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
6 votes
1 answer
809 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, ...
6 votes
1 answer
952 views

How to calculate the state given by two qubits?

Let's say two qubits are both in $|+\rangle$ state. We need to find $a_1$, $a_2$, $a_3$, and $a_4$ in $|\phi\rangle = a_1|00\rangle + a_2|01\rangle + a_3|10\rangle + a_4|11\rangle$, how do we find ...
6 votes
1 answer
285 views

Do entangled measurements across multiple copies help in state distinguishability?

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 ...
  • 1,011
6 votes
1 answer
138 views

Is it possible to extract $x_1$ and $x_2$ from $|\phi\rangle=\frac1{\sqrt2}(|x_1,0^n\rangle+|0^n,x_2\rangle)$ with non-negligible probability?

Let $\left\vert \phi\right\rangle=\frac 1{\sqrt2}\left\vert x_1,0^n\right\rangle+\frac1{\sqrt2}\left\vert 0^n,x_2\right\rangle$ be a $2n$-bit quantum state for some unknown $x_1,x_2\in\{0,1\}^n$. My ...
  • 63
6 votes
1 answer
136 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 ...
  • 3,737
6 votes
0 answers
82 views

Are there separable states $\rho$ with separable pure decompositions requiring $\operatorname{rank}(\rho)^2$ components?

In What separable $\rho$ only admit separable pure decompositions with more than $\mathrm{rank}(\rho)$ terms?, examples were given of separable states $\rho$ with separable decompositions requiring ...
  • 20.8k
6 votes
0 answers
83 views

Postselection and hardness of estimating amplitudes

Let $A$ be a class of quantum circuits such that \begin{equation} \text{Post}A = \text{Post}BQP, \end{equation} where $\text{Post}$ indicates post-selection. Is only this amount of information ...
  • 1,011
6 votes
0 answers
145 views

Encoding bosonic degrees of freedom

A well-known way of encoding $N$ levels of a harmonic (bosonic) oscillator is as follows: \begin{equation} |n\rangle = |1\rangle^{\otimes n} \otimes |0\rangle^{\otimes N-n+1} \quad,\qquad ...
  • 1,658
6 votes
0 answers
140 views

How exactly is the stated composite state of the two registers being produced using the $R_{zz}$ controlled rotations?

This is a sequel to How are two different registers being used as "control"? I found the following quantum circuit given in Fig 5 (page 6) of the same paper i.e. Quantum Circuit Design for ...
5 votes
3 answers
3k views

What is a "maximally mixed state"?

What is meant by maximally mixed states? Does this mean that there are partially mixed states? For example, consider $\rho_{GHZ} = \left| {GHZ} \right\rangle \left\langle {GHZ} \right|$ and $\rho_W =...
  • 255
5 votes
3 answers
800 views

Why is the transpose of a density matrix positive and trace preserving?

With density matrix $\rho=\sum_{a,b=0}^1\rho_{a,b}|a\rangle\langle b|$ and it's transpose $\rho^T=\sum_{a,b=0}^1\rho_{a,b}|b\rangle\langle a|$. How to confirm that $\rho^T$ is positive and trace ...
  • 505
5 votes
4 answers
3k views

How to check if 2 quantum bits are orthogonal?

How would you check if 2 qubits are orthogonal with respect to each other? I need to know this to solve this problem: You are given $2$ quantum bits:$$ \begin{align} |u_1\rangle &= \cos\left(\...
5 votes
2 answers
613 views

Expected value of a Haar random quantum state multiplied by a unitary

Consider a quantity \begin{equation} \mathbb{E}\big[\langle z|\rho|z\rangle\big], \end{equation} where $\rho = |\psi \rangle \langle \psi|$ is a Haar-random state $n$-qubit quantum state and $z$ is ...
  • 1,011
5 votes
2 answers
742 views

Is it possible to represent a qubit using latitude and longitude?

I've watched several videos explaining qubits but I can't yet understand why they are typically represented as a pair of probabilities. The videos explain it's more accurate to understand them as ...
5 votes
2 answers
404 views

What is the computational complexity in initializing a quantum register?

I'm trying to figure out what is the computational complexity of initializing a quantum register of N qubits. For my research, I have used the initialize method of qiskit, in which you set the ...
5 votes
5 answers
842 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.:...
  • 65
5 votes
2 answers
420 views

Why does measuring one qubit after the other in this entangled system alter the result?

