Skip to main content

Questions tagged [quantum-state]

Questions about or related to quantum states. Consider using the density-matrix tag when relevant.

Filter by
Sorted by
Tagged with
7 votes
3 answers
3k views

How can infinite information be theoretically encoded or stored in a single qubit?

I've just gotten started with Nielsen and Chuang's text, and I'm a little stuck. They mention that theoretically, it would be possible to store an infinite amount of information in the state of a ...
agiri's user avatar
  • 484
7 votes
2 answers
4k views

How to show a density matrix is in a pure/mixed state?

Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
bhapi's user avatar
  • 869
7 votes
4 answers
2k views

What is the quantum circuit to prepare a Bell state?

I was watching some lectures on qubits. They were talking about how to generate a Bell state. They described it as follows: Prepare state 00: $$\left |0 \right> \otimes \left |0 \right>$$ Apply ...
RMS's user avatar
  • 173
7 votes
3 answers
2k views

What is the difference between $\vert 0 \rangle + \vert 1 \rangle$ and $\vert 0 \rangle \langle 0 \vert + \vert 1 \rangle \langle 1 \vert$?

In a discussion with Jay Gambetta on the QISKit Slack channel, Jay told me that "T2 is the time that $\vert 0 \rangle + \vert 1 \rangle$ goes to $\vert 0 \rangle \langle 0 \vert + \vert 1 \rangle \...
Adrien Suau's user avatar
  • 4,987
7 votes
2 answers
2k views

Why are Bell states the maximally entangled ones?

I just want to know why actually Bell states are examples of maximally entangled states and significance of that "maximal" term. Is there anything for proving that?
user avatar
7 votes
3 answers
462 views

Forming states of the form $\sqrt{p}\vert 0\rangle+\sqrt{1-p}\vert 1\rangle$

I'm curious about how to form arbitrary-sized uniform superpositions, i.e., $$\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}\vert x\rangle$$ for $N$ that is not a power of 2. If this is possible, then one can ...
Sam Jaques's user avatar
  • 2,076
7 votes
2 answers
205 views

A two qubit state in a special form

How can a pure two-qubit state $|\psi\rangle = a |00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$, be written in the following form \begin{equation} |\psi_{\alpha}\rangle = \sqrt{\alpha}|01\...
Tobias Fritzn's user avatar
7 votes
4 answers
1k views

How is a single qubit fundamentally different from a classical coin spinning in the air?

I had asked this question earlier in the comment section of the post: What is a qubit? but none of the answers there seem to address it at a satisfactory level. The question basically is: How is a ...
Sanchayan Dutta's user avatar
7 votes
2 answers
564 views

Confusion regarding projection operator

Suppose we have a qutrit with the state vector $|\psi\rangle = a_0|0\rangle + a_1|1\rangle + a_2|2\rangle$, and we want to project its state onto the subspace having the basis $\{|0\rangle,|2\rangle\}$...
alphauser's user avatar
7 votes
2 answers
338 views

How can quantum error correction correct small rotations/continuous errors?

I'm having trouble understanding the so-called "digitization of errors" argument in QEC. Suppose I have to encode my logical qubit into $n$ physical qubits to do error correction. I will use ...
Joan's user avatar
  • 73
7 votes
1 answer
4k views

What is the probability of finding the second qubit as $0$ in the state $|\psi\rangle=\frac1{\sqrt2}|00\rangle+\frac12|10\rangle-\frac12|11\rangle $?

