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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SWAP test and density matrix distinguishability

Let us either be given the density matrix \begin{equation} |\psi\rangle\langle \psi| \otimes |\psi\rangle\langle \psi| , \end{equation} for an $n$ qubit pure state $|\psi \rangle$ or the maximally ...
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How is transformation for measurement in an arbitrary basis derived?

I started with Qiskit today and find it very exciting. As a first question I want to understand how to measure an arbitrary state $|\Psi\rangle$ not in the basis of ...
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How to write the covariance matrix of a quantum gaussian state as a function of photon numbers?

Assume having a one-mode quantum Gaussian state with quadrature observable vector $\hat r = [\hat q , \hat p ] $ and covariance matrix $\sigma$. According to definition [1]: \begin{equation} \sigma = \...
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Why does quantum distinguishability ensure no faster-than-light communication?

On page 56-57 in Nielsen and Chuang, for a proposed scenario, it's said that: if Bob had access to a device that could distinguish the four states $|0\rangle$, $|1\rangle$, $|+\rangle$, $|−\rangle$ ...
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Why is there there an infinite number of possible bases but only a finite number of measurement outcomes

The statement in the question comes from the answer to the following question on Reddit, if more context is needed: And how does a perfect quantum computer ensure that we know for sure that the state ...
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Working with higher energy states in Qiskit Experiments

I'm looking at the source code for the Qiskit Experiments module and see that the EFRabi class subclasses the Rabi class with the intent of calibrating rotations between the 1 and 2 states instead of ...
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Can changing reference frames generate entanglement in identical particles?

Suppose we have a pair of qubits, physically realised as two spin-half particles in some separable pure state $|\psi\rangle$, separated by a distance '$l$', large enough to be regarded as ...
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Conditional lower bound on approximate stabilizer rank of magic states

I am currently reading about the approximate stabilizer rank and properties of the same. I will quote the definitions from this paper. The stabilizer rank of a quantum state $|\psi\rangle$ is the ...
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What is the computational complexity in initializing a quantum register?

I'm trying to figure out what is the computational complexity of initializing a quantum register of N qubits. For my research, I have used the initialize method of qiskit, in which you set the ...
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how can i find ground state of final hamiltonian

decompose the evolution operator into a sequence of steps using the Trotter-Suzuki formula unitary operator is the evolution operator from 0 to T , k is a large integer so that τ = T /k is a small ...
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How can extract reduced dynamics of a bipartite system from unitary evolution in quite

Let us assume that I have a bipartite system $A\otimes B$ and an initial product state undergoing some evolution $H^{AB} = H^A+H^B+V^{AB}$, which is time independent. I want to simulate the reduced ...
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When can pairs of states be transformed into other pairs of states via unitary mapping?

The states $|+\rangle, |-\rangle$ can be mapped to $|0\rangle, |1\rangle$ by a simple rotation. But if I now have other states ($|\psi_0\rangle, |\psi_1\rangle$) which are not orthogonal, does a ...
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How can we measure a quantum system when the sum of amplitudes-squared does not equal one?

How can we measure a quantum system if the sum of amplitudes-squared does not equal one? For example, if we want to measure $|a\rangle = 0.25|0\rangle + 0.25|1\rangle$, how can we measure it?
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Find a unitary to prepare state $|0\rangle$ to a specific vector

I am working with Variational Quantum Linear Solver (VQLS) algorithm, where it needs to prepare a control_b circuit. Assume b is 1d with $ 2^n $ elements in it. $$ {\bf Ax = b} \tag{1}$$ I need to ...
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Proof for Cardinality of the Clifford Group

In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ...
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Do entangled measurements across multiple copies help in state distinguishability?

Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one. There is an optimal measurement to ...
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How does Equation (9) follows from these definitions?

How does Equation (9) follows from these definitions? This is my working out for the first identity. It appears that the identity is not true. For the second identity, I used sympy to check. The ...
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Is the CNOT in the standard three-qubit circuit for the GHZ state necessary?

This is a very basic question about the GHZ state. I know the standard construction: A Hadamard on one qubit, and then CNOT gates with targets on all the other ones. However, why can't I just have $n$...
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Are projective measurements the only optimal measurements to discriminate between two states?

Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one. There is an optimal measurement to ...
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How do we ensure that the states input to a quantum algorithm are what we want them to be?

