15 votes
Accepted

What is a stabilizer state?

Let $\mathcal{G}_n$ denote the Pauli group on $n$ qubits. An $n$-qubit state $|\psi\rangle$ is called a stabilizer state if there exists a subgroup $S \subset \mathcal{G}_n$ such that $|S|=2^n$ and $A|...
  • 14.8k
14 votes
Accepted

What are min and max overlaps of a maximally entangled state with a separable state?

The minimum overlap is zero and the maximum overlap is $\frac{1}{d}$. The overlap is a linear function of $\rho$ and the set $S$ of separable states is convex, so the overlap is both minimized and ...
  • 14.8k
12 votes
Accepted

What is the longest time a qubit has survived with 0.9999 fidelity?

Well, for the longest coherence time ever, I'm finding this Science from 2013 entitled Room-Temperature Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28, which indicates ...
  • 3,349
10 votes
Accepted

What does fidelity mean?

It might be worth mentioning the physical motivation for these definitions and the concept of fidelity itself. Unlike the classical computers we all know and love, quantum computers are ...
  • 316
9 votes
Accepted

What is the difference between the "Fubini-Study distances" $\arccos|\langle\psi|\phi\rangle|$ and $\sqrt{1-|\langle\psi|\phi\rangle|}$?

Recall the law of cosines for two unit vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb R^2$: $$ \|\mathbf{u}-\mathbf{v}\|^2 = 2-2\cos\theta, $$ where $\theta$ is the angle between the vectors. ...
9 votes
Accepted

How to calculate the average fidelity of an amplitude damping channel

An elementary method is to simply carry out the integration $$ \begin{align} \overline{F} &= \int\langle\psi|\mathcal{N_\gamma}(|\psi\rangle\langle\psi|)|\psi\rangle d\psi\\ &=\int\langle\psi|...
  • 14.8k
8 votes
Accepted

Closeness of purifications of states

No dimension-independent bound is possible. Consider states $\rho_A$ and $\sigma_A$ that are close in $p$-norm (for $p>1$) but have relatively low fidelity. Specifically, assume $$ \|\rho_A - \...
  • 4,598
7 votes
Accepted

What would be an ideal fidelity measure to determine the closeness between two non unitary matrices?

When you ask about an 'ideal' fidelity measure, it assumes that there is one measure which inherently is the most meaningful or truest measure. But this isn't really the case. For unitary operators, ...
7 votes

How can I calculate the inner product of two quantum registers of different sizes?

I guess you're looking at equations (130) and (131)? So, here, you have $|\psi\rangle=(|0\rangle|a\rangle+|1\rangle|b\rangle)/\sqrt{2}$ and $|\phi\rangle=|a| |0\rangle+|b| |1\rangle$. When it says to ...
  • 48.2k
6 votes

Purpose of using Fidelity in Randomised Benchmarking

Nielsen and Chuang in their book "Quantum Computation and Quantum Information" have section (Chapter 9) on distance measures for quantum information. Surprisingly they say in Section 9.3 " How well ...
  • 591
6 votes
Accepted

How to limit the error probability in large scale quantum computers

It is true that fidelity decays exponentially in the course of quantum computation. This is indeed a major limitation of NISQ computers that imposes a stringent "depth budget". In order to ...
  • 14.8k
6 votes
Accepted

How to find the distance between a given $\rho$ and the nearest pure state(s)?

Recall that for any Hermitian operator $A$ and any unit vector $|\psi\rangle$ the real number $\langle \psi|A|\psi\rangle$, known as the Rayleigh quotient, is bounded by the largest eigenvalue $\...
  • 14.8k
6 votes

If $\rho,\sigma$ are classical-quantum states, can the fidelity $F(\rho,\sigma)$ be expressed in terms of $F(\rho_i,\sigma_i)$?

Observe that, for any collection of matrices $A_i$, we have $$\sqrt{\sum_i |i\rangle\!\langle i|\otimes A_i} = \sum_i |i\rangle\!\langle i|\otimes \sqrt{A_i}, \\ {\rm Tr}\left(\sum_i |i\rangle\!\...
  • 19.6k
6 votes
Accepted

Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

Background If $v_1, v_2, \dots, v_n$ is an orthonormal basis in the inner product space $V$, then any vector $u\in V$ can be expressed as a linear combination $$ u = \alpha_1 v_1 + \alpha_2 v_2 + \...
  • 14.8k
6 votes
Accepted

Approximate Cloning

Given the constraints you impose (why those are appropriate constraints is perhaps another discussion), I think you're over-complicating things. Without loss of generality, you can assume $$ |\alpha_0\...
  • 48.2k
6 votes

Does a fidelity of $\mathcal{F}(U_1|0\rangle, U_2|0\rangle)=1$ imply that $U_1=U_2$?