Assuming two qubits start in the state: $|\psi\rangle = \frac{1}{\sqrt 2}|00\rangle + \frac{1}{2}|10\rangle- \frac{1}{2}|11\rangle $ What is the probability of measuring the second qubit as 0? And ...
Lizzo's user avatar
  • 319
7 votes
2 answers
114 views

Symmetry of tensor product w.r.t. Vazirani 2-qubit video

Quantum Computing (QC) pioneer Vazirani has graciously long provided some nice videos on an intro to QC. E.g. in "2 qubit gates + tensor product" (2014) he introduces the tensor product w.r.t. QC ...
vzn's user avatar
  • 255
7 votes
1 answer
751 views

Correct Formulation of N&C Exercise 4.11 and other textbooks misquoting

Inspired by the comments in this question How to approximate $Rx$, $Ry$ and $Rz$ gates?, there is the errata for question 4.11 pg 176 in N&C. The original form states that for any non parallel $m$ ...
Sam Palmer's user avatar
7 votes
2 answers
8k views

How to check if 2 qubits are entangled? [duplicate]

I know that 2 qubits are entangled if it is impossible to represent their joint state as a tensor product. But when we are given a joint state, how can we tell if it is possible to represent it as a ...
Archil Zhvania's user avatar
7 votes
2 answers
573 views

Why is the decomposition of a qubit-qutrit Hamiltonian in terms of Pauli and Gell-Mann matrices not unique?

If I have the $X$ gate acting on a qubit and the $\lambda_6$ gate acting on a qutrit, where $\lambda_6$ is a Gell-Mann matrix, the system is subjected to the Hamiltonian: $\lambda_6X= \begin{pmatrix}0 ...
user1271772 No more free time's user avatar
7 votes
1 answer
143 views

What is the stabilizer rank of the W state?

The $ n $ qubit $ W $ state is defined here https://en.wikipedia.org/wiki/W_state The stabilizer rank of a quantum state $|\psi\rangle$ is the minimal $r$ such that \begin{equation} |{\psi}\rangle = \...
Ian Gershon Teixeira's user avatar
7 votes
2 answers
256 views

What is the smallest quantum circuit to produce two-qubit state (a,b,b,b)?

How can I synthesis a two-qubit quantum state of the state vector (a,b,b,b) using basic quantum-gate circuit (arbitrary single-qubit rotation and controlled $Z$ gate)? And further, can I know a given ...
cmc's user avatar
  • 173
7 votes
1 answer
2k views

What is the matrix for a SWAP operation on two qubits?

Say we want to swap qubits $a$, $b$ in the same register, where $a,b \in \left \{ 0, 1,\cdots, n-1 \right \}$. What would be the corresponding matrix. For those interested, I'm curious about this ...
LPenguin's user avatar
  • 184
7 votes
1 answer
2k views

Understanding a quantum algorithm to estimate inner products

While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here, in the Appendix, the author/s have included a section on quantum inner product estimation. Consider ...
IntegrateThis's user avatar
7 votes
2 answers
846 views

How is the quantum relative entropy $S(\rho\|\sigma)$ defined when $\sigma$ is a pure state?

I want to evalualte the quantum relative entropy $S(\rho|| \sigma)=-{\rm tr}(\rho {\rm log}(\sigma))-S(\rho)$, where $\sigma=|\Psi\rangle\langle\Psi|$ is a density matrix corresponding to a pure state ...
Confinement's user avatar
7 votes
1 answer
859 views

Solving linear systems represented by NxN matrices with N not power of 2

As far as I have seen, when it comes to solving linear systems of equations it is assumed to have a matrix with a number of rows and columns equal to a power of two, but what if it is not the case? ...
FSic's user avatar
  • 879
7 votes
1 answer
797 views

Quantum circuit for computing fidelity

Suppose we use Uhlmann-Jozsa fidelity $$ F(\rho, \sigma):=\left(\mathrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2. $$ Can we construct a quantum circuit that helps us calculate the fidelity of ...
raycosine's user avatar
  • 870
7 votes
1 answer
436 views

Are projective measurements the only optimal measurements to discriminate between two states?

Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one. There is an optimal measurement to ...
BlackHat18's user avatar
  • 1,363
7 votes
1 answer
202 views

Bob applies a projector - what happens to eigenvalues of Alice's reduced state?

Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global ...
user1936752's user avatar
  • 3,075
7 votes
1 answer
2k views

What is the intuition behind "states with support on orthogonal subspaces"?

