Entanglement monotones are nonnegative functions whose values do not increase under local operations of classical communications. Here, I explore novel entanglement monotones, primarily the Extropy and Dual Entropy, and its implementation in the Wolfram Quantum Framework Paclet.
Initialize Quantum Framework Paclet
Initialize Quantum Framework Paclet
PacletInstall[CloudObject["https://wolfr.am/DevWQCF"],ForceVersionInstall->True]<<Wolfram`QuantumFramework`
Out[]=
PacletObject
Novel Entanglement Monotones: Extropy & Dual Entropy
Novel Entanglement Monotones: Extropy & Dual Entropy
Overview
Overview
Purpose: Here, I attempt to construct two new monotone measure: The Dual Entropy and Extropy. I first provide the classical definitions and then offer the quantum version of these monotone measures. The references are given below.The classical version of extropy is given by -(1-)log(1-) where are probabilities associates with a state • The quantum version of extropy is given by -Tr[(1-ρ)log(1-ρ)]. • Property: The entropy and the extropy of a binary distribution are identical. The classical version of dual entropy is given by -log()+(1-)log(1-) • The quantum version of dual entropy is given by -Tr[ρlog(ρ)+(1-ρ)log(1-ρ)] • The dual entropy = von Neumann entropy + extropy. These two monotones were discussed recently on the arXiv (see the references below).
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classical extropy
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quantum extropy
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dual entropy
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quantum dual entropy
Novel Entanglement Monotones
Novel Entanglement Monotones
Let’s first define a set of representative quantum states to study the monotones. Here, PsiMinus, W, and the GHZ states come preconfigured in the QF paclet. In addition, I demonstrate how to define a Generalized Bell State (GBS) by mapping the basis coefficients in a bipartite system to a circle.
GBS=QuantumState[<|{0,1}->Cos[θ],{1,0}->-Sin[θ]|>,2,2];states={QuantumState["PsiMinus"],QuantumState["W"],QuantumState["GHZ"],GBS};TableForm[Transpose[{{"","W","GHZ","Generic Bell"},#["Formula"]&/@states}],TableHeadings{None,{"state","formula"}}]
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Ψ
Out[]//TableForm=
state | formula |
- Ψ | |01〉 2 |10〉 2 |
W | |001〉 3 |010〉 3 |100〉 3 |
GHZ | |000〉 2 |111〉 2 |
Generic Bell | Cos[θ]|01〉-|10〉Sin[θ] |
The states considered above are representative of bipartite systems (PsiMinus and GBS) or tripartite systems (W and GHZ). We can perform a QuantumPartialTrace over any qubit and obtain a reduced set of states. I chose to trace out the second qubit {2} but one can relax this choice.
reducedstates=Outer[Map[QuantumPartialTrace[#1,{2}],{"DensityMatrix"}][[1]]&,states];TableForm[Transpose[{#["Formula"]&/@states,MatrixForm/@reducedstates}],TableHeadings{None,{"Original state","Reduced denisty matrix after tracing out 2nd qudit"}}]
Out[]//TableForm=
Original state | Reduced denisty matrix after tracing out 2nd qudit | ||||||||||||||||
|01〉 2 |10〉 2 |
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|001〉 3 |010〉 3 |100〉 3 |
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|000〉 2 |111〉 2 |
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Cos[θ]|01〉-|10〉Sin[θ] |
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Extropy
Extropy
Here, I compute the Extropy -Tr[(1-ρ)log(1-ρ)] for the reduced states.
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quantum extropy
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Extropy[θ_,ρ_]:=FullSimplify[-Tr[(IdentityMatrix[Length[ρ]]-ρ).MatrixLog[IdentityMatrix[Length[ρ]]-ρ]],Assumptions0<=θ<=π/2]
The extropy is a nice monotone measure as one does not have to find the eigenvalues of each density matrix as can be seen with the W and GHZ states. For these states, one cannot compute the entanglement entropy via a matrix representation as shown below
Assuming[θ∈Reals,FullSimplify[Table[-Tr[reducedstates[[i]].MatrixLog[reducedstates[[i]]]],{i,1,Length[reducedstates]}]]]
Out[]=
Log[2],-TrSparseArray.MatrixLogSparseArray,-TrSparseArray.MatrixLogSparseArray,Log[]+Log[]
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Cos[θ]
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Sec[θ]
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Csc[θ]
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Sin[θ]
Instead, one needs to calculate the eigenvalues and remove the cases where the eigenvalue vanishes as can be shown below
Assuming[θ∈Reals,FullSimplify[Table[-DeleteCases[Eigenvalues[reducedstates[[i]]],0].Log[DeleteCases[Eigenvalues[reducedstates[[i]]],0]],{i,1,Length[reducedstates]}]]]
Out[]=
Log[2],Log,Log[2],Log[]+Log[]
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2/3
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Cos[θ]
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Sec[θ]
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Csc[θ]
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Sin[θ]
However, one can immediately compute the extropy in a nice vectorized implementation
Table[Extropy[θ,reducedstates[[i]]],{i,1,Length[reducedstates]}]
Out[]=
Log[2],Log,Log[2],-2(Log[Cos[θ]]+Log[Sin[θ]])
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2/3
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Cos[θ]
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Sin[θ]
Main Points:
- Can compute the extropy for states after tracing out one qubit.
