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 close or far apart are the distributions generated by two Haar random states?

Consider two $n$ qubit Haar-random quantum states $|\psi\rangle$ and $|\phi\rangle$. Let $D_{|\psi\rangle}$ and $D_{|\phi\rangle}$ be the two probability distributions (over $n$-bit strings) obtained ...
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Changing the sign of relative phase

Say I have a qubit in the state (ignoring normalization) $$|\phi\rangle = \alpha|0\rangle + e^{i\alpha}\beta|1\rangle.$$ How can I invert the sign of its phase, thus making it $$\alpha|0\rangle + e^{-...
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What does it mean “the N uses of classical-quantum channel”?

I was reading a paper "Quantum Polar codes by M.M wilde", where he discusses the N uses of the channel in the classical-quantum channel setting. What does he mean by "multiple channel ...
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In shadow tomography, how is the state reconstructed from its shadows?

I'm reading Huang et al. (2020) (nature physics), where the authors present a version of Aaronson's shadow tomography scheme as follows (see page 11 in the arXiv version): We want to estimate a state $...
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47 views

Inequality in overlap of quantum states

For quantum states $\vert\psi_1\rangle, \vert\psi_2\rangle, \vert\phi\rangle$, is it true that $$\tag{1}\langle \phi\vert\psi_1\rangle\langle\psi_1\vert\phi\rangle\langle \phi\vert\psi_2\rangle\langle\...
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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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Is the trace distance between multipartite states invariant under permutations?

Consider two multipartite states $\rho_{A_1A_2..A_L}$ and $\sigma_{A_1A_2..A_L}$ in $\mathcal{H}_{A_1} \otimes\mathcal{H}_{A_2} \otimes...\mathcal{H}_{A_L} $. For an arbitrary permutation $\pi$ over $\...
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Does it make sense to sum two Choi operators?

I am very new to the theory of the Choi representation of quantum processes and I am learning it all by myself from research papers (especially this https://arxiv.org/abs/1111.6950) as I didn't find ...
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Can we characterise how correlated the expectation values associated with a pair of observables are?

Consider a state $\rho$ and two observables $P$ and $Q$. Is there a good way to characterise how correlated the associated expectation values are? Be it in terms of mutual information or something ...
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Two qubit state + Depolarizing channel = Bell diagonal state?

In multiple sources, e.g. RGK, KGR, it is stated (without proof) that if you take any two qubit state and send it through a depolarizing channel, the resulting state would be a Bell-diagonal state. ...
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How does the CPTP constraint reflect on the matrix representation of a qubit channel in the Pauli basis?

Let us write the possible states of a qubit in the Bloch representation as $$\newcommand{\bs}[1]{{\boldsymbol{#1}}}\rho_{\bs r}\equiv \frac{I+\bs r\cdot\bs \sigma}{2},$$ where $\bs\sigma=(\sigma_1,\...
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What is the meaning of measuring a Bell state with Pauli operators?

There will be a certain value of getting the probability when measuring any Bell's state with Pauli operators such as observable X, Y, or Z. What is the meaning behind all this measurement? the result ...
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How to relate the result of measuring Pauli operators on Bell states with the quantum entanglement? [closed]

how to relate the result of measuring Pauli operators such as observable X and observable Y on the Bell's state with the quantum entanglement? In which way to say that there are correlated to each ...
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What are the possible results of measuring $X$ and $Z$ on the state $|01\rangle+|10\rangle$?

When calculating the probability of getting +1 on X-basis on the first qubit of Bell's state $|01\rangle+|10\rangle$, the result is 1/2 with the state after measurement |++⟩ while the probability of ...
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How to apply a operator to qubit system on the basis of current state of system?

Suppose I have three different operators $U_1, U_2,U_3$. Now, these three operators will be applied if my current state of the system is $|\psi_0\rangle,|\psi_1\rangle $ and $|\psi_2\rangle$ ...
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What is the probability of finding the second qubit as $0$ in the state $|\psi\rangle=\frac1{\sqrt2}|00\rangle+\frac12|10\rangle-\frac12|11\rangle $?

