Questions tagged [textbook-and-exercises]

Applies to questions of primarily educational value - styled in the format similar to that found in textbook exercises. This tag should be applied to questions that are (1) stated in the form of an exercise and (2) at the level of basic quantum information textbooks.

Filter by
Sorted by
Tagged with
1
vote
2answers
21 views

How are eigenvectors and eigenvalues expressed in the Bloch sphere?

I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues ...
1
vote
1answer
41 views

Proof of quantum data processing inequality in N&C on pg 566

On page 566, it states that using $S(\rho^{'})-S(\rho,\varepsilon) \ge S(\rho)$ and combining this with $S(\rho) \ge S(\rho^{'})-S(\rho,\varepsilon))$, we get $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
1
vote
2answers
59 views

Meaning behind obtaining a hermitian operator for measurement in another basis?

If $$P_{+} = |+\rangle\langle+|=\frac{1}{2}(|0\rangle\langle0|+|0\rangle\langle1|+|1\rangle\langle0| +|1\rangle\langle1|)$$ and $$P_{-} = |-\rangle\langle-|=\frac{1}{2}(|0\rangle\langle0|-|0\rangle\...
1
vote
1answer
68 views

What does the unitary $[|0\rangle\langle 0|\otimes I+|1\rangle\langle1|\otimes(|1\rangle\langle 0|+|0\rangle\langle1|)]\otimes I$ represent?

Consider the following unitary defined for a system $A$ interacting with a bipartite system $BB^\prime$ $$U_{AB} = \Big[|0\rangle \langle 0|_{A} \otimes \mathbf{I}_{B} + |1\rangle \langle 1|_{A} \...
3
votes
3answers
127 views

How is it possible to guess what state the qubit was in by measuring it?

Let's say that the qubit is in the state $\psi = \alpha|0\rangle+\beta|1\rangle$. We want to find out the values $\alpha$ and $\beta$. If we measure it in, say, the standard basis, then the outcome we ...
2
votes
7answers
189 views

Show that $I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$ for all states $\rho$

I am trying to show that for any qubit state p, the following holds: $$I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$$ I have tried different manipulations,...
1
vote
1answer
86 views

Prove that $A\preceq B$ implies $A=\Psi(B)$ for some channel $\Psi$

Define $\newcommand{\PP}{\mathbb{P}}\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\tr}{\operatorname{tr}}\newcommand{\ketbra}[1]{\lvert #1\rangle\!\langle #1\rvert}\PP_\psi\equiv\ketbra\psi$, and ...
2
votes
2answers
48 views

Can Eve perform this operation?

I am a beginner in quantum computing. Please consider the following scenario: Suppose Alice wants to send $\frac{1}{\sqrt{N}}\sum_{j=0,1,2,..N-1} |j\rangle$ to Bob. Eve has intercepted the state ...
7
votes
4answers
251 views

How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?

How can I demonstrate on the exponential part equality of the Hadamard matrix: $$H=\frac{X+Z}{\sqrt2}\equiv\exp\left(i\frac{\pi}{2}\frac{X+Z}{\sqrt2}\right).$$ In general, how can I demonstrate on: $\...
1
vote
1answer
40 views

How do I measure a single qubit in a two-qubit state?

Let us suppose that I have the state \begin{equation} \frac{1}{\sqrt2}(\alpha|0\rangle|+\rangle+\beta|1\rangle|-\rangle) \end{equation} and I choose to measure the first qubit in the basis $\{(1/\...
2
votes
2answers
50 views

How does Bell measurement work in the teleportation?

I'm a complete beginner and one of the first things I was taught was the teleportation protocol. In the protocol, the party sending its state (which we call say $|\phi\rangle$) makes a Bell ...
-1
votes
2answers
91 views

Nielsen and Chuang: Demonstration of equation 2.12

Reproduced from Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition) in page 64: We've seen that matrices can be regarded as linear operators. [...] Suppose $...
0
votes
1answer
39 views

What is the Kraus representation of quantum-to-classical channels?

As discussed in Watrous' book, quantum-to-classical channels are CPTP maps whose output is always fully depolarised. These can always be written as $$\Phi_\mu(X) = \sum_a \langle X,\mu(a)\rangle E_{a,...
0
votes
1answer
22 views

Compute the negativity of maximally entangled bipartite states

The entanglement negativity $\mathcal N(\rho)$ of a (bipartite) state $\rho$ is defined as the absolute value of the sum of the negative eigenvalues of the partial transpose of a state, or ...
2
votes
0answers
67 views

Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$?

Given the state of a system as $\rho_s$ and that of the ancilla (pointer) as $\rho_a$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective ...
2
votes
2answers
102 views

Why do we divide by $\sqrt2$ in the qubit states $\lvert\pm\rangle=\frac{1}{\sqrt2}(\lvert0\rangle\pm\lvert1\rangle)$?

