Applications of matrix theory to quantum coherenceshb.skku.edu/_res/mao2016/etc/SPlosker.pdf ·...

Applications of matrix theory to quantum coherence

Sarah Plosker

Department of Mathematics and Computer ScienceBrandon University

Joint work with J. Chen (University of Maryland), N. Johnston (Mount Allison University),and C.-K. Li (College of William and Mary)

MAO July 3–6, 2016

S. Plosker (Brandon) http://arxiv.org/pdf/1601.06269.pdf Quantum Coherence 1 / 21

Outline

1 Preliminaries and MotivationClassical versus Quantum—What’s the Difference?Physical MotivationMathematical Framework

2 Measuring Coherence

3 The trace distance of coherence of a pure state

4 Maximally coherent states under the trace norm of coherence

5 Conclusion

Classical versus Quantum—What’s the Difference?

In short, entanglement and superpositions.

Superpositions?

When multiple quantum systems interact with each other, the resource ofinterest is typically entanglement.

When there is no interaction between different systems, the resource ofinterest is instead coherence, or the amount that a state is in asuperposition of a given set of mutually orthogonal states.

Superpositions are essentially linear combinations of basis states (with theadditional property that the coefficients αi ∈ C in the linear combinationsatisfy

∑i |αi |2 = 1)

In the absence of measurement, electron spin is in a superposition of thetwo states ↑ / ↓ (even without entanglement).

Also, Schrodinger’s cat.

Superpositions?

∑i |αi |2 = 1)

Superpositions?

∑i |αi |2 = 1)

Superpositions?

∑i |αi |2 = 1)

Superpositions?

∑i |αi |2 = 1)

Physical Motivation

One of the major goals in quantum information theory is to find effectiveways of quantifying the amount of “quantumness” within a givensystem—that is, how much the system differs from any possible classicalmechanical description of it.

Some Notation

Restrict attention to unit vectors

Unit vectors x ∈ Cn represent quantum information in our quantumsystem Cn and are called quantum states

States are either pure (represented by a unit vector x ∈ Cn or,equivalently, its outer product xx∗, which is a rank-one projection) ormixed (represented by a density matrix ρ =

∑i pixx

∗, where {pi}forms a probability distribution).

Density matrices (including outer products associated to pure states)are precisely the trace-one, positive (semi-definite) matrices.

Some Notation

∑i pixx

Some Notation

∑i pixx

Mathematical Framework

Fix an orthonormal basis {vi}ni=1 of Cn. An incoherent state is any densitymatrix that is diagonal in this basis—that is, any matrix of the form∑n

i=1 pivivi∗ where 0 ≤ pi ≤ 1 for all i and

∑ni=1 pi = 1.

Note: Incoherent = no coherence. That is, diagonal density matrices canbe seen as “classical” in that there are no superpositions.

We consider the distance of a given state to the closest incoherent statevia a coherence measure:

C (ρ) = minδ∈I

d(ρ− δ)

where I is the set of all incoherent states in the quantum system and d isa distance measure defined on the system.

—this is highly basis-dependent!

∑ni=1 pi = 1.

C (ρ) = minδ∈I

d(ρ− δ)

∑ni=1 pi = 1.

C (ρ) = minδ∈I

d(ρ− δ)

∑ni=1 pi = 1.

C (ρ) = minδ∈I

d(ρ− δ)

Measuring Coherence

The two most widely-known coherence measures are:

the `1-norm of coherence:

C`1(ρ) := minδ∈I‖ρ− δ‖`1

= minδ∈I

n∑i ,j=1

|ρ− δ|ij =∑i 6=j

|ρij |,

the relative entropy of coherence:

Cr(ρ) := S(ρdiag)− S(ρ),

where S(ρ) = −tr(ρ log2 ρ) is the von Neumann entropy of the stateρ and ρdiag is the state obtained from ρ by deleting all off-diagonalentries.

Measuring Coherence

The two most widely-known coherence measures are:

the `1-norm of coherence:

C`1(ρ) := minδ∈I‖ρ− δ‖`1

= minδ∈I

n∑i ,j=1

|ρ− δ|ij =∑i 6=j

|ρij |,

the relative entropy of coherence:

Cr(ρ) := S(ρdiag)− S(ρ),

where S(ρ) = −tr(ρ log2 ρ) is the von Neumann entropy of the stateρ and ρdiag is the state obtained from ρ by deleting all off-diagonalentries.

