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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How to derive $|0\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$?

When learning measurement basis, my teacher told us $|0\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$ and said that we can derive it ourselves. Along this, he also mentioned $|+\rangle=\frac{1}{\...
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Does QRNG generates True Random numbers?

I read an article that claims, QRNG can produce a true random number. So I wonder, how could they prove that this is a true random numbers generator? In fact, imagine I look at my memory state and ...
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54 views

How is a classical bipartite state written in quantum notation?

As in the title, is a classical bipartite state on $AA'$ given by $$\sum_{ij} p_{ij} \vert i\rangle\langle i\vert_A \otimes \vert j\rangle\langle j\vert_{A'}$$ with $\sum_{ij}p_{ij} = 1$. In ...
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How to generate a quantum circuit from the quantum state $|1000\rangle+|0100\rangle+|0010\rangle+|0001\rangle$?

I am trying to understand the steps of how make a state preparation circuit from a quantum state. For making my question more clearer, for example, for the state is $\frac{|00\rangle+|11\rangle}{\...
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86 views

What are the constraints for the coefficents of the basis states in quantum computing?

$\newcommand{\ket}[1]{\left|#1\right>}$ It's known that the Kolmogorov axioms characterise a probability distribution: Probability of an event is a non-negative real number. The sum of all ...
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1answer
77 views

How to normalise when the probability of measurement is zero?

In one of the answers to this question on measuring one qubit it is explained that given a general two-qubit state $$ |\psi\rangle = \begin{bmatrix} \alpha_{00} \\ \alpha_{01} \\ \alpha_{10} \\ \...
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How to determine the direction of arrows on a Bloch sphere from a vector/ket representation?

The following graphs contain the vector/ket representation of the vector on the bloch sphere. I don't understand how to determine the position of the vector on the bloch sphere based on those vector/...
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Can one always find purifications which preserve equality of statistical mixtures?

When pure states $|\psi_1⟩$, $|\psi_2⟩$ and $|\phi_1⟩$, $|\phi_2⟩$ in $\mathcal{H}_A \otimes \mathcal{H}_B$ have identical statistical mixtures $$\frac{1}{2}(|\psi_1⟩⟨\psi_1| + |\psi_2⟩⟨\psi_2|) = \...
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How do get $\rho_{BA}$ if I have $\rho_{AB}$

If Alice and Bob share the state: $$\left| {{\psi _{AB}}} \right\rangle = \sin \theta \left| {10} \right\rangle + \cos \theta \left| {01} \right\rangle $$ then $\rho_{AB}$ can be obtained as: $${\...
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Meaning of a pound sign (#) on a Bloch sphere

For the following Bloch sphere representation of a qubit, what does the highlighted symbol mean? I'm not sure if it means anything or it's just for showing that it's a sphere, not a circle.
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138 views

Is there an algorithm that can decide if a state is a mixed state or a pure state?

Given we are using the computational basis,is there a quantum algorithm that can decide if an arbitrary input state $\vert A\rangle$ ( using $N$ qubits) is a pure state or a mixed state? $\vert A\...
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What is the complexity of loading $n$ inputs using a qRAM?

I am interested in reading data from a real database and I find qRAM has the effect of loading data: $\sum\phi\left|i\right>\left|0\right>\rightarrow\sum\phi\left|i\right>\left|d_i\right>$,...
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What are the conditions ensuring a two-qubit density matrix is positive semidefinite?

I've seen some papers writing $$\rho=\frac{1}{4}\left(\mathbb{I} \otimes \mathbb{I}+\sum_{k=1}^{3} a_{k} \sigma_{k} \otimes \mathbb{I}+\sum_{l=1}^{3} b_{l} \mathbb{I} \otimes \sigma_{l}+\sum_{k, l=1}^{...
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What is the “quantum phase” of a quantum state?

On this page IMBQ docs, until the sentence '..and since the global phase of a quantum state is not detectable..' I follow everything. However 'quantum phase' is introduced without any explaination? ...
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1answer
62 views

Unitary Transformations for Schmidt Decomposition

$\newcommand{\ket}[1]{|#1\rangle}$ Suppose a pure state $\ket{\psi}$ has a Schmidt decomposition given by $\ket{\psi^{SD}}$, which can be obtained via the diagonalization of the reduced density matrix ...
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How to understand combination states vs pure/mixed states?

