Questions tagged [quantum-state]
Questions about or related to quantum states. Consider using the density-matrix tag when relevant.
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Are Absolutely Maximally Entangled states maximally entangled under all entanglement monotones?
In Ref. [1] absolutely maximally entangled (AME) states are defined as:
An $\textrm{AME}(n,d)$ state (absolutely maximally entangled state) of $n$ qudits of dimension $d$, $|\psi\rangle \in \mathbb{...
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Fidelity concentration bound for random stabilizer states
Let $|\Phi\rangle$ be a normalized vector in $\mathbb{C}^d$ and let $|\psi\rangle$ be a random stabilizer state. I am trying to compute the quantity
$$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \...
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Get state vector of a single qubit in a circuit in Qiskit
I have two quantum circuits, and I would like to compare state vector of the first qubit and check if equals, what is the best way to do that in qiskit ?
Let's say I have :
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What is the intuition behind "states with support on orthogonal subspaces"?
I'm sure I don't fully understand support, but I am having trouble seeing how it connects to things like density operators. I have an idea that it means, according to Wikipedia:
"In mathematics, ...
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Why is the probability vector of a uniformly random state $\sum_i\alpha_i|i\rangle$ uniformly random only if $\alpha_i\in\mathbb C$?
In these lecture notes by Scott Aaronson, the author states the following (towards the end of the document, just before the Linearity section):
There's actually another phenomenon with the same "...
glS♦
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One-qubit gate results in QISKit
I found it odd that the result of the action of identity gate (namely a $2\times2$ identity matrix) on a pure state $|0\rangle$ (namely the vector corresponding to the $2\times1$ matrix $\begin{...
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Are there measuring standards (and units) for the identification of qubits?
The representation of bits in different technological areas:
Normal digital bits are mere abstractions of the underlying electric current through wires. Different standards, like CMOS or TTL, assign ...
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Most efficient way for general state generation
Assume we are given an $n$-qubit system and complex numbers $a_0, \ldots, a_{m-1}$ with $m = 2^n$. Assume further we start with the initial state $|0 \ldots 0\rangle$ and want to make the ...
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Closest quantum state with a fixed marginal: Analytical solution?
Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
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Are there separable states $\rho$ with separable pure decompositions requiring $\operatorname{rank}(\rho)^2$ components?
In What separable $\rho$ only admit separable pure decompositions with more than $\mathrm{rank}(\rho)$ terms?, examples were given of separable states $\rho$ with separable decompositions requiring ...
glS♦
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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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What is a "maximally mixed state"?
What is meant by maximally mixed states? Does this mean that there are partially mixed states?
For example, consider $\rho_{GHZ} = \left| {GHZ} \right\rangle \left\langle {GHZ} \right|$ and $\rho_W =...
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Is there anything practical that can be done with a single qubit?
Is there anything practical that can be done with a single qubit? And by "practical," I mean a problem that can be solved or information that can be stored.
I realize that one practical thing that ...
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Aren't qubits just ternary?
Qubits have 3 states: 1, 0, and 1 and 0 at the same time. If a qubit can have 3 states, then how come they are seen as different from ternary computing, which also has 3 states?
Is it that the 3 ...
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Is effective quantum cloning possible, given that any classical function can be implemented as a quantum circuit?
As in Compiling a classical function to a quantum circuit in practice, as far as my understanding goes, it is known that any classical function can be implemented as a quantum circuit. So given $f(x)=...
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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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Maximum number of "almost orthogonal" vectors one can embed in Hilbert space
In a Hilbert space of dimension $d$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is small but finite compared ...
user2669
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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 do I get the Unitary matrix of a circuit without using the 'unitary_simulator'?
I am using jupyter notebook and qiskit. I have a simple quantum circuit and I want to know how to get the unitary matrix of the circuit without using 'get_unitary' from the Aer unitary_simulator. i.e.:...
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How to compute the average value $\langle X_1 Z_2\rangle$ for a two-qubit system?
Show that the average value of the observable $X_1Z_2$ in a two-qubit system measured in the state $(|00\rangle + |11\rangle)/\sqrt{2}$ is zero.
How would we approach this question? I understand that ...
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In Stinespring dilation, can we always use a mixed state as the ancilla?
The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\...
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Is a qubit always in superposition?
I am introduced to ancilla qubits which are usually initialized to $\vert 0 \rangle$. It seems that an ancilla qubit is equivalent to the $0$ bit in classical computing as it will evaluate to $\vert 0 ...
