Entanglement increase from local interactions with
  not-completely-positive maps
Entanglement increase from local interactions with not-completely-positive maps
Thomas F. Jordan∗
Physics Department, University of Minnesota, Duluth, Minnesota 55812
Anil Shaji†
The University of New Mexico, Department of Physics and Astronomy, 800 Yale Blvd. NE, Albuquerque, New Mexico 87131
E. C. G. Sudarshan‡
The University of Texas at Austin, Center for Statistical Mechanics, 1 University Station C1609, Austin Texas 78712
Simple examples are constructed that show the entanglement of two qubits being both increased and decreased
by interactions on just one of them. One of the two qubits interacts with a third qubit, a control, that is never
entangled or correlated with either of the two entangled qubits and is never entangled, but becomes correlated,
with the system of those two qubits. The two entangled qubits do not interact, but their state can change from
maximally entangled to separable or from separable to maximally entangled. Similar changes for the two qubits
are made with a swap operation between one of the qubits and a control; then there are compensating changes of
entanglement that involve the control. When the entanglement increases, the map that describes the change of
the state of the two entangled qubits is not completely positive. Combination of two independent interactions that
individually give exponential decay of the entanglement can cause the entanglement to not decay exponentially
but, instead, go to zero at a finite time.
Keywords: Entanglement, Quantum information
I. INTRODUCTION
We construct simple examples here that show the entan-
glement of two qubits being both increased and decreased by
interactions on just one of them. In our first and basic step,
taken in Sec. II, we have one of the two qubits interact with a
third qubit, a control, that is never entangled or correlated with
either of the two entangled qubits and is never entangled, but
becomes correlated, with the system of those two qubits. In
Sec. III, we do this for each of the two entangled qubits, and
consider the combination of the two interactions, with sepa-
rate control qubits that are not correlated and do not interact
with each other. The two entangled qubits do not interact, but
their state can change from maximally entangled to separable
or from separable to maximally entangled. Similar changes
for the two qubits are made with a swap operation between
one of the qubits and a control; then there are compensating
changes of entanglement that involve the control. This is de-
scribed in Sec. II.A.
Whenever the entanglement increases, and in some cases
where the entanglement decreases, the map that describes the
change of the state of the two entangled qubits is not com-
pletely positive and does not apply to all states of two qubits.
It all depends on whether there are correlations with the con-
trols at the beginning of the interval for which the dynamics is
considered. The maps are described in Sec. IV and discussed
in Sec. V. The completely positive maps that decrease the en-
tanglement have already been described [1].
When the interaction of each qubit with its control by itself
∗email: tjordan@d.umn.edu
†email: shaji@unm.edu
‡email: sudarshan@physics.utexas.edu
gives exponential decay of the entanglement, the combination
of the two interactions gives exponential decay at the rate that
is the sum of the rates for the individual interactions, when the
two interactions are made the same way. Making them differ-
ently can cause the entanglement to not decay at that rate or
at any single rate. Instead, the entanglement goes to zero at a
finite time; the state becomes separable and remains separable
at later times. This is described in Sec. III.A. Similar behavior
has been observed in more physically interesting and mathe-
matically complicated models [2, 3, 4].
These examples are built on the same framework, but to a
very different design, from those we made for Lorentz trans-
formations that entangle spins [5]. There the momenta that
played the roles of controls were purposely correlated. Here
the controls are kept independent. The framework makes the
operations transparent by describing the qubit states with den-
sity matrices written in terms of Pauli matrices, so you can
see the Pauli matrices being rotated by the interactions. States
are shown to be separable by writing out the density matrices
explicitly as sums of products for pure states. For each in-
teraction here, the map that makes the change of the density
matrix for the entangled qubits is described by a simple rule
that particular Pauli matrices in the density matrix are multi-
plied by a number; equivalently, the map of the state of the
entangled qubits is described by a rule that particular mean
values are multiplied by a number.
