# Questions tagged [mathematics]

Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.

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### Quantum states are unit vectors… with respect to which norm?

The most general definition of a quantum state I found is (rephrasing the definition from Wikipedia) Quantum states are represented by a ray in a finite- or infinite-dimensional Hilbert space over ...
469 views

### Automatic compilation of quantum circuits

A recent question here asked how to compile the 4-qubit gate CCCZ (controlled-controlled-controlled-Z) into simple 1-qubit and 2-qubit gates, and the only answer given so far requires 63 gates! The ...
351 views

### Rigorous security proof for Wiesner's quantum money

In his celebrated paper "Conjugate Coding" (written around 1970), Stephen Wiesner proposed a scheme for quantum money that is unconditionally impossible to counterfeit, assuming that the issuing bank ...
315 views

### Quantum Algorithm for God's Number

God's number is the worst case of God's algorithm which is a notion originating in discussions of ways to solve the Rubik's Cube puzzle, but which can also be applied to other combinatorial puzzles ...
303 views

### Graphical Calculus for Quantum Circuits

So far I have read a little bit about zx-calculus & y-calculus. From Reversible Computation: The zx-calculus is a graphical language for describing quantum systems. The zx-calculus is an ...
732 views

### How can I calculate the inner product of two quantum registers of different sizes?

I found an algorithm that can compute the distance of two quantum states. It is based on a subroutine known as swap test (a fidelity estimator or inner product of two state, btw I don't understand ...
569 views

### Decomposition of a unitary matrix

Following is an excerpt from QCQI: I can understand that this matrix satisfies a unitary matrix. Also, intuitively, I am able to understand it. However, what is the proof that any given Unitary ...
252 views

### What are the constraints on a matrix that allow it to be “extended” into a unitary?

DaftWulie's answer to Extending a square matrix to a Unitary matrix says that extending a matrix into a unitary cannot be done unless there's constraints on the matrix. What are the constraints?
73 views

### When would I consider using an outer product of quantum states, to describe aspects of a quantum algorithm?

I know the inner product has a relationship to the angle between two vectors and I know it can be used to quantify the distance between two vectors. Similarly, what's an use case for the outer ...
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### How do you represent the output of a quantum gate in terms of its basis vectors?

I'm stuck while trying to understand the Hadamard Gate in a more linear algebra understanding. (I understand the algebraic way). This is because I want to program a simulation of a quantum computer. ...
45 views

### Clarification of a procedure to compute the product of the exponential of two matrices

In trying to understand a method outlined here (page 3, subroutine 1). Consider $$R_3 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} .$$ Let $A$ be a ...
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### Quantum teleportation of a state, from one of two bases

I'm watching Christian Schaffner's talk on quantum position-based cryptography (link here) and have a question about a particular application of teleportation. At about the 16:40 mark, he seems to ...
95 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 ...
113 views

### Definition of locality in Bell experiments

Continuing from my previous question on Brunner et al.'s paper; so given a standard Bell experimental setup: where independent inputs $x,y \in \{0, 1\}$ decide the measurement performed by Alice &...
379 views

### Nielsen & Chuang Exercise 2.1 - “Linear dependence: example” [closed]

Reproduced from Exercise 2.1 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition): Show that $(1, −1)$, $(1, 2)$ and $(2, 1)$ are linearly dependent. ...
48 views

### Equivalent determinant condition for Peres-Horodecki criteria

The Peres-Horodecki criteria for a 2*2 state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable. According to this paper (...
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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 ...
469 views

### 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 ...
116 views

### 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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### Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere

The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia: Quantum ...
150 views

### Grover operator as a rotation matrix

I have seen that it is possible to represent the Grover iterator as a rotation matrix $G$. My question is, how can you do that exactly? So we say that $|\psi\rangle$ is a superposition of the states ...
171 views

### Extending a square matrix to a unitary matrix

Suppose we have a square matrix $M$ of size $n\times n$. It is given that any element $M_{ij}$ of $M$ is a real number and satisfies $0 \leq M_{ij} \leq 1$, $\forall$ $i,j$. No other property for $M$ ...
70 views

### Ways in which $\frac{1}{\sqrt 2} (|00\rangle + |11\rangle)$ can be expressed as $\frac{1}{\sqrt 2} (|uu\rangle + |vv\rangle)$

I want to find out what values $|u\rangle$ and $|v\rangle$ can take if I want to write $$\frac{1}{\sqrt 2} (|00\rangle + |11\rangle)$$ as $$\frac{1}{\sqrt 2} (|uu\rangle + |vv\rangle).$$ Say |u\...
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### What are the standard eigenvalues in $\mathbb{C^2}\otimes\mathbb{C^2}$?

In $\mathbb{C^2}$, we generally take $+1$ and $-1$ as the standard eigenvalues, that's what Pauli-X, Pauli-Z measurements, etc will give us. Is there a similar standard while measuring in the Bell ...
157 views

### What can I deduce about $f(x)$ if $f$ is balanced or constant?

$\newcommand{\qr}{|#1\rangle}$Question. Can you check whether this is correct? Also, given the analysis below, what is the domain of and co-domain of $f(\qr{x})$? I think it is $V^4 \to W^4 : f$ ...
232 views

### Projection operator on Time evolution Operator

From a 9×9 Hamiltonian lying 9D space, I choose a certain subspace of 4D for designing a two qubit gate. Now the original unitary time evolution operator also lies in 9D space and it's a 9×9 size ...