# Questions tagged [trace-distance]

In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions. (Wikipedia)

73 questions
Filter by
Sorted by
Tagged with
91 views

42 views

219 views

133 views

### Closeness of unitary dilations of CPTP maps

Let $\Phi_1,\Phi_2 \colon S(\mathcal{H}) \to S(\mathcal{H})$ be CPTP maps on the same Hilbert space $\mathcal{H}$ which are $\varepsilon$-close in diamond norm, and let $U_1,U_2$ be respective unitary ...
205 views

### How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?

The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order. In the proof given in Box 11.2, ...
1 vote
67 views

### Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?

In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says: A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i, The ...
76 views

### Upper bounding the trace distance between a noisy and noiseless quantum state

Consider a quantum state $$\rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \\ \end{pmatrix}.$$ Now, consider the effect of the amplitude damping noise $\mathcal{N}$ of ...
50 views

### Upper bound on trace distance of subsystems based on full system

If we have the following upper bound on the sum of trace distances: $$\frac{1}{N} \sum_{a, b}||p_1(a | b) \rho_{ab} - p_2(a | b) \sigma_{ab}|| \le \epsilon,$$ where $p_1$ and $p_2$ are two ...
59 views

### References that use trace distance to calculate quality of teleportation

State fidelity is the most used measure to compute similarity of input and output states in articles dealing with a quantum teleportation. For my research, I would like to know whether are there ...
276 views

217 views

### How many measurements are needed to distinguish two fixed density matrices?

Suppose there are two fixed density matrices $\rho_1$ and $\rho_2$ are prepared for equal probability. Can we say something about the minimum number of measurements required to distinguish the two ...
42 views

### Bounds relating min-fidelity and induced one-norm

Consider two CPTP maps $M_{A\rightarrow B}$ and $N_{A\rightarrow B}$. Let $\Phi = M - N$. To distinguish between the two maps, there are several measures but here I want to compare two of them. The ...
123 views

### Why is state discrimination possible to infidelity $\delta$ using $n=\Theta(1/\delta)$ states?

In (Haah et al. 2015), in the first section, the authors study the asymptotic behaviours of fidelity and trace distance between $\rho^{\otimes n}$ and $\sigma^{\otimes n}$ for some given pair of ...