Assuming two qubits start in the state: $|\psi\rangle = \frac{1}{\sqrt 2}|00\rangle + \frac{1}{2}|10\rangle- \frac{1}{2}|11\rangle $ What is the probability of measuring the second qubit as 0? And ...
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Question about Haar random quantum states

Let $|\psi\rangle$ be a $n$ qubit Haar-random quantum state. I am trying to show that in the limit of large $n$, for each $z_{i} \in \{0, 1\}^{n}$, $$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\...
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$3 \rightarrow 1$ QRAC encoding for XOR functions

I'm currently working on QRAC and was wondering if there's an encoding protocol in $3 \rightarrow 1$ such that the receiver is able to retrieve any one of the XOR combinations of the bits, along with ...
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What is this equation for coin operator is trying to do in this quantum walk for Non-regular graph? This coin operator is called Fourier coin

I am reading the following paper: Discrete-time quantum walk on complex networks for community detection by Kanae Mukai We define the Coin operator $C$ by: $C=C_1\otimes C_2....C_n$ , We define coin ...
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Measurement of 1 qubit in a two qubit system

I created two qubits in IBM quantum experience and measured the second qubit without applying any gate. The result I got had a computational basis of 00 and 10. How does the measurement of the second ...
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How to perform the unitary transformation $U|i,j_1\rangle = 1/\sqrt{k}(|i,j_1\rangle+|i,j_2\rangle+|i,j_3\rangle…+|i,j_k\rangle)$?

Is the following unitary transformation possible? If so, what will be the value of $U$? $$U|i,j_1\rangle = 1/\sqrt{k}(|i,j_1\rangle+|i,j_2\rangle+|i,j_3\rangle...+|i,j_k\rangle)$$ Here, $i$ is a node ...
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Quick question about Two-qubit SWAP gate from the Exchange interaction

I am reading the following paper: Optimal two-qubit quantum circuits using exchange interactions. I have a problem with the calculation of the unitary evolution operator $U$ (Maybe it is stupid): I ...
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What is a unitary operator that makes all the amplitudes all negative on the arbitrary state of $n$ qubits?

What is a unitary operator that makes all the amplitudes all negative on the arbitrary state of $n$ qubits? For example suppose, $n=2$, the arbitrary state is: $a_1|00\rangle+a_2|01\rangle-a_3|10\...
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Produce a quantum state with its density matrix an identity matrix up to an constant

For a n-qubit quantum state $|\psi\rangle=\displaystyle\sum_{i=0}^{2^N-1}|i\rangle$, by definition it's density matrix is $|\psi\rangle\langle\psi|=\displaystyle\sum_{i,j=0}^{2^N-1}|j\rangle\langle i|$...
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Probabilities of entangled state. Quantum measurement [duplicate]

I confused about how to calculate the PROBABILITIES and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be ...
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Prove that for a general tri-partite state $\rho_{ABE}$, $H(AB) = H(E)$

How do I prove that for a general tri-partite state $\rho_{ABE}$, the following holds: $$ H(\rho_{AB}) = H(\rho_{E}), H(\rho_{AE}) = H(\rho_{B}), $$ where, $H$ is the Von Neumann entropy. Would ...
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What is the result of measuring $\sigma_x$ on the state $|01\rangle+|10\rangle$?

I confused about how to calculate the probabilities and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be ...
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Which Gate set can be used to perform the function $U|0\rangle =\frac{1}{\sqrt{n}}\sum_{i=0}^{n}(|i\rangle)$?

I want to perform the following operation: $$U|0\rangle =1/\sqrt{n}\sum_{i=0}^{n}(|i\rangle).$$ I know that Hadmard gate can give me the superposition of states $|0\rangle$ and $|1\rangle$. But it can ...
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How to encode $|i \rightarrow j \rangle$ using binary string?

We define the quantum state on a complex network in the form, $$\mid\psi(t)\rangle=\sum_{i=1}^{N}\sum_{j=1}^{k_i}\psi_{i,j}(t)\lvert i\to j\rangle, $$ where $N$ is the total number of nodes, the ...
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Can the (universal) state inversion operator be physically realized?

I was trying to solve an exercise from Vazirani's course "Qubits, Quantum Mechanics and Computers": A mathematically nice, but unphysical, way to detect entanglement is to use the state ...
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Quantum Boltzmann machine: How do you sample from the Boltzmann distribution on a quantum computer?

