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 justifies using tensor product of matrices for parallel gates when applying them to an entangled state?

In my university we defined tensor product of two matrices $A$, $B$ as a matrix $A \otimes B$ such that for any vectors $\left| \phi \right>$, $\left| \psi \right>$ the following is satisfied: $$...
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Entropy relations in classical-quantum states with conditioned independence

Consider a classical-quantum (pure) state $\rho_{AEBC}$ where $A,C$ are classical registers. Suppose $\rho$ can be written in the following form: $$\rho = \sum_{c} \alpha_c \vert c\rangle\langle c|\...
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Sampling Haar over two systems

Say $M$ is a matrix acting on $C^r \otimes C^s$. $X$ is the system of dimension $r$, and $Y$ is the system of dimension $s$. With $|\psi\rangle$ sampled from Haar, how can we show that $$ \int (...
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Applying controlled unitary operations during quantum phase estimation

I am trying to understand Shor's algorithm for a personal research project. I am currently going through quantum phase estimation, and have came accross something I'm struggling to understand in the ...
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General Ehrenfest Theorem applied to N-qubit system operator

Please advise if the following short calculation of the derivative of the expectation value of an all spin Pauli-$\hat{Y}$ operator (acting on a $N$-qubit system) is consistent: The general Ehrenfest ...
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What is the Stinespring dilation of $T\otimes I$ for some CPTP map $T$?

Let $T: \mathcal{H}_A \rightarrow \mathcal{H}_B$ be a CPTP map with Stinespring extension $U: \mathcal{H}_{A} \rightarrow \mathcal{H}_{B} \otimes \mathcal{H}_E$. That is $U$ is an isometry such that ...
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Relationship between entanglement and complex vector space

In the article Quantum Algorithm Implementations for Beginners I found the following sentence Entanglement makes it possible to create a complete $2^n$ dimensional complex vector space to do our ...
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Does the 4x4 matrix $|00\rangle\!\langle00|+|11\rangle\!\langle11|$ have a decomposition?

Can the diagonal matrix $$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0& 0 \\0&0&0&0 \\ 0&0&0&1 \end{pmatrix}$$ be written as a tensor product $A\otimes B$...
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Why do unitaries act on maximally entangled states as $(U\otimes I)|\phi\rangle=(I\otimes U^T)|\phi\rangle$?

I was reading about teleportation which had the following Bell counterpart in $N$ dimensions $|{\phi}\rangle=\sum_{i=0}^{N-1}|i\rangle\otimes|i\rangle$. The next line was $$(U\otimes I)|\phi\rangle=\...
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Is factoring of a product state unique?

Suppose I have a product state of two qubits (i.e. a vector of size 4x1). Given it is separable (no entanglement), is this separation unique?
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Two-qubit Bell measurement matrix where the two qubits are not contiguouis

In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
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Transformation matrix for a two-qubit operation where there are more than two qubits

I have an OPENQASM program that performs entanglement swapping. It has five qubits: the data qubit and four link qubits. It works, but I want to see the details of the Bell measurement transformation. ...
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What is the form of a unitary $U$ that preserves the marginals on a given state, $\text{Tr}_A(U\rho_{AB} U^\dagger) = \rho_B$?

Suppose for some quantum state $\rho_{AB}$ and unitary $U_{AB}$, one has $$\text{Tr}_A(U\rho U^\dagger) = \rho_B$$ does this imply that $U_{AB} = U_A\otimes I_B$? Also, the same question as above, but ...
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How to measure a correlated operator $Z_1Z_2$?

I was reading this articl and I am stuck trying to understand equation $(60)$, which reads $$\langle\psi|\Lambda_{1,2}(X)Z_1\Lambda_{1,2}(X)|\psi\rangle=\langle\psi|Z_1Z_2|\psi\rangle$$ where $\Lambda(...
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How to apply 2x2 matrix transformation gates upon 1x4 tensor product columns of two tensored qubits?

The tensor product of two qubits yields a 4-row column vector. Once tensored, how are we supposed to apply common gates such as NOT on just one of the qubits, an operation which expects 2x2 or at ...
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why is $H^{⊗2}$ used to denote the parallel action of two Hadamard gates?