Measuring Coherence

Other coherence measures that have recently been proposed:

the trace distance of coherence:

CTr(ρ) := minδ∈I‖ρ− δ‖Tr = min

δ∈I

n∑i=1

|λi (ρ− δ)|,

where λi (ρ− δ) are the eigenvalues of the matrix ρ− δ and ‖ · ‖Tr isthe trace norm

the robustness of coherence:

CR(ρ) := minτ

{s ≥ 0

∣∣∣ ρ+ sτ

1 + s∈ I},

where the minimum is over all density matrices τ .Note CR(xx∗) = C`1(xx∗) (Napoli et al, 2016).

Measuring Coherence

Other coherence measures that have recently been proposed:

the trace distance of coherence:

CTr(ρ) := minδ∈I‖ρ− δ‖Tr = min

δ∈I

n∑i=1

|λi (ρ− δ)|,

where λi (ρ− δ) are the eigenvalues of the matrix ρ− δ and ‖ · ‖Tr isthe trace norm

the robustness of coherence:

CR(ρ) := minτ

{s ≥ 0

∣∣∣ ρ+ sτ

1 + s∈ I},

where the minimum is over all density matrices τ .Note CR(xx∗) = C`1(xx∗) (Napoli et al, 2016).

How good are these measures?

The defining properties of a proper coherence measure have recently beenidentified (Baumgratz-Cramer-Plenio, 2014); for example, a state ρ shouldhave zero coherence under the proposed measure if and only if ρ isincoherent, and the proposed measure should be convex:∑

pnC (ρn) ≥ C (∑

pnρn) for any set of states {ρn} and any pn ≥ 0 with∑pn = 1.

The `1-norm of coherence, relative entropy of coherence, and robustness ofcoherence have all been shown to be proper coherence measures, and ithas been shown that the trace distance of coherence is a proper measureof coherence when restricted to qubit states or X states (though thegeneral case remains open).

How good are these measures?

The defining properties of a proper coherence measure have recently beenidentified (Baumgratz-Cramer-Plenio, 2014); for example, a state ρ shouldhave zero coherence under the proposed measure if and only if ρ isincoherent, and the proposed measure should be convex:∑

pnC (ρn) ≥ C (∑

pnρn) for any set of states {ρn} and any pn ≥ 0 with∑pn = 1.

The `1-norm of coherence, relative entropy of coherence, and robustness ofcoherence have all been shown to be proper coherence measures, and ithas been shown that the trace distance of coherence is a proper measureof coherence when restricted to qubit states or X states (though thegeneral case remains open).

2× 2 setting

If ρ is 2× 2 and D ∈ I minimizes Ctr (ρ), then D = ρdiag and henceCtr (ρ) = C`1(ρ) (Rana-Parashar-Lewenstein, 2016).

In their words, “finding an analytic form even for pure qutrits is almostintractable”. Evidence was given to suggest that a simple closed-formformula might not exist.

2× 2 setting

If ρ is 2× 2 and D ∈ I minimizes Ctr (ρ), then D = ρdiag and henceCtr (ρ) = C`1(ρ) (Rana-Parashar-Lewenstein, 2016).

In their words, “finding an analytic form even for pure qutrits is almostintractable”. Evidence was given to suggest that a simple closed-formformula might not exist.

The trace distance of coherence of a pure state

We wish to characterize CTr(xx∗), where x ∈ Cn is an arbitrary pure state

(unit vector).

We give an “almost formula” for the trace distance of coherence of a purestate: we show that it is given by one of n different formulas (dependingon the state), and which formula is the correct one can be determinedsimply by checking log2(n) inequalities.

We wish to characterize CTr(xx∗), where x ∈ Cn is an arbitrary pure state

(unit vector).

We give an “almost formula” for the trace distance of coherence of a purestate: we show that it is given by one of n different formulas (dependingon the state), and which formula is the correct one can be determinedsimply by checking log2(n) inequalities.

Note that there is a diagonal unitary U and a permutation matrix P suchthat PUx is a unit vector having non-negative entries x1 ≥ · · · ≥ xn ≥ 0 indescending order. We then have

‖xx∗ − δ‖ = ‖PU(xx∗ − δ)U∗Pt‖

for any δ ∈ I. So, we may replace x by PUx without loss of generality.

