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Abstract

An overview of the various different way to quantify coherence via quantum resource theories.

Keywords:Resource TheoriesCoherencespeakableunspeakableincoherent operations.

The content of this article is taken from Marvian & Spekkens (2016), see there for more detail on everything discussed.

Coherence Overview

Mathematically, for a state ρ \rho to have coherence with respect to a basis, the density matrix of ρ \rho written with respect to that basis must have off diagonal terms. This means that with respect to this basis, there exists superpositions between the different basis states.

Details

Consider the states ∣ψ1⟩,∣ψ2⟩∈H2\ket{\psi_1}, \ket{\psi_2} \in \mathcal{H}^2 that can be written with respect to the computational basis as

∣ψ1⟩=∣0⟩,  ∣ψ2⟩=12(∣0⟩+∣1⟩),\ket{\psi_1} = \ket{0}, ~ ~ \ket{\psi_2} = \frac{1}{\sqrt{2}} \big( \ket{0} + \ket{1} \big),
(1)

such that ∣ψ2⟩\ket{\psi_2} is in a superposition of the computational basis states.

Now, consider the density matrices of these two states with respect to the computational basis,

ρ1=∣ψ1⟩⟨ψ1∣,=∣0⟩⟨0∣,=(1000),\begin{align*} \rho_1 &= \ket{\psi_1}\bra{\psi_1}, \\ &= \ket{0}\bra{0}, \\ &= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \end{align*}
(2)

and

ρ2=∣ψ2⟩⟨ψ2∣,=12(∣0⟩⟨0∣+∣0⟩⟨1∣+∣1⟩⟨0∣+∣1⟩⟨1∣),=12(1111).\begin{align*} \rho_2 &= \ket{\psi_2}\bra{\psi_2}, \\ &= \frac{1}{2} \big( \ket{0}\bra{0} + \ket{0}\bra{1}+ \ket{1}\bra{0}+ \ket{1}\bra{1} \big), \\ &= \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}. \end{align*}
(3)

Therefore, being a superposition of states with respect to a basis means having coherence with respect to that basis.

Whilst already implied above it is worth explicitly noting that coherence is basis dependent. Given ∣ψ2⟩=∣+⟩\ket{\psi_2} = \ket{+}, it can be seen that ∣ψ2⟩\ket{\psi_2} has no coherence with respect to the basis defined by the eigenvector of Pauli XX but does have coherence with respect to the computational basis (basis defined by Pauli ZZ).

Speakable vs Unspeakable Coherence

If only the amount of coherence one has is the useful quantity (the resource) then the coherence is speakable. If the basis and the sub-spaces in which the coherence exists is important then the coherence is unspeakable.

Consider the following two states,

∣ψ1⟩=12(∣0⟩+∣1⟩),   ∣ψ2⟩=12(∣0⟩+∣2⟩).\ket{\psi_1} = \frac{1}{\sqrt{2}} \big( \ket{0} + \ket{1} \big), ~ ~ ~ \ket{\psi_2} = \frac{1}{\sqrt{2}} \big( \ket{0} + \ket{2} \big).
(4)

If the label inside the ket is irrelevant to the task, such as in quantum computation, then the two states would be equally resourceful. If, instead, the labels do matter, such as in quantum phase estimation, then the coherence is speakable, and these two states are not equally resourceful.

Resource Theory of Unspeakable Coherence

Framework

This resource theory is defined with respect to translationally-covariant operations.

Firstly, consider an observable LL and the continuous transformations defined by the family of unitary operators

UL(x)=e−ixL: x ∈ R1.U_{L}(x) = e^{-ixL} : ~x~\in~\mathbb{R}^1.
(5)

A state ρ∈H\rho \in \mathcal{H}, has coherence with respect to the eigenspaces of LL if and only if it is asymmetric with respect to the continuous transformations generated by LL,

UL(x)ρUL(x)†≠ρ ∀ x ∈ R1.U_{L}(x) \rho U_{L}(x)^{\dagger} \neq \rho ~ \forall ~x~\in~\mathbb{R}^1.
(6)

On the contrary, a state therefore has no coherence with respect to the eigenspaces of LL if and only if it is symmetric with respect to the continuous transformations generated by LL,

UL(x)ρUL(x)†=ρ ∀ x ∈ R1.U_{L}(x) \rho U_{L}(x)^{\dagger} = \rho ~ \forall ~x~\in~\mathbb{R}^1.
(7)

The state ρ \rho must be block diagonal with respect to the eigenspaces of L L for it to be symmetric under UL(x) U_{L}(x) , which is the definition of having no coherence with respect to the eigenspaces of LL.

Free States

The set of free states, F\mathfrak{F}, are those states which contain no resource - hence, in this case, they are termed the incoherent states. The set of free states is therefore

F={ρ:UL(x)ρUL(x)†=ρ ∀ x ∈ R1},\mathfrak{F} = \{ \rho : U_{L}(x) \rho U_{L}(x)^{\dagger} = \rho ~ \forall ~x~\in~\mathbb{R}^1 \},
(8)

which are the states that have no coherence with respect to the eigenspaces of LL.

It can be seen that

ρ∈F⟺[L,ρ]=0,\rho \in \mathfrak{F} \Longleftrightarrow [L, \rho] = 0,
(9)

meaning the set of free states are those that commute with the generator of the continuous transformations.

