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Questions tagged [trace]

Use this tag when having questions concerning expressions with the trace of a matrix/operator.

The trace of a matrix A is defined as the sum of the diagonal values, as described on Wikipedia.

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How to take partial trace?

$L$ is a linear operator acting on hilbert space $V$ of dimension $n$, $L: V \to V$. The trace of a linear operator is defined as sum of diagonal entries of any matrix representation in same input and output basis of $V$. But if $L$ is a linear…
advocateofnone
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What information does the trace of a matrix give?

I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant will tell about the volume of the space enclosed…
Ayushi
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Lowering/raising metric indexes

So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing. Consider Minkowski spacetime. The trace of a matrix $A$ can be written in terms of the Minkowski metric as…
QuantumBrick
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Is a partial trace cyclic?

I want to know if a partial trace keeps the cyclic property of the trace. The partial trace is defined as $$ tr_B: \mathcal{B}_1(\mathcal{H}_A\otimes \mathcal{H}_B) \longrightarrow \mathcal{B}_1(\mathcal{H}_A) \\[3mm] \text{such that} \\[3mm] tr_B…
João Bravo
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Calculating $\mathrm{Tr}[\log \Delta_F]$

I am stuck with this problem for quite sometime. I have a propagator in the momentum representation (from this Phys.SE question), which looks like $$ \widetilde\Delta_F(p) = \frac{1}{(p^0)^2-\left(\left(n\pi/L\right)^2+m^2\right)+i\epsilon} $$ I…
user35952
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Why is the trace of the outer product of two states equal to the inner product of the two states?

Why is it that, given two quantum states $|\psi_1\rangle$, $|\psi_2\rangle$, $$\mathrm{Tr}(|\psi_1\rangle\langle\psi_2|) = \langle\psi_2|\psi_1\rangle \quad $$ I went through the equation with the qubit space $\mathcal{H}=\{|0\rangle,|1\rangle\}$…
redpanda2236
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Question about the Dirac notation for partial trace

I saw the following definition for the partial trace operator: $\rho_A=\sum_k \langle e_k|\rho_{AB}|e_k\rangle$, where $e_k$ is basis for the state space of system $B$. From what I know, in the Dirac notation, the meaning of $\langle v|A|u\rangle$…
John Smith
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A limit of a particular Quantum Fidelity

I have the following problem. Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More precisely assume that $$ \mathbf{\hat{\rho}}(t):=…
Hldngpk
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Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact algebras) and the generators scaled such…
nox
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Reasoning Check: Trace of squared mixed-state density matrix

It's often written in the QI literature that, for a density operator $\rho$, if $\text{Tr}\left[\rho^{2}\right] < 1$, then $\rho$ describes a mixed state. However, I haven't seen any proofs of this except in the case where the states are in $\rho$…
miggle
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Trace of a Tensor

What is the significance of defining the trace of a tensor as $g^{\alpha\beta} R_{\alpha\beta}$ instead of $R_{\alpha\alpha}$ on a Riemannian manifold?
Ramesh Chandra
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Deriving the Spinor Completeness Relation without using a Representation

Reference: DAMTP problem set 3, question 5 but ignore the spinor solutions given. To preface, this has taken up 1 entire day and a further 2 afternoons of work so I will just list the most promising attempts rather than clutter half a notebook…
Alexander McFarlane
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Dimension and Basis properties of $SU(N)$

$SU(N)$ is the group of special unitary matrices of dimension $N$, i.e., the set of all unitary ($U^{\dagger}U=I$) $N\times N$ matrices with $\det(U)=1$. For $N=2$, these matrices are spanned by the identity and the Pauli matrices, i.e. we can…
TMG
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Understanding the partial trace and deriving $\langle l|R_{B}|k\rangle = \text{Tr}((\mathbb{I}_{A} \otimes |k\rangle \langle l|)(R_{AB}))$

By definition according to the notes I am looking through: The partial trace $\text{Tr}_A:L(H_A \otimes H_B) \rightarrow L(H_B)$ is the unique map that satisfies: $$\text{Tr}(L_B \cdot \text{Tr}_A(R_{AB})) = \text{Tr}((\mathbb{I} \otimes…
Relative0
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Operator traces in Kontsevich quantization

In quantization, one studies maps from functions on the phase space to operators acting on the Hilbert space. Let's fix one such map and call it $Q$. Deformation quantization is based on the idea that $Q$ can be studied indirectly, by endowing the…
Prof. Legolasov
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