The algorithm

Suppose x = (x1, . . . , xn)t is a unit vector with entries x1 ≥ · · · ≥ xn ≥ 0.Let s` =

∑`j=1 xj , m` =

∑nj=`+1 x

2j , and p` = s2` − 1− `m` for

` ∈ {1, . . . , n}. There is a maximum integer k ∈ {1, . . . , n} satisfying

xk > qk :=1

√p2k + 4kmks

). (1)

The unique best approximation of xx∗ in I with respect to the trace norm(and the operator norm) is D = diag(d1, . . . , dk , 0, . . . , 0) ∈ I with

dj =xj − qksk − kqk

for 1 ≤ j ≤ k .

Furthermore,

CTr(xx∗) = ‖xx∗ − D‖Tr = 2(qksk + mk), and

‖xx∗ − D‖ = qksk + mk .

Although it might seem somewhat time-consuming at first to find thevalue of k described by the theorem, the proof of the theorem shows thatif qk < xk then qj < xj for all j < k. Thus we can search for k via binarysearch, which requires only log2(n) steps, rather than searching through alln possible values of k. MATLAB code that implements this algorithm isable to compute CTr(xx

∗) for pure states x ∈ C1,000,000 in under onesecond on a standard laptop computer.

A pure state x ∈ Cn is maximally coherent if all of its entries have equalabsolute value: |x1| = · · · = |xn| = 1/

Recently it has been suggested that the maximum value of a propermeasure of coherence should be attained exactly by the maximallycoherent states. This property is known to hold for the relative entropy ofcoherence, the `1-norm of coherence, and the robustness of coherence.

Theorem

For all (potentially mixed) states ρ, we have CTr(ρ) ≤ 2− 2/n.Furthermore, equality holds if and only if ρ = xx∗, where x is a maximallycoherent state.

This theorem provides further evidence that the trace norm is indeed aproper measure of coherence.

A pure state x ∈ Cn is maximally coherent if all of its entries have equalabsolute value: |x1| = · · · = |xn| = 1/

Recently it has been suggested that the maximum value of a propermeasure of coherence should be attained exactly by the maximallycoherent states. This property is known to hold for the relative entropy ofcoherence, the `1-norm of coherence, and the robustness of coherence.

Theorem

For all (potentially mixed) states ρ, we have CTr(ρ) ≤ 2− 2/n.Furthermore, equality holds if and only if ρ = xx∗, where x is a maximallycoherent state.

This theorem provides further evidence that the trace norm is indeed aproper measure of coherence.

Relationship between the `1-norm of coherence and therelative entropy of coherence

Conjecture in

Theorem in

But also in

Theorem (Rana-Parashar-Lewenstein, 2016)

For every pure state x,

C`1(xx∗) ≥ max{Cr (xx∗), 2Cr (xx∗) − 1}.

The `1-norm coherence of a pure state is never smaller than its relativeentropy of coherence.

Does this hold for mixed states as well?

Theorem (Rana-Parashar-Lewenstein, 2016)

For every pure state x,

C`1(xx∗) ≥ max{Cr (xx∗), 2Cr (xx∗) − 1}.

The `1-norm coherence of a pure state is never smaller than its relativeentropy of coherence.

Does this hold for mixed states as well?

What have we done?

We derived an explicit expression for the trace distance of coherenceof a pure state as well as the closest incoherent state to a given purestate with respect to the trace distance.

We showed that the trace distance of coherence of an arbitrary (pureor mixed) state is maximized iff the state under consideration is amaximally coherent state.

We gave a more explicit proof to the conjecture relating the `1-normof coherence and the relative entropy of coherence

Future Work

One natural question that arises from this work is whether or not ouralgorithm can be used to show that the trace distance of coherence isstrongly monotonic under incoherent quantum channels (anotherproperty of a proper coherence measure), at least when it is restrictedto pure states.

C`1(ρ) ≥ max{Cr (ρ), 2Cr (ρ) − 1}

for all states ρ.

Applications of matrix theory to quantum coherenceshb.skku.edu/_res/mao2016/etc/SPlosker.pdf ·...

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