I'm sure I don't fully understand support, but I am having trouble seeing how it connects to things like density operators. I have an idea that it means, according to Wikipedia: "In mathematics, ...
BADatThis's user avatar
7 votes
1 answer
216 views

Fidelity concentration bound for random stabilizer states

Let $|\Phi\rangle$ be a normalized vector in $\mathbb{C}^d$ and let $|\psi\rangle$ be a random stabilizer state. I am trying to compute the quantity $$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \...
BlackHat18's user avatar
  • 1,363
7 votes
2 answers
3k views

Get state vector of a single qubit in a circuit in Qiskit

I have two quantum circuits, and I would like to compare state vector of the first qubit and check if equals, what is the best way to do that in qiskit ? Let's say I have : ...
user12910's user avatar
  • 451
7 votes
1 answer
391 views

Why is the probability vector of a uniformly random state $\sum_i\alpha_i|i\rangle$ uniformly random only if $\alpha_i\in\mathbb C$?

In these lecture notes by Scott Aaronson, the author states the following (towards the end of the document, just before the Linearity section): There's actually another phenomenon with the same "...
glS's user avatar
  • 25.6k
7 votes
1 answer
496 views

One-qubit gate results in QISKit

I found it odd that the result of the action of identity gate (namely a $2\times2$ identity matrix) on a pure state $|0\rangle$ (namely the vector corresponding to the $2\times1$ matrix $\begin{...
Mathist's user avatar
  • 495
7 votes
1 answer
354 views

Are there measuring standards (and units) for the identification of qubits?

The representation of bits in different technological areas: Normal digital bits are mere abstractions of the underlying electric current through wires. Different standards, like CMOS or TTL, assign ...
AG-M's user avatar
  • 381
7 votes
1 answer
395 views

Most efficient way for general state generation

Assume we are given an $n$-qubit system and complex numbers $a_0, \ldots, a_{m-1}$ with $m = 2^n$. Assume further we start with the initial state $|0 \ldots 0\rangle$ and want to make the ...
tobias's user avatar
  • 171
7 votes
1 answer
151 views

Threshold and practical requirements for initial state preparation?

At the beginning of a quantum computational process we generally want to start in a perfectly known initial state, and evolve from there. This cannot be done perfectly, for fundamental reasons, but I ...
agaitaarino's user avatar
  • 3,827
7 votes
1 answer
195 views

In qubit/qudit terms, where is the experimental limit between S=3/2 and 2·S=1/2?

This question is inspired by "What is the difference between a qudit system with d=4 and a two-qubit system?", as an experimental follow-up. Consider for illustration these two particular cases: ...
agaitaarino's user avatar
  • 3,827
7 votes
1 answer
227 views

Closest quantum state with a fixed marginal: Analytical solution?

Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
user1936752's user avatar
  • 3,075
6 votes
3 answers
2k views

Is there anything practical that can be done with a single qubit?

Is there anything practical that can be done with a single qubit? And by "practical," I mean a problem that can be solved or information that can be stored. I realize that one practical thing that ...
vy32's user avatar
  • 641
6 votes
2 answers
2k views

Aren't qubits just ternary?

Qubits have 3 states: 1, 0, and 1 and 0 at the same time. If a qubit can have 3 states, then how come they are seen as different from ternary computing, which also has 3 states? Is it that the 3 ...
jort57's user avatar
  • 83
6 votes
3 answers
2k views

Is effective quantum cloning possible, given that any classical function can be implemented as a quantum circuit?

As in Compiling a classical function to a quantum circuit in practice, as far as my understanding goes, it is known that any classical function can be implemented as a quantum circuit. So given $f(x)=...
Paulske's user avatar
  • 457
6 votes
2 answers
978 views

What would be the meaning of an $i$ in a qubit state $i\alpha|0\rangle+\beta|1\rangle$?

