Bounds on Negativity of Superpositions
Bounds on Negativity of Superpositions
Yong-Cheng Ou and Heng Fan
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
The entanglement quantified by negativity of pure bipartite superposed states is studied. We
show that if the entanglement is quantified by the concurrence two pure states of high fidelity to one
another still have nearly the same entanglement. Furthermore this conclusion can be guaranteed by
our obtained inequality, and the concurrence is shown to be a continuous function even in infinite
dimensions. The bounds on the negativity of superposed states in terms of those of the states being
superposed are obtained. These bounds can find useful applications in estimating the amount of
the entanglement of a given pure state.
PACS numbers: 03.67.Mn, 03.65.Ta, 03.65.Ud
Quantum entanglement plays an important role both
in many aspects of quantum information theory[1] and in
describing quantum phase transition in quantum many-
body systems[2, 3]. As such characterization quantifi-
cation of quantum entanglement is a fundamental issue.
Consequently the legitimate measures of entanglement
are desirable as a first step. The existing well-known bi-
partite measure of entanglement with an elegant formula
is the concurrence derived analytically by Wootters[4]
and the entanglement of formation[5, 6] is a monoton-
ically increasing function of the concurrence. In general
for a multipartite or higher-dimensional system it is a
formidable task of quantifying its entanglement since it
needs complicate convex-roof extension. In the last 10
years some important properties of quantum entangle-
ment were found, one of which is the monogamy prop-
erty described by Coffman-Kundu-Wootters inequality in
terms of concurrence [7]. In our previous work we have
shown that the monogamy inequality can not general-
ize to higher-dimensional systems[8] and established a
monogamy inequality in terms of negativity giving a dif-
ferent residual entanglement[9].
On the other hand, quantum entanglement is a direct
consequence of the superposition principle. It is an in-
teresting physical phenomenon that the superposition of
two separable states may give birth to an entangled state,
on the contrary, the superposition of two entangled states
may give birth to a separable state. The relation between
the entanglement of the state and the entanglement of the
individual terms that by superposition yield the state has
been studied, where the entanglement is quantified by the
von Neumann entropy[10] and the concurrence[12]. Re-
cently it was generalized to the superposition of more
than two components[13]. If the entanglement is quan-
tified by negativity, it would be interesting to establish
the analogous relation and obtain the bound of entangle-
ment for the superposition state. In this paper, we first
show that, by contrast to the von Neumann entropy, the
concurrence is a continuous function even in infinite di-
mensions. We deduce an inequality to guarantee this
property. Next we give the bounds of the negativity of
the superposition state. The discussion and conclusion
are presented in the end.
The authors in[10] have shown that two states of high
fidelity to one another may not have the same entan-
glement, i.e., |〈ψ|φ〉|2 → 1 may not generally result in
E(ψ) → E(φ), where E is the von Neumann entropy.
For a bipartite pure state |Φ〉AB the von Neumann en-
tropy is defined as
E(ΦAB) ≡ S(TrB|Φ〉AB〈Φ|) = S(TrA|Φ〉AB〈Φ|), (1)
where S(ρ) = −Tr(ρ log ρ), and the concurrence is de-
fined as
C(ΦAB) ≡
2 (1− Trρ2
, (2)
where ρA = TrB|Φ〉AB〈Φ| with the eigenvalues µi. How-
ever, if we employ the concurrence to quantify the entan-
glement, |〈ψ|φ〉|2 → 1 must result in C(ψ) → C(φ). Let
us see their example letting
|φ〉AB = |00〉, (3)
|ψ〉AB =
1− ǫ|φ〉AB +
[|11〉+ |22〉+ · · ·+ |dd〉]. (4)
It is obviously true that E(φAB) = C(φAB) = 0, while
according to [10] the von Neumann entropy of the state
|ψ〉AB is
E(ψAB) ≈ ǫ log2 d→ ∞, (5)
specially when d is as large as we expect. It follows from
Eq.(2) that the concurrence of the state |ψ〉AB give us
the result
C2(ψAB) = 2
2ǫ− ǫ2 − ǫ
→ 0, (6)
when ǫ is adequately small. By contrast to E(ψAB) in
Eq.(5), C2(ψAB) in Eq.(6) is independent of d. Note that
when ǫ is small the two states have high fidelity |〈ψ|φ〉|2 =
1 − ǫ → 1. Comparing Eq.(5) to Eq.(6), we can draw a
http://arxiv.org/abs/0704.0757v1
conclusion that if the entanglement is quantified by the
concurrence two states of high fidelity to one another still
have nearly the same entanglement.
