Entangling and disentangling capacities of nonlocal maps
Entangling and disentangling capacities of nonlocal maps
Berry Groisman
Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.
Entangling and disentangling capacities are the key manifestation of the nonlocal content of a
quantum operation. A lot of effort has been put recently into investigating (dis)entangling capacities
of unitary operations, but very little is known about capacities of non-unitary operations. Here we
investigate (dis)entangling capacities of unital CPTP maps acting on two qubits.
I. INTRODUCTION
Entanglement content is one of the fundamental ways
to characterize nonlocal quantum resources (nonlocal
states and operations). For pure bipartite states the ul-
timate measure of entanglement, the von Neumann en-
tropy of entanglement, had been recently discovered [1].
A universal measure of entanglement for mixed states
had not been found yet and different measures are used
depending on the operational context. Nevertheless, the
important feature of all entanglement measures of states
is that their values are directly inferred using the param-
eters of a state itself.
Similarly to mixed states, the entanglement content
of quantum operations can be characterized in different
ways, e.g. via the amount of entanglement necessary to
generate that operation or via the amount of entangle-
ment the operation is able to produce/destroy (the so
called entangling/disentangling capacities). This article
is concerned with the two latter measures.
Unlike the amount of entanglement in a state, the
(dis)entangling capacities of an operation do not have
an operational interpretation on their own. They mani-
fest themselves via the change of the entanglement of a
particular state that the operation acts upon. And the
operation has to act on a specific state (the “optimal”
state) in order to realize its (dis)entangling capacity in
full. Thus, the straightforward way to calculate these
quantities is to maximize the change of entanglement over
all possible initial states.
Substantial progress have been made recently in inves-
tigating (dis)entangling capacities of unitary operations.
The capacities of two-qubit unitary operations were ex-
plicitly calculated [2, 3, 4]. It was also shown that single-
shot capacities are equal to asymptotic capacities [3, 5, 6].
Some results for higher dimensions were also obtained [7].
However, extending these techniques to systems of higher
dimensionality seems to be a very difficult task. Even in
the two-qubit case the capacities of a general unitary op-
eration have been calculated numerically, no analytical
technique is known.
In all real situations an experimentalist never deals
with a perfect unitary in the laboratory. And it is need-
less to say that calculating capacities of non-unitary op-
erations, i.e. nonlocal quantum maps, is even a bigger
challenge.
In this article we consider nonlocal completely positive
trace preserving (CPTP) maps of the form
τ(ρ) →
pkUkρU
, (1)
where Uk are unitary transformations. Maps of this type
are often called random unitary processes, and they are
doubly stochastic. The map (1) may arise, for example,
if the desired unitary transformation can be implemented
successfully only with certain probability, while another
unitary is realized in the case of failure. A continuous
version of the map (1) may arise naturally in experiment
if parameters of a desired unitary transformation are sub-
ject to a noise (the case of Gaussian noise will be analyzed
in detail in Section IVC). The scope of this article covers
the case of τ that act on two qubits. We calculate single-
shot (dis)entangling capacities of τ in some particular
cases.
The structure of the article is as follows. In Sec. II
the definition(s) of (dis)entangling capacities of unitaries
are presented and some recent results concerning two-
qubit unitaries. Sec. III generalizes the definition of
(dis)entangling capacities for non-unitaries. Some nu-
merical results for (lower bounds on) (dis)entangling ca-
pacities for discreet and continuous mixtures of unitaries
are presented in Sec. IV.
II. (DIS)ENTANGLING CAPACITY OF A
UNITARY: DEFINITIONS AND SOME RELATED
RESULTS
Consider an unitary operation UAB that acts on a
tensor product Hilbert space HA ⊗ HB of two spatially
separated particles A and B. If UAB can not be de-
composed into a tensor product of local unitaries, i.e.
UAB 6= VA⊗WB, then we say that UAB is nonlocal. Un-
like local unitaries, nonlocal unitaries have an ability to
produce or destroy entanglement. This ability is usually
characterized by the entangling, E↑(U), and the disen-
tangling, E↓(U), capacities, i.e. by the maximal increase
(decrease) of entanglement that can be achieved when
U acts on quantum states. To quantify these capacities
we have to choose appropriate measures of entanglement.
The most sensible choice is to use the entanglement of for-
mation [11] as a measure of entanglement of the initial
state ρ, and the distillable entanglement [12] as a measure
http://arxiv.org/abs/0704.1042v1
of entanglement of the final state UρU †. The reason for
this asymmetric choice is purely operational one. What
counts is the amount of resources (pure maximally en-
tangled states) needed to create ρ (asymptotically) and
the amount of pure-state entanglement one will be able
to extract from UρU †, again asymptotically. Thus the
most general definition is
E↑(U) = max
[D(UρU †)− EF (ρ)],
E↓(U) = max
[EF (ρ)−D(UρU †)],
where the maximization is over all possible states ρ
(mixed and pure) accessible to U . The Hilbert space of
an accessible ρ is not necessarily restricted to HA ⊗HB.
It turns out to be the case that some U create more en-
tanglement if the original particles are entangled with
local ancillary particles [2, 3]. It also appears to be the
case that the maximization in Eq. (2) can be restricted
to pure-states only [3], therefore, the definition (2) can
be simplified as
E↑(U) = max
[E(U |ψ〉) − E(|ψ〉)]
E↓(U) = max
[E(|ψ〉) − E(U |ψ〉)],
where E is an entanglement measure for pure state
(Throughout this paper we will use the von Neumann
entropy of entanglement as the most appropriate mea-
sure). This obviously simplifies the job significantly.
Let us briefly recall the main results for A and B being
two-level particles, qubits.
Any UAB acting on qubits can be decomposed as [2, 10]
UAB = [VA ⊗ VB] e
α=x,y,z
α [WA ⊗WB] , (4)
where π/4 ≥ ξx ≥ ξy ≥ |ξz | ≥ 0. The middle term sand-
wiched by local unitaries is called the canonical decom-
position of U . Any U can be transformed to its canon-
ical form by sandwiching it with Hermitian conjugates
of corresponding local unitaries. That means that the
canonical form is genuinely nonlocal part of U - every-
thing else is local. The beauty of this results is that
out of 15 real parameters that parameterize a general
two-qubit unitary only three are necessary to describe
its nonlocal nature. It simplifies considerably the clas-
sification of nonlocal unitaries. For the purpose of our
discussion three classes can be identified; namely, the
Controlled-NOT(CNOT)-class (ξx 6= 0, ξy = ξz = 0),
the DoubleCNOT-class (ξx 6= 0, ξy 6= 0, ξz = 0), and the
SWAP-class (all three ξα are not equal zero) [15, 16]. The
names reflect the fact that the corresponding “mother”
unitary transformation (i.e. with ξα = π/4 for α 6= 0)
belongs to that class.
The main results for qubits are [2, 3]:
(a) E↑(U) = E↓(U).
(b) For CNOT-class the optimal state, i.e. the state
that satisfies definition (3), lives solely in the Hilbert
space of particles A and B (no ancillas are needed) and
takes the form
|ψopt〉 = cosα|0〉A|0〉B ± i sinα|1〉A|1〉B, (5)
where ± correspond to E↑ and E↓ respectively. Thus
all U from that class achieve their capacity by acting on
pure states with the same Schmidt basis (only values of
Schmidt coefficients differ depending on the value of ξx).
The values α = f(ξx) can be