Questions tagged [quantum-state]

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

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General representation for GHZ states in any orthonormal basis

We know that if we consider a Bell state for example $$ |\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}} $$ Then we can write this state in some other orthonormal basis in the same form. Like:...
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No-cloning theorem and distinguishing between two non-orthogonal quantum states revisited

There are a couple of posts on this question, but I think they are not satisfactory. The question is Nielsen and Chuang's QCQI, Exercise 1.2, page 57, which asks "Explain how a device which, upon ...
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2 answers
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How to understand intuitively the Grover diffusion operator?

According to the tutorial https://qiskit.org/textbook/ch-algorithms/grover.html I understand the mathematical principle of diffusion operator: $$ \begin{equation} \begin{split} U_s&=2\left|s\right\...
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How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived?

Let's say I have a density matrix of the following form: $$ \rho := \frac{1}{2} (|a \rangle \langle a| + |b \rangle \langle b|), $$ where $|a\rangle$ and $|b\rangle$ are quantum states. I saw that ...
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How to create the logical $|0_L\rangle$ state for the Steane's 7-qubit code?

I don't know how to prepare using Qiskit the following state in order to implement a Steane's 7-qubit code circuit (I omit the normalization factor): \begin{align*} |0_L\rangle =&|0000000\rangle+|...
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How to tell if the ground states of two Hamiltonians are solutions of the same optimization problem?

Let's say, that we have an optimization problem in the form: $$ \min_x f(x) \\ g_i(x) \leq 0, i = 1, ..., m \\ h_j(x) = 0, j = 1, ..., p, $$ where $f(x)$ is an objective function, $g_i(x)$ are ...
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What is the $\left| 22\right>$ state?

I came across with a problem that involves $2$ quantum trits in state $\left| 22 \right>.$ What is it's tensor product interpretation and a matrix interpretation?
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What are the best-known lower bounds on the number of measurements required for quantum state tomography?

I'm very curious to know more about bounds of number of measurements (or number of independent copies of state) required to reconstruct full density matrix $\rho$ such that it is $\epsilon$-close (...
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What is the Stinespring dilation of $T\otimes I$ for some CPTP map $T$?

Let $T: \mathcal{H}_A \rightarrow \mathcal{H}_B$ be a CPTP map with Stinespring extension $U: \mathcal{H}_{A} \rightarrow \mathcal{H}_{B} \otimes \mathcal{H}_E$. That is $U$ is an isometry such that ...
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What is the form of a unitary $U$ that preserves the marginals on a given state, $\text{Tr}_A(U\rho_{AB} U^\dagger) = \rho_B$?

Suppose for some quantum state $\rho_{AB}$ and unitary $U_{AB}$, one has $$\text{Tr}_A(U\rho U^\dagger) = \rho_B$$ does this imply that $U_{AB} = U_A\otimes I_B$? Also, the same question as above, but ...
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Anticoncentration for two independent random quantum circuits in parallel

Consider two Haar random $n$ qubit unitaries, $U_1$ and $U_2$. Consider the quantum state $$|\psi\rangle = (U_1 \otimes U_2) |0^{2n}\rangle. $$ Let $p_x = |\langle x| \psi \rangle|^{2}$, for $x \in \{...
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Setting initial state in Qiskit unitary simulator

I'm getting started in IBM quantum lab for quantum computing. My task is to put quantum state $|0\rangle$ on the 1st qubit and state $|1\rangle$ on second one. I tried using this method to initialize ...
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If $\text{supp}(\rho_{AB})\subset \text{supp}(\sigma_{AB})$, is $\text{supp}(\rho_{A})\subset \text{supp}(\sigma_{A})$?

For any linear operator $A$, the support of $A$ is the orthogonal complement of its kernel. Hence when we say, $supp(A)\subset supp(B)$, we have that for any vector $v$ in the kernel of $B$ i.e. $Bv = ...
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Creating a specific cluster state

I have a state $$\dfrac{1}{2}(|00000\rangle+|00111\rangle+|11101\rangle+|11010\rangle).$$ How does one create this state? In general, how does one create for instance an $n$-bit cluster state, is ...
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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 ...
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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?

