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Questions tagged [density-matrix]

For questions about density matrices and related concepts and ideas, e.g. procedures for computing properties of quantum states from their density matrices.

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Can results about pure states be extended to convex sums of pure states?

Most results discussed in quantum computing textbooks are about pure states. I'm curious whether these result can be extend to convex sum of pure states. In the case of qubit states, pure states are ...
Noreen's user avatar
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Complexity of Partial transpose operation

Let us take a general bipartite system $H = H_a \otimes H_b $, with $d_a$ = dim $H_a$, and similarly $d_b$ = dim $H_b$. My question is: what is the complexity of the partial transpose operation on a ...
Soumyadeep sarma's user avatar
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Tests of entanglement between one and many qubits

I have a 5-qubit state $|\psi \rangle$, which has a physical interpretation that the middle qubit is the "impurity" with spin $s_{imp} = 1/2$. The rest 4 qubits highlight the presence of ...
Soumyadeep sarma's user avatar
1 vote
2 answers
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Can non-linear operations be implemented as a circuit on a quantum computer?

Suppose I have a Quantum circuit, which gives an output state $|\psi \rangle$ let's say. I wish to obtain the reduced density matrix by tracing out subsystem B, i.e. $\rho = |\psi \rangle \langle \psi ...
Soumyadeep sarma's user avatar
2 votes
1 answer
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Density matrix $\rho = I/2$ implies an ensemble of orthonormal states

Suppose that a density matrix $\rho = I/2$ is obtained as a description of an ensemble of two pure states. How can I show that the ensemble must then be of the form: $$ \{(|\psi\rangle, 1/2), (|\psi^\...
Olly Britton's user avatar
4 votes
2 answers
77 views

Why do minimal ensemble decompositions for $\rho$ contain $|\psi⟩\in{\rm supp}(\rho)$ with probability $1/\langle\psi|\rho^{-1}|\psi⟩?$

I came across the following exercise (2.73) in Nielsen & Chuang and am trying to understand it intuitively. Here is my reasoning of what is going on: The purpose of this exercise: Let’s say we are ...
researcher101's user avatar
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2 answers
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What is meant with "different ensembles can give rise to the same density matrix?"

I am reading the Nielsen & Chuang section on density matrices and I don't understand the example given to demonstrate a concept. Here is what I am reading: First, they said these two different ...
researcher101's user avatar
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Reasoning behind unitary freedom in the ensemble for density matrices theorem

Although my question has the same title of a different question, it is not a duplicate. I am asking a different question. I don't care why it made it into the book. Here is a theorem from Nielsen &...
researcher101's user avatar
2 votes
2 answers
388 views

How to calculate the Schmidt decomposition of a state without SVD

I have this state of two qubits here: $$ |\psi_{AB}\rangle = \frac{1}{2}(|0\rangle_A |0\rangle_B + |1\rangle_A |1\rangle_B + |1\rangle_A |0\rangle_B - |0\rangle_A |1\rangle_B) $$ Which means that the ...
Alessandro Romancino's user avatar
3 votes
3 answers
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Interpretation of a circuit that yields the same result for initializations $|+\rangle$ and $|-\rangle$

How can I interpret a quantum circuit that results in the same state for the initialization $\newcommand{\ket}[1]{|#1\rangle}\newcommand{\bra}[1]{\langle #1|}\ket{+}$ and $\ket{-}$? For example, the ...
upe's user avatar
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What is the density matrix of a GHZ state when onle a qubit is in a decoherence channel?

Suppose Alice, Bob and Rob share a GHZ state. Now consider Rob's qubit is in a bit-flip channel. How to obtain the density matrix in this senario? Also i would be glad to get some articles adrresing ...
reza's user avatar
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Particle number expectation value in QuTip

I am learning now to use QuTiP by going through their documentation site. I am trying to understand what does the argument - particle number expectation value in thermal density matrix do? How does it ...
CuriousMind's user avatar
2 votes
2 answers
103 views

Can different density matrices have 100% fidelity with a given pure state?

