Is phase factor negligible in fidelity of quantum states?

One well-known fidelity is defined as $(Tr\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^2$. And for pure states, fidelity is always in the form $|\langle\psi|\phi\rangle|^2$.As we know, in the context of two-qubit quantum computation, we cannot tell the difference between $|\psi\rangle$ and $e^{ia}|\psi\rangle$($a$ is real and perhaps $|\psi\rangle$ need to be a pure state). And the definitions of fidelity above are all modes of some complex values. So does that means the phase factor of quantum states is negligible in calculating the fidelity? And coul...Read more

No-cloning theorem and distinguishing between two non-orthogonal quantum states

I'm currently reading Quantum Computation and Quantum Information and I'm not sure if I correctly understand this exercise (on page 57) : Exercise 1.2: Explain how a device which, upon input of one of two non-orthogonal quantum states $\left|\psi\right>$ or $\left|\phi\right>$ correctly identified the state, could be used to build a device which cloned the states $\left|\psi\right>$ and $\left|\phi\right>$, in violation of the no-cloning theorem. Conversely, explain how a device for cloning could be used to distinguish non-orthogon...Read more

Stabilizer state verification and specification from state vector

Given an arbitrary $n$-qudit state vector $|\psi\rangle =\sum_i c_i| i \rangle \in \mathbb{C}_d^n$ for some orthonormal basis $\{|i\rangle\}$, what is the most efficient way one can:Verify whether the state is a stabilizer state (i.e. can be defined by $n \leq m \leq 2n$ stabilizer generators, with equality for $d=2$), and if so,Find the state's stabilizer generators (in the form of some tensor product of local Pauli matrices)....Read more

State of a system after the second qubit of a Bell state sent through a bit flip error channel

The second qubit of a two-qubit system in the Bell state$$|\beta_{01}\rangle= \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$$ is sent through an error channel which introduces a bit flip error with probability $p$. I want to calculate the state of the system after the second qubit exits the error channel. I know how to do this for a single qubit, but there isn't anything in my notes about doing it on a single qubit of a two-qubit system (or on two qubits for that matter).This question is from a past exam paper (my exam is next week) could anyone sh...Read more

entanglement - Quantum secret Sharing using GHZ state paper

I am reading a paper on Quantum Secret Sharing Quantum Secret Sharing using GHZ statesI have doubts regarding the initial phase of the paper, which are: Let me state what things I read and understoodAlice, Bob, Charlie share a GHZ state $|\psi\rangle=\dfrac{|000\rangle+|111\rangle}{\sqrt{2}}$ and are given one particle each from this state, well this is quite clear to me, proceeding nextThey each choose at random whether to measure theirparticle in the $x$ or $y$ direction They then announcepublicly in which direction they have made a measurem...Read more

Why does the "Phase Kickback" mechanism work in the Quantum phase estimation algorithm?

I've probably read the chapter The quantum Fourier transform and its applications from Nielsen and Chuang (10 th anniversary edition) a couple of times before and this took this thing for granted, but today, when I looked at it again, it doesn't seem obvious to me at all! Here's the circuit diagram for the Phase estimation algorithm:The first register having $t$ qubits is supposedly the "control register". If any of the qubit in the first register is in state $|1\rangle$ the corresponding controlled unitary gate gets applied to the second regis...Read more

measurement - Evaluate the given quantum circuit

The input state on the qubit is $|1\rangle\otimes(2|0\rangle+|1\rangle)$. I am assuming this means that the first qubit has the state $|1\rangle$, and the second qubit has the state $2|0\rangle+|1\rangle$. I could be wrong though. My question is, given the quantum circuit is shown below, what is the probability of the measurement outcomes, as well as the outcome measurement, and finally the new state after measurement. If there is anything that needs to be further specified please let me know. Thank you....Read more

measurement - Measuring first qubit of Bell state

Let's consider the following Bell state:$$\lvert \Phi^+\rangle = \frac{1}{\sqrt{2}} (\lvert00\rangle + \lvert11\rangle)$$What would happen if I measure the first qubit in the standard basis and keep the other one intact? Is the following computation correct?$$\langle 0I\lvert\Phi^+\rangle\langle\Phi^+\lvert0I\rangle = \frac{1}{2}\lvert0\rangle\langle0\lvert$$I is the two dimensional identity matrix. And$$\langle 1I\lvert\Phi^+\rangle\langle\Phi^+\lvert1I\rangle = \frac{1}{2}\lvert1\rangle\langle1\lvert$$Shouldn't have I gotten the probability o...Read more

Cloning quantum states with a device that distinguishes between two non-orthogonal quantum states

I'm aware that this is basically a duplicate question, but I don't have any rep in this community so I can't comment on it, and I don't think I should "answer" that question with my own question:No-cloning theorem and distinguishing between two non-orthogonal quantum states Exercise 1.2: Explain how a device which, upon input of one of two non-orthogonal quantum states $|ψ⟩$ or $|ϕ⟩$ correctly identified the state, could be used to build a device which cloned the states $|ψ⟩$ and $|ϕ⟩$, in violation of the no-cloning theorem. Conversely, expla...Read more

simulation - How to compactly represent multiple qubit states?

Since access to quantum devices capable of quantum computing is still extremely limited, it is of interest to simulate quantum computations on a classical computer. Representing the state of $n$ qubits as a vector takes $2^n$ elements, which greatly restricts the number of qubits one can consider in such simulations.Can one use a representation1 that is more compact, in the sense that it uses less memory and/or computational power than the simple vector representation? How does it work?While easy to implement, it is clear that the vector repres...Read more

quantum state - Proof of no-cloning

I was reading a proof of No-cloning theorem, there are a couple of steps that are not clear to me, but the book does not give explanation for them. So here it is:Theorem: It is impossible to create an identical copy of an arbitrary unknown quantum state.Proof (by contradiction): Suppose that there exists a unitary $C$ (that copies an arbitrary unknown q-state). Then: .This is the first thing I have trouble with, why do we take only one state ($\left| 0 \right>$ state) to prove that it is impossible to copy any arbitrary state into $any$ qubi...Read more

quantum state - Entangling marked qubit with part of the register in Grover's search

Can the marked qubit be associated only with half of the register? I mean if I can ask the oracle if the $2$ low qubits from a $4$ qubit register have a certain value. I ask this to know if I can bit-append for example names and telephones into one register, so I can entangle the marked qubit with CNOT gates only with one half of register qubits which are the telephones, so when I seek a telephone the oracle amplifies the state associated with the qubit marked. There will be $N\times N$ states in the register initialized all to $0$ except the $...Read more