Allowed Operations

The allowed operations of the resource theory are those that are translationally-covariant. A quantum operations E\mathcal{E} is translationally-covariant with respect to the transformation UL(x)U_{L}(x) if

UL(x)∘E=E∘UL(x),\mathcal{U}_{L}(x) \circ \mathcal{E} = \mathcal{E} \circ \mathcal{U}_{L}(x),
(10)

where UL(x)(⋅)=UL(x)(⋅)UL(x)†\mathcal{U}_{L}(x)(\cdot) = U_{L}(x)(\cdot)U_{L}(x)^\dagger. This means that one can apply the operation before or after a transformation UL(x)U_{L}(x) and get the same outcome.

The broadest definition of allowed operations is that they are the set of operations that maps the free states to the free states. This can be seen to be true for translationally-covariant operations by considering ρ∈F\rho \in \mathfrak{F} and letting E\mathcal{E} be a translationally-covariant operation, then

UL(x)∘E(ρ)=E∘(UL(x))(ρ)=E(ρ) ∀ x ∈ R1,\mathcal{U}_{L}(x) \circ \mathcal{E}(\rho) = \mathcal{E} \circ \big(\mathcal{U}_{L}(x)\big)(\rho) = \mathcal{E}(\rho)~\forall~x~\in~\mathbb{R}^1,
(11)

as (UL(x))(ρ)=ρ ∀ x ∈ R1\big(\mathcal{U}_{L}(x)\big)(\rho) = \rho ~\forall~x~\in~\mathbb{R}^1. Therefore, E(ρ)∈F\mathcal{E}(\rho) \in \mathfrak{F} as it is asymmetric under UL(x)U_{L}(x). Hence, translationally-covariant operations map free states to free states.

Finally, the allowed operations can then be defined as

O={E:UL(x)∘E=E∘UL(x) ∀ x ∈ R1}.\mathfrak{O} = \{ \mathcal{E}: \mathcal{U}_{L}(x) \circ \mathcal{E} = \mathcal{E} \circ \mathcal{U}_{L}(x)~\forall~x~\in~\mathbb{R}^1 \}.
(12)

Free Dilations:

All allowed operations, E∈O\mathcal{E} \in \mathfrak{O}, have a free dilation, meaning they can be written as

E(ρ)=tr[VAB(ρA⊗σB)VAB†]\mathcal{E}(\rho) = \textrm{tr} \big[ V_{AB} \big(\rho_A \otimes \sigma_B) V^\dagger_{AB} \big]
(13)

where σ∈F\sigma \in \mathfrak{F} and VABV_{AB} is a translationally-covariant isometry.

Measurements:

Consider an operation E∈O\mathcal{E} \in \mathfrak{O} that maps from a non-trivial input space to a trivial output space i.e, a measurement defined by an effect EE such that

E(⋅)=tr[E(⋅)],  0≤E≤I.\mathcal{E}(\cdot) = \textrm{tr}\big[E(\cdot)\big], ~ ~ 0 \leq E \leq \mathbb{I}.
(14)

In this case, one gets the a similar conditions as for states with

UL†(x)(E)=E ∀ x ∈ R1  if  E∈O.\mathcal{U}^{\dagger}_{L}(x)\big(E \big) = E ~ \forall~x~\in~\mathbb{R}^1 ~~ {\rm if} ~ ~ \mathcal{E} \in \mathfrak{O}.
(15)

As before, this condition is true if and only if [L,E]=0[L,E]=0.

A POVM - a collection of effects that sums to the identity - is said to be a translationally-covariant POVM if and only if each of its effects commutes with LL.

Unitaries:

If E∈O\mathcal{E} \in \mathfrak{O} is a unitary operation, E(⋅)=V(⋅)V†\mathcal{E}(\cdot) = V(\cdot)V^\dagger, then

[UL(x),V]=0 ∀ x ∈ R1.[U_{L}(x), V] = 0 ~\forall~x~\in~\mathbb{R}^1.
(16)

which is equivalent to [L,V]=0[L, V] = 0, meaning VV is block diagonal with respect to the eigenspaces of LL.

Resource Measures

The spectral decompostion of LL is

L=∑kλkΠk,L = \sum_k \lambda_k \Pi_k,
(17)

where Πk \Pi_k is the projector onto the eigenspace of LL associated to the eigenvalue λk \lambda_k .

The dephasing channel with respect to the operator LL is then

D(ρ)=∑kΠk(ρ)Πk.\mathcal{D}(\rho) = \sum_k \Pi_k (\rho) \Pi_k.
(18)

The following functions are resource measures of unspeakable coherence. This means they are at least faithful and monotonic under allowed operations.

Relative Entropy
Trace Norm
Trace Norm 2

let ρ∈H\rho \in \mathcal{H}.

The relative entropy of asymmetry is defined as

A(ρ)≔S(ρ∣∣D(ρ))=S(D(ρ))−S(ρ),A(\rho) \coloneqq S(\rho \vert \vert \mathcal{D}(\rho)) = S(\mathcal{D}(\rho)) - S(\rho),
(19)

where S(⋅∣∣⋅)S(\cdot \vert \vert \cdot) is the quantum relative entropy and S(⋅)S(\cdot) is the Von Neumann Entropy.

References
  1. Marvian, I., & Spekkens, R. W. (2016). How to quantify coherence: Distinguishing speakable and unspeakable notions. Physical Review A, 94(5). 10.1103/physreva.94.052324
Resource Theories
Quantifying Resources
Resource Theories
Glossary