I am trying to understand fidelity a bit better, to do so consider the bell state: $$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle-|10\rangle\right),$$ the density matrix associated with this state ...
Adrien Amour's user avatar
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Qiskit: Density matrix's dimension is too large

My circuit is with 18 qubits. I want to find density matrix by calling DensityMatrix.from_instruction(qc) but then I get an error "Maximum supported dimension for an ndarray is 32, found 36"....
tomek's user avatar
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Exponential Quantum Speedup for the Traveling Salesman Problem - where is the catch?

Such an article claims that an NP-complete problem can be solved efficiently. Is it real? I noticed that they prepare a state $|0\rangle\langle0|+|1\rangle\langle1|$ on an ancilla, which is impossible ...
Ron Cohen's user avatar
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Why is the trace distance between two density matrices not always $0$?

If $|A|_{tr}=Tr(\sqrt{A^\dagger A})$ then surely $$ |\rho_1-\rho_2|_{tr}=Tr(\sqrt{(\rho_1-\rho_2)^\dagger (\rho_1-\rho_2)}) $$ $$ =Tr(\sqrt{(\rho_1^\dagger -\rho_2^\dagger)(\rho_1-\rho_2)}) $$ $$ =Tr(\...
mrepic1123's user avatar
2 votes
1 answer
68 views

What's the Schmidt decomposition of $|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$?

$|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle) $ I absolutely cannot figure out the Schmidt decomposition of this state. I have looked at a ton of ...
qityhd's user avatar
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If states are close together does there always exist a channel close to the identity mapping one to the other?

Question: Given states $\rho,\omega\in\mathbb C^{n\times n}$ and $\varepsilon>0$ such that $\rho$ and $\omega$ are $\varepsilon$-close in trace norm does there exist a channel $\Phi$ with $\Phi(\...
Frederik vom Ende's user avatar
3 votes
1 answer
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How to prove the inclusion relation $\text{Im} (\rho) \subseteq \text{Im} (\rho[X] \otimes \rho[Y])$ about density operators?

For $\rho \in \mathrm{D}(\mathcal{X} \otimes \mathcal{Y})$ denoting an arbitrary state of the pair $(\mathrm{X}, \mathrm{Y})$, how to prove the fact $\text{Im} (\rho) \subseteq \text{Im} (\rho[X] \...
Aimin Xu's user avatar
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Is $\rho = \sum_{j} p_j|n_j\rangle\langle n_j|$ a valid construction for any mixed state?

I have a mixed state $\rho$ and its hamiltonian $H$. Firstly, I find the eigenvalues $\{p_j\}$ of $\rho$, and orthonormal basis of $H$. I write $\rho$ in terms of $H$'s eigenstates and $\rho$'s ...
Việt Nguyễn's user avatar
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1 answer
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Efficient Clifford simulation and entropy of reduced density matrices

Suppose I have a Clifford circuit $C$ and I want to estimate the entanglement entropy of a subset of two qubits, say, $\{q_0, q_1\}$, i.e. the quantity $$S(\rho_{q_0 q_1}) = - \text{Tr}[\rho_{q_0 q_1} ...
jth's user avatar
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1 answer
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relationship between helstrom operators acting on different pairs of quantum states

Let $\rho_1, \rho_2, \rho_3, \rho_4$ be arbitrary single-qubit density matrices. Let $A$ be an Hermitian operator and its spectral decomposition as $A = \sum_i \lambda_i \lvert i \rangle \langle i \...
user185671631's user avatar
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27 views

Better optimization of bounds on sums of Pauli strings?

I'm trying to bound a quantity $||\sum_i \alpha_i P_i ||$ where the $P_i$ are arbitrary Pauli strings, $||.||$ is the operator norm (max eigenvalue) and $\alpha_i$ are arbitrary real coefficients. If ...
Hans Schmuber's user avatar
1 vote
0 answers
58 views

State tomography in Qiskit on a subset of qubits of real QPU

Could anyone please explain how should I carry out a state tomography on a real device in Qiskit (version 0.43.2)? I have access to devices with 127 qubits, but I want to perform a simulation using ...
Andrea's user avatar
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1 answer
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Density Matrix for a Quantum Circuit with Clifford Gates and a $T$ Gate in Qiskit

I am trying to analyze the impact of a single $T$ gate within a quantum circuit that primarily consists of Clifford gates. My goal is to understand the $T$ gate's role in $T$-design and Anti-...
Asim Sharma's user avatar
2 votes
1 answer
31 views