I am reading through the following article https://arxiv.org/abs/1601.02036. Eq. (22) describes one of the terms of the gradient of the log-likelihood cost function, which can be estimated using ...
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Is the set of two-qubit absolutely separable states convex?

Companion question on MathOverflow Let us order the four nonnegative eigenvalues, summing to 1, of a two-qubit density matrix ($\rho$) as \begin{equation} 1 \geq x \geq y \geq z \geq (1-x-y-z) \geq 0. ...
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Depolarization of density operator with zeros in diagonal

I suppose a quantum state with density matrix like the following is not valid. $$ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. $$ Now, let's say I have a valid density operator representing ...
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How can I construct a 2-qubit state using single qubit gates and CNOT gate?

How can I construct the below 2-qubit state using suitable single qubit gates (maximum 3) and one CNOT gate starting with state $|00\rangle$? $$ |\omega\rangle=\frac{1}{3}(2|00\rangle+|01\rangle+2|11\...
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Qiskit get qubits from statevector

How do I get the amplitude of each qubit, like plot_bloch_multivector(), but not output the tensor product of all qubits. In this case I need output ...
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What does the notation $|\psi(0)\rangle = |0\rangle|n=0\rangle$ mean?

Let us take the initial state with the particle located at the origin $|n=0\rangle$ and the coin state with spin up $|0\rangle$. So, $$ |\psi(0)\rangle = |0\rangle|n=0\rangle, $$ where $|\psi(0)\...
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How can I find a quantum channel connecting two arbitrary quantum states?

Given two arbitrary density matrices $\rho, \sigma\in \mathcal{H}$ (they have unit trace and are positive), how do I go about finding a possible quantum channel $\mathcal{E}$ such that $\mathcal{E}(\...
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Loading a random distribution using controlled-Y rotations

In the paper by Stamatopoulos et al., the authors say that it is possible to load a distribution on a three qubit state to obtain: In Qiskit finance this is performed using the uncertainty model ...
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Cirq - Measure Density Matrix Function Getting First Element

Hello I am using measure density matrix function like that: ...
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Is it wrong to say that $a$ and $b$ are the square roots of the detection probabilities in a qubit state $|\psi \rangle = a|0 \rangle +b|1 \rangle $?

Is it wrong to say in $a$ and $b$ are the square roots of the probability of the qubit being in the state 0 and 1 when measured for a qubit in the state $|\psi \rangle = a|0 \rangle +b|1 \rangle $? ...
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What is the difference between MPS and DMRG? [closed]

How can matrix product states simulate quantum circuits? I heard matrix product states come from DMRG (density matrix renormalization group). Here are my questions. What is the difference between MPS ...
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How to represent the state vector form of a qubit in density matrix representation? [duplicate]

While I'm studying state vector and density matrix. I wonder how to write qubit state as density matrix. qubit state can be represented with state vector form. But how about density matrix?
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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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Quantum State for Non-Regular Graphs

While studying quantum walk on graphs, I am able to find how d-regular graphs state can be constructed in quantum domain. But not able to find anything on non-regular graph. For ex-: how this graph ...
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Quantum analogues of information theoretic measures: are log probabilities replaced with the density matrix?

Below is a question and an answer. How does quantum information relate to, diverge from or reduce to Shannon information, which used log probabilities? What people are more often interested in are ...
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Cirq-Measuring a State with Rotation Matrix

I have this state: $$p |\text{GHZ}\rangle \langle \text{GHZ}| + (1-p)\rho$$ And after creating this state I have this code lines: ...
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Alternative definition of the coherent information of a quantum channel

Let $T: M_n \to M_n$ be a quantum channel. If I understand Definition 13.5.1 of the book "Quantum information theory" of Wilde, the coherent information $Q(T)=\max_{\phi_{AA'}} I(A \rangle B)...
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Trace distance bound after partial trace

Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.} I know that the trace distance ...
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How to verify teleportation was successful in this circuit?

In this circuit (link), q0 is Alice's qubit and q1, q2 are entangled qubits given to Alice ...
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Why does describing a quantum state take an infinite amount of classical information? [duplicate]

In chapter 1 of Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, I came across this paragraph on quantum teleportation, Intuitively, things look pretty bad for ...

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