- Matches the entanglement entropy.
- Can compute the extropy for states after tracing out one qubit.
- Matches the entanglement entropy.
Dual Entropy
Dual Entropy
Here, I compute the dual entropy -Tr[ρlog(ρ)+(1-ρ)log(1-ρ)] for a pure Bell state:
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quantum dual entropy
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DualEntropy[θ_,_]:=FullSimplify[-(DeleteCases[Eigenvalues[],0].Log[DeleteCases[Eigenvalues[],0]]+Tr[(IdentityMatrix[Length[]]-).MatrixLog[IdentityMatrix[Length[]]-]]),Assumptions0<=θ<=π/2]
Table[DualEntropy[θ,reducedstates[[i]]],{i,1,Length[reducedstates]}]
Out[]=
Log[4],Log,Log[4],-4(Log[Cos[θ]]+Log[Sin[θ]])
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Cos[θ]
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Sin[θ]
Main Points:
- The Dual Entropy is twice value as the entanglement entropy in the case of binary distributions.
- Suffers from a similar problem to the entanglement entropy as it needs eigenvalues in certain cases.
- The Dual Entropy is twice value as the entanglement entropy in the case of binary distributions.
- Suffers from a similar problem to the entanglement entropy as it needs eigenvalues in certain cases.
Monotone Showcase: Generalized Bell State
Monotone Showcase: Generalized Bell State
Next, let’s compare all monotones for the generalized Bell state. - The entanglement monotones a priori should not all agree on the generalized Bell state except at the maximally mixed state and separable state.
entropyMatrix=Assuming[θ∈Reals,FullSimplify[Outer[QuantumEntanglementMonotone[#1,{{1},{2}},#2]&,states,{"Concurrence","LogNegativity","EntanglementEntropy","RenyiEntanglementEntropy"}]]]//MatrixForm;Show[Plot[{Extropy[θ,reducedstates[[4]]],DualEntropy[θ,reducedstates[[4]]]},{θ,0,π/2},PlotLegends{"(Unnormalized) Extropy","(Unnormalized) Dual Entropy"},FrameLabel{Style[θ,14],Style["S"states[[4]]["Formula"],14]},PlotTheme"Scientific",PlotLabel"Entanglement Monotones Showcase",PlotStyle{{Thick,Dashed},{Thick,Dashed}}],Plot[Evaluate[QuantityMagnitude[entropyMatrix[[1,4]]]],{θ,0,π/2},PlotLegends{"Concurrence","LogNegativity","Entanglement Entropy","Renyi Entropy"},FrameLabel{Style[θ,14],Style["S"states[[4]]["Formula"],14],Style["Entanglement Monontones",14]},PlotTheme"Scientific",PlotStyle{{},{},{},{Thick,Dashed}}]]
Main Points:
1. All monotone measures agree at the minimum and maximum entanglement locus for a generalized Bell state.
2. New monotone measures require normalization however.
1. All monotone measures agree at the minimum and maximum entanglement locus for a generalized Bell state.
2. New monotone measures require normalization however.
Conclusions:
Conclusions:
Both the dual entropy and the extropy can be shown to be entanglement monotones -- specifically nonnegative functions whose values do not increase under local operations and classical communication. We checked these properties in various examples. Furthermore,
References:
References:
1. A new entanglement measure based dual entropy, Xue Yang, Yan-Han Yang, Li-Ming Zhao, Ming-Xing Luo, e-Print: https://arxiv.org/abs/2204.07407 [quant-ph].
2. Extropy: Complementary Dual of Entropy, Lad, Frank ; Sanfilippo, Giuseppe ; Agrò, Gianna, e-Print: https://arxiv.org/abs/1109.6440 [cs]
2. Extropy: Complementary Dual of Entropy, Lad, Frank ; Sanfilippo, Giuseppe ; Agrò, Gianna, e-Print: https://arxiv.org/abs/1109.6440 [cs]