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Question on state distinguishability
Consider the following protocol.
We are given either
$|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ or $|\phi\rangle = \alpha_{0} |0\rangle + \alpha_{1}|1\rangle$ where $\alpha_{0}^{2}$ ...
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How is a single qubit fundamentally different from a classical coin spinning in the air?
I had asked this question earlier in the comment section of the post: What is a qubit? but none of the answers there seem to address it at a satisfactory level.
The question basically is:
How is a ...
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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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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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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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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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Can $|\Psi\rangle\simeq\sum_k |u_k\rangle|v_k\rangle$ be maximally entangled even if $\{|u_k\rangle\}_k,\{|v_k\rangle\}_k$ are not orthonormal?
Let $\newcommand{\ket}[1]{\lvert#1\rangle}\{\ket{u_k}\}_k,\{\ket{v_k}\}_k\subset\mathcal H$ be orthonormal bases in an $N$-dimensional space. It then follows that the state
$$\ket\Psi = C\sum_{k=1}^N \...
glS♦
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Symmetry in Conditional Phase Shift Gates and Realizing CNOT through HCZH
Why are conditional phase shift gates, such as CZ, symmetrical? Why do both the control and target qubit pick up a phase?
Furthermore, assuming that they are symmetrical, when using a CNOT gate as an ...
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Do no-cloning and no-deletion theorems follow one from the other?
I am very new to Quantum Information. I had the following question on two no-go theorems:
The No cloning theorem states there is no unitary operator $U$ such that $U|\psi\rangle|0\rangle=|\psi\rangle|\...
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Is 216 qumodes photonic quantum processor equivalent to 216 qubits superconducting quantum processor, in terms of computational power?
Xanadu just launched borealis, 216 qumodes photonic quantum computer, this week.
https://xanadu.ai/blog/beating-classical-computers-with-Borealis
Its number of qubits is very interesting because it ...
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Does a photonic quantum computer control a single photon?
Does a photonic quantum computer control a single photon and use it to represent a single qubit?
I think ion trapped quantum computers use a single ion to represent a qubit. I would like to know how a ...
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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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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 do we physically initialize qubits in a Quantum register?
In quantum algorithms we need to initialize the qubits at the start of our algorithm in some quantum register. Suppose that if we are working with a four qubit quantum register we can initialize the ...
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Incorrectly Calculating Probability Amplitudes for 3-qbit Circuit
I’m trying to calculate the probability amplitudes for this circuit:
My Octave code is:
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Schmidt decomposition for tripartite system $ABC$ with vanishing mutual information between $A$ and $C$
Suppose I have a tripartite system $ABC$ in a pure state $|\psi_{ABC}\rangle$ with mutual information $I(A:C)=0$. This implies that the reduced density matrix $\rho_{AC}$ factorizes as $\rho_{AC} = \...
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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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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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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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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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Preparing Bell state $(1/\sqrt{2}) (|01\rangle + |10\rangle)$ in Qiskit
I'm working through the Qiskit textbook right now, and wanted to complete part 1 of exercise 3.4, which asks me to use qiskit to produce the Bell state:
$$\frac{|01\rangle + |10\rangle}{\sqrt{2}}$$
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Confusion about the output distribution of Haar random quantum states
Consider a Haar random quantum state $|\psi \rangle$. I was confused between two facts about $|\psi \rangle$, which appear related:
Consider the output distribution of a particular $n$-qubit $|\psi \...
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What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?
I have trouble understanding two equations in the Nielsen & Chuang textbook. Suppose we perform a measurement described by the operator $M_m$. If the initial state is $|\psi_i\rangle$, then the ...
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What is the matrix for a SWAP operation on two qubits?
Say we want to swap qubits $a$, $b$ in the same register, where $a,b \in \left \{ 0, 1,\cdots, n-1 \right \}$. What would be the corresponding matrix.
For those interested, I'm curious about this ...
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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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Does the circuit with qubit-wise CZ gates compute the inner product of two states? If not, is there another circuit that does?
I've been searching for a quantum algorithm to compute the the inner product between two $n$-qubit quantum states, namely $\langle\phi|\psi\rangle$, which is in general a complex number.
One can get $|...
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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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Understanding a quantum algorithm to estimate inner products
While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here, in the Appendix, the author/s have included a section on quantum inner product estimation.
Consider ...
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