I have a very basic question. I have found qubits are represented as complex vectors. I get it totally. I understand bracket notation and vector\matrix algebra. However, I cannot move further from ...
2
votes
3answers
92 views

Can a Kraus representation act as the identity on any operator?

In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A, $$\...
2
votes
1answer
51 views

Calcuate $\langle x | D | y \rangle$ for arbitrary $x,y \in \{0,1\}^n$

We are considering Grover's algorithm with a search space of size $2^n$ for an arbitrary integer $n$ for arbitrary $n$, and a unique marked element $x_0$. Question: Calculate $\langle x | D | y \...
3
votes
3answers
118 views

Is the tensor product of two states commutative?

I'm reading "Quantum Computing Expained" of David McMahon, and encountered a confusing concept. In the beginning of Chapter 4, author described the tensor product as below: To construct a ...
2
votes
1answer
58 views

How do you embed a POVM matrix in a Unitary?

In QuantumKatas Measurement Task 2.3 - Peres-Wooter's Game, we are given 3 states A,B and C. We construct a POVM of these states. But how do we convert that POVM into a Unitary that we can apply. ...
2
votes
2answers
61 views

Find the Kraus operators of a combined amplitude and phase damping channel

I am going through the paper Surface code with decoherence: An analysis of three superconducting architectures and I have a doubt about how the authors get what they refer to as the combined channel ...
1
vote
1answer
84 views

How does the graphical notation used to denote doubly-controlled gates work?

$\qquad$ $\qquad$ What is the difference between solid and hollow? How to express the corresponding matrix of these figures? In addition, if they are not adjacent, what should be done in the middle of ...
1
vote
1answer
25 views

In quantum teleportation what is the need for the extra X/Z gate after classical information is sent from Alice to Bob?

In the Qiskit textbook, at step 4, Bob who has received the classical bits from Alice then needs to apply a X and/or Z gate depending on what the classical bits received are, why is that?
1
vote
1answer
34 views

What is the representative matrix for a measurement in the Bell-state basis?

I have a few questions about measurement in Bell-state basis. In particular, if $Z = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$ is for a measurement on the computational basis, then what ...
2
votes
2answers
153 views

How do you implement a negative controlled gate using the regular controlled gate?

I have been reading a paper about perfect error correction codes, and when the circuit is described, the author uses some negative controlled gates, that is: The gate is applied if the control is $|0\...
2
votes
2answers
100 views

Prove that the depolarizing channel is completely positive

In two dimensions, for a density operator $\rho$ and probability $\lambda$, a depolarizing channel can be written as: $$\mathcal{E}(\rho) = (1-\lambda) \frac{\mathbb{I}}{2} + \lambda\rho$$ In ...
4
votes
1answer
64 views

What is the unitary operator realizing a given CPTP operator

Complete Positive Trace Preserving Map (CPTP) operator is the most general operation that can be performed on a quantum system. This post mentioned that a CPTP operator is nothing but a unitary ...
4
votes
2answers
1k views

How to find the operator sum representation of the depolarizing channel?

In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity ...
1
vote
0answers
44 views

Step-by-step passages in calculation

I would like to better understand some passages in a paper (Appendix A): Properties of Tensor Product Bilinearity: $A\otimes(B+ C) = A \otimes B + A \otimes C $ Mixed-product property: $(A\otimes B)(...
1
vote
2answers
133 views

How to find the matrix representation of an operator from its action on a basis?

First, I apologize if something is poorly written but English is not my first language. I know that these exercises have been solved in this question. But I do not agree. Inner product and concrete ...
3
votes
1answer
52 views

Problem about entanglement swapping

I have been learning about the concept of entanglement swapping and found an equation mentioned in the textbook, Mathematics Of Quantum Computing: An Introduction written by Wolfgang Scherer. At ...
0
votes
1answer
23 views

Why does the entanglement negativity equal (in magnitude) the sum of the negative eigenvalues?

The entanglement negativity, introduced in (Vidal and Werner 2002), is defined as $$\mathcal N(\rho) \equiv \frac{\|\rho^{T_B}\|_1-1}{2}.$$ It is mentioned there that this equals the sum of the ...
3
votes
1answer
79 views

How to apply the Schmidt Decomposition to a Bell state?

I am trying to understand the Schmidt Decomposition, currently in my QC class. We had a tutorial where we were told if $|\psi\rangle$ is a pure state of a composite system A then there exists $|i_A\...
2
votes
1answer
37 views

Why is the function $f_s(x)=\sum_i x_i s_i \pmod 2$ balanced?

