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Wolfram`QuantumFramework`

QuantumEntanglementMonotone

QuantumEntanglementMonotone [qs,bipart,t]
	computes the entanglement monotone using the measure or metric t on the quantum state qs between the subsystems of the bipartition list bipart.

QuantumEntanglementMonotone [qs,m,bipart,t]
	uses the QuantumMeasurementOperator m on the second subsystem, for the measurement-dependent measures "Discord" and "MutualInformationJ" .

Details and Options

▪

The following measures or metrics are supported:


	"Concurrence"	the generalized concurrence, exact for two qubits and for pure states of any dimension
	"Negativity"	the negativity (Peres-Horodecki criterion)
	"LogNegativity"	the logarithmic negativity (Peres-Horodecki criterion)
	"EntanglementEntropy"	the von Neumann entropy of a reduced subsystem, for a pure state
	{"RenyiEntropy",α}	the Renyi-α entropy of a reduced subsystem, for a pure state
	"Realignment"	the realignment metric: the summed singular values of the realigned density matrix, minus one
	"ConcurrenceVector"	the generalized concurrence vector, whose norm is the concurrence
	"MutualInformationI"	the quantum mutual information I(A:B)=S( ρ A )+S( ρ B )-S( ρ AB )
	"MutualInformationJ"	the one-sided classical correlation J for a given measurement on the second subsystem
	"Discord"	the quantum discord relative to a given measurement, I-J

▪

The bipartition list bipart has the form

{{

,…},{

,…}}

. For example,

{{1,2},{3}}

splits the state into the subsystem

{1,2}

and the subsystem

{3}

. If no bipartition list bipart is given,

qs["Bipartition"]

is performed first, which splits the system into its first subsystem and the rest (that is,

{{1},{2,3,…}}

▪

The measure or metric t is

"Concurrence"

by default, unless specified otherwise.

▪

For

"Concurrence"

QuantumEntanglementMonotone

uses the Rungta-Buzek-Caves-Hillery-Milburn generalization of the Hill-Wootters concurrence (Phys. Rev. A 64, 042315 (2001)), which applies in any dimension. It is exact for two qubits and for pure states of any dimension, and a lower bound on the convex-roof concurrence for a higher-dimensional mixed state, where (unlike the two-qubit case) it is basis-dependent and not invariant under local unitaries. The value is the norm of the concurrence vector, whose components are

max(0,

∑

i≥2

)

with

the singular values of

ω=



, where



⊗

)

(

⊗

)

is the spin-flipped state built from a pair of antisymmetric

𝔰𝔲(d)

generators. For a bipartition of dimensions

the concurrence vector has

(

-1)

(

-1)

components. For a pure state its norm reduces to

2(1-Tr[

])

, with

the reduced density matrix of one part.

▪

If no α is given for

"RenyiEntropy"

, it is set to

α=1/2

. The alias

"RenyiEntanglementEntropy"

behaves identically.

▪

The

"EntanglementEntropy"

and

{"RenyiEntropy",α}

measures quantify entanglement only for a pure global state. For a genuinely mixed input they return

Indeterminate

and issue a

QuantumEntanglementMonotone::mixedentropy

message, since the reduced von Neumann or Renyi entropy of a mixed state counts classical mixing rather than entanglement. Use

"Negativity"

"LogNegativity"

, or

"Concurrence"

for a mixed state.

▪

"Discord"

returns

I(A:B)-J

for the measurement supplied as the second argument (a

QuantumMeasurementOperator

), or a projective computational-basis measurement on the second subsystem when none is given. It is therefore the discord for that fixed measurement, an upper bound on the Ollivier-Zurek quantum discord, which minimizes over all local measurements (equivalently maximizes

). The value depends on the chosen measurement, and the computational-basis default carries no special physical status. Likewise

"MutualInformationJ"

is the classical correlation for the given measurement, whose maximum over measurements is the Ollivier-Zurek classical correlation. (References: Ollivier & Zurek, Phys. Rev. Lett. 88, 017901 (2001); Henderson & Vedral, J. Phys. A 34, 6899 (2001).)

Examples

(15)

Basic Examples

(5)

Compute the concurrence of a quantum state:

In[1]:=