Depolarizing channel on GHZ-state

Consider a GHZ-state $|\psi\rangle =\frac{1}{\sqrt{2}}(|0\rangle^{n}+|1\rangle^n)$, and consider a depolarizing channel that maps a density matrix $$\rho\to(1-\lambda)\rho + \frac{\lambda}{2^d}I.$$ ...
nippon's user avatar
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2 votes
1 answer
51 views

Existence of a two-outcome measurement $M$ such that the induced distributions differs between different density matrices

Let $\rho \neq \sigma$ be density matrices. I want to show that there exists a two-outcome measurement $M$ such that the induced distributions $M(\rho)$ and $M(\sigma)$ differ. From what I learned, ...
Gabi G's user avatar
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6 votes
3 answers
430 views

Can a CPTP map increase the purity of a state?

I am wondering if there exist CPTP maps $T$ such that the purity of a quantum state $\rho$ can increase, i.e. $$ \text{tr} ( T ( \rho )^2 ) \geq \text{tr} ( \rho ^2). $$ If so, what are the conditions ...
Rell's user avatar
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3 votes
1 answer
86 views

Can I obtain the pure state corresponding to a density matrix from its main diagonal?

Suppose we have a bipartite pure state as follows: $$|\psi\rangle=a_1|00\rangle+a_2|01\rangle+a_3|10\rangle+a_4|11\rangle\,.$$ Then, the density matrix is as follows: $$|\psi\rangle\langle\psi|=\left( ...
reza's user avatar
  • 733
1 vote
1 answer
143 views

What is the formula for the matrix representation of a general controlled gate?

Suppose I have $n$-qubit circuit. I have a single-qubit gate (e.g. a Pauli gate) at qubit $a$ and it is controlled by the qubit $b$. What is the matrix representation for this controlled gate? The ...
user1747134's user avatar
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0 answers
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How to analyze a system in nonthermal equilibrium?

In quantum information theory, density matrix is one of the main resource for analyzing a system. I know in general how to obtain density matrix of a system but there is a case that still i dont know ...
reza's user avatar
  • 733
2 votes
1 answer
114 views

Minimizing $1 - \text{Tr}(\Phi(\rho,U)^2)$

I am looking for a computationally efficient way to minimize the following function. Let $$\Phi(\rho, U) = \text{Tr}_2(U\rho U^\dagger)$$ be a reduced density matrix where $\rho = \overline{\rho}_1 \...
Silly Goose's user avatar
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0 answers
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Multimode unitary channel in terms of action on characteristic function

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
Phil K.'s user avatar
1 vote
2 answers
67 views

Is $\text{Tr}(\text{Tr}_\mathcal{E}(\rho)) = \text{Tr}(\rho)$?

Let $\rho$ be a density matrix over some composite Hilbert space $\mathcal{H}_S \otimes \mathcal{H}_{\mathcal{E}}$. Is partial trace full trace preserving? I.e., is $$\text{Tr}(\text{Tr}_\mathcal{E}(\...
Silly Goose's user avatar
1 vote
1 answer
37 views

Can we de-decohere an open system?

Can a mixed state become pure due to its interaction with a vast environment? Certainly, a strange proposal, and yet let's take a diagonal matrix representing a mixed state, say $$\begin{pmatrix} p_{1}...
Loading - 146 Complete's user avatar
2 votes
1 answer
40 views

How is the expression $\frac{\|\rho^{T_B}\|-1}{2}$ obtained from the definition of negativity?

In quantum information theory, negativity is defined as summation of the absolute values of negative eigenvalues of the partial transposed density matrix. The expression of negativity is given as $$ \...
Anindita Sarkar's user avatar
1 vote
1 answer
51 views

If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD

Def: The partial transpose of a linear operator $\rho_{AB}$ over a Hilbert space $H_A \otimes H_B$ w.r.t A is defined for a linear operator $\rho_{AB}=\rho_A \otimes\rho_B$ as $\rho^{T_A}_{AB}=\rho_A^...
some_math_guy's user avatar
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0 answers
81 views

Show that the Choi of a tensor product is the tensor product of the Chois

I have the following problem. Let $N:L(H_A)\rightarrow L(H_A)$ be a quantum superoperator. The quantum state corresponding to this operator (via Choi-Jamiolkowski Isomorphism) is $\Gamma_A^{N}=id\...
Piotr Masajada's user avatar
6 votes
1 answer
321 views

Is it possible to derive a Schmidt decomposition for a mixed state?

