Questions tagged [quantum-state]

Quantum systems can mathematically be described by their 'quantum state'. When the system is closed/isolated, the state is 'pure' and can be written as a sum (i.e. 'superposition') of basis vectors. When the system is a subsystem of an open system, the state is instead usually 'mixed' and cannot be written as a pure state, so has to be written as a density matrix. Consider using the density-matrix tag when relevant

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What is the eigenvalue distribution of arbitrary unitary matrices?

I had a question regarding the nature of the eigenvalue distribution of unitary matrices. Searching for the answer I found that the unitary matrices which are sampled randomly have a defined ...
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Distinguishing $\frac{| 0 \rangle + e^{i\theta} |1 \rangle}{\sqrt{2}} $ from $| 0 \rangle/|1 \rangle$ with probability $1/2 + \epsilon$

I am given one copy of one of two quantum states - $\frac{| 0 \rangle + e^{i\theta} | 1 \rangle}{\sqrt{2}} $, for some unknown fixed $\theta$. One of $| 0 \rangle/|1 \rangle$ - don't know which one, ...
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How to decompose given 4x4 matrix to one and two qubit unitary matrices?

I have matrix $B=\begin{bmatrix}0&&0&&0&&0\\0&&1&&0&&0\\0&&0&&2&&0\\0&&0&&0&&3\end{bmatrix}$. By doing $...
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What are the possible initial states that can be prepared in a lab for use in a quantum computation?

So here's something that's been bothering me. Given the time evolution of the wavefunction can only be unitary or discontinuous as a process of the measurement. So let the observables for our ...
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How is this expression for a GHZ state obtained in the nature paper by Pan et al. (2000)?

Can someone tell me how the authors of the paper "Experimental test of quantum nonlocality" (Nature link to abstract) have rewritten their equation 1 in terms of equation 2 and 3?
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How can a density matrix be prepared on a quantum register?

I am currently trying to implement the VQSE algorithm. There the biggest eigenvalues and their corresponding eigenvectors of a density matrix $\rho$ are computed. In contrast to VQE, the matrix $\rho$ ...
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Properties of frames in quasiprobability representation

Let $\mathbb{C}^{d}$ be a complex Euclidean space. Let $\mathsf{H}(\mathbb{C}^{d})$ be the set of all Hermitian operators, mapping vectors from $\mathbb{C}^{d}$ to $\mathbb{C}^{d}$. I had some ...
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If $\rho,\sigma$ are classical-quantum states, can the fidelity $F(\rho,\sigma)$ be expressed in terms of $F(\rho_i,\sigma_i)$?

Let $\rho = \sum_i \vert i\rangle\langle i\vert \otimes \rho_i$ and $\sigma = \sum_i\vert i\rangle\langle i\vert\otimes\sigma_i$ where we are using the same orthonormal basis indexed by $\vert i\...
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The expectation of a measurement of qubit 2 after qubit 1 has been measured

In section 1.2.4 (page 13) of these lecture notes http://users.cms.caltech.edu/~vidick/teaching/fsmp/fsmp.pdf, it says \begin{aligned}\left\langle\psi\left|X_{1}^{0} Z_{2} X_{1}^{0}\right| \psi\right\...
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Why does $(XZ\otimes I)|\Phi^+\rangle$ equal the Bell state $|\Psi^-\rangle$?

I'm slightly confused by the solution provided below by a suggested solution online to convert |$\phi^+$⟩ to |$\psi^-$⟩. I tried doing the operation XZ but I got $\frac{1}{\sqrt2}$(|10⟩-|01⟩) instead ...
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Learning k positions of a Boolean function with a quantum computer

Consider a Boolean function with multiple outputs $f: \{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$, and consider being given oracle access to the function $f$. Let us denote the oracle by $O_f$. For an $x \...
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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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Circuit to transform $|0\rangle$ into $\alpha|0\rangle + \beta|1\rangle$ for any $\alpha, \beta$

Hi I'm new to QC and doing some katas in Q#. I got stuck on this excercise and would appreciate help: Quantum circuit to get following state qubit: $\alpha|0\rangle + \beta|1\rangle$ when the input is ...
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How do we change the basis of a given qubit state?

