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Questions tagged [quantum-state]

In quantum physics, quantum state refers to the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, i.e. for the outcome of each possible measurement on the system. A mixture of quantum states is again a quantum state. (Wikipedia)

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Why do we need reversibility

Suppose we have qubit $|a\rangle$. and we want to implement quantum addition say adding $|a\rangle$ and $|a\rangle$. When drawing the circuit for this operation one of the outputs that we get is ...
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Quantum representation of cube

Let say i have a square matrix of size $2^n\times 2^n$ with entries being 8 bit integers, where $2^n\times 2^n=b\times b\times b=2^l\times 2^l\times 2^l$, then if i want to represent that matrix in ...
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Encoding bosonic degrees of freedom

A well-known way of encoding $N$ levels of a harmonic (bosonic) oscillator is as follows: \begin{equation} |n\rangle = |1\rangle^{\otimes n} \otimes |0\rangle^{\otimes N-n+1} \quad,\qquad ...
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Finding separable decompositions of bipartite X-states using the methodology of Li and Qiao

Two recent papers of Jun-Li Li and Cong-Feng Qiao (arXiv:1607.03364 and arXiv:1708.05336) present "practical schemes for the decomposition of a bipartite mixed state into a sum of direct products of ...
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What is the physics behind a quantum classifier for labeling quantum states? [on hold]

I am trying to understand what physics is found behind labeling quantum states on the Ising Model.
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How to show a density matrix is in a pure/mixed state?

Say we have a single qubit with some density matrix, for example lets say we have the density matrix $\rho=\begin{pmatrix}3/4&1/2\\1/2&1/2\end{pmatrix}$. I would like to know what is the ...
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Associating quantum states with decimal numbers

Let's say I have a vector $\vec{v}=(1,0)$ and a state $|10\rangle$, and a decimal number $3$. I know that I can associate the decimal $3$ with the vector $\vec{v}$, but can I also associate the state $...
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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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How to superpose two composite qubit states?

Assuming we have two sets of $n$ qubits. The first set of $n$ qubits is in state $|a\rangle$ and second set in $|b\rangle$. Is there a fixed procedure that generates a superposed state of the two $|a\...
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Preparing odd integers using quantum computation

This is just a basic question. I need to output odd integers till $15$, i.e $1,3,5,7,9,11,13,15$. Since $15$ requires $4$ bits, I prepare initial state by using Hadamard gate on initial $|0\rangle$, i....
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Understanding this description of teleportation

In the context of quantum teleportation, my lecturer writes the following (note that I assume the reader is familiar with the circuit): If the measurement of the first qubit is 0 and the measurement ...
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Notation for two qubit composite product state

In my lecture notes on quantum information processing my lecturer gives an example of composite systems as $|\phi\rangle=|0\rangle |0\rangle=|00\rangle$. I understand that if we have two qubits then ...
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What are the individual probabilities after √SWAP gate?

Say, qubit $\left|a\right\rangle = \alpha_1|0\rangle + \beta_1|1\rangle$ and $|b\rangle = \alpha_2|0\rangle + \beta_2|1\rangle$. After $\sqrt{\text{SWAP}}$(a,b) what are new probability amplitudes of ...
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Are X-state separability and PPT- probabilities the same for the two-qubit, qubit-qutrit, two-qutrit, etc. states?

On p. 3 of "Separability Probability Formulas and Their Proofs for Generalized Two-Qubit X-Matrices Endowed with Hilbert-Schmidt and Induced Measures" (https://arxiv.org/abs/1501.02289), it is ...
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Hidden shift problem as a benchmarking function

I encountered the hidden shift problem as a benchmarking function to test the quantum algorithm outlined in this paper (the problem also features here). There are two oracle functions $f$, $f'$ : $...
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How to construct a quantum circuit for the following state transformation?

