Measurable Concurrence of Mixed States

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by  Chang-shui Yu

Exactly Measurable Concurrence of Mixed States
Chang-shui Yu, C. Li, and He-shan Song∗
School of Physics and Optoelectronic Technology,
Dalian University of Technology, Dalian 116024, P. R. China
(Dated: February 2, 2008)
We,forthefirsttime,showthatbipartiteconcurrenceforrank2mixedstatesofqubitsiswritten
by an observable which can be exactly and directly measurable in experiment by local projective
measurements,providedthatfourcopiesofthecompositequantumsystemareavailable. Inaddition,
for a tripartite quantum pure state of qubits, the 3-tangle is also shown to be measurable only by
projective measurements on the reduced density matrices of a pair of qubits conditioned on four
copies of thestate.
8
0 PACSnumbers: 03.67.Mn,42.50.-p
0
2
I. INTRODUCTION of mixed states can be obtained by effective experimen-
n
tal estimations. However, both the two estimations in
a
J Quantum entanglement has been realized to be a use- factonlyprovideobservablelowerboundsofconcurrence
with purity dependent tightness for mixedstates instead
0 ful physical resource for quantum information process-
1 ing. The quantification of entanglement has attracted of the exact concurrence. One has to accept the fact
that the exactly measurable entanglement for a general
increasing interests in recent years [1]. However, entan-
] mixed state is still a challenge. Here we only take the
h glement per se is not an observable in strict quantum
first step to focus on concurrence for rank 2 bipartite
p mechanicalsense. Thatis to say,up till now,no directly
mixed states of qubits. The exact value instead of lower
- measurable observable corresponds to entanglement of a
t bound is given by a Hermite operator (observable) and
n given arbitrary quantum state, owing to the unphysical
a quantum operations in usual entanglement measure [2], shown to be directly measurable in experiment based on
u localprojectivemeasurements,providedthat four copies
for example, the complex conjugation of concurrence [3]
q of the tripartite quantum pure state are available. It
and the partial transpose of negativity [4,5]. A general
[ is quite interesting that, for a given tripartite quantum
approach to characterize entanglement in experiment is
1 quantum state tomography, by which one needs to first purestateofqubits,its3-wayentanglementmeasurei.e.,
v 3-tangle can be directly measured only by the local pro-
reconstruct the density matrix by measuring a complete
6 jective measurements on the reduced density matrices of
set of observables [6-8] and then turn to the mathemat-
7 any two qubits without the requirement. of performing
ical problem of evaluating some entanglement measure.
5
any operation on the third qubit.
1 But this approach has been implemented successfully in
The paper is organizedas follows. In Sec. II, we show
. experimentwhichisonlysuitableforsmallquantumsys-
1 that the exact concurrence is directly and locally mea-
tems because the number ofmeasuredobservablesgrows
0 surable in experiment, and that the exact 3-tangle for
rapidlywith the dimensionofthe system. Entanglement
8
a tripartite quantum pure state of qubits is also mea-
0 witnesses are also effective for the detection of entangle-
surable conditioned on fourfold copy, respectively. The
: ment [9] if priori knowledge on the states is available,
v conclusion is drawn in Sec. III.