I'm reading this paper (Link to pdf) about a test of entanglement with three particles. I wanted to ask if there is any mathematical shortcut to express one quantum state on another basis like the ...
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What does it mean that a qubit is a triple $(H,X,Z)$ with $H$ Hilbert space and $X,Z$ Pauli operators?

In this paper, http://users.cms.caltech.edu/~vidick/teaching/fsmp/fsmp.pdf, it gives the definition of a qubit as follows: A qubit is a triple $(H, X, Z)$ consisting of a separable Hilbert space H and ...
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Prove that a Bell state is invariant under the single-qubit gate acting on both qubits

I have a Bell state ${\Psi}^{-}= \frac{1}{\sqrt2} (|01\rangle - |10\rangle).$ How can I prove that this state is invariant (up to a global phase), when doing the same unitary $U$ on each qubit? That ...
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quantum label classification using qiskit

I am generating points for classification. Some will be above the main diagonal, while others will be below (blue or red). ...
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If the partial traces $\rho_A,\rho_B$ are pure, does it imply that $\rho$ is a product state?

Suppose $\rho$ is some bipartite state such that the partial traces $\rho_A={\rm Tr}_B\rho$ and $\rho_B={\rm Tr}_A\rho$ are both pure. Does this necessarily imply that $\rho$ is a product state? This ...
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Regrouping the terms in expression 1.31 in Quantum Computing and Quantum Information, Nielsen and Chuang

I'm trying to reproduce the passage from expression 1.31 to 1.32 in the book Quantum Computing and Quantum Information, by Michael Nielsen and Isaac Chuang. Expression 1.31 is: $$|\psi_2\rangle = \...
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How does superposition apply to quantum computing?

beginner here. I've always heard explanations on quantum computing, all about superposition, entanglement, etc. But how does superposition actually apply to quantum computing? Yeah, its "in ...
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Minmax theorem for optimization over isometries and states

I have the following minmax problem and I am wondering if the order of the minimum and maximum can be interchanged and if yes, why? Let $\|\cdot\|_1$ be the trace norm defined as $\|\rho\|_1 = \text{...
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Normalize and encode real data in a way that encode collinear vectors with different values

Now, I am working on a quantum supervised learning problem and I have a problem with amplitude encoding. Before being encoded, a vector $(a_1, a_2,\dots,a_n)$ must be normalized in such a way that $\...
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Permutation invariant states have permutation invariant purifications - proof?

I don't remember where I came across the statement but I'm pretty sure it is true and am interested in understanding why it holds. For any $n-$ register state $\rho^n \in H^{\otimes n}$ that is ...
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What is the Schmidt decomposition for the 00 + 01 state?

If I try to write the two-qubit state $$ |\psi \rangle = \frac{|0 \rangle |0 \rangle + |0 \rangle |1 \rangle}{\sqrt{2}}$$ as $$ |\psi \rangle = \lambda_0 |\phi_0 \rangle |\phi_0 \rangle + \lambda_1 |\...
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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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Is the composite state necessarily mixed when there are all but exactly one mixed state?

Let $\rho_A$ be a mixed state (density operator). If we wish, we can purify it and get a $\rho_{AB}$ that is pure on some composite Hilbert space $H_{AB}$. And by the question I just ask, How to prove ...
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Quantum based lottery using W-State and spatial separation

I'm thinking of a use case of building a quantum based lottery. Using a W-state with spatial separation (see this question) the circuit is build at one location and afterwards the n qubits are ...
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Entanglement distribution of W-State over different locations

I would like to create a quantum system with the gates for a W state where each qubit is at a different location. Entanglement distribution has been proven in several research articles. I'm new to ...
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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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Does the definition of separability of pure states require the components of the summands to be pure?

Does the definition of separability of pure states require the components of the summands to be pure? More precisely, let $\rho$ be a pure state (i.e., $\rho=|\phi\rangle\langle\phi|$) on the space $...
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Why are the probabilities $|\alpha|^2$ and $|\beta|^2$ when measuring in the computational basis?