In various quantum algorithms like quantum Fourier transform we see that our input states are forced to be specified states. But we know that the main property of quantum computers is that the state ...
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Compute ${\rm tr}(a_k a_{k'}\rho)$ with $\rho=e^{-\beta H}/Z(\beta)$ Gibbs state and $a_k$ ladder operators

Consider a harmonic oscillator with hamiltonian $H=\sum_k\omega_k a_k^\dagger a_k$ and a state $\rho=\frac{e^{-\beta H}}{Z(\beta)}$ where $Z(\beta)=\text{tr}[{e^{-\beta H}}]$. The quantity $$A:=\sum_{...
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Quantum communications and "knowledge" of receiving a qubit

Question from a computer scientist (not a physicist). Imagine two nodes on a quantum network. Alice sends Bob a qubit in some state of superposition over a quantum channel. Is Bob able to sense that ...
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How large can we make the fidelity between mixed states by allowing unitaries?

For pure states, it is known that one can always find a unitary that relates the two i.e. for any choice of states $\vert\psi\rangle$ and $\vert\phi\rangle$, there exists a unitary $U$ such that $U\...
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Why in quatum coin toss, when starting from state $|1\rangle$, we get $|-\rangle$?

I was reading qiskit's text book, there I found that for a Double quantum coin toss, we have negative probability amplitude for $|1\rangle$ state when we starts from $|1\rangle$ state. Link : https://...
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How to make sense of phases of individual qubits in the context of multiple entangled qubits?

I hear that phase information of qubits is important and you can clearly see the phase of a qubit when represented on the Bloch sphere. But, I am not sure what to think of the phase of individual ...
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Interpretation of the unitaries involved in the eigenvalue decomposition of a density operator

If $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle \psi_{i}|$, this ensemble doesn't require $\langle \psi_{i}|\psi_{j}\rangle$=0. Given that $\rho$ is positive semi-definite, by the spectral theorem it ...
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Are nearly all pure two-qubit state entangled?

I am using the code below, utilizing QETLAB's RandomStateVector(4) and IsPPT, to generate a random state and to judge whether the state is entangled or separable: ...
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Is there any possibility to draw graphs for a maximally entangled state?

I am working on quantum entanglement swapping and I have derived that: $|\psi\rangle=\frac{1}{2}(|0001\rangle+|0010\rangle+|0100\rangle+|1000\rangle)$ Is there any possibility that I can make a ...
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Relation between symmetric subspaces and $n$-exchangeable density matrices

Let us consider $n$ elements, each taken from the set $\{1, 2, \ldots, d\}$ and let $S_n$ be the set of all permutations on these $n$ elements. Define a permutation operator on the set of $n$ qudits ...
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Find expectation value of the observable $X_1\otimes Z_2$ for a maximally entangled two-qubit system

In this exercise I need to find the expectation value of the observable $M=X_1 \otimes Z_2$ for two qubit system measured in the state $\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$. I know that $E[M]=\...
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What happens measuring the first qubit of a $GHZ$ state in the basis $\{|+\rangle, |-\rangle\}$?

This exercise ask me to explain what happens when the first qubit is measured in the diagonal base ($|+\rangle,|-\rangle$), considering this state: $|GHZ\rangle=\dfrac{1}{\sqrt{2}} (|000\rangle+|111\...
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Average output state of random quantum circuits

Let $|\psi\rangle = C_1 |0^{n}\rangle$ be a quantum state, such that $C_1$ is a Haar random unitary circuit. Consider a density matrix $\rho$ as follows \begin{equation} \rho_1 = \mathbb{E}[|\psi\...
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How to access/ read $\alpha$ and $\beta$ values of a certain qubit in a multi-qubit system in a for loop?

In order to rotate the qubit state I need first to be able to read the values of $\alpha$ and $\beta$ and based on their values I will apply the Q-Gate for rotating the states of the single Qubits in ...
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Intuition for why the normalization for states in quantum parallelism is the same

On page 31 of Quantum Computation and Quantum information by Nielsen and Chuang, it is said that: Consider the circuit shown in Figure 1.17, which applies $U_f$ to an input not in the computational ...
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Why are Bell states the maximally entangled ones?

I just want to know why actually Bell states are examples of maximally entangled states and significance of that "maximal" term. Is there anything for proving that?
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How can we keep Schrödinger's cat alive?