I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following: $$ F(\rho, \sigma) := \...
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Nielsen & Chuang Exercise Question on CSS code

I was reading the CSS ( Steane Code) from the Nielsen & Chuang book. It asked in Ex. 10.27 to prove that: suppose $C_1$ and $C_2$ are $[n,k_1]$ and $[n,k_2]$classical linear codes such that $C_2\...
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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-...
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1 answer
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Understanding quantum circuit diagrams: a circuit that compares two states $|YX\rangle$ and $|AB\rangle$

I have a quantum circuit which I would like to understand, which compares two standard basis states $|YX\rangle$ and $|AB\rangle$. It operates on the corresponding bits in each of the two states: i.e.,...
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Multi-photon states in photonic quantum computing?

Within photonic quantum computing, one of the ways to represent information is the dual-rail representation of single-photon states ($c_0|01\rangle \ + \ c_1|10\rangle$). Is it possible to utilize ...
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Ways in which $\frac{1}{\sqrt 2} (|00\rangle + |11\rangle)$ can be expressed as $\frac{1}{\sqrt 2} (|uu\rangle + |vv\rangle)$

I want to find out what values $|u\rangle$ and $|v\rangle$ can take if I want to write $$\frac{1}{\sqrt 2} (|00\rangle + |11\rangle)$$ as $$\frac{1}{\sqrt 2} (|uu\rangle + |vv\rangle).$$ Say $$|u\...
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How does using a superposition of 0 and 1 improve the processing capabilities of a quantum computer compared to classical computers? [closed]

Whenever I learn about quantum computing and qubits, it always talks about the superposition principle and that the qubits can be in both states 0 and 1 simultaneously, thus claiming that quantum ...
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Is there a way to convert a superposition $\sum_i a_i |x_i\rangle$ into $\sum_i |a_i,x_i\rangle$?

I am wondering if there is a way to convert a superposition $$\left|\phi\right>=\sum_{i}a_i\left|x_i\right>$$ into $$\left|\phi'\right>=\frac{1}{|{\rm norm}|}\sum_{i}\left|a_i,x_i\right>,$$...
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What is a bipartite quantum state?

I'm very confused for the definition bipartite quantum states. If it's just quantum states defined in $H_1 \otimes H_2$, then if $H_1$ and $H_2$ both are just 1 qubit system, then the bipartite ...
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What is the quantum analogue of $P_{XY} = P_{Y|X}P_X$

A standard trick in probability manipulation is to take some joint distribution $P_{XY}$ and express it as $P_{Y|X}P_X$. This trick is useful because when one looks at things like the ratio of $\frac{...
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Why can't we simulate a Qubit using classical computer?

I am completely a noob in terms of quantum computing, have watched several videos to understand what Quantum computers are trying to achieve. I am a programmer of classical computers. We have a ...
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What is the role of entanglement in quantum-computational speed-up?

The way I see it, there are three main quantum properties utilized in quantum computing - superposition, quantum interference, and quantum entanglement. I'm looking to understand which one is ...
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Is a black-box gate whose output is conditional on the value of an input amplitude possible?

Suppose we have a qubit in the state $|q\rangle = a |0\rangle + b |1\rangle$, and another ancilla qubit $= |0\rangle$. I wish to have the following black-box gate: ...
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Entanglement properties of $SU(8)$ quantum circuits vs nearest-neighbor $SU(4)$ quantum circuits

For this question, fix three qubits $q_1, q_2, q_3$. I'll use the notation $U_{123} \in SU(8)$ to denote an arbitrary quantum circuit/unitary on the three qubits, and $U_{12}, U_{23} \in SU(4)$ to ...
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1 answer
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Trace distance between mixed state and pure state vs trace distance between their purifications

Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that $$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$ where $\|A\...
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Upper bounding a permutation invariant state

Let $\rho_{A^n}$ be a permutation invariant quantum state on $n$ registers i.e. $\pi(A^n)\rho_{A^n}\pi(A^n) = \rho_{A^n}$ for any permutation $\pi$ among the $n$ registers. If we trace out $n-1$ ...
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HHL algorithm, How can I get result from register $|b\rangle$?

From the paper A survey on HHL algorithm: From theory to application in quantum machine learning , I use qasm code from here. I try to follow the example in page 7. with Ax = b and the answer x ...
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1 answer
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Prove that the conditional entropy of a classical-quantum state is non-negative

Let $\rho_{XA}$ be a classical-quantum state, i.e., $\rho_{XA} = \sum_{x} p(x) |x\rangle \langle x| \otimes \rho_A^x$. How to prove that the conditional von Neumann entropy $S(X|A) = S(\rho_{XA}) - ...
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Quantum teleportation and the reality of quantum states

This question is perhaps philosophical but it's been confusing me. Suppose Alice is teleporting some qubit state $|\phi\rangle$ to Bob via the quantum teleportation protocol. After Alice applies the ...
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How can the Holevo bound be used to show that $n$ qubits cannot transmit more than $n$ classical bits?