Our examples show that statements like “entanglement
should not increase under local operations and classical com-
munication” [6, 7] are not generally true outside the set of
local operations considered in the original proof [6]. In our
examples, each control qubit interacts with only one of the
two entangled qubits. In this sense, the quantum operations
are local. Correlation with a control at the beginning of the
interval for which the dynamics is considered can give local
operations that increase entanglement.
http://arxiv.org/abs/0704.0461v2
mailto:tjordan@d.umn.edu
mailto:shaji@unm.edu
mailto:sudarshan@physics.utexas.edu
II. ONE INTERACTION
We consider the entanglement of two qubits, A and B. We
use Pauli matrices Σ1,Σ2,Σ3 for qubit A, and Pauli matrices
Ξ1,Ξ2,Ξ3 for qubit B. We let qubit A interact with a third
qubit, which we call C. We think of C as a control. By inter-
acting with qubit A, it will control the entanglement of qubits
A and B. We work with states represented by orthonormal
vectors |α〉 and |β〉 for C. We consider a state of the three
qubits represented by a density matrix
Π = ρ⊗ 1
11C (2.1)
with ρ the density matrix for the state of qubits A and B.
We follow common physics practice and write a product of
operators for separate systems, for example a product of Pauli
matrices Σ andΞ for qubitsA andB, simply as ΣΞ, not Σ⊗Ξ.
Occasionally we insert a ⊗ for emphasis or clarity. We write
11A, 11B , 11C , but we do not put labels A and B on the Σj and
Ξk. The single statement that the Σj are for qubit A and the
Ξk are for qubit B eliminates the need for continual use of
both A and B lalels and ⊗ signs.
Suppose ρ is one of the density matrices
(11 ± Σ1Ξ1 ± Σ2Ξ2 − Σ3Ξ3). (2.2)
Both ρ+ and ρ− represent maximally entangled pure states
for the two qubits. They are Bell states. The state of zero total
spin is represented by ρ− and the state obtained from that by
rotating one of the spins by π around the z axis is represented
by ρ+.
For a rotation W , let DA(W ) be the 2 × 2 unitary rotation
matrix made from the Σj so that
DA(W )
ΣDA(W ) = W (Σ) (2.3)
where W (Σ) is simply the vector Σ rotated by W . Let W (φ)
be the rotation by φ around the z axis, and let DA(φ) be
DA(W (φ)).
We consider an interaction between qubits A and C de-
scribed by the unitary transformation
U = DA(φ)|α〉〈α| +DA(−φ)|β〉〈β| (2.4)
or, in Hamiltonian form,
U = e−iφH (2.5)
H = Σ3
(|α〉〈α| − |β〉〈β|). (2.6)
This changes the density matrix ρ for qubits A and B to
ρ′ = TrC
(U ⊗ 11B)Π(U ⊗ 11B)†
DA(φ)ρDA(φ)
DA(−φ)ρDA(−φ)†. (2.7)
For ρ± this gives
ρ′± =
[11 ± (Σ1 cosφ+Σ2 sinφ)Ξ1
± (−Σ1 sinφ+Σ2 cosφ)Ξ2 − Σ3Ξ3]
[11 ± (Σ1 cosφ− Σ2 sinφ)Ξ1
± (Σ1 sinφ+Σ2 cosφ)Ξ2 − Σ3Ξ3]
[11 ± (Σ1Ξ1 +Σ2Ξ2) cosφ− Σ3Ξ3]
= ρ± cos
2(φ/2) + ρ∓ sin
2(φ/2). (2.8)
A. From maximally entangled to separable and back
We focus first on the case where φ is π/2. Then both ρ+
and ρ− are changed to
[11 − Σ3Ξ3]
(11 − Σ3)
(11 + Ξ3)
(11 +Σ3)
(11 − Ξ3). (2.9)
The density matrix ρ for a maximally entangled state is
changed to the density matrix ρ′ for a separable state that is
a mixture of just two products of pure states. The inverse of
the unitary dynamics of qubits A and R takes ρ′ back to ρ; it
changes a separable state to a maximally entangled state.
The dynamics continuing forward also changes this separa-
ble state to a maximally entangled state. As φ goes from π/2
to π, the density matrix ρ′± changes from that of Eq. (2.9) to
ρ′± = ρ∓. (2.10)
There can be revivals of entanglement between two qubits
when there is no interaction between them, as well as when
[8] there is.
Changes in the state of qubits A and B from maximally
entangled to separable and back to maximally entangled can
also be made very simply with a swap of states[9] between A
and C. This can be done with a unitary operator U ⊗ 11B with
U a unitary operator for qubits A and C that acts on a basis
of product state vectors simply by interchanging the states of
A and C. There is interaction between qubits A and C only;
qubit B is not involved.