Suppose I have the following circuit where q0 and q1 are measured one after the other. The simulation results state that the state 00 occurs 75% of the time, and the state 11 occurs 25% of the time. ...
5 votes
2 answers
475 views

How to write the three-qubit GHZ state in the Pauli basis?

How can one write the GHZ state defined in Ket notation as $|\psi\rangle= \frac{1}{\sqrt{2}} \left(|000\rangle + |111\rangle\right)$, in terms of Pauli matrices $\sigma_{1},\sigma_{2},\sigma_{3}$?
  • 578
5 votes
2 answers
278 views

Why can the most general state of a qubit be written as $|\Psi\rangle=\cos(\frac\theta2)|0\rangle+e^{i\phi}\sin(\frac\theta2)|1\rangle$?

Why we can express a most general qubit as $|\Psi\rangle = \cos{\left(\frac{\theta}{2}\right)}|0\rangle + e^{i \phi} \sin{\left(\frac{\theta}{2}\right)} |1\rangle$? Is there any formal proof for this?
5 votes
3 answers
334 views

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

I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state: $$ \min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
  • 5,305
5 votes
2 answers
1k views

Is there a circuit to compare two quantum states?

Lets have two quantum states (single qubits ones for simplicity) $|\psi\rangle$ and $|\phi\rangle$: $$ |\psi\rangle = \alpha_\psi|0\rangle+\mathrm{e^{i\varphi_\psi}}\beta_\psi|1\rangle $$ $$ |\phi\...
5 votes
3 answers
273 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 ...
  • 4,764
5 votes
2 answers
901 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: $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$...
  • 648
5 votes
3 answers
754 views

How can pure state ensemble decompositions not be unique?

Apparently, the decomposition of a state into an ensemble of pure states is not unique. I can't understand why, as if I understood correctly a "pure state ensemble decomposition" is just the ...
  • 1,016
5 votes
4 answers
127 views

Derivation of the identity $\sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right)$

For measurement, we know $$\langle M \rangle = \sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right).$$ My question is, how can we go ...
  • 1,303
5 votes
3 answers
544 views

Writing state $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as separate qubits (qiskit textbook)

While going through the IBM qiskit textbook online, I came across the following question in section 2.2: Write the state: $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as two separate ...
5 votes
2 answers
111 views

How instantaneous is state preparation in a quantum register, if all possible superpositions are to be initialized equally?

Before the start of a quantum algorithm qubits need to be initialized into a quantum register. How fast can a quantum register of length $n$ be initialized in a way that all possible superpositions of ...
  • 181
5 votes
3 answers
367 views

Calculating measurement result of quantum swap circuit

Consider the following circuit, where $F_n$ swaps two n-qubit states. If the inital state is $|0\rangle \otimes |\psi\rangle \otimes |\phi\rangle = |0\rangle|\psi\rangle|\phi\rangle$, the state ...
  • 297
5 votes
2 answers
152 views

Why are commuting density operators said to be "classical states"?

In quantum information it is commonly said that a set of states $S=\{ \rho_i \}_i$ is classical if $[\rho_m, \rho_n] = 0, \,\forall \rho_m,\rho_n \in S$. This is meant in the sense that all observed ...
  • 53
5 votes
2 answers
854 views

Why are $d^2$ dimensions required to describe a density matrix?

A density matrix is defined as: $$\sum p_i |\psi_i \rangle \langle \psi_i|$$ If the dimensionality of each $|\psi_i \rangle$ is $d$, why does it take $d^2$ dimensions to represent a density matrix? (...
5 votes
2 answers
304 views

Quantum proof for the group non-membership problem

Group non-membership problem: Input: Group elements $g_1,..., g_k$ and $h$ of $G$. Yes: $h \not\in \langle g_1, ..., g_k\rangle$ No: $h\in \langle g_1, ..., g_k\rangle$ Notation: $\langle g_1, ..., ...
5 votes
1 answer
335 views

Why does quantum distinguishability ensure no faster-than-light communication?

On page 56-57 in Nielsen and Chuang, for a proposed scenario, it's said that: if Bob had access to a device that could distinguish the four states $|0\rangle$, $|1\rangle$, $|+\rangle$, $|−\rangle$ ...
  • 649
5 votes
2 answers
362 views

Proof for Cardinality of the Clifford Group

In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ...
5 votes
3 answers
190 views

Intuition for why $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ can be written as $\frac{|++\rangle+|--\rangle}{\sqrt{2}}$

In analyzing measurement of $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ in the local $|+\rangle$, $|−\rangle$ basis, through algebra manipulation, the initial state is first written as $\frac{|++\rangle+|...
  • 649
5 votes
2 answers
201 views

Can every bipartite state be written as $\rho_{AB} = \sum_{ij} c_{ij}\sigma_A^i\otimes \omega_B^j$?

