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20 votes
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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|...
Adam Zalcman's user avatar
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14 votes
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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 ...
Adam Zalcman's user avatar
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12 votes
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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 ...
auden's user avatar
  • 3,459
12 votes
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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|...
Adam Zalcman's user avatar
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11 votes
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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 ...
MMK1137's user avatar
  • 366
11 votes
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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. ...
Chris Ferrie's user avatar
9 votes
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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 - \...
John Watrous's user avatar
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8 votes
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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, ...
Niel de Beaudrap's user avatar
8 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 ...
DaftWullie's user avatar
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7 votes
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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{\...
glS's user avatar
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7 votes
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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 ...
Adam Zalcman's user avatar
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7 votes
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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 + \...
Adam Zalcman's user avatar
  • 22.9k
7 votes

What is fidelity in quantum computing?

Fidelity can refer to any measure of similarity between two things. This can include quantum states (statevectors or density matrices), quantum gate operations (unitary matrices), or more general ...
jchadwick's user avatar
  • 451
6 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 : $$ |\psi\rangle = \frac{1}{\sqrt{2}} (|0,a\rangle + |1,b\rangle) $$ $$ |...
cnada's user avatar
  • 4,754
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 ...
snulty's user avatar
  • 591
6 votes
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

Both definitions are used and authors usually make it clear which one they mean. Wikipedia also points this out under the Alternative Defintion section.
user1936752's user avatar
  • 2,997
6 votes
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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 ...
Adam Zalcman's user avatar
  • 22.9k
6 votes
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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 $\...
Adam Zalcman's user avatar
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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\!\...
glS's user avatar
  • 25.2k
6 votes
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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\...
DaftWullie's user avatar
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6 votes
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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 ...
Martin Vesely's user avatar
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 ...
DaftWullie's user avatar
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5 votes
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Can we combine the square roots inside the definition of the fidelity?

Square root is not differentiable at 0, so that cyclic property cannot be applied While $\rho\sigma$ has the same non-negative eigenvalues as $\sqrt\rho\sigma\sqrt\rho$, it's not self-adjoint. Non ...
Danylo Y's user avatar
  • 7,332
5 votes
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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 - |...
Adam Zalcman's user avatar
  • 22.9k
5 votes
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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 ...
Norbert Schuch's user avatar
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}{\...
Aman's user avatar
  • 503
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 ...
BFG's user avatar
  • 81
5 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 ...
glS's user avatar
  • 25.2k
5 votes

Can we combine the square roots inside the definition of the fidelity?

This result is indeed correct and gets rediscovered from time to time. We rediscovered it last year and got it published at Phys Rev A 107, 012427 (2023) (free version at arXiv:2211.02623). But it's ...
Jonathan Jones's user avatar
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 &...
John Watrous's user avatar
  • 6,087

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