I think you mean that if $\mathcal{F}=1$, the two states are identical. Now to your question: is it true that $$ \mathcal{F}(U_1|0\rangle,U_2|0\rangle)=0\implies U_1=U_2. $$ This is not true. You need ...
  • 48.2k
5 votes

What does fidelity mean?

Simply it is the distance (similarity measure) between two quantum states, for example the fidelity between $|0\rangle$ and $|1\rangle$ is less than the fidelity between $|0\rangle$ and $\frac{1}{\...
  • 473
5 votes

Are there disadvantages in using the inner product between states instead of the fidelity?

The quantity $\text{Tr}(\sqrt{A}\sqrt{B})$ that you defined there is actually referred to as the "just-as-good fidelity" (see 1801.02800) because it does have a relationship with the trace distance ...
  • 81
5 votes
Accepted

Quantum circuit for computing fidelity

Prohibited device Such a circuit $C$ would enable faster-than-light communication and therefore does not exist. Suppose Alice and Bob share a Bell pair $|\psi\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |...
  • 14.8k
5 votes
Accepted

Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?

The answer is no, as the following counter-example reveals. Let $\varepsilon\in(0,1)$ and define $$ \rho_0 = \begin{pmatrix} \frac{1+\varepsilon}{2} & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 &...
  • 4,598
5 votes
Accepted

Fidelity of extensions of states

Let's start with the second question. There is nothing special about an extension $\sigma_{AR}^{\ast}$ that allows it to be optimal for the right-hand side of (1); any extension $\sigma_{AR}$ of $\...
  • 4,598
5 votes
Accepted

Simulate a quantum channel with a certain fidelity

In the first case, when the details of the physical process do not matter, you can choose any quantum channel type that can achieve your target average fidelity. Depolarizing channel may be a good ...
  • 14.8k
5 votes

How are eavesdroppers detected when using BB84 in the presence of noise?

The standard noisy approach is not to try to determine the presence of an eavesdropper as such, but to create a final key where, even if there is an eavesdropper, you can still be confident that the ...
  • 361
5 votes
Accepted

What use cases are there for 127 qubit QPUs?

I think that the main reason behind is to tackle technical difficulties connected with building huge number of qubits. Having hundred of qubits brings about issues with interconnection, connections to ...
4 votes
Accepted

Intuitive role of the polar decomposition in proof of Uhlmann's theorem for fidelity

(I will give the argument with formulas for now, hopefully I find time for some pictures later.) Let $|m\rangle$ be the (unnormalized) maximally entangled state. Then, a purification of $\rho$ is ...
4 votes

Are there disadvantages in using the inner product between states instead of the fidelity?

A few thoughts: It mostly depends on what you are trying to quantify. The inner product of states, $\text{Tr}(\rho\sigma)$, is used to quantify the distance in state space. More precisely, the ...
  • 19.6k
4 votes
Accepted

Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem

I'll give a couple of methods to do this: (Using matrix inequalities) The idea is to use CS inequality in the form $$\newcommand{\tr}{\operatorname{Tr}}\lvert \sum_{ij}A_{ij}^* B_{ij}\rvert\le\sqrt{\...
  • 19.6k
4 votes

How can I calculate the inner product of two quantum registers of different sizes?

Actually, there should be a minus. There is a mistake in the paper. Wittek uses a minus in his (expensive) book. Indeed say : $$ |\psi\rangle = \frac{1}{\sqrt{2}} (|0,a\rangle + |1,b\rangle) $$ $$ |...
  • 4,574
4 votes

What is the longest time a qubit has survived with 0.9999 fidelity?

Answer: Fidelity of 0.9999 at 1.08 seconds in 2013: http://science.sciencemag.org/content/342/6160/830.full?ijkey=uhZaDNPnwgTdA More details: The $T_2$ was 180 minutes, or 3 hours. What about the ...
  • 12.1k
4 votes
Accepted

How to calculate the fidelity of a certain gate of a IBMQ device in Qiskit using randomized benchmarking/tomography?

Fidelity is a single-number measure of how good a gate is. Since there are many ways that a gate can go wrong, there are multiple ways that the fidelity can be defined. The exact answer to your ...

Only top scored, non community-wiki answers of a minimum length are eligible