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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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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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
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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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Almost perfect quantum encryption of a mixed state using only $n + O(\log(n) +\log(\frac{1}{\epsilon}))$ shared bits

Alice holds a state $\psi$ of $n$-qubits, and wants to send it to Bob using a single quantum message. Bob and Alice share only $n + O(log(n) +log(\frac{1}{\epsilon}))$ random bits, for some value $\...
Gabi G's user avatar
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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
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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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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
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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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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
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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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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
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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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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
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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
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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
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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
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1 answer
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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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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
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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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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
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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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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
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1 answer
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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
36 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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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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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
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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
208 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
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1 answer
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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
151 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
1 vote
1 answer
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QISKIT: ValueError: too many subscripts in einsum DensityMatrix()

I am trying to compute the entanglement entropy of a partition of a quantum system on qiskit. To do this, I call the function DensityMatrix(). If I go above 10 sites (e.g. 12), I get an error like: ...
Alberto Zorzato's user avatar
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1 answer
261 views

Derivation of Choi-Jamiolkowski isomorphism

I'm following the course Mathematical Methods of Quantum Information Theory by Reinhard Werner. In lecture 6, he gave a derivation of Choi-Jamiolkowski isomorphism, and I'm struggling to understand ...
Manuel E's user avatar
2 votes
1 answer
44 views

How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?

I am confused as to how we get $M_{0} = \sqrt{1-p} I$ and the following $M_{1}, M_{2}, M_{3}$. The above notes say that we should partially trace over the environment in the $|{0}\rangle, |{1}\rangle, ...
am567's user avatar
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2 answers
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Why is a density matrix an orthogonal projector?

Suppose I have a density matrix like $\rho = \frac{1}{2}[I + \hat{n}\vec{\sigma}]$. The claim is that $\rho$ is an orthogonal projector for the state $|+\rangle$ in an arbitrary direction $\hat{n}$. ...
Physkid's user avatar
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1 vote
1 answer
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I have two Choi matrix I suspect be equivalent. Can I manipulate them?

I am performing a process tomography over a protocol I suspect to be equivalent to the $CS$ gate. To compare basic operators I usually compute the Choi matrix of the target gate -- which in this case ...
Daniele Cuomo's user avatar
1 vote
0 answers
39 views

Can we use a Werner state for quantum teleportation? [duplicate]

Some background: The quantum teleportation protocol requires first that Alice and Bob share an entangled state, say a Bell state $|\psi^{+}\rangle_{AB}$. There is another state $|\psi\rangle_{A'}$ to ...
Physkid's user avatar
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1 vote
0 answers
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How is Pauli twirling so powerful?

So the Pauli twirling approximation gives us a quantum channel $\Phi$ that transforms a density matrix $\rho$ to: $\Phi(\rho)\mapsto\sum_{i=0}^3 \sigma^i \rho \sigma^i,$ where $\sigma^0 = \mathbb{I}, \...
JoJo P's user avatar
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General Bell state expression: What condition for mixture of Bell states to be entangled?

Convention: $|qubit_{A}, qubit_{B}\rangle$ The general Bell state equation: $|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k\oplus b\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\...
Physkid's user avatar
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-2 votes
1 answer
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Finding entanglement in matrix that is a sum of 4 Bell states

A general Bell state: $|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\rangle + (-1)^{a}|1,1 \oplus b \rangle]$ $|\beta(0,0)\rangle = \frac{1}{2}[|00\rangle \langle 00| + |00\rangle \langle 11| +...
librarian_'s user avatar
1 vote
2 answers
51 views

Does separability of a matrix implies the matrix is a density matrix?

Suppose I have a matrix that is unknown whether it is a density matrix and assume that finding the eigenvalues of it is difficult because the matrix is expressed generally. However, suppose that this ...
Physkid's user avatar
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-1 votes
2 answers
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Define a traceless part of $\rho$ [closed]

I saw in a paper: $|\bar{\rho}\rangle\rangle=|\rho\rangle\rangle-|\hat{I}\rangle\rangle / 2^{n / 2}$ for the $4^n$-dimensional vector representing the traceless part of $\rho$. https://arxiv.org/abs/...
karry's user avatar
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3 votes
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calculate the reduced density matrix of a 2 qubit state and compare the eigenvalues

So I have the exercise to apply a Cz gate to the following 2 Qubit state $|a\rangle \otimes |b\rangle = (a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)\\\\$ Afterwards, I ...
Ruebli's user avatar
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5 votes
1 answer
142 views

Moments of Pauli coefficients of Haar-random states

I want to evaluate the quantity $\sum_{P\in \rm{P}^n}\text{Tr}^{\alpha}(\rho P)$, where $P$ is an element of n-qubit Pauli group $\rm{P}^n$ and $\rho$ is a density matrix of a Haar random state. It is ...
Feng Pan's user avatar
1 vote
1 answer
57 views

How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = \text{Tr}_{2}[\rho_{12}] = \sum_{i}\langle i_{2} | \rho | i_{2} \rangle$?

Let the system be a 2 - qubit system and let $\rho_{12}$ be a density matrix of some state for this 2 - qubit system. How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\...
Physkid's user avatar
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4 votes
1 answer
103 views

How do I find the reduced density matrix of a system where two people share one qubit and have one qubit of their own?

I have the following problem and have attempted to find a solution to it, but to no avail. Alice and Bob have one qubit each, say $|\psi\rangle$ with Alice and $|\phi\rangle$ with Bob. They also share ...
requiemman's user avatar
2 votes
1 answer
123 views

Given that for every valid density matrix $\rho$, $\text{Tr}(M\rho) = 1$; what can we conclude about matrix $M$?

My intuition says that $M$ has to be the identity matrix, but I am not able to show it rigorously. I tried playing around using spectral decomposition. If $$ \rho = \sum_i \lambda_i |\lambda_i \rangle ...
FDGod's user avatar
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2 votes
2 answers
106 views

How to compute marginal probabilities of Alice's qubit (in density operator language)?

Let $| \psi \rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{2}|01\rangle + \frac{\sqrt{3}}{4} |10\rangle + \frac{1}{4}|11\rangle$ be a state vector describing a closed quantum mechanical system. ...
Physkid's user avatar
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1 vote
1 answer
121 views

Statevector from Density matrix of non-pure state

I have a state vector of a 16 qubit system. I want to get the wave function (in the form of a state vector) for half and quarter of this system. When I try to make a ...
VladislavOkatev's user avatar
1 vote
0 answers
81 views

What does the product of two density matrices represent physically?

A quantum state, pure or mixed, can be described by a density matrix that encodes the Bloch vector $\hat{m}$ analog of a quantum state like $\rho = \frac{1}{2}[\mathbb{I} + \hat{m}.\vec{\sigma}]$. Let ...
Physkid's user avatar
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3 votes
2 answers
186 views

Prove the fidelity equals $F( \rho , \sigma) = |\langle \psi_{\rho} | \psi_{\sigma}\rangle|^2$ for pure states

I am trying to learn by myself quantum computing and information and I have a very simple question concerning the demonstration of the following equality: $F( \rho , \sigma) = |\langle \psi_{\rho} | \...
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