Why is the tensor product used here, what's its meaning? I learned tensor products as an operation between 2 matrices, and have an effect such as the follows: How does the tensor product above relate ...
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Negative Probability — Reality vs Description [closed]

I understand that quantum physics supports the concept that the probability of a qubit collapsing into (say) 1, can be negative or positive… and that quantum computing uses this as a feature, adding ...
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Confusion regarding the tensor product usage in book

I have recently started with quantum computing, and I've found great book about it - Learn Quantum Computing with IBM Quantum Experience, which explains a lot of things in quite a simple language. ...
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Python shorthand for tensor product (Kronecker product)

When using numpy or tensorflow in Python, we can simply write C = A @ B for matrix ...
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Tensor product and Dirac notation

Can someone shows me how to proof this equality: $\frac{1}{\sqrt2}(\alpha|000⟩+\alpha|011⟩ + \beta|100⟩ + \beta|111⟩ )$ = $ \frac{1}{2\sqrt2}[(|00⟩+|11⟩) \otimes (\alpha|0⟩+\beta|1⟩) + (|01⟩+|10⟩) \...
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How the single qubit unitary (U) calculates when apply a gate to only one qubit at a time?

Qiskit Textbook, Chapter 2, Section 2.2. Single Qubit Gates on Multi-Qubit Statevectors (here). In here, they have described that: If we want to apply a gate to only one qubit at a time (such as in ...
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For a bipartite operator $M\in L(H_{AB})$, suppose $0\leq M\leq \mathbb{I}$. Prove $M^{AB}\leq M^A\otimes \mathbb{I}$

As stated in the title, let $M$ be a linear operator on a finite bipartite Hilbert space. Suppose $0\leq M^{AB}\leq \mathbb{I}$ and $0\leq M^A,M^B\leq\mathbb{I}$, where $M^A=\mathrm{Tr}_B\left(M^{AB}\...
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Is it possible to retrieve $|\psi_1\rangle,|\psi_2\rangle$ from their tensor product $|\psi_1\rangle\otimes|\psi_2\rangle$?

Consider two quantum states$$\left| \psi_1 \right> = \alpha \left|0\right> + \beta\left|1\right>$$ and $$\left| \psi_2 \right> = \gamma \left|0\right> + \delta\left|1\right>$$ Now ...
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Understanding the quantum circuit for the quantum adder Toffoli gate

I am trying to understand the toffoli operation for the quantum adder below: (especially for the second toffoli gate) but I am stuck in understanding the calculation to get the correct outputs. The ...
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In the hidden subgroup problem for finite Abelian groups, where does the state $\frac{1}{\sqrt{|G|}}\sum_{g\in G} |g,0\rangle$ come from?

I am new to the concept of HSP. Previously, I saw how to solve hidden subgroup problem over $\mathbb{Z}_2^n$, which was Simon's algorithm. Over there the first step was to apply $H^{\otimes n}$, which ...
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What is the "quantum mean value problem"?

What is the "Quantum mean value problem"? A definition I found was that it is "estimating the expected value of the tensor product observable on the output state of a quantum circuit&...
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How would I apply rotations to both qubits in a 2 qubit system?

Say I have the two qubit system $\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}$. I have two 2x2 unitary gates, one is a rotation ...
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How to create an observable: 'Identity \tensor Pauli gate' in Cirq

I am working on an implementation of the RQAOA algorithm on the Maxcut problem in Cirq. My graph G has n vertices. And after running a QAOA circuit with n qubits I obtain a state gammabeta (a vertical ...
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Can we express $\mathrm{tr}_A((A\otimes B)\rho_{AB})$ in terms of $A$, $B$, $\rho_A$ and $\rho_B$?

For a density matrix $\rho_{AB}$ and some operators $A, B$, is there a way to express $$\text{Tr}_A((A\otimes B)\rho_{AB})$$ using the reduced states $\rho_A$ and $\rho_B$ and operators $A$ and $B$? ...
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If you apply a unitary transformation to an entangled state, is it still entangled?