Applied to an initial state described by Eqs. (2.1) and (2.2),
where qubits A and B are maximally entangled, this swap op-
eration gives a separable state for A and B. Applied a second
time, it restores the initial state where A and B are maximally
entangled. For qubits A and B, this is similar to what hap-
pens when φ goes from 0 to π/2 to π. For the three qubits,
it is different. The swap operation does not change the com-
plete inventory of entanglements for the three qubits. It just
moves the entanglements around. In particular, C becomes
maximally entangled with B. We will see, in Secs. II.C and
D, that the interaction described by Eqs. (2.4), (2.5) and (2.6)
does change the complete inventory of entanglements for the
three qubits. When the state of qubits A and B changes from
maximally entangled to separable and back to maximally en-
tangled, there are no compensating changes of other two-part
entanglements. In particular, qubit C never becomes entan-
gled with anything.
B. Concurrence
The change of entanglement is smaller when φ does not
change by π/2. ¿From Eq. (2.8) we have
ρ′± =
[11±(Σ1Ξ1+Σ2Ξ2) cosφ+(Σ1Ξ1)(Σ2Ξ2)], (2.11)
after rewriting the last term. This shows that for both ρ′
ρ′− the eigenvalues are
(1 + cosφ),
(1 − cosφ), 0, 0 (2.12)
because Σ1Ξ1 and Σ2Ξ2 each have eigenvalues 1 and −1 and
together they make a complete set of commuting operators:
their four different pairs of eigenvalues label a basis of eigen-
vectors for the space of states for the two qubits. The Wooters
concurrence [10] is a measure of the entanglement in a state
of two qubits. It is defined by
C(ρ) ≡ max
(2.13)
where ρ is the density matrix that represents the state and
λ1, λ2, λ3, λ4 are the eigenvalues, in decreasing order, of
ρ Σ2Ξ2 ρ
⋆ Σ2Ξ2, with ρ
⋆ the complex conjugate that is ob-
tained by changing Σ2 and Ξ2 to −Σ2 and −Ξ2. From
Eq. (2.11) we have
ρ′± Σ2Ξ2 (ρ
⋆ Σ2Ξ2 = ρ
⋆ (Σ2Ξ2)
2=(ρ′±)
2 (2.14)
so for ρ′± the
λi are the eigenvalues of ρ
± and the concur-
rence is
C(ρ′±) = | cosφ|. (2.15)
We can consider the change of entanglement as φ changes
through any interval. When | cosφ| decreases, the entangle-
ment decreases. When | cosφ| increases, the entanglement
increases.
C. Two-part entanglements
The only two-part entanglements are when qubit A is in
one part and qubit B is in the other. There is entanglement
between qubitA and the subsystem of two qubits B and C and
between qubit B and the subsystem of two qubits A and C, as
well as between qubits A and B. There is never entanglement
between the state of qubit C and the state of the subsystem of
two qubits A and B. The density matrix
(U ⊗ 11B)Π(U ⊗ 11B)†=
DA(φ)ρ±DA(φ)
†|α〉〈α|
DA(−φ)ρ±DA(−φ)†|β〉〈β|
(2.16)
is always a mixture of two products of pure states. The re-
duced density matrix for the subsystem of qubits A and C,
obtained by taking the trace over the states of qubit B, is just
11A ⊗ 11C/4, and the reduced density matrix for qubits B and
C, obtained by taking the trace over the states of qubit A,
is 11B ⊗ 11C/4. There is never entanglement or correlation
between qubits A and C or between qubits B and C. The
reduced density matrices for the individual single qubits are
just 11A/2, 11B/2, and 11C/2. The only subsystem density ma-
trix that carries any information is the density matrix ρ for
the qubits A and B, which is changed by the interaction with
qubit C. The entropy of the subsystem of qubits A and B can
increase or decrease, but there is no change of entropy for any
other subsystem or for the entire system of three qubits.
D. Three-part entanglement
There is three-part entanglement. The state represented by
the density matrix (2.16) is called biseparable because it is
separable as the state of a system of two parts, with C one
part and the subsystem of two qubits A and B the other part.
