Questions tagged [quantum-state]

Questions about or related to quantum states. Consider using the density-matrix tag when relevant.

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6
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1answer
520 views

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 ...
6
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2answers
302 views

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 $f$, $f'$ : $...
6
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3answers
1k views

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 ...
6
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2answers
180 views

What is the smallest quantum circuit to produce two-qubit state (a,b,b,b)?

How can I synthesis a two-qubit quantum state of the state vector (a,b,b,b) using basic quantum-gate circuit (arbitrary single-qubit rotation and controlled $Z$ gate)? And further, can I know a given ...
6
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2answers
579 views

Incorrectly Calculating Probability Amplitudes for 3-qbit Circuit

I’m trying to calculate the probability amplitudes for this circuit: My Octave code is: ...
6
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2answers
168 views

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 ...
6
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1answer
416 views

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: ...
6
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1answer
199 views

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 ...
6
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1answer
75 views

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 $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ ...
6
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1answer
246 views

What is a Haar random quantum state?

Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it.
6
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2answers
205 views

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 $2^n$. ...
6
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247 views

Quantum relative entropy with respect to a pure state

I want to evalualte the quantum relative entropy $S(\rho|| \sigma)=-{\rm tr}(\rho {\rm log}(\sigma))-S(\rho)$, where $\sigma=|\Psi\rangle\langle\Psi|$ is a density matrix corresponding to a pure state ...
6
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120 views

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 $|b\rangle$, ...
6
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2answers
544 views

How to translate matrix back into Dirac notation?

In Circuit composition and entangled states section of Wikipedia's article on Quantum logic gates the final result of a combined Hadamard gate and identity gate on $|\Phi^{+}\rangle$ state is: $ M \...
6
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1answer
177 views

What does W stand for in the W entangled state?

For the $|W_3\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$, what does W stand for? Does it refer to an author name? Anyone knows a reference? Thanks
6
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1answer
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$n$ qubit vs. a $d=2^n$ qudit states and measurements

The pure states of a qudit inhabit the $\mathbb{CP}(d-1)$ manifold. Is it true that the pure states of $n$ qubits live on the $\mathbb{CP}(2^n-1)$ manifold? If the answer to the first question is yes,...
6
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1answer
339 views

Is the set of classical-quantum states convex?

I read about the classical-quantum states in the textbook by Mark Wilde and there is an exercise that asks to show the set of classical-quantum states is not a convex set. But I have an argument to ...
6
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3answers
439 views

Use of change of phase gates

As explained in this this answer, when we have different (relative) phases between two states, those two states will yield the same probabilities when measured in the same basis but different ...
6
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1answer
112 views

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, ...
6
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1answer
159 views

Bob applies a projector - what happens to eigenvalues of Alice's reduced state?

Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global ...
6
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1answer
113 views

Why does full state reconstruction require at least $N+1$ MUBs?

Consider an $N$-dimensional space $\mathcal H$. Two orthonormal bases $\newcommand{\ket}[1]{\lvert #1\rangle}\{\ket{u_j}\}_{j=1}^N,\{\ket{v_j}\}_{j=1}^N\subset\mathcal H$ are said to be Mutually ...
6
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2answers
81 views

Non maximally entangled states for QKD

Why aren't non maximally entangled states produced and used in quantum key distribution schemes? What would be the advantage/disadvantage to use such states rather than maximally entangled ones?
6
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168 views

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|) = \...
6
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1answer
816 views

How to calculate the state given by two qubits?

Let's say two qubits are both in $|+\rangle$ state. We need to find $a_1$, $a_2$, $a_3$, and $a_4$ in $|\phi\rangle = a_1|00\rangle + a_2|01\rangle + a_3|10\rangle + a_4|11\rangle$, how do we find ...
6
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1answer
126 views

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 ...
6
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2answers
602 views

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 : ...
6
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1answer
378 views

Stabilizer state verification and specification from state vector

Given an arbitrary $n$-qudit state vector $|\psi\rangle =\sum_i c_i| i \rangle \in \mathbb{C}_d^n$ for some orthonormal basis $\{|i\rangle\}$, what is the most efficient way one can: Verify whether ...
6
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1answer
128 views

Threshold and practical requirements for initial state preparation?

At the beginning of a quantum computational process we generally want to start in a perfectly known initial state, and evolve from there. This cannot be done perfectly, for fundamental reasons, but I ...
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99 views

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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71 views

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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130 views

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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127 views

How exactly is the stated composite state of the two registers being produced using the $R_{zz}$ controlled rotations?