I have a certain transformations that goes as follows: Given $A=|abc\rangle$, $B=|xyz\rangle$, now I have cases as: $$\text{if }c=1,z=1, b\oplus y=1 \implies \text{flip}(x)$$ $$\text{if }c=1, z=1 \...
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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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Summing states of two qubit registers

I'm addressing the implementation with gates of an algorithm where there is the need of creating a qubit register $|\Psi\rangle$ starting from two input qubit registers $|a\rangle$ and $|b\rangle$, ...
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How do I embed classical data into qubits?

How do I embed classical data into qubits? I have a classical data [0 1] and I want to encode it as quantum amplitude to a superposition? What are the gates used to achieve that? I am a beginner to ...
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Meaning of “diagonal to the computational basis”

I came across the term "diagonal to the computational basis" in my reading recently. I'm not entirely sure what this term means. I know that a diagonal matrix is one with only non-zero elements on the ...
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How to understand the operators for watermarking schemes?

Note: Cross-posted on Physics SE. I am reading a research article based on quantum image watermarking (PDF here). The authors have defined some unitary transforms for the watermarking schemes, which ...
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How to complete this teleportation circuit? How to create a copy of $|\psi〉$?

This is a quantum circuit. M represents the act of making a measurement on the first two qubits. The circuit is supposed to transfer the state $|\psi\rangle = a |0\rangle + b |1\rangle$ ($a, b \in \...
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Swap Test for vector difference - how are different sized inputs combined?

I'm working on a similar problem of that raised by Aman in Inner product of quantum states Concerning the use of Swap Test for calculating the difference of two vectors. An example of the original ...
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How can I invert the least significant bit of a certain term of a superposed state?

Say I have $$\dfrac{1}{\sqrt{2}}\bigl(|1\rangle|221\rangle|0\rangle + |3\rangle|73\rangle|2\rangle\bigr).$$ How can I change that into $$\dfrac{1}{\sqrt{2}}\bigl(|1\rangle|221\rangle|1\rangle + |...
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How to implement such a gate where one input bit is zero and the other is one and the output should be one?

I have created 10 qubits, where I have set 2 in superposition by applying Hadamard gate to each of the 2 qubits. So the 2 qubits state can be $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$. I ...
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How to measure superposition coefficients to determine state?

There was a problem at the Winter 2019 Q# codeforces contest (that is now over), which I cannot find a mathematical solution for. The problem goes like this: You are given 3 qubits that can be in one ...
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What does it mean that copying a state is impossible but creating a copy of by entangling is possible?

Why is it that copying an unknown qubit is impossible but creating a copy of the standard computational basis is possible by entangling it to existing qubits? And why one is possible and the other isn'...
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EPR states with permuted qubits

Suppose I prepare following state consisting of (for example) three EPR pairs: $$\lvert\Psi\rangle = \frac{\lvert00\rangle+\lvert11\rangle}{\sqrt{2}}\otimes\frac{\lvert00\rangle+\lvert11\rangle}{\...
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POVM three-qubit circuit for symmetric quantum states

I have been reading this paper but don't yet understand how to implement a circuit to determine in which state the qubit is not for a cyclic POVM. More specifically, I want to implement a cyclic POVM ...
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What does an observable in a different basis mean physically?

So when you have an observable which is the measurement operator acting on the state you get a different result than in a different basis. What does it mean in physical terms? Does that mean writing ...
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Why isn't the circuit performing a measurement in the Bell basis?

Nielsen and Chuang (on page 188 exercise 4.33) says that the circuit including CNOT and Hadamard is performing a measurement in the Bell basis. But I can't see how. The matrix representing the ...
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Purity of mixed states as a function of radial distance from origin of Bloch ball

@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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For CNOT gate with control qubit set to 1, the measured state of the second qubit unexpectedly depends on the measurement

I have the following circuit in qiskit. It is a simple CNOT operation with the control qubit set to 1, by running the control qubit through an X-gate. However, when I measure the 2nd qubit, I get the ...
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Why is a density operator defined the way it's defined?