It is not separable as the state of a system of three parts A, B,
and C. The density matrix (2.16) is not a mixture of products
of density matrices for pure states of the individual qubits A,
B, and C. If it were, its partial trace over the states of C, the
reduced density matrix that represents the state of the subsys-
tem of the two qubitsA andB, would be a mixture of products
for pure states of A and B. That happens only when cosφ is 0.
In that case, we can see that the density matrix (2.16) is not a
mixture of products for pure states of the individual qubits A,
B, and C because its partial transpose obtained by changing
Ξ2 to −Ξ2 is not a positive matrix.
In the classification of three-part entanglement for qubits,
biseparable states are between separable states and states that
involve W or GHZ entanglement [11, 12, 13]. Let Π1,
Π2, Π3 be Pauli matrices for the qubit C such that |α〉〈α|
is (1/2)(11 +Π3) and |β〉〈β| is (1/2)(11 −Π3). Bounds from
Mermin witness operators say that for separable or bisepara-
ble states
−2 ≤ 〈ΣjΞjΠj−ΣjΞkΠk−ΣkΞjΠk−ΣkΞkΠj〉 ≤ 2 (2.17)
for j, k = 1, 2, 3 and j 6= k; a mean value outside these
bounds is a mark of W or GHZ entanglement [14]. In our
examples, these mean values are always 0. A mean value
〈|GHZ〉〈GHZ|〉 larger than 3/4 for the projection operator
onto the GHZ state,
|GHZ〉 = 1√
|0〉|0〉|0〉+ 1√
|1〉|1〉|1〉, (2.18)
is a mark of GHZ entanglement; it can not be larger than 3/4
for a W state [13]. A mean value 〈|GHZ〉〈GHZ|〉 larger than
1/2 is a mark of a W state; it can not be larger than 1/2 for
a biseparable state [13]. In our examples, 〈|GHZ〉〈GHZ|〉
is always 0. A mean value 〈|W 〉〈W |〉 larger than 2/3 for the
projection operator onto the W state,
|W 〉 = 1√
|1〉|0〉|0〉+ 1√
|0〉|1〉|0〉+ 1√
|0〉|0〉|1〉, (2.19)
is a mark of W entanglement; it can not be larger than 2/3 for
a biseparable state [13]. In our examples,
〈|W 〉〈W |〉 = 1
(1 ± cosφ). (2.20)
This mean value does not involve either entanglement or cor-
relation of the qubitC; it would be the same if both |α〉〈α| and
|β〉〈β| in the density matrix (2.16) were replaced by (1/2)C ,
the completely mixed density matrix for C.
For any φ, the density matrices (2.16) for the two cases +
and − are changed into each other by the local unitary trans-
formation that changes the Pauli matrices for one of the qubits
A or B by rotating its spin by π around the z axis. As a func-
tion of φ, the mean value 〈|W 〉〈W |〉 changes in opposite di-
rections for the + and − cases. So will any mean value for the
states described by the density matrices (2.16), if it changes at
For the states described by the density matrices (2.16), the
only nonzero mean values that involve the qubit C are
〈Σ1Ξ2Π3〉 = ∓ sinφ
〈Σ2Ξ1Π3〉 = ± sinφ. (2.21)
These would be the same if they were calculated with only the
|α〉〈α| part or only the |β〉〈β| part of the density matrix (2.16).
In fact, they are the same as 〈Σ1Ξ2〉〈Π3〉 and 〈Σ2Ξ1〉〈Π3〉
calculated for one of those parts. Their values do not require
either entanglement or correlation of C.
III. TWO INTERACTIONS
If a control were coupled similarly to qubit B as well, then
cosφ would be replaced by cosφA cosφB in the next to last
line of Eq. (2.8) and in Eqs.(2.11) and (2.15). If the coupling
of qubit B is made with a rotation around the x axis instead
of the z axis, then the next to last line of Eq. (2.8) becomes
ρ′± =
[11 ± Σ1Ξ1 cosφA ± Σ2Ξ2 cosφA cosφB
−Σ3Ξ3 cosφB ]. (3.1)
Rewriting the last term and looking at eigenvalues in terms of
Σ1Ξ1 and Σ2Ξ2 as before yields the concurrence
C(ρ′±)=
max[0, | cosφA|+| cosφA cosφB|+| cosφB|−1].