See title. If this is not true, is there a counter example? If it is not true, does it hold true for certain combinations of unitaries and entangled states?
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How does the sum of two operators act on a two-level system of qubits?

I am confused how the sum of N operators will act on an N-level system of qubits. Here, lets say N=2 so the state is $|00⟩_{CD}$. Then how will this operator $ X_{C} + Z_{D} ⊗ I_{C} + X_{D}$ act on ...
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4 votes
2 answers
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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 the order of the tensor product in $|\phi\rangle\otimes|\chi\rangle=|\chi\rangle\otimes|\phi\rangle$ relevant? [duplicate]

I am reading this book “Quantum Computing Explained” by David McMahon. I found the following statement on page 74 Note that the order of the tensor product is not relevant, meaning $|\phi\rangle \...
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How do I represent my 3-qubit state in the computational basis?

I have taken the tensor product of $|0\rangle \otimes |-\rangle \otimes |+\rangle$ which resulted in the matrix $$\begin{bmatrix} 1/2\\ 1/2 \\ -1/2 \\ -1/2 \\ 0 \\ 0\\ 0\\ 0\\ \end{bmatrix}.$$ How ...
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Does $\mathrm{tr}(A \otimes B) = \mathrm{tr} (A) \otimes \mathrm{tr}(B)$ hold for partial trace?

I was reading this question from this site answered by DaftWullie. I would like to request you to read the question there. The answer says However, in this particular case, the calculation is much ...
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Are the two ways of interpreting the expression $(|a\rangle\otimes|b\rangle)(\langle c|\otimes\langle d|)(|e\rangle\otimes |f\rangle)$ equivalent?

Reading Nielsen and Chuang, I am under the impression that a linear operator on the tensor product can be written in two ways: \begin{equation} (\left|a\right> \otimes \left|b\right>)(\left<c\...
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Writing state $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as separate qubits (qiskit textbook)

While going through the IBM qiskit textbook online, I came across the following question in section 2.2: Write the state: $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as two separate ...
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Show that the two circuits are equivalent mathematically

This exercise wants me to prove the equivalence of the two circuits using their mathematical representations. Circuit 1: Circuit 2: Circuit 1 (q1 CNOT ...
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Stinespring dilation: Size of environment

Let $\mathcal{E}_{A\rightarrow B}$ be a quantum channel and consider its $n-$fold tensor product $\mathcal{E}^{\otimes n}_{A^n\rightarrow B^n}$. Any isometry $V_{A\rightarrow BE}$ that satisfies $\...
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Discrepancy in inner product between tensor products

I have noticed one identity in case of tensor product from this post. But I can't understand why it is true. $\langle v_i| \otimes \langle w_j| \cdot |w_k\rangle \otimes |v_m\rangle = \langle v_i|v_m\...
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Simulate Hamiltonians with Pauli operations (controlled time evolution)

I had a question last week regarding the simulation of Hamiltonians composed of the sum of Pauli products: How can I simulate Hamiltonians composed of Pauli matrices? I'm having a follow-up question: ...
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Trace distance of two classical-quantum state with hashing

Let's say I have a classical-quantum(cq) state $\rho_{XE}$, where the classical part $(X)$ is orthogonal. It's trace distance from another uniform density operator is defined to be: $$ \frac{1}{2}||\...
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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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9 votes
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Quantum tensor product closer to Kronecker product?

Coming more from a computer science background, I never really studied tensor products, covariant/contravariant tensors etc. So until now, I was seeing the "tensor product" operation mostly ...
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6 votes
4 answers
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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 ...
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Changing the Basis

I am attempting to use a VQE algorithm to find the ground state of a deuterium nucleus by applying a constructed hamiltonian to an ansatz state with one parameter created by a circuit. While I am ...
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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 ...
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6 votes
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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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3 answers
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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 ...
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Grover oracle result: vectors (0,1) & (0,1) => two Hadamards => product of two H results => CZ = (.5, .-5, -.5, -.5)

According to the Grover's algorithm section in the IBM Quantum Experience, if I have two qubits in the "one" state (vectors (0,1) and (0,1)), and I apply a Hadamard gate to each of them, and then ...
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