It is indeed that the difference of the von Neumann en-
tropy between two pure states of fixed dimension can be
bounded using Fannes’ inequality[11], while the von Neu-
mann entropy is not a continuous function and no such
bound applies in infinite dimensions. However, as we will
show here, a similar bound still works if the entanglement
is quantified by the concurrence and the concurrence is
a continuous function even in infinite dimensions. In or-
der to explain our above viewpoint we present the fol-
lowing Theorem which is similar to the original Fannes’
inequality except that the entanglement is quantified by
the concurrence.
Theorem 1. Suppose ρAB and σAB are density matri-
ces of two bipartite pure states in arbitrary dimensions.
For the trace distance T (ρA, σA) ≡ Tr|ρA − σB| between
ρA = TrBρAB and σA = TrBσAB we have
|C2(ρAB)− C2(σAB)| ≤ 4T (ρA, σA). (7)
Proof. Let r1 ≥ r2 ≥ · · · ≥ rd be the eigenvalues of
ρA, in decreasing order, and s1 ≥ s2 ≥ · · · ≥ sd be the
eigenvalues of σA, also in decreasing order. According
to[1], it follows that
|ri − si| ≤ T (ρA, σA). (8)
From the observation of the definition of the concurrence
in Eq.(2), we can rewrite the left-hand-side of Eq.(7) as
∣∣C2(ρAB)− C2(σAB)
∣∣ = 2
∣∣∣∣∣
(r2i − s2i )
∣∣∣∣∣
∣∣r2i − s2i
|ri + si||ri − si|
|ri − si|. (9)
The second formula is obtained from the observation that
|a+ b+ · · ·+ k| ≤ |a| + |b| + · · · + |k| for any complex
quantities a, b, · · ·, k. In the derivation of the last formula
we have taken into account the fact that ri + si ≤ 2
since each eigenvalue of ri and si is not greater than one.
Combining Eqs.(8) and (9) can give Eq.(7). Thus the
proof is completed.
From the Theorem 1 it can be seen that the difference
of the concurrences of two pure states is a function of
fidelity and can be bounded by Eq.(7). What’s more, by
contrast to the von Neumann entropy[10] the concurrence
is a continuous function and such a bound still works in
infinite dimensions. Note that whether a similar bound
in Eq.(7) holds for the negativity is still open. In the next
paragraphs we are devoted to deducing the bounds on the
negativity of any bipartite pure state as a superposition
of two terms |Γ〉AB = α|Ψ〉+ β|Φ〉.
Before embarking on this study, we first recall some
basic definitions of the negativity. As for detecting en-
tangled state in higher-dimensional Hilbert space, Peres-
Horodecki criterion based on partial transpose[15, 16] is
a convenient method. Given a density matrix ρ in a bi-
partite pure system of A and B, the partial transpose
with respect to A subsystem is described by (ρTA)ij,kl =
(ρ)kj,il and the negativity is defined as
N = 1
(‖ρTA‖ − 1). (10)
The trace norm ‖R‖ is given by ‖R‖ = Tr
RR†. Note
that N > 0 is the necessary and sufficient condition for
entangled bipartite pure states.
There are two key ingredients to obtain the bounds
of the negativity for bipartite superposition pure states.
One is that the negativity can be expressed by means of
Schmidt coefficients of a pure state. Suppose that a pure
m⊗ n(m ≤ n) quantum state has the standard Schmidt
form |ψ〉AB =
µi|aibi〉, where
µi(i = 1, · · · ,m)
are the Schmidt coefficients, ai and bi are the orthogonal
basis in HA and HB, respectively. For the pure bipartite
state we can derive ‖ρTA‖ =
[18], and there-
fore Eq.(10) can be reexpressed as
N = 1
 . (11)
In order for the later use we can transform Eq.(11) into
= 2N + 1. (12)
The other is the Theorem[17], which states that for any
two Hermitian matrix H and K defined in Cn×n,
µi(H) + µ1(K) ≤ µi(H +K) ≤ µi(H) + µn(K), (13)
holds, where µi(·) are the eigenvalues in increasing order.
If µ1(K) ≥ 0, from Eq.(13) it is easy to check that
µi(H) ≤
µi(H +K) ≤
µi(H) +
µn(K), (14)
holds also. Then Eq.(14) will be used repeatedly in what
follows.