I do not know if the question is not too easy, but I'll put it here, because I'm interested in it. So the state of a qubit is often stated in this form: $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$...
P_Gate's user avatar
  • 678
6 votes
2 answers
2k views

If Alice measures a qubit and doesn't tell Bob the result, what's Alice's state from Bob's perspective?

Suppose Alice has a qubit $|\phi\rangle=\alpha|0\rangle+\beta|1\rangle$ and measures it. Bob knows the initial state but not the result of her measurement. So after the measurement, Alice knows what ...
mp12853's user avatar
  • 63
6 votes
3 answers
635 views

When can pairs of states be transformed into other pairs of states via unitary mapping?

The states $|+\rangle, |-\rangle$ can be mapped to $|0\rangle, |1\rangle$ by a simple rotation. But if I now have other states ($|\psi_0\rangle, |\psi_1\rangle$) which are not orthogonal, does a ...
Johny Dow's user avatar
  • 157
6 votes
5 answers
1k views

How do I get the Unitary matrix of a circuit without using the 'unitary_simulator'?

I am using jupyter notebook and qiskit. I have a simple quantum circuit and I want to know how to get the unitary matrix of the circuit without using 'get_unitary' from the Aer unitary_simulator. i.e.:...
Jared's user avatar
  • 75
6 votes
3 answers
300 views

Terminology: what do $|i\rangle$ and $|\mbox{-}i\rangle$ represent?

$|0⟩$ and $|1⟩$ are usually referred as the computational basis. $|+⟩$ and $|-⟩$, the polar basis. What about $|i\rangle$ and $|\mbox{-}i\rangle$? And collectively? Orthonormal states? References are ...
luciano's user avatar
  • 5,813
6 votes
3 answers
2k views

Reverse Quantum Computing: How to unmeasure a qunit

After taking some measure, how can a qunit be "unmeasured"? Is unmeasurement (ie reverse quantum computing) possible?
user820789's user avatar
  • 3,312
6 votes
2 answers
2k views

How to compute the average value $\langle X_1 Z_2\rangle$ for a two-qubit system?

Show that the average value of the observable $X_1Z_2$ in a two-qubit system measured in the state $(|00\rangle + |11\rangle)/\sqrt{2}$ is zero. How would we approach this question? I understand that ...
Alk's user avatar
  • 163
6 votes
2 answers
309 views

In Stinespring dilation, can we always use a mixed state as the ancilla?

The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\...
Jacob Drori's user avatar
6 votes
2 answers
892 views

Is a qubit always in superposition?

I am introduced to ancilla qubits which are usually initialized to $\vert 0 \rangle$. It seems that an ancilla qubit is equivalent to the $0$ bit in classical computing as it will evaluate to $\vert 0 ...
M. Al Jumaily's user avatar
6 votes
3 answers
658 views

Maximum number of "almost orthogonal" vectors one can embed in Hilbert space

In a Hilbert space of dimension $d$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is small but finite compared ...
user avatar
6 votes
2 answers
528 views

Is it true that for a quantum algorithm to be efficient it must feature a highly entangled state at some point?

I'm wrapping my head around how and why quantum computers can provide advantage over classical. A basic and naive argument is that the dimension of the Hilbert space of $n$ qubits grows as $2^n$. ...
Nikita Nemkov's user avatar
6 votes
1 answer
250 views

Question on state distinguishability

Consider the following protocol. We are given either $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ or $|\phi\rangle = \alpha_{0} |0\rangle + \alpha_{1}|1\rangle$ where $\alpha_{0}^{2}$ ...
NewUser2020's user avatar
6 votes
3 answers
781 views

Interpretation of the unitaries involved in the eigenvalue decomposition of a density operator

If $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle \psi_{i}|$, this ensemble doesn't require $\langle \psi_{i}|\psi_{j}\rangle$=0. Given that $\rho$ is positive semi-definite, by the spectral theorem it ...
GaussStrife's user avatar
  • 1,115

1 2
3
4 5
34