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.
128
questions
1
vote
2
answers
38
views
What's $(\langle 0|\otimes I)(|00\rangle + |11\rangle)$ simplified?
It's a rather simple question. I think I am confused by the fact that using $\langle0|$ on a qubit doesn't result in another qubit. So I'm not sure if I should interpret $\langle 0|$ as the $1\times2$ ...
- 13
1
vote
2
answers
21
views
Recover local systems from composite systems
Define $A,B$ as two linear operator of two local systems. Define $C:= A \otimes B$ as the composite systems. How to recover $A$ and $B$ given $C$? For example, we set
\begin{align}
A=\left[\begin{...
- 175
0
votes
0
answers
40
views
Tensor product of the state of a system after partial measurement
I am trying to solve the question below:
While solving the post measurement state, I understand we can take the 1st and last qubit common using tensor product if they are the same(1st part of the ...
- 13
0
votes
1
answer
28
views
Where am I going wrong in my understanding of qubit associativity?
I am studying the basics of quantum computing math and am confused about qubit associativity. As I understand it, in quantum math, multiple qubits are represented as the tensor product of the qubits ...
- 11
0
votes
1
answer
22
views
Qiskit.opflow can't conbine Pauli Tensor sum
I have the problem when I run my code
...
0
votes
1
answer
43
views
How to find the matrix representation of a given many-qubit Hamiltonian?
I have the following Hamiltonian
H = - Z1Z2 - Z2Z3 - Z1Z3 - 6(Z1 + Z2 + Z3)
Here, Z1, Z2, Z3 represent the Pauli-Z operators acting on qubits 1, 2, and 3, ...
- 495
0
votes
0
answers
28
views
Locally commuting operators and positivity on tensor product spaces
I am reading this paper https://arxiv.org/abs/quant-ph/0501020 and two questions have been arisen for me: 1. In page 3 (left column) has been written: "Hence it follows that if $\tilde{S}_{k}^{(...
- 23
0
votes
1
answer
56
views
Tensor product of Pauli strings?
We define
\begin{equation}
\sum_{i=1}^{4^l} P_i \otimes P_i,
\sum_{i=1}^{4^m} Q_i \otimes Q_i,
\end{equation}
where $P_l$ is the $n$ qubit Pauli string and $Q_m$ is the $m$ qubit Pauli string.
Does ...
- 175
3
votes
1
answer
59
views
Independence in state prepared by independently drawn Haar random gates
Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and
$$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by
$$\sigma = V \rho V^{*}, $$
...
- 1,013
2
votes
1
answer
45
views
How to write CNOT within a tensor product expression of operations on the whole system
If I have an $n$ qubit register and I act on the $k^{\mathrm{th}}$ qubit with an arbitrary operator $\hat{G}$, I can write the operation on the whole register as the operation,
$$\underbrace{\mathbb{1}...
- 155
9
votes
1
answer
250
views
How does the number of copies affect the diamond distance?
Suppose we are given two maps $\Phi$ and $\Psi$ such that
$$\|\Phi-\Psi\|_{\diamond}\leqslant\varepsilon.$$
What can we say about $\left\|\Phi^{\otimes t}-\Psi^{\otimes t}\right\|_{\diamond}$? Is it ...
- 4,243
0
votes
1
answer
45
views
How to transform a multipartite state in tensor form into a bipartite state?
I have a 5-party state represented as a tensor of dimensions (2,2,2,2,2). We want to transform it to a bipartite state such that the parties 3 and 4 are on one side and parties 1,2,5 are on the other.
...
- 127
-1
votes
1
answer
83
views
How do I get this FRQI equation? [duplicate]
I have been working with FRQI and there is this equation in a paper is given. Can anyone explain how they get that just by multiplying by $\mathcal{H}$ ?
$$\mathcal{H}\left(|0\rangle^{\otimes2n+1}\...
- 123
1
vote
2
answers
84
views
Generating and executing large Pauli rotations in Python
I am interested in generating collective Pauli X, Y and Z spin operators for the purpose of rotating $2^N$ dimensional state vectors $|\psi\rangle$ (in the computational basis) for a quantum protocol. ...
- 747
7
votes
1
answer
59
views
Is there a CPTP map that takes $\rho_{AB}$ to $\rho_A\otimes\rho_B$?
Given some joint state $\rho_{AB}$, one can find either the marginal state $\rho_A$ or the marginal state $\rho_B$ through a CPTP map. The proof being that partial tracing is indeed CPTP.
Is a CPTP ...
- 2,417
1
vote
1
answer
27
views
How can I conceptualize virtual indices used in the time evolving block decimation (TEBD) algorithm?
I am trying to work with the Heisenberg $XX$ model which the Hamiltonien is given by $\hat{H} = -J \sum_i \left(\hat{X}_i\hat{X}_{i+1} + \hat{Y}_i\hat{Y}_{i+1}\right), \quad J > 0.$ Using $s^+ = |{...
1
vote
1
answer
36
views
Matrix representation for biproduct mixed states
Nielsen and Chuang [10e, p. 74] introduce the Kronecker product $A\otimes_K B$ as a matrix representation of the tensor product $A\otimes B$ of the operators $A$ and $B$ (for clarity I use a subscript ...
- 113
0
votes
1
answer
80
views
Applying CNOT operator to specific qubits in a composite system
In the given problem statement, How do I apply the fourth operation i.e. how to apply a $CNOT_{c=3,t=1}$ to a 3-bit composite system:
Approach:
First, each bit is set to the state 0. Therefore ...
- 251
2
votes
1
answer
76
views
Encoding arbitrary quantum gates using qubits
Given an arbitrary 3-qubit state $\sum_{xyz} c_{xyz}|xyz\rangle$, is there a circuit (possibly with measurement) that creates the state $\sum_{xy} c_{xyy}|x\rangle$, up to a normalization constant?