Can every bipartite quantum state (including entangled ones) be written in the following way $$\rho_{AB} = \sum_{ij} c_{ij}\sigma_A^i\otimes \omega_B^j$$ where $\sigma_A^i$ and $\omega_B^j$ are ...
5 votes
2 answers
490 views

SWAP gate on 2 qubits in 3 entangled qubit system

Suppose I had 3 entangled qubits and I wanted to apply a SWAP gate on the first and third qubit. Because it's entangled I can't decompose it into individual states, and because the qubits are not ...
5 votes
2 answers
110 views

Prove that $\rho_{AB} \leq |B|(\rho_A\otimes I_B)$ for any bipartite state $\rho_{AB}$

I'm trying to prove the following statement but am lost on how to show it. For a quantum state $\rho_{AB}$ with marginal $\rho_A$, how can one show that $$ \rho_{AB} \leq|B|(\rho_A\otimes I_B)$$ where ...
  • 51
5 votes
2 answers
598 views

Can quantum circuits/operations have truth tables?

In the caption for the following figure, the word "truth table" is put inside a quotation. I am wondering if this means that the truth table the caption refers to isn't exactly a real truth ...
  • 649
5 votes
2 answers
237 views

How can quantum interference happen in real world if a wave function does not have any physical meaning?

I understood quantum interference as a heart of quantum computing, because it enables two possibilities to cancel each other. Quantum algorithms utilize this property to reduce the probability of ...
5 votes
2 answers
811 views

Preparing Bell state $(1/\sqrt{2}) (|01\rangle + |10\rangle)$ in Qiskit

I'm working through the Qiskit textbook right now, and wanted to complete part 1 of exercise 3.4, which asks me to use qiskit to produce the Bell state: $$\frac{|01\rangle + |10\rangle}{\sqrt{2}}$$ ...
5 votes
1 answer
337 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 $m$ ...
5 votes
2 answers
178 views

Relationship between entanglement and complex vector space

In the article Quantum Algorithm Implementations for Beginners I found the following sentence Entanglement makes it possible to create a complete $2^n$ dimensional complex vector space to do our ...
5 votes
1 answer
207 views

What is the superop simulator in Qiskit for?

I'm trying to understand what the use case of a superop simulator would be. My understanding is that density matrix is generally more resource intensive than state vector, but it has additional ...
5 votes
1 answer
735 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 ...
  • 164
5 votes
2 answers
153 views

Are all pure entangled states `robust'?

Let $\mathcal{H}_A \otimes \mathcal{H}_B$ be the tensor product of two finite dimensional Hilbert spaces, let $d = \operatorname{dim}(\mathcal{H}_A \otimes \mathcal{H}_B)$ and let $| \psi \rangle \in \...
  • 4,441
5 votes
1 answer
774 views

How to check a qubit's state?

Is it possible to write an algorithm that returns a single bit that represents if a qubit is in superposition or not, without compromising the qubit's wavefunction? Example: ...
  • 53
5 votes
1 answer
431 views

Compute the output of the quantum teleportation circuit

Sender and receiver use the teleportation protocol, where the sender teleports a quantum state $\left| \varphi \right>=\alpha\left| 0 \right> + \beta \left|1\right>$ to the receiver. I want ...
  • 398
5 votes
1 answer
318 views

Understanding this description of teleportation

In the context of quantum teleportation, my lecturer writes the following (note that I assume the reader is familiar with the circuit): If the measurement of the first qubit is 0 and the measurement ...
  • 869
5 votes
2 answers
210 views

How to understand the operators for watermarking schemes?

Note: Cross-posted on Physics SE. I am reading a research article based on quantum image watermarking (PDF here). The authors have defined some unitary transforms for the watermarking schemes, which ...
  • 51
5 votes
2 answers
508 views

Many-Worlds Interpretation and GHZ States

I'm working through a problem set, and I've come across the following problem: In this problem, you'll explore something that we said in class about the Many-Worlds Interpretation of quantum ...
  • 257

1 2 3
4
5
26