We know, Schrödinger's cat inside the box is in the equal superposition state of both alive and dead. We can express its state as $$|\text{cat}_\phi\rangle= \frac{|\text{alive}\rangle+e^{i\phi}|\text{...
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How to calculate the Hartree-Fock energy in Qiskit?

I am wondering how to calculate the Hartree-Fock energy efficiently using Qiskit. For one can get the Hartree-Fock state efficiently in Qiskit, however, it seems that it is not so obvious to get the ...
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What can be said about the non-negativity of the relative entropy of $S(\rho_{AB}||\rho_{B})$?

Taking $\rho_{AB}=\rho_{A}\otimes \rho_{B}$, where $S(\rho_{A})$ and $S(\rho_{B})$ aren't 0, it's easy to see that $$S(\rho_{AB}||I \otimes \rho_{B})=-S(\rho_{A})-S(\rho_{B})+S(\rho_{B})=-S(\rho_{A}).$...
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Is there a convention for denoting $Y$ eigenstates?

Two common shorthands for eigenstates of the $Z$ operator are $\{|0\rangle,|1\rangle\}$ and $\{|1\rangle,|-1\rangle\}$, where in the first case we have $Z|z\rangle=(-1)^z|z\rangle$ and in the second ...
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Get the probability amplitudes of the 0 and 1 states of a single qubit in Qiskit considering a system of multiple qubits

I am looking for a method to get the values of $\alpha$ and $\beta$ probability amplitudes of each single qubit in a multiple qubit system. Is that possible? As you can see in the image, I have a 4-...
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How to efficiently construct quantum circuits of oracles in multi-target quantum search?

In standard Grover's quantum search with only one target or its extension of multi-target quantum search, one of the two key parts is to quantize the boolean function $$f(x):\{0,1,\cdots,N-1\}\...
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Cliffordness of the qutrit Hadamard gate

Consider a simple generalization of the Hadamard gate to qutrits, defined as follows. \begin{equation} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ \frac{1}{\sqrt{2}} &...
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Can we do error correction for entangled particles?

There are several quantum error correction techniques, such as 3-qubit bit-flip code, and Shor’s 9-qubit code. 3-qubit bit-flip code is a straightforward technique for correcting a single error (...
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Different statevectors with different quantity of complex numbers for same single-qubit states in Qiskit

I have several questions, but before ask, I want to write some theoretical. As we know, we can represent any single-qubit quantum state by the next representation: $$ |\psi\rangle=c_0|0\rangle+c_1|1\...
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What are min and max overlaps of a maximally entangled state with a separable state?

Let $A,B$ be Hilbert spaces of dimension $d$. Let $\rho$ be some separable quantum state of the composite system $AB$. Given a maximally entangled state: $$\vert\phi\rangle = \frac{1}{\sqrt{d}}\sum_{i=...
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Intuition for why $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ can be written as $\frac{|++\rangle+|--\rangle}{\sqrt{2}}$

In analyzing measurement of $\frac{|00\rangle+|11\rangle}{\sqrt{2}}$ in the local $|+\rangle$, $|−\rangle$ basis, through algebra manipulation, the initial state is first written as $\frac{|++\rangle+|...
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What does "measurement destroys information" mean?

I am reading a paper on quantum cryptography. The author used two facts: quantum- information cannot be copied and Furthermore, measurements destroy information... For the first statement, I came ...
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What happens if a Pauli $X$ gate is applied to part of a Bell state?

I have started to learn about the mathematics behind ebits and I have a question. Assume $\color{red}{\text{Alice}}$ and $\color{blue}{\text{Bob}}$ share the following ebit: $\begin{align}\vert\Phi^+ \...
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Saving statevector on more than one location in a quantum circuit in Qiskit

So, I'm fairly new to Qiskit, and I've been playing around and following the tutorials from the Qiskit textbook. However, there is one thing I fail to understand/implement: for a quantum circuit with ...
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Derivation for the result of performing the Hadamard transform on $|0\rangle^{\otimes n}$ being $2^{-n/2}\sum_x|x\rangle$

It's said that the result of performing the Hadamard transform on n qubits initially in the all |0> state is $$ \frac{1}{\sqrt{2^n}}\sum_x|x\rangle $$ where the sum is over all possible values of x....

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