As ...
- 123
3
votes
1
answer
57
views
How to write down product operators acting on non-adjacent subsystems?
Given the following fusion gate (type-2) which is projecting 2 qubits to an even state
$$F_{ZZ}=(\langle00|+\langle|11|)$$
I would like to find the operator for the bigger space. For example, if I ...
- 1,124
2
votes
1
answer
126
views
What is the tensor product expression for the following quantum circuit? [duplicate]
Qiskit generates the following matrix for this 3-qubit CNOT circuit.
Can anyone explain how do we get this mathematically ?
This is the Quantum Circuit
This is the Output of Unitary Simulator
- 21
2
votes
0
answers
75
views
Ordered Tensoring of Operators
Novice warning: I have a strong feeling I am simply misunderstanding something fundamental!
Setup:
Classically -
I taken in some structured input and assign to each qubit $q_i$ in a register $Q = \{...
- 21
1
vote
1
answer
56
views
Calculating the product between two bra vectors
Let |Y> and |S> be two unitary (quantum gates) such that
...
- 251
1
vote
2
answers
125
views
Inner product of multiple qubit registers
I have read the following statement in some lecture notes:
The inner product of two n-qubit registers is taken by mirrored qubit
pairs. Example: $\lvert ABC\rangle$ and $\lvert abc \rangle$ leading
...
- 115
7
votes
1
answer
248
views
How to check if a $n$-qubit unitary is the tensor product of single-qubit unitaries
Let's assume I give you the expression of a unitary matrix acting on two qubits that is:
$$U=\sum_{i} A_i \otimes B_i$$
for some operators $A_i$ and $B_i$.
Is there a simple criterion allowing you to ...
- 1,340
1
vote
2
answers
90
views
Basis for permutation invariant states
It is known that the maximally entangled qubit states form a basis (the Bell basis). Let $\Phi$ be the canonical maximally entangled state i.e.
$$\Phi = \left(\frac{\vert 00\rangle + \vert 11\rangle}{\...
- 2,417
1
vote
1
answer
53
views
Raise tensor product to float power in qiskit
I am trying to implement the gate
$(X \otimes X)^\alpha$
where $X$ is the standard Pauli-X gate, $\otimes$ is the tensor product and $\alpha$ is a real number. Is there a way to implement this in a ...
- 87
3
votes
2
answers
99
views
What justifies using tensor product of matrices for parallel gates when applying them to an entangled state?
In my university we defined tensor product of two matrices $A$, $B$ as a matrix $A \otimes B$ such that for any vectors $\left| \phi \right>$, $\left| \psi \right>$ the following is satisfied:
$$...
- 33
1
vote
0
answers
48
views
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:
$$\rho = \sum_{c} \alpha_c \vert c\rangle\langle c|\...
- 449
2
votes
1
answer
49
views
Sampling Haar over two systems
Say $M$ is a matrix acting on $C^r \otimes C^s$. $X$ is the system of dimension $r$, and $Y$ is the system of dimension $s$.
With $|\psi\rangle$ sampled from Haar, how can we show that
$$
\int (...
- 153
2
votes
2
answers
84
views
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 ...
- 23
0
votes
1
answer
63
views
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 ...
- 747
5
votes
1
answer
95
views
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 ...
- 449
5
votes
2
answers
191
views
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 ...
- 137
0
votes
2
answers
153
views
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 $A\otimes B$...
- 578
2
votes
1
answer
83
views
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
$$(U\otimes I)|\phi\rangle=\...
- 1,390
0
votes
1
answer
65
views
Is factoring of a product state unique?
Suppose I have a product state of two qubits (i.e. a vector of size 4x1). Given it is separable (no entanglement), is this separation unique?
- 363
1
vote
1
answer
101
views
Two-qubit Bell measurement matrix where the two qubits are not contiguouis
In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
- 663
2
votes
1
answer
105
views
Transformation matrix for a two-qubit operation where there are more than two qubits
I have an OPENQASM program that performs entanglement swapping. It has five qubits: the data qubit and four link qubits. It works, but I want to see the details of the Bell measurement transformation. ...
- 663
5
votes
2
answers
61
views
What is the form of a unitary $U$ that preserves the marginals on a given state, $\text{Tr}_A(U\rho_{AB} U^\dagger) = \rho_B$?
Suppose for some quantum state $\rho_{AB}$ and unitary $U_{AB}$, one has
$$\text{Tr}_A(U\rho U^\dagger) = \rho_B$$
does this imply that $U_{AB} = U_A\otimes I_B$?
Also, the same question as above, but ...
- 2,417
3
votes
1
answer
81
views
How to measure a correlated operator $Z_1Z_2$?
I was reading this articl and I am stuck trying to understand equation $(60)$, which reads
$$\langle\psi|\Lambda_{1,2}(X)Z_1\Lambda_{1,2}(X)|\psi\rangle=\langle\psi|Z_1Z_2|\psi\rangle$$ where $\Lambda(...
- 187
0
votes
1
answer
175
views
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 ...
- 101
1
vote
1
answer
83
views
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 ...
- 659
0
votes
1
answer
100
views
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 ...
3
votes
2
answers
184
views
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. ...
- 145
4
votes
1
answer
415
views
Python shorthand for tensor product (Kronecker product)
When using numpy or tensorflow in Python, we can simply write
C = A @ B
for matrix ...
- 161
-2
votes
2
answers
236
views
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⟩) \...
- 195
2
votes
1
answer
473
views
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 ...
- 43
4
votes
2
answers
81
views
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}\...
- 113
3
votes
2
answers
250
views
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 ...
- 67