(3.2)
When cosφA is 1, these Eqs.(3.1) and (3.2) describe the result
obtained when there is only the interaction of qubit B made
with a rotation around the x axis. If neither cosφA nor cosφB
is 1, the concurrence becomes zero, the state separable, be-
fore cosφA or cosφB is zero. The interactions of qubits A
and B with their controls change maximally entangled states
to separable states. The inverses change separable states to
maximally entangled states. In the following subsection, we
describe the density matrices that show explicitly that the sep-
arable states are mixtures of products of pure states.
A. Exponential decay
To describe exponential decay of entanglement we let
cosφA = e
−ΓAt, cosφB = e
−ΓBt (3.3)
by letting each interaction be modulated by a time-dependent
Hamiltonian H(t) that is related to the Hamiltonian H of
Eqs.(2.4) and (2.5) by
H(t) = H
= HΓcotφ, (3.4)
where φ and Γ are φA and ΓA or φB and ΓB . The same result
could be produced in different ways. The interactions could be
with large reservoirs instead of qubit controls [2, 3, 4]. Each
qubit A or B could interact with a stream of reservoir qubits
[15]. Here we are interested in the way the entanglement is
changed by the combination of the two interactions. That de-
pends only on the changes in the density matrix ρ for qubits A
and B, not on the nature of the controls and the interactions.
Maps that make the changes in ρ will be described in the next
section.
If there is only the interaction of qubit A with qubit C, the
concurrence is e−ΓAt. If there is only interaction of qubit B
with its control, the concurrence is e−ΓBt. If there are both
and both are made with rotations around the z axis, the con-
currence is e−ΓAte−ΓBt. If there are both and the interaction
of qubit B with its control is made with a rotation around the
x axis, the concurrence is
C(ρ′±)=
max[0, e−ΓAt+e−ΓAte−ΓBt+e−ΓBt−1]. (3.5)
This concurrence (3.5) is zero when
e−ΓAt + e−ΓAte−ΓBt + e−ΓBt = 1. (3.6)
Then the state is separable; it is a mixture of six products of
pure states: from Eqs.(3.1) and (3.3)
ρ′± =
e−ΓAt
(11 +Σ1)
(11 ± Ξ1)
e−ΓAt
(11 − Σ1)
(11 ∓ Ξ1)
e−ΓAte−ΓBt
(11 +Σ2)
(11 ± Ξ2)
e−ΓAte−ΓBt
(11 − Σ2)
(11 ∓ Ξ2)
e−ΓBt
(11 +Σ3)
(11 − Ξ3)
e−ΓBt
(11 − Σ3)
(11 + Ξ3). (3.7)
The state remains separable at later times; when the sum of
the exponential decay factors is less than 1, the density ma-
trix is a mixture in which just a multiple of the density ma-
trix 1/4 for the completely mixed state is added to the terms
of Eq. (3.7). This change of maximally entangled states to
separable states can be described without reference to expo-
nential decay by continuing to use cosφA and cosφB instead
of e−ΓAt and e−ΓBt. Similar behavior involving exponential
decay has been observed in more physically interesting and
mathematically complicated models [2, 3, 4].
IV. MAPS
The maps that make the changes in the density matrix ρ for
qubits A and B could be described in different ways using
various matrix forms. That is not needed here. Writing ρ in
terms of Pauli matrices provides a very simple way to describe
the maps. For any density matrix
〈Σj〉Σj +
〈Ξk〉Ξk +
j,k=1
〈ΣjΞk〉ΣjΞk
(4.1)
for qubits A and B, the result of the interaction of qubit A
with qubit C, described by Eq. (2.7), is that in ρ, in both the
Σj and ΣjΞk terms,
Σ1 −→ Σ1 cosφA, Σ2 −→ Σ2 cosφA; (4.2)
the result of the interaction of qubit B with its control is that
Ξ1 −→ Ξ1 cosφB, Ξ2 −→ Ξ2 cosφB (4.3)
if the interaction is made with a rotation around the z axis;
and the result of the interaction of qubit B with its control is
that in ρ
Ξ2 −→ Ξ2 cosφB, Ξ3 −→ Ξ3 cosφB (4.4)
if the interaction is made with a rotation around the x axis.
The terms with sin φ cancel out because there is an equal mix-
ture of parts with φ and parts with −φ.