This is a sequel to How are two different registers being used as "control"? I found the following quantum circuit given in Fig 5 (page 6) of the same paper i.e. Quantum Circuit Design for ...
5
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3answers
485 views

Why is the transpose of a density matrix positive and trace preserving?

With density matrix $\rho=\sum_{a,b=0}^1\rho_{a,b}|a\rangle\langle b|$ and it's transpose $\rho^T=\sum_{a,b=0}^1\rho_{a,b}|b\rangle\langle a|$. How to confirm that $\rho^T$ is positive and trace ...
5
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3answers
284 views

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 ...
5
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4answers
2k views

How to check if 2 quantum bits are orthogonal?

How would you check if 2 qubits are orthogonal with respect to each other? I need to know this to solve this problem: You are given $2$ quantum bits:$$ \begin{align} |u_1\rangle &= \cos\left(\...
5
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3answers
398 views

Unentangling a qubit from a system: can we convert $\alpha|000\rangle+\beta|111\rangle$ into $\alpha|00\rangle+\beta|11\rangle$?

Let's say I start with the following arbitrary qubits: $$ \color{red}{\vert Q_1 \rangle = \alpha_1 \vert 0 \rangle + \beta_1 \vert 1 \rangle}\\ \color{green}{\vert Q_2 \rangle = \alpha_2 \vert 0 \...
5
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2answers
334 views

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 ...
5
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2answers
399 views

Why does measuring one qubit after the other in this entangled system alter the result?

Suppose I have the following circuit where q0 and q1 are measured one after the other. The simulation results state that the state 00 occurs 75% of the time, and the state 11 occurs 25% of the time. ...
5
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3answers
243 views

Terminology: what do $|i\rangle$ and $|\mbox{-}i\rangle$ represent?

$|0⟩$ and $|1⟩$ are usually referred as the computational basis. $|+⟩$ and $|-⟩$, the polar basis. What about $|i\rangle$ and $|\mbox{-}i\rangle$? And collectively? Orthonormal states? References are ...
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830 views

What would be the meaning of an $i$ in a qubit state $i\alpha|0\rangle+\beta|1\rangle$?

I do not know if the question is not too easy, but I'll put it here, because I'm interested in it. So the state of a qubit is often stated in this form: $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$...
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4answers
121 views

Derivation of the identity $\sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right)$

For measurement, we know $$\langle M \rangle = \sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right).$$ My question is, how can we go ...
5
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2answers
92 views

How instantaneous is state preparation in a quantum register, if all possible superpositions are to be initialized equally?

Before the start of a quantum algorithm qubits need to be initialized into a quantum register. How fast can a quantum register of length $n$ be initialized in a way that all possible superpositions of ...
5
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1answer
806 views

What is a stabilizer state?

I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'. "The number of repetitions depends on the ...
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2answers
59 views

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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2answers
96 views

Why are commuting density operators said to be "classical states"?

In quantum information it is commonly said that a set of states $S=\{ \rho_i \}_i$ is classical if $[\rho_m, \rho_n] = 0, \,\forall \rho_m,\rho_n \in S$. This is meant in the sense that all observed ...
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2answers
614 views

Why are $d^2$ dimensions required to describe a density matrix?

A density matrix is defined as: $$\sum p_i |\psi_i \rangle \langle \psi_i|$$ If the dimensionality of each $|\psi_i \rangle$ is $d$, why does it take $d^2$ dimensions to represent a density matrix? (...
5
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388 views

Non-uniqueness of pure states ensemble decomposition

Apparently, the decomposition of a state into an ensemble of pure states is not unique. I can't understand why, as if I understood correctly a "pure state ensemble decomposition" is just the ...
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2answers
205 views

Quantum proof for the group non-membership problem

Group non-membership problem: Input: Group elements $g_1,..., g_k$ and $h$ of $G$. Yes: $h \not\in \langle g_1, ..., g_k\rangle$ No: $h\in \langle g_1, ..., g_k\rangle$ Notation: $\langle g_1, ..., ...
5
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3answers
143 views

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+|...
5
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2answers
96 views

Can every bipartite state be written as $\rho_{AB} = \sum_{ij} c_{ij}\sigma_A^i\otimes \omega_B^j$?

Can every bipartite quantum state (including entangled ones) be written in the following way $$\rho_{AB} = \sum_{ij} c_{ij}\sigma_A^i\otimes \omega_B^j$$ where $\sigma_A^i$ and $\omega_B^j$ are ...

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