It is relatively simple to derive the Schmidt decomposition of a pure state $|{\psi}\rangle \in H_A \otimes H_B$ with the SVD decomposition theorem. There are plenty of examples (lecture notes, books, ...
JMark's user avatar
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4 votes
1 answer
84 views

confusion on the LCU method regarding the normalization

Let $A = \sum_{k} a_k U_k$ where $a_k$ are real, positive coefficients $U_k$ are unitary matrices. I have realized that $\sigma = A \rho A$ can be implemented on a quantum computer by using the LCU ...
Hailey Han's user avatar
-1 votes
1 answer
55 views

What are the eigenvalues of a state in thermal equilibrium?

Suppose the density matrix $\rho$ with eigenvalues $k_{i}$. Now consider the density matrix $\rho$ in a thermal equilibrium with temperature $T$. Let's show the density matrix with $\rho(T)$ in this ...
reza's user avatar
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1 vote
1 answer
51 views

Simplification of a generic quantum state

We are given a generic 2-qubit density matrix $$\rho=\frac{1}{4}\left[I_4+\Sigma_i a_i \sigma_i \otimes I_2 + \Sigma_i b_i I_2 \otimes \sigma_i + \Sigma_{i,j} c_{ij} \sigma_i \otimes \sigma_j\right]$$ ...
Anindita Sarkar's user avatar
3 votes
1 answer
46 views

Can a generic 2-qubit state be unitarily converted into one of the form $I_2\otimes I_2+\lambda\sigma_z\otimes\sigma_z$?

Suppose I have a general 2-qubit state written in a basis consisting of tensor products of Pauli matrices: $\rho=\frac{1}{4}\left[I_2\otimes I_2+\Sigma_{i} a_i \sigma_i\otimes I_2+\Sigma_{i} b_i I_2\...
Anindita Sarkar's user avatar
2 votes
1 answer
43 views

What is the best way to model a polarizer?

If I have a photon reaching a polarizer, I can think of a polarizer as an operator of $P=a^\dagger_Va_V$ where $a^\dagger_V$ creates a photon with vertical polarization (V). However, on the other ...
Mauricio's user avatar
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0 votes
1 answer
49 views

Obtaining the reduced density matrices for both subsystems of a bipartite system [duplicate]

If we have a single copy of a bipartite quantum system with density matrix $\rho$, is it possible to extract the reduced density matrices of the constituent subsystems separately, i.e. to achieve the ...
Bard's user avatar
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1 vote
0 answers
35 views

Proving CLDM is in QMA, In particular why is it possible to assume that the given input is a product of copies in the soundness section?

I'm wondering about a specific proof for Consistency of Local Density Matrices (CLDM) $ \in $ QMA appearing in "QMA-hardness of Consistency of Local Density Matrices with Applications to Quantum ...
Dudu Ponar's user avatar
0 votes
1 answer
143 views

Initialize circuit by density matrix (i.e. mixed state) by Cirq, Qiskit, QuTip

I want to do a simulation involving: mid-circuit measurement (i.e. based on the measurement result of some qubits to append further gates on other qubits; e.g., Pauli error correction in entanglement ...
Showhands's user avatar
2 votes
1 answer
209 views

Finding the eigenvalues of a qutrit state

I am interested in the state: $\frac{1}{\sqrt{2}} (\left|11\right> + \left|22\right>)$ If I find the density matrix of this, I find the $9 \times 9$ matrix $\rho$. If I want to find the reduced ...
am567's user avatar
  • 597
1 vote
1 answer
76 views

How to find density matrix of 3 qubit W state?

Given a state in bra-ket notation as $|\psi\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. What is the density matrix of this state written using Pauli's spin operator?
Jatin Ghildiyal's user avatar
4 votes
1 answer
159 views

Is possible to write a separable state as a finite or countable infinite sum of product states?

Let us consider the tensor product of two finite Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_1$. According to Watrous book, the set of separable states is the convex hull of the set of product ...
raskolnikov's user avatar

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