A parity function $f_s:\{0,1\}^{n}\rightarrow\{0,1\}$, for some $s\in \{0,1\}^n$, is a function of the form $f_s(x) = x \cdot s$, where the inner product is taken modulo 2. Show that $f_s$ is a ...
2
votes
1answer
75 views

How to compute the tensor product of the depolarizing channel with the identity?

Consider two quantum systems A and B, B goes through a depolarizing noise channel, while A is not changed, i.e., they go through the channel $\mathbb{I}_A \otimes \mathcal{E_{\text{depol}}} $. If the ...
2
votes
3answers
59 views

Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates

Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates and write down the corresponding circuit. Recall from Quantum Information Theory that $Z=HXH$. As $CNOT$ is a ...
3
votes
1answer
39 views

Can $U(\rho_1\otimes\rho_2)U^\dagger$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally mixed state?

Given two qubit states $\rho_1$ and $\rho_2$. By applying some unitary $U$ we get $\rho = U(\rho_1 \otimes \rho_2)U^\dagger$. Can $\rho$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally ...
2
votes
3answers
173 views

How to spot the matrix representation of the quantum NOT operation

Applying the above construction to AND we get the map $(x1,x2,y) \rightarrow (x1,x2,y⊕(x1∧x2))$ for $x1,x2,y \in \{0,1\}$. The unitaryoperator which implements this is then simply the map $|x1〉|x2〉|y&...
3
votes
1answer
109 views

Estimate of the absolute value of the probability amplitude of |0⟩ in the superposition

You are given a source of an unknown qubit state. You make measurements in the computational basis, that is, your measurement is $\{|0\rangle \langle0|,|1\rangle \langle 1|\}$. You observe that you ...
2
votes
2answers
27 views

Find the probability of a measurement outcome in terms of the coefficients of the state

Suppose we have a quantum state $|\psi \rangle$ of $n$ qubits, where $|\psi\rangle=\sum_{x∈\{0,1\}^n}\alpha_x |x\rangle$,and we measure the first qubit of $|\psi\rangle$ in the computational basis. ...
0
votes
1answer
53 views

Understanding global phase

I have looked at the following: What is the difference between a relative phase and a global phase? In particular, what is a phase? Global and relative phases of kets in QM Global phases and ...
1
vote
0answers
30 views

How do you decompose an arbitrary quantum state into its corresponding projection subspaces such that their direct sum is the quantum state?

I understand that every Hilbert space $H$ can be decomposed into two mutually orthogonal subspaces $H_1$ and $H_2$ whose direct sum is $H$. Therefore, every vector $v\in H$ can be decomposed into $...
1
vote
0answers
57 views

Example of Simple phase Change Using 2 Qubits [closed]

I need to manipulate phases in 2 qubits. I will eventually create 34 distinct phase sets to map to an alphabet I built. I see https://github.com/oreilly-qc/oreilly-qc.github.io/blob/master/samples/...
4
votes
2answers
160 views

Procedures and intuition for designing simple quantum circuits?

I'm working my way through one of the quantum circuits sections in Nielsen and Chuang and I'm struggling to get a feel for the basics of circuit construction. For example, one of the exercises is as ...
1
vote
1answer
54 views

What is the physical meaning of the Hamiltonian $H = \alpha ( |01 \rangle \langle10| + | 10 \rangle \langle 01| )$?

In natural basis $| 0 \rangle = \begin{pmatrix} 1 \\0 \end{pmatrix}$, $| 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, what physical situation/model does the following Hamiltonian represent: $H = ...
1
vote
1answer
30 views

Quantum operation to get rid of small but nonzero eigenvalues

Updated and edited question: Let $N_{\delta}:P(\mathcal{H}_A)\rightarrow P(\mathcal{H}_B)$ be a completely positive trace nonincreasing map from the set of positive semidefinite operators in $\...
3
votes
1answer
159 views

Unable to use qiskit_textbook module

When I tried to use the Qiskit textbook module, it threw the following error: Command was: ...
3
votes
2answers
140 views

How to decompose a unitary single qubit gate by universal quantum gate set?

How to decompose a unitary single qubit gate? I have read some paper or books, which told me a unitary single qubit gate could be decomposed by universal quantum gates set. For example {phase gate, ...
2
votes
2answers
74 views

What are the possible qubit states?

For a quantum state on the form $$|\psi \rangle = \alpha |0 \rangle + \beta |1 \rangle$$ which possible qubit states can you construct from this? I know that $\alpha$ and $\beta$ must satisfy $$|\...
0
votes
2answers
47 views

Find the number of elements in the Schmidt decomposition of a pure state

Consider a pure state $\boldsymbol{\eta} \in \mathcal{H}_{AB}$. There exist orthonormal sets $\{\alpha_1, \alpha_2 \dots \alpha_i\} \subset \mathcal{H}_A$ and $\{\beta_1, \beta_2 \dots \beta_i\} \...