which depends on the detected states. Quite recently,
i
X some approaches have been reported for the determina-
r tion of entanglement in experiment [2,10-12,14-16]. The II. MEASURABLE CONCURRENCE
a
most remarkableare the new formulationof concurrence
[13] in terms of copies of the state which led to the first
We first show that the concurrence of a rank-2 mixed
direct experimental evaluation of entanglement [12] and
state is exactly measurable. The concurrence of a given
someanalogouscontributions[14]tomultipartiteconcur-
rank 2 mixed state ofa pair of qubits ρ is given[3,17]
rence. Theseareonlyrestrictivetopurestates. However, AB
by
in reality quantum states are never really pure, hence it
is very necessary to consider how to determine the en- C(ρ )= λ λ , (1)
AB 1 2
tanglement of a mixed state in experiment. | − |
where λ are the square roots of the eigenval-
In this paper, we study the exact and measurable en- i
ues of ρ ρ˜ in decreasing order with ρ˜ =
tanglement of mixed states. Recently concurrence in AB AB AB
(σ σ )ρ∗ (σ σ ). After a simple algebra,one has
terms ofcopies ofstates was generalizedto bipartite [15] y ⊗ y AB y ⊗ y
andmultipartite mixedstates[16]suchthatconcurrence C2(ρ )=Tr(ρ ρ˜ ) τ, (2)
AB AB AB
−
where
τ = 2 [Tr(ρ ρ˜ )]2 Tr (ρ ρ˜ )2 (3)
∗Electronicaddress: [email protected] r n AB AB − h AB AB io
2
is half of the 3-tangle [18,19] of the tripartite quantum the different copies are exchanged. It happened that
A
purestateofqubits suchthatρ isthereduceddensity can always be written by
AB
matrix. It is easy find that
1
= (M M N N ),
A B A B
Tr(ρ ρ˜ )= Tr[(ρ ρ ) ], (4) A 2 ⊗ − ⊗
AB AB AB AB
⊗ B
p
where M = M and N = N , M and N denote the
with =4PA1A2 PB1B2, where A B A B i i
B − ⊗ − observables(Hermite operators)of the four copies of the
ith subsystem. In principle, one can always obtain the
1
P−(imin) = √2 |0iim|1iin −|1iim|0iin , (5) projectors of Mi and Ni by their eigenvalue decompo-
(cid:0) (cid:1) sitions and perform corresponding local projective mea-
surements on the fourfold subsystems. We have found
denotingtheprojectorontotheanti-symmetricsubspace
thatbothM andN arerank2,hence canbedirectly
Him ∧ Hin of Him ⊗ Hin where i = A,B corresponds measured byi8 projeictors. That is to saAy, the exact con-
to the subsystems, and m,n=1,2,3,4 marks the differ-
currencecanbe directly measuredby 10 localprojective
ent copies of ρ . Tr(ρ ρ˜ ) has been written in the
AB AB AB
measurements (plus ).
formoftheexpectationvalueofthe self-adjointoperator B
In fact one can reduce the number of projective mea-
whichismeasuredbylocalprojectivemeasurementson
B surements further. After a simple derivation, one can
twocopiesofρ ,henceitismeasurable. Theremaining
AB find that M can always be written by
aretoshowthatTr (ρ ρ˜ )2 isalsodirectlymeasur- i
AB AB
able in experiment.h i √2
M = Pi1i2 Pi3i4 Pi1i3 Pi2i4 Pi1i4 Pi2i3 .
Consider a decomposition of i 2 − ⊗ − − − ⊗ − − − ⊗ −
(cid:0) (10(cid:1))
ρAB =Xi |ψiiABhψi|=Xi pi|ϕiiABhϕi|, (6) TmheuassuMremAe⊗ntMs (B9coobrsreersvpaobnldess)t.oC9ongsrioduerpsthoef ipnrvoajreicatnivcee
of the exchange of different copies, one can find that
with pi = 1 and |ψiiAB = √pi|ϕiiAB, MA⊗MB canbe reducedto 2 groupsofprojectivemea-
Pi surements as
Tr (ρ ρ˜ )2 can be given by
AB AB
1
h i M M = 3PA1A2 PA3A4 PB1B2 PB3B4
Tr (ρ ρ˜ )2 A⊗ B 2(cid:16) − ⊗ − ⊗ − ⊗ −
AB AB 2PA1A2 PA3A4 PB1B3 PB2B4(11.)
= Trh ψ 1 ψ iΣ ψ∗ 2 ψ∗ Σ ψ 3 ψ Σ ψ∗ 4 ψ∗ Σ − − ⊗ − ⊗ − ⊗ − (cid:17)
| ii h i| j j | ki h k| | li h l|
Xijkl (cid:12) (cid:11) (cid:10) (cid:12) Tr(ρABρ˜AB) is also included in the two measurements.