For the negativity of the arbitrary superposition state
let us first see the simplest case in which two bi-
partite states we are superposing, Φ1 and Ψ1, are
biorthogonal[10], i.e., Φ1Ψ
= Ψ1Φ
= 0[12]. Since the
matrix representation of a reduced density matrix will
be used, we explain the corresponding notations in the
following. For the pure state |Φ〉AB defined in m ⊗ n
dimensions, generally it can be considered as a vector:
|Φ〉AB = [a00, a01, · · · , a0m, a10, a11, · · · , amn]T with the
superscript T denoting transpose operation. With the
matrix notation, the reduced density matrix reads
ρA = ΦΦ
†, (15)
whose eigenvalues are µi appearing in Eq.(11).
Theorem 2. Suppose that two biorthogonal pure states
Φ1 and Ψ1, which are defined in m ⊗ n(n ≤ m) di-
mensions. The negativity of their superposed states
Γ1 = αΦ1 + βΨ1 with |α2|+ |β|2 = 1 satisfies
2|α|2N (Φ1) + 2|β|2N (Ψ1)− 1
≤ N (αΦ1 + βΨ1) ≤
2|α|2Ñ (Φ1) + 2|β|2Ñ (Ψ1)− 1
, (16)
where
Ñ (Φ1) = N (Φ1) +
µn(Ψ1)[2N (Φ1) + 1]
|α| +
n2|β|2µn(Ψ1)
2|α|2 ,
Ñ (Ψ1) = N (Ψ1) +
µn(Φ1)[2N (Ψ1) + 1]
|β| +
n2|α|2µn(Φ1)
2|β|2 .
Proof. From Eq.(15) the reduced density matrix of the state Γ1 can read
= |α|2Φ1Φ†1 + |β|2Ψ1Ψ
+ αβ∗Φ1Ψ
+ α∗βΨ1Φ
. (17)
The biorthogonal condition with Φ1Ψ
= 0 and Ψ1Φ
0 makes Eq.(17) reduce to
= |α|2Φ1Φ†1 + |β|2Ψ1Ψ
. (18)
Substituting Eq.(18) into the left inequality of Eq.(13)
we have
|α|2µi(Φ1Φ†1) + |β|2µ1(Ψ1Ψ
) ≤ µi(Γ1Γ†1). (19)
Since Ψ1Ψ
is positive semidefinite, µ1(Ψ1Ψ
) ≥ 0. Thus
Eq.(19) becomes
|α|2µi(Φ1Φ†1) ≤ µi(Γ1Γ
). (20)
Taking the square root of both sides in Eq.(20) and the
sum of
µi(·) over all index i, we have
µi(Φ1Φ
µi(Γ1Γ
). (21)
In a similar way, substituting Eq.(18) into the right in-
equality of Eq.(14) and taking the sum of
µi(·) over all
index i, we have
µi(Γ1Γ
) ≤ |α|
µi(Φ1Φ
) + n|β|
µn(Ψ1Ψ
Substituting Eqs.(21) and (22) into Eq.(12), respectively,
we can obtain
|α|2N (Φ1) +
|α|2 − 1
≤ N (αΦ1 + βΨ1)
≤ |α|2Ñ (Φ1) +
|α|2 − 1
.(23)
If we replace the matrix |α|2Φ1Φ†1 with |β|2Ψ1Ψ
Eqs.(20) and (21), i.e., equivalently exchange the ma-
trixes H and K in Eq.(14), finally we can also obtain
|β|2N (Ψ1) +
|β|2 − 1
≤ N (αΦ1 + βΨ1)
≤ |β|2Ñ (Ψ1) +
|β|2 − 1
.(24)
Then combining Eqs.(23) and (24) gives Eq.(16). Thus
the proof is completed.
Note that the lower bound in Eq.(16) can provide a
nonzero value only when 2|α|2N (Φ1) + 2|β|2N (Ψ1) > 1.
Next we show an example to illustrate the validity of our
bound. Consider the state
|φ〉AB = α|ϕ〉AB + β|ψ〉AB , (25)
|ϕ〉AB =
|00〉+ 1√
|11〉, (26)
|ψ〉AB =
|22〉+ 1√
|33〉, (27)
where α = β = 1/
2. It is easy to check that |ϕ〉AB and
|ψ〉AB are biorthogonal, N (|φ〉AB) = 3/2, N (|ϕ〉AB) =
N (|ψ〉AB) = 1/2, and µ4(|ϕ〉AB) = µ4(|ψ〉AB) = 1/2.
