Questions tagged [tensor-product]

A tensor is an abstract object generalising a scalar or vector and can be represented by a number, a 1D array, 2D matrix or higher order generalisations thereof. A tensor product is a product defined on these tensors yielding other tensors or a method to define or represent tensors. If appropriate, also use the [mathematics] tag.

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What does the notation $U(B,\beta) = \prod_{j =1}^n e^{-i \beta \sigma_j^x} $ mean in the context of QAOA?

In the article Quantum Observables for continuous control of the Quantum Approximate Optimization Algorithm via Reinforcement Learning, the following notation is used to describe an Unitary operation ...
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What is a local operator?

I have a sort of basic question. I think an operator that acts on $n$-partite states is defined (up to permutation of parties) to be local if it can be written as $$A = A_1 \otimes_{i=2}^n \mathbb{I}...
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2 qubit gate operation on multi qubit systems

Considering a 3 qubit system, what does the matrix operation will look like if I apply CNOT on qubit 1 and qubit 2 and then apply CNOT on qubit 1 and qubit 3?
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Is kronecker product identifiable?

I have a unitary matrix $U$ and a quantum state $\vert \Psi \rangle$ such that $$ U \vert \Psi \rangle = e^{i \theta} \vert \Psi \rangle.$$ I also know that my unitary matrix and my quantum state can ...
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119 views

Quantum Principal Component analysis by Seth Lloyd

I am currently reading the paper quantum principal component analysis from Seth Lloyd's article Quantum Principal Component Analysis There is the following equation stated. I know from the ...
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How to obtain the tensor-product of two quantum operations (superoperators) explicitly?

I have an amplitude damping channel, denoted as a superoperator $\mathcal{E}$ with operator elements \begin{matrix} E_1=\begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-r} \end{pmatrix},\quad ...
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How should I interpret $|2\rangle|3\rangle$?

I am a beginner at QC. I was going through the basics of multi-qubits I encountered a state $|2\rangle|3\rangle$. I want clarification on the following points: Can I write $|2\rangle$ as $|10\...
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Clarification of bra-ket notation [duplicate]

How do I get from equation 1.31 to equation 1.32? It seems like some terms are changing.
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Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$

I've been trying to figure this out for a while and I'm totally lost. My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$ So far ...
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Understanding the action of operators on vectors in tensor product spaces

I'm studying Quantum Computing: A Gentle Introduction. On page 33, Section 3.1.2, after defining tensor product with 3 properties (distribution over addition on both left and right, scalar on both ...
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Is there an algorithm for determining if a given vector is separable or entangled?

I'm trying to understand if there is some sort of formula or procedural way to determine if a vector is separable or entangled – aka whether or not a vector of size $m$ could be represented by the ...
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How to factor the output of a CNOT acting on the input $|-,+\rangle$

I am trying to implement the Deutsch oracle in classical computer, using direction from this talk. There is this slide where they show how the CNOT gate modify 2 Hadamard transformed Qubits: While ...
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The correct set of measurement operators on a mutiple qubit system

I was wondering if the complete set of measurement operators for a state : $|\phi \rangle=c_{00}|00\rangle+c_{01}|01\rangle+c_{10}|10\rangle+c_{11}|11\rangle$ Would be given by : $P_0\otimes I=|00\...
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Derive one equation from the other

Equation 1.31 in Quantum Computation and Quantum Information a textbook by Isaac Chuang and Michael Nielsen is as follows, $\left|\psi_2 \right> = \frac{1}{2}[\alpha(\left|0 \right>+\left|1 \...
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Which representation describes the composite Hilbert space?

Very often in the standard textbooks on quantum mechanics, one finds that the joint Hilbert space of two systems is given by the tensor product of the individual Hilbert spaces. That is, if $H_1$ and ...
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Kronecker product and multiplication operation on qubit states

It may look a silly question but anybody of you knows what's: $$(|0\rangle+|1\rangle)\otimes(|0\rangle+|1\rangle)$$ (x: Kronecker operator) $$(|0\rangle+|1\rangle)*(|0\rangle+|1\rangle)$$ (*: ...
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How do I write a tensor product of conditional gates in matrix form?

I am writing a program where I need to find the eigenstates of an operator that is a Kronecker product of conditional quantum gates. I am wondering how I would compute this product in matrix form as ...
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How to interpret $-\rvert1\rangle \otimes \rvert1\rangle = -\rvert11\rangle$?

I'm having trouble accepting, intuitively, that $-\rvert1\rangle \otimes \rvert1\rangle = -\rvert11\rangle = \rvert1\rangle \otimes -\rvert1\rangle$. It's my understanding that $ -\rvert1\rangle$ ...
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Writing the notation when gates act on non successive registers

Suppose I have registers $|a\rangle^{l}|b\rangle^{l} |c\rangle^{l}$ and want an adder mod $l$ gate between the $a$ and $c$ registers. Let $R$ be the adder mod $l$ gate. So is this the correct ...
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Kronecker notation of an operator

Suppose I have the state $|A\rangle=|x\rangle^l\otimes |y\rangle^l \otimes |z\rangle^l \otimes |0\rangle_x^l\otimes |0\rangle_y^l\otimes |0\rangle_z^l$. I perform the transformation between the $|x\...
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Writing the transformation matrix for the following in terms of Kronecker products of elementary 2-qubit gates

I have a set of transformations that transforms $|11001\rangle\to |10101\rangle$ which is basically keeping the leftmost qubit as it is and then it is just the CNOT between the successive qubits, I ...
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What is the tensorial representation of the quantum swap gate?