(cid:12) (cid:12)
N can be explicitly by
= ψ 1 ψ 2P ψ 2 ψ 3 ψ 3 ψ 4P ψ 4 ψ 1 i
i j j k k l l i
h | h | | i | i h | h | | i | i
Xijkl N =√3 Ψ Ψ Ψ¯ Ψ¯ , (12)
i
| ih |−
= ψ 1 ψ 2 ψ 3 ψ 4(P P) ψ 2 ψ 3 ψ 4 ψ 1 (cid:0) (cid:12) (cid:11)(cid:10) (cid:12)(cid:1)
h i| h j| h k| h l| ⊗ | ji | ki | li | ii where (cid:12) (cid:12)
Xijkl
= Tr ⊗4i=1ρiAB(P⊗P)SWAP =Tr ⊗4i=1ρiABA , (7) Ψ = 1 0011 +e2π3i 0101 +e−2π3i 0110
(cid:2) (cid:3) (cid:2) (cid:3) | i √6(cid:16)| i | i | i
with
+e−2π3i 1001 +e2π3i 1010 + 1100 (13)
(P P)SWAP +SWAP†(P P) | i | i | i(cid:17)
= ⊗ ⊗ , (8)
A 2 is the four-qubit Dicke state [21] with two excitations if
neglecting the relative phases and Ψ¯ is the conjugate
Ψ . However, so far we can not fi(cid:12)nd(cid:11)a proper decom-
P=P P (9) | i (cid:12)
−⊗ − positionin terms of two-qubitprojectorsfor Ni (Strictly
speaking, we can not find such decompositions that re-
where P without superscripts and subscripts means a
− duce the number of observables). That is to say, the
general projector onto the anti-symmetric subspace of
projectors of N have to be performed on a four-qubit
two qubits, Σ=σ σ , SWAP is defined as i
y ⊗ y space as a whole. In this sense, only 6 groups of local
projective measurements are enough for concurrence.
ψ 2 ψ 3 ψ 4 ψ 1 =SWAP ψ 1 ψ 2 ψ 3 ψ 4,
| ji | ki | li | ii | ii | ji | ki | li More interestingly, from eq. (3) one can find that τ
(3-tangle) for a tripartite pure state of qubits can be di-
andall ψ denotebipartitepurestatewithsubscirptsAB
| i rectly measured only by local projective measurements
omitted. The superscripts of ψ in eq. (7) mark the dif-
| i on the subsystems of two qubits. Consider a tripartite
ferentcopiesofρ . Thelast”=”ineq. (7)followsfrom
AB quantum pure state of qubits Ψ with the reduced
the fact that Tr (ρABρ˜AB)2 should not be changed if density ρAB, then ρAB is obvio|usilAyBrCank-two. Based on
h i
3
ref. [18,19], 3-tangle of Ψ can be given by eq. (3). ing efforts.
| iABC
The above procedure implies eq. (3) is measurable, i.e.
3-tangle is directly measurable if four copies of ρ are
AB
available. Different from our previous work [20] that re-
quires projective measurements on all subsystems, only III. CONCLUSION AND DISCUSSION
projective measurements on two subsystems (reduced
density matrix of two qubits) are enough, even though
Wehaveshownthattheexactbipartiteconcurrencefor
so far projection measurements on four-qubit space are
rank2 mixedstates ofqubits insteadofa lowerboundis
required.
directly and locally measurable in experiment, provided
We have expressed concurrence of rank-two mixed
that four copies of the states of interests are available.
states by Hermite operators which can be decomposed
Inparticular,itisveryinterestingthat3-tangleforatri-
into local projectors. This shows in principle that the
partite quantum pure state of qubits has been shown to
exact concurrence can be directly and locally measured
bemeasurableexperimentallyifonlythefourfold copyof
in experiment. However, as mentioned above, it re-
bipartitereduceddensitymatrixisavailable,whichisdif-
quires projective measurements on the whole space of
ferentfromourpreviouswork[20]. Furthermore,because
four qubits, which means the interactions of four qubits.
a state has to be prepared repeatedly in order to obtain
Take a photon experiment as an example, it might need
reliable measurement statistics in any experiment [2], a
the interference of four photons. In the current experi-
fourfold copy of a state should be feasible in principle,
ment,togetherwithfourcopyofastate,itmightbevery
which implies the observation of mixed-state entangle-
difficulttorealize,hencepresentschememightintroduce
ment may be feasible. However, practical experiments
no advantage in current experiment. But we can safely
may lead to four different ”copies” of a state and influ-
saythatonlytwoqubitinteractioncannotbeenoughfor
ence the fidelity, which requires that one has to perform
the really applicable quantum computer. In this sense,
necessary error analysis based on different experimental
the current work should be a valuable contribution for
realization [2].
the further research. Inaddition, itisstillanopenprob-
IV. ACKNOWLEDGEMENT.
lem whether the observables given here are derived from
an optimal decomposition such that the number of ob-
servablesareminimalandwhetherthereexistsomeother This workwassupportedbythe NationalNaturalSci-
decompositions of N or which lead to a direct contri- ence Foundation of China, under Grant Nos. 10747112,
i
A
butiontothecurrentexperiments. This isourforthcom- 10575017and 60703100.
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