Accordingly from Eq.(16) we obtain the lower and upper
bounds
0 < N (|φ〉AB) =
< 4, (28)
which work well.
Finally we directly present the main Theorem of this
paper, in which the two states being superposed can be
biorthoganal, orthogonal, or nonorthogonal.
Theorem 3. Suppose that two arbitrary normalized
pure states Φ2 with rank r1 and Ψ2 with rank r2, which
are defined in any dimensions. The negativity of their
superposed states Γ2 = αΦ2 + βΨ2 with rank r3 and
|α2|+ |β|2 = 1 satisfies
2‖α|Φ2〉+ β|Ψ2〉‖2N (αΦ2 + βΨ2) ≤ 2|α|2Ñ (Φ2) + 2|β|2Ñ (Ψ2)− ‖α|Φ2〉+ β|Ψ2〉‖2 + 1, (29)
where
Ñ (Φ2) = N (Φ2) +
µn(Ψ2)[2N (Φ2) + 1]
|α| +
r2|β|2µn(Ψ2)
2|α|2 ,
Ñ (Ψ2) = N (Ψ2) +
µn(Φ2)[2N (Ψ2) + 1]
|β| +
r2|α|2µn(Φ2)
2|β|2 ,
where r = max{r1, r2, r3}.
Proof. Consider the matrix
M = |α|2Φ2Φ†2 + |β|2Ψ2Ψ
, (30)
which can be rewritten as
‖Γ2‖2
Γ̂2(Γ̂2)
)†, (31)
where Γ−
= αΦ2 − βΨ2, Γ̂2 = Γ2/‖Γ2‖, and Γ̂−2 =
‖. Thus Eqs.(13) shows that
|α|2µi(Φ2Φ†2) + |β|2µ1(Ψ2Ψ
≤ µi(M) ≤ |α|2µi(Φ2Φ†2) + |β|2µn(Ψ2Ψ
), (32)
‖Γ2‖2
Γ̂2Γ̂
≤ µi(M) ≤
‖Γ2‖2
Γ̂2Γ̂
.(33)
Since µ1(Ψ2Ψ
) ≥ 0 and µ1
≥ 0, observing
the left inequality of Eq.(33) and the right inequality in
Eq.(32) we have
‖Γ2‖√
Γ̂2Γ̂
≤ |α|
µi(Φ2Φ
) + |β|
µn(Ψ2Ψ
Substituting Eqs.(34) into Eq.(12) we have
‖α|Φ2〉+ β|Ψ2〉‖2N (αΦ2 + βΨ2)
≤ 2|α|2Ñ (Φ2)−
‖α|Φ2〉+ β|Ψ2〉‖2
+ |α|2. (35)
Likewise, if we replace the two matrixes |α|2Φ2Φ†2 with
|β|2Ψ2Ψ†2 in Eq.(32), we can obtain
‖α|Φ2〉+ β|Ψ2〉‖2N (αΦ2 + βΨ2)
≤ 2|β|2Ñ (Ψ2)−
‖α|Φ2〉+ β|Ψ2〉‖2
+ |β|2. (36)
Combining Eqs.(35) and (36) gives Eq.(29). Thus the
proof is completed.
Since there exists a extra term of the maximal eigen-
value in the second inequality in Eq.(33), generally it
is difficult to achieve the universal formula for the lower
bound of the negativity in this case. But it is our interest
in the future work.
In conclusion, we have shown that if the entanglement
is quantified by the concurrence two pure states of high
fidelity to one another still have nearly the same en-
tanglement and obtained an inequality that can guaran-
tee that the concurrence is a continuous function even
in infinite dimensions. However, whether the similar
property can apply to the negativity is still open. The
bounds on the negativity of superposed states in terms
of those of the states being superposed were obtained.
So far some bounds of the wildly-studied measures of
entanglement like the von Neumann entropy[10], the
concurrence[12] and the negativity in this paper for the
superposition states have been provided. In view of that
the concurrence can be directly accessible in laboratory
experiment[19], these bounds can find useful applications
in estimating the amount of the entanglement of a given
pure state.
The author Y.C.O. was supported from China Post-
doctoral Science Foundation and the author H.F. was
supported by ’Bairen’ program NSFC grant and ’973’
program (2006CB921107).
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http://arxiv.org/abs/quant-ph/0702127
http://arxiv.org/abs/quant-ph/0701188