The inequality $\chi \le H(X)$ gives the upper bound on accessible information. This much is clear to me. However, what isn't clear is how this tells me I cannot transmit more than $n$ bits of ...
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Represent a pure state in terms of 2 antipodal points on the Bloch sphere

I recently had an assignment where the question is based on the assumption that we can write any pure state qubit $|\phi \rangle$ as: $$|\phi \rangle = \gamma |\psi\rangle + \delta |\psi^\perp ...
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Physical Interpretation of Pauli Matrices as Polarization Check

We know that the Pauli matrices are: $$\sigma_x = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \sigma_y = \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}, \sigma_z = \begin{bmatrix}1 & ...
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How to find the reduced density matrix of a four-qubit system?

I have the state vector $|p\rangle$ made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to find the reduced density matrix ...
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Quantum teleportation with "noisy" entangled state

This is actually an exercise from Preskill (chapter 4, new version 4.4). So they are asking about the fidelity of teleporting a random pure quantum state from Bob to Alice, who both have one qubit of ...
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Connection between the definitions of concurrence for a two-qubit states

The concurrence for a state $\rho$ as defined here is \begin{equation} C(\rho) = {\rm max}\{0, \lambda_1-\lambda_2-\lambda_3-\lambda_4\}. \end{equation} Where $\lambda_i$ are the eigenvalues of matrix ...
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What does equality of partial traces, ${\rm Tr}_1\rho={\rm Tr}_1\sigma$, say about a pair of states $\rho,\sigma$?

Let $\rho,\sigma$ be a pair of bipartite quantum states such that ${\rm Tr}_1\rho={\rm Tr}_1\sigma$. What does this tell us about $\rho,\sigma$? More precisely, is there a way to write more explicitly ...
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Eigenvalues of a quantum state after partial tracing

I am interested in the smallest nonzero eigenvalue of a quantum state. Does this eigenvalue always increasing after a partial trace i.e. the smallest nonzero eigenvalue of $\rho_A$ is always larger ...
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How to decompose a multi-target controlled gate?

I'm trying to replicate with qiskit the results of this paper in which basically they implement a quantum version of the Principal Component Analysis applying Quantum Phase Estimation algorithm in ...
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1 answer
663 views

Quantum PCA State Preparation

In Quantum Algorithm Implementations for Beginners is an example of the Quantum PCA with an given 2 x 2 covariance matrix $\sum$. The steps for state preparation are given in the paper. The steps are: ...
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How does a $2 \pi$ pulse in Cirac Zoller give a -1 sign to the state?

I understand the first step in the Cirac-Zoller controlled-phase gate; about how to move the state from the electronic state to the vibrational mode state. However, I am unable to understand how a $2\...
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What exactly is a phase vector?

The following $2\times 2$ matrix $$ P = \begin{bmatrix} e^{i\theta} & 0 \\ 0 & e^{i\phi} \end{bmatrix} $$ represents a quantum gate because it's a unitary matrix. If we multiply $P$ by ...
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1 answer
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Does applying the transformation $\sum\alpha_{jk}|j,f(k)\rangle\mapsto\sum\omega_N^{-jk}\alpha_{jk}|j,f(k)\rangle$ require computing $f^{-1}$?

Suppose that I have a bijective function $f: \mathbb{Z}_N → Y$ where $Y$ is a finite set. Suppose that $f$, but not its inverse, can be computed efficiently classically. I would like to apply the ...
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How many samples are required to estimate the probabilities of a state?

Suppose that we have a quantum state of the form: $$|\psi\rangle = \sqrt{p}|0\rangle + \sqrt{1-p}|1\rangle$$ In order to get an estimate of the probability of reading $|0\rangle$ or $|1\rangle$, we ...
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How do physical implementations of Z gate selectively affect $\lvert1\rangle $ basis vector?

The Pauli Z gate inverts the phase of $\lvert1\rangle $ while leaving $\lvert0\rangle$ unaffected. When I think about how $\lvert1\rangle $ and $\lvert0\rangle$ are physically realized, however, as ...
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Why does $x\sqrt{1-x^2}$ enhance the ability to approximate analytical functions in quantum circuit learning?

In this paper Quantum Circuit Learning they say that the ability of a quantum circuit to approximate a function can be enhanced by terms like $x\sqrt{1-x^2}$ ($x\in[-1,1])$. Given inputs $\{x,f(x)\}$, ...
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