It's stated that the density operator is: $$\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.$$ But I don't understand why this is the way both in mixed state and pure state. ...
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The meaning of measurements in different bases

There are other similar questions. But I don't understand the answers. Suppose I express $a|0⟩+b|1⟩$ in the form $\frac{c}{\sqrt2}(|0⟩+|1⟩)+\frac{d}{\sqrt 2}(|0⟩−|1⟩)$ where $a,b,c,d∈\mathbb C$. ...
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What's a vector in the format of the Bloch Sphere?

The Bloch sphere isn't so intuitive for me. But I am not sure how you are supposed to manipulate it using vectors and matrices. How do you actually represent a vector on it? Is it $(\cos(a) , e^i\...
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The relationship between entanglement of vector states to matrix operations

I don't understand something which is I believe pretty fundamental. It's said that an operation represented by a matrix A is an entanglement if A can't be written as a tensor product of other matrices....
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If a qubit is in $|1\rangle$ state, is it possible to prove that it is in $|1\rangle$ state?

Measuring is clearly not the answer here. Is there some trick to prove it is $|1\rangle$ state?
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What is the meaning of the state $|1\rangle-|1\rangle$?

I don't understand it. But suppose you have a state of two qubits in superposition and it looks like $|10\rangle-|11\rangle$. Now you have in your circuits two gates one is a controlled-NOT from the ...
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Quantum machine learning after Ewin Tang

Recently, a series of research papers have been released (this, this and this, also this) that provide classical algorithms with the same runtime as quantum machine learning algorithms for the same ...
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Concurrence for a two qubit state

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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A two qubit state in a special form

How can a pure two-qubit state $|\psi\rangle = a |00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$, be written in the following form \begin{equation} |\psi_{\alpha}\rangle = \sqrt{\alpha}|01\...
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Is it possible to create a superposition in IBMQ QISkit which has probability amplitudes $|a|\neq |b|$?

For example, we can create a single qubit state with a polar angle of $\pi/2$ with the Hadamard gate. But, can we create a state such as this, $$\Psi = \cos(\pi/8)|0\rangle + \sin(\pi/8)|1\rangle$$ ...
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Why does $|P_U − P_V |$ equal $\langle \psi |U^{\dagger} M U|\psi\rangle −\langle \psi |V^{\dagger} M V |\psi\rangle$?

In QC and QI by Chuang and Nielsen, they state that the $P_U$ of operation $U$ acting on $\psi$ can be reached by $\langle \psi |U^{\dagger} M U |\psi\rangle$. Where $P_U$ (or $P_V$) is the ...
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If $|\psi\rangle, U|\psi\rangle$ are known, how many pairs of such qubits are required to find the operator $U$?

Assume that we know a quantum state and the result of applying an unknown unitary $U$ on it. For example, if the quantum states are pure qubits, we know $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ ...
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Do the probability amplitudes describe the probability of projective measurement?

I don't really know the difference between projective measurement and regular measurement or POVM. Until now, wherever I read about the subject I saw that the amplitude describes the probability of ...
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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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Forming states of the form $\sqrt{p}\vert 0\rangle+\sqrt{1-p}\vert 1\rangle$

I'm curious about how to form arbitrary-sized uniform superpositions, i.e., $$\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}\vert x\rangle$$ for $N$ that is not a power of 2. If this is possible, then one can ...
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Better Way Of Separating Two CQ-States

I have this cq-state: $$\frac{1}{2} \times (|0\rangle \langle0|_A \otimes \rho^0_E + |1\rangle \langle1|_A \otimes \rho^1_E)$$ Where Alice (A) is classical and an adversary Eve (E) has some ...
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Rabi oscillations with different energy differences

Please check this Rabi oscillations image1: Now, let's say I want to create a superposition of E1 and E2, so I started an EM field tuned to a frequency related to E2-E1. But, let's say there is ...