The changes in the state of qubits A and B can be described
equivalently by maps of mean values that describe the state:
the result of the interaction of qubit A with qubit C, described
by Eq. (2.7), is that
〈Σ1〉 −→ 〈Σ1〉 cosφA
〈Σ2〉 −→ 〈Σ2〉 cosφA
〈Σ1Ξk〉 −→ 〈Σ1Ξk〉 cosφA
〈Σ2Ξk〉 −→ 〈Σ2Ξk〉 cosφA (4.5)
for k = 1, 2, 3; the result of the interaction of qubit B with its
control is that
〈Ξ1〉 −→ 〈Ξ1〉 cosφB
〈Ξ2〉 −→ 〈Ξ2〉 cosφB
〈ΣjΞ1〉 −→ 〈ΣjΞ1〉 cosφB
〈ΣjΞ2〉 −→ 〈ΣjΞ2〉 cosφB (4.6)
for j = 1, 2, 3 if the interaction is made with a rotation around
the z axis; and the result of the interaction of qubit B with its
control is that
〈Ξ2〉 −→ 〈Ξ2〉 cosφB
〈Ξ3〉 −→ 〈Ξ3〉 cosφB
〈ΣjΞ2〉 −→ 〈ΣjΞ2〉 cosφB
〈ΣjΞ3〉 −→ 〈ΣjΞ3〉 cosφB (4.7)
for j = 1, 2, 3 if the interaction is made with a rotation around
the x axis.
When φA and φB change over intervals from initial values
φAi and φBi to final values φAf and φBf , the cosφA and
cosφB factors in the maps are replaced by cosφAf/ cosφAi
and cosφBf/ cosφBi. If either of these factors is larger than
1, the map is not completely positive and does not apply to all
density matrices ρ for qubitsA andB. This happens whenever
the entanglement increases. It also happens in cases where the
concurrence (3.2) decreases, when one of cosφA and cosφB
increases and the other decreases and there is more decrease
than increase. The completely positive maps that decrease the
entanglement have already been described [1].
V. RECONCILIATION
Entanglement being increased by local interactions may
seem surprising from perspectives framed by experience in
common situations where it is impossible. Entanglement is
not increased by a completely positive map of the state of two
qubits produced by an interaction on one of them. The in-
teraction will produce a completely positive map if it is with
a control whose state is initially not correlated with the state
of the two qubits, as in Eq. (2.1). In our examples, that hap-
pens only when the initial value of φ is 0 or a multiple of π.
Otherwise, the state of the control is correlated with the state
of the two qubits as in Eq. (2.16). When a subsystem is ini-
tially correlated with the rest of a larger system that is being
changed by unitary Hamiltonian dynamics, the map that de-
scribes the change of the state of the subsystem generally is
not completely positive and applies to a limited set of subsys-
tem states [16, 17]. We see this in our examples whenever the
entanglement increases and in some cases when the entangle-
ment decreases.
The map depends on both the dynamics and the initial cor-
relations. It describes the effect of both in one step. Com-
pletely positive maps are what you get in the simplest set-up
where you bring a system and control together in independent
states and consider the effect of the dynamics that begins then.
Dynamics over intervals where the maps are not completely
positive can be expected to play roles in more complex set-
tings. We should not let expectations for completely positive
maps prevent us from seeing things that can happen.
Our perspective is enlarged when we look beyond the map
and include the dynamics. We can see the dynamics and the
initial preparation as two related but separate steps. We can
consider the effect of the dynamics, whatever the preparation
may be.
Local interactions that increase entanglement are com-
pletely outside a perspective that is limited to pure states. An
interaction on one of the qubits can not change the entangle-
ment at all if the state of the two qubits remains pure [18].
The entanglement of a pure state of two qubits depends only
on the spectrum of the reduced density matrices that describe
the states of the individual qubits, which is the same for the
two qubits. If that could be changed by an interaction on one
of the qubits, there could be a signal faster than light. In our
examples, the state of the two qubits is pure only when it is
maximally entangled. In our examples, the spectrum of the
density matrices for the individual qubits never changes, and
gives no measure of the entanglement.
Acknowledgments
We are grateful to a referee for very helpful suggestions,
including the comparison with a swap operation. Anil Shaji
acknowledges the support of the US Office of Naval Research
through Contract No. N00014-03-1-0426.
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