlled input $n$-qubit state.
2. Evolution: Prescribed sequence of Unitary Single-qubit and Two-qubit logic quantum gates which give the evolution of the initial quantum state.
3. Output: Readout (measurement) of the state of the $n$ qubits.