I've learned that representing a combination of two states, I simply need to take the tensor product of the states. For example: $$\left|\Psi\right>=\alpha_0\left|0\right>+\beta_0\left|1\right&...
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Error correction on $n$ qubits all in same state except for a few

This might be a straightforward problem for you guys; it would be helpful if you can explain it in simple language. I have $n$-qubits given as $$\frac{1}{\sqrt{2}} \left(|0\rangle+ e^{\iota\theta_{1}}|...
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How to perform a plot histogram for a circuit?

I have created a circuit and I don't know how to plot a histogram. I tried to plot a histogram but it gives me output for 0000 case only, how to get to know the probability for all of the cases. The ...
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What is the best way to extend a state $\rho_S$ to a tensor product of spaces ${\cal H}_S\otimes{\cal H}_A?$

Let $\Phi_S$ be an operator acting on a space $\mathcal H_S$. If we introduce an ancilla $A$, the total space becomes $\mathcal H_S\otimes \mathcal H_A$ and I can naturally extend the operator $\Phi_S$...
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On existence of orthonormal basis for each subsystem in Separable state [closed]

A separable state in $\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ is given by $$\rho_{s}=\sum_{\alpha,\beta}p(\alpha,\beta)|\alpha\rangle\!\langle\alpha|\otimes|\beta\rangle\!\langle\beta|.$$ Now, my ...
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What are the conditions under which an unknown quantum state is learnable with arbitrary precision?

Assume that we have an unknown quantum state and we need to learn that unknown state with arbitrary precision. Under what conditions can we learn the unknown state with arbitrary precision? One ...
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Restrictions on Quantum Gates and Pure Partially Traced Output States

Suppose we have a quantum circuit that contains an arbitrary number of quantum gates and takes as an input more than a single qubit, say three. What are the restrictions on the quantum gates and the ...
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102 views

Find the conditions under which the state $|\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle$ is unentangled

Show that the state $ |\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle $ is unentangled if $a \in \{ 0,1,...,2^n - 1\} $ and $|\phi\rangle$ can be expressed in the form $ \...
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Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
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44 views

How can I simulate the Avg CNOT Error on IBMQ Backends?

I want to know exactly how to estimate the Avg CNOT Error rate for IBMQ Backends? For instance, I tried to estimate the Avg CNOT Error rate for Belem Backend; I randomly prepared the 00,01,10,11 ...
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What is this pulse experiment showing?

The following results in the figure were achieved using Qiskit pulse by doing the following pulse sequence; { $\frac{\pi}{2}$, delay $\tau$, $\pi$, delay $\tau$, $\frac{\pi}{2}$} The figure is the ...
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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 ...
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Do sequences of operations (including measurements) applied to different halves of an entangled pair always commute?

Let us say $A$ has one half of an entangled qubit pair, and $B$ has the other half. $A$ may be able to perform any type of operation on their half of the pair, such as unitary operations, entangling ...
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How do you quantify the figurative 'cost' of a quantum circuit

Many gates are not available on a real computer and therefore the circuit must be transpiled into a specific set of gates. I have seen this equation below which is used to to determine the 'cost' of a ...
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Is the quantum mutual information variance bounded from above?

The relative entropy variance between two quantum states $\rho$ and $\sigma$ is defined to be $$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$ where $D(\rho\|\sigma)$...
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53 views

Circuit state preparation using amplitude encoding

I am following an example of preparing an input state using amplitude encoding from this book. How to calculate $\beta_1^1$ using given formula above? In my understanding, $\beta_1^1 = 2\arcsin(\frac{\...
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How do I find the state of each qubit at the end of the circuit?

I have this Quantum Fourier Transform (QFT) and I want to know how to find the final state of each qubit if q0, q1, ...
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107 views

Lower bounds on the number of measurements outcomes required for quantum state tomography

It seems that in order to reconstruct a quantum state, a large number of measurements is typically used. Are there any known theoretical lower bounds on the number of measurements required to ...
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1answer
136 views

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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1answer
65 views

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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1answer
85 views

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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1answer
59 views

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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44 views

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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1answer
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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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81 views

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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81 views

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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146 views

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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