# 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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### 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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### 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 ...
2k views

### 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 ...
90 views

### 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 ...
2k views

### 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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### 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 $\{|0\rangle,|2\rangle\}$...
111 views

### 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 ...
149 views

### 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 $-\rvert1\rangle$ ...
133 views

### 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 ...
777 views

### 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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### 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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### 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 ...
814 views

### 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 ...
128 views

### Is the tensor product of two states commutative?

I'm reading "Quantum Computing Expained" of David McMahon, and encountered a confusing concept. In the beginning of Chapter 4, author described the tensor product as below: To construct a ...
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### Bell state preparation

I was watching some lectures on qubits. They were talking about how to generate a Bell state. They described it as follows: Prepare state 00: $$\left |0 \right> \otimes \left |0 \right>$$ Apply ...
105 views

### 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 ...
174 views

### 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. ...
63 views

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 $|10\... 1answer 94 views ### How to calculate tensor product for the magic square The magic square game is a two-player pseudo-telepathy game that was presented by Padmanabhan Aravind, who built on work by Mermin. In the magic square we have ones in columns (odd number) and rows (... 5answers 1k views ### What circuit or operation corresponds to the tensor product? What Clifford gate circuit operating on states$|\psi_1\rangle$and$|\psi_2\rangle$prepares the state$|\Psi\rangle=|\psi_1\rangle \otimes |\psi_2\rangle$? 3answers 76 views ### Show that a$CZ$gate can be implemented using a$CNOT$gate and Hadamard gates Show that a$CZ$gate can be implemented using a$CNOT$gate and Hadamard gates and write down the corresponding circuit. Recall from Quantum Information Theory that$Z=HXH$. As$CNOT$is a ... 2answers 119 views ### Writing the notation when gates act on non successive registers Suppose I have registers$|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$gate. So is this the correct ... 2answers 101 views ### What is a local operator? I have a sort of basic question. I think an operator that acts on$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}... 1answer 219 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. Suppose that on is ... 1answer 240 views ### 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^... 1answer 311 views ### 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 ... 1answer 81 views ### 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 ... 1answer 249 views ### 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 ... 1answer 152 views ### 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 ... 1answer 304 views ### 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 ... 2answers 108 views ### 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 ... 0answers 160 views ### 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. ... 2answers 102 views ### Is there a good way to mathematically write a CNOT operation over non-neighboring qubits in a circuit? [duplicate] I was wondering if there is any way to present the CNOT matrix as we usually present single qubit operations$$... 1 \otimes NOT \otimes 1 ...$$I know that for adjacent qubits in a circuit we can ... 2answers 82 views ### Is kronecker product identifiable? I have a unitary matrix U and a quantum state \vert \Psi \rangle such that$$ U \vert \Psi \rangle = e^{i \theta} \vert \Psi \rangle.$$I also know that my unitary matrix and my quantum state can ... 1answer 90 views ### What does the notation U(B,\beta) = \prod_{j =1}^n e^{-i \beta \sigma_j^x}  mean in the context of QAOA? In the article Quantum Observables for continuous control of the Quantum Approximate Optimization Algorithm via Reinforcement Learning, the following notation is used to describe an Unitary operation ... 1answer 133 views ### Understanding the action of operators on vectors in tensor product spaces I'm studying Quantum Computing: A Gentle Introduction. On page 33, Section 3.1.2, after defining tensor product with 3 properties (distribution over addition on both left and right, scalar on both ... 2answers 133 views ### How do I write a tensor product of conditional gates in matrix form? I am writing a program where I need to find the eigenstates of an operator that is a Kronecker product of conditional quantum gates. I am wondering how I would compute this product in matrix form as ... 2answers 198 views ### Tensor product between operators If the state of one qubit can be described by a ray in \mathbb{C}^2, then the combined state of an n-qubit system can be described by a ray in (\mathbb{C}^2)^{\otimes n}=\mathbb{C}^{2 n}. ... 1answer 96 views ### Kronecker product and multiplication operation on qubit states It may look a silly question but anybody of you knows what's:$$(|0\rangle+|1\rangle)\otimes(|0\rangle+|1\rangle)$$(x: Kronecker operator)$$(|0\rangle+|1\rangle)*(|0\rangle+|1\rangle)$$(*: ... 2answers 314 views ### What is the tensorial representation of the quantum swap gate? I need to write the tensorial representation of the Controlled Swap Gate, what I have written is \operatorname{CSWAP}=|0\rangle\langle0|\otimes I\otimes I+|1\rangle\langle1|\otimes U, where U is ... 1answer 60 views ### 2 qubit gate operation on multi qubit systems Considering a 3 qubit system, what does the matrix operation will look like if I apply CNOT on qubit 1 and qubit 2 and then apply CNOT on qubit 1 and qubit 3? 1answer 42 views ### Derive one equation from the other Equation 1.31 in Quantum Computation and Quantum Information a textbook by Isaac Chuang and Michael Nielsen is as follows, \left|\psi_2 \right> = \frac{1}{2}[\alpha(\left|0 \right>+\left|1 \... 0answers 78 views ### Writing the transformation matrix for the following in terms of Kronecker products of elementary 2-qubit gates I have a set of transformations that transforms |11001\rangle\to |10101\rangle which is basically keeping the leftmost qubit as it is and then it is just the CNOT between the successive qubits, I ... 1answer 85 views ### Proof that 2^n \times 2^n operator be decomposed in terms of 2 \times 2 operators What is the proof that any 2^n\times 2^n quantum operator can be expressed in terms of the tensor product of n number of 2\times 2 quantum operators acting on a single qubit space each? 1answer 1k views ### Proof of no-cloning I was reading a proof of No-cloning theorem, there are a couple of steps that are not clear to me, but the book does not give explanation for them. So here it is: Theorem: It is impossible to create ... 1answer 53 views ### Better Way Of Separating Two CQ-States I have this cq-state:$$\frac{1}{2} \times (|0\rangle \langle0|_A \otimes \rho^0_E + |1\rangle \langle1|_A \otimes \rho^1_E)$$Where Alice (A) is classical and an adversary Eve (E) has some ... 1answer 41 views ### How do I compute the output of quantum circuit involving multiple gates? I'm new in quantum computing, I have this question. Qubits x and y are in \mathbb{C}^2 (column vector) and A, B are unitary matrices (A 8x8 and B 4x4 matrix). If I'm not wrong the input ... 1answer 56 views ### Composition of tensor product I don't have much confidence with density matrices, and I would like to be sure about a property of composition of tensor products operations. Specifically,$$ \sum_i \sum_j |a_i\rangle|b_i\rangle\...
I'm trying to understand how to apply tensor products on a 3 qbit systems (or well at least 2 qbits). Let's take a basic example: where $$\lvert \psi \rangle = \lvert q2q1q0\rangle$$ with q2 being ...