In measurement in the computational basis, I was being told that it is a way to extract information from a qubit, and it outputs a classical bit. For the quantum state $\alpha |0\rangle + \beta |1\...
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How to calculate inner product of quantum states with other method than swap test? [duplicate]

In connection to this question, I am wondering how to calculate value $\langle \psi|\phi \rangle$ for arbitrary quantum states $|\psi\rangle$ and $|\phi\rangle$. A swap test is able to return only $|\...
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Relation between approximate counting and sampling

Consider the following statement of Stockmeyer counting theorem. Given as input a function $f:\{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$ and $y \in \{0, 1\}^{m}$, there is a procedure that runs 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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What is the significance of the phase angle? [duplicate]

I've the following circuit which gives an output of 1 with a phase angle of 3π/4. When we measure the circuit all we get is the ...
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How to write the eigenvectors of a mixture of two pure states?

Let $|\psi_1\rangle,|\psi_2\rangle$ be two pure states. Assume $\langle\psi_1|\psi_2\rangle\neq0$, and consider the convex combination $$\rho\equiv p_1 |\psi_1\rangle\!\langle\psi_1| + p_2 |\psi_2\...
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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 $2^n$. ...
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In the swap test, how is the final probability $P(0)$ calculated?

Does anyone know much about quantum dot product: Lets say: $$|\psi \rangle = \frac{|0\rangle_1|\overrightarrow{x_i}\rangle_2 + |1\rangle_1|\overrightarrow{x_j}\rangle_2}{\sqrt 2}$$ $$|\phi \rangle = \...
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How to find the $A_i$ in the matrix product state representation?

From what I understand, MPS is just a simpler way to write out a state, compared to the density matrix. But how do I get those $A_i$ matrices? From all the examples I read, people just somehow "...
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General expression for a state that is close in trace distance to a pure state

Suppose we are given that a quantum state $\rho$ is close in trace distance to a pure state $\vert\psi\rangle\langle\psi\vert$. That is $$\|\rho - \vert\psi\rangle\langle\psi\vert\|_1 \leq \varepsilon,...
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Expectation value of a quantum circuit [closed]

The expectation value of an operator $A$ is defined by this equation $\langle A \rangle_\psi = \sum_j a_j |\langle \psi | \phi_j \rangle|^2 $. My first question is does it mean that the expectation ...
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What are spin-coherent states?

Trying to understand the paper; https://arxiv.org/pdf/1702.02577.pdf and ran into "spin-coherent" states. I wonder those are.
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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 ...
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Given $|\psi\left(t\right)\rangle=\sum_{n=0}^{N}c_{n}\left(t\right)|N_{1}\rangle_{a}|N_{2}\rangle_{b}$, what is the expression for $c_n(t)$?

We have: $$|\psi\left(t\right)\rangle=e^{-iH_{NL}t}|N\rangle_{a}|0\rangle_{b}$$ Also $$|\psi\left(t\right)\rangle=\sum_{n=0}^{N}c_{n}\left(t\right)|N_{1}\rangle_{a}|N_{2}\rangle_{b}$$ I would like to ...
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What's free evolution for a period T?

I am currently studying a model of a quantum (atomic) clock. And in this paper, I came across the term "Free evolution for a period T": Free evolution for a period T where a phase ...
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Quantum State Tomography from IQ plane data

Background: I am given to understand that the steps of Quantum State Tomography (QST) are as follows for a single qubit: The qubit is in the state $\psi=a_0|0\rangle+a_1|1\rangle$ with density matrix ...
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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| \...
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Can I use Grover's algorithm on overlapping sets of qubits?

Let's say I have 3 qubits: $q_1,q_2,q_3$. I want to apply Grover's algorithm on q1,q2, such that q1,q2 $\neq$ 10 and do the same for q2,q3, so that q2,q3 $\neq$ 11. The final possible combinations of ...
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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 ...
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What is the IQ plane?

I struggle to find any information on Nielsen and Chuang or similar texts on the exact definition of the so-called IQ plane (I think this is a notion closely related to solid state quantum computers ...

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