I need to write the tensorial representation of the Controlled Swap Gate, what I have written is $\operatorname{CSWAP}=|0\rangle\langle0|\otimes I\otimes I+|1\rangle\langle1|\otimes U$, where U is ...
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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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Quantum Toffoli gate equation

I was reading a research article on quantum computing and didn't understand the tensor notations for the unitary operations. The article defined two controlled gates. Let $U_{2^m}$ be a $2^m \times 2^...
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Composing the CNOT gate as a tensor product of two level matrices

I don't understand, why is the control not gate used so often. As far as I understand it, if you apply two 2 level operations on two qubits then you get a 4 x 4 matrix by the tensor product. So how ...
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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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Proof that $2^n \times 2^n$ operator be decomposed in terms of $2 \times 2$ operators

What is the proof that any $2^n\times 2^n$ quantum operator can be expressed in terms of the tensor product of $n$ number of $2\times 2$ quantum operators acting on a single qubit space each?
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Partial trace over a product of matrices - one factor is in tensor product form

$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$ I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
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Clock matrix vs matrix clock

In the process of research leading up to my previous question, I found out about matrix, vector & logical clocks. The citation in the aforementioned question mentions clock and shift matrices. ...
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1answer
764 views

Proof of no-cloning

I was reading a proof of No-cloning theorem, there are a couple of steps that are not clear to me, but the book does not give explanation for them. So here it is: Theorem: It is impossible to create ...
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Why is the state of multiple qubits given by their tensor product?

How did we derive that the state we get by $n$ qubits is their tensor product? You can use $n=2$ in the explanation for simplicity.
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What circuit or operation corresponds to the tensor product?

What Clifford gate circuit operating on states $|\psi_1\rangle$ and $|\psi_2\rangle$ prepares the state $|\Psi\rangle=|\psi_1\rangle \otimes |\psi_2\rangle$ ?
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What role does the non-commutativity of the tensor product play in experimental quantum computation?

We know that $H_A\otimes H_B\neq H_B\otimes H_A$ (in general). Theoretically, we know the formalism and what observables to construct from the two compositions possible, but we never talk about both ...
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1answer
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Confusion over tensor products in sympy.physics.quantum.qubit (in Python)

I am working with sympy.physics.quantum.qubit to help teach myself more about quantum computing. I'm confused about how best to simplify two ket expressions that ...
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178 views

Tensor product between operators

If the state of one qubit can be described by a ray in $\mathbb{C}^2$, then the combined state of an $n$-qubit system can be described by a ray in $(\mathbb{C}^2)^{\otimes n}=\mathbb{C}^{2 n}$. ...
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1answer
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How to calculate tensor product for the magic square

The magic square game is a two-player pseudo-telepathy game that was presented by Padmanabhan Aravind, who built on work by Mermin. In the magic square we have ones in columns (odd number) and rows (...
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Simplifying Quantum Tensor products with coefficients

$\newcommand{\ket}[1]{\lvert#1\rangle}$I am trying to show equality of two intermediate steps in the rearrangement of the Quantum Fourier transform definition, but I do not know how to rearrange the ...
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How is the joint state of these qubits derived?

Can someone show to me the steps to derive the joint state at the bottom of this image, please? I tried to follow his explanation but I didn't get the same results… This is taken from the lecture ...
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1answer
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What is the $\left| 22\right>$ state?

I came across with a problem that involves $2$ quantum trits in state $\left| 22 \right>.$ What is it's tensor product interpretation and a matrix interpretation?
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Confusion regarding projection operator

Suppose we have a qutrit with the state vector $|\psi\rangle = a_0|0\rangle + a_1|1\rangle + a_2|2\rangle$, and we want to project its state onto the subspace having the basis $\{|0\rangle,|2\rangle\}$...
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How do I show that a two-qubit state is an entangled state?

The Bell state $|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle )$ is an entangled state. But why is that the case? How do I mathematically prove that?
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Tensor product properties used to obtain Kraus operator decomposition of a channel

I work on a Quantum Information Science II: Quantum states, noise and error correction MOOC by Prof. Aram Harrow, and I do not understand which property of tensor products is used in one of the ...
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How to keep track of entanglements when emulating quantum computation?

I am trying to build a quantum computation library as my university project. I am still learning all the aspects of the Quantum Computing field. I know there are efficient libraries already for ...
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What do we mean by the notation $\lvert \mathbf{x}, 0\rangle$?

In quantum computation, a common operation performed between two quantum states is the tensor product, which allows us to create a new and higher-dimensional state from two lower-dimensional states. ...
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Symmetry of tensor product w.r.t. Vazirani 2-qubit video

Quantum Computing (QC) pioneer Vazirani has graciously long provided some nice videos on an intro to QC. E.g. in "2 qubit gates + tensor product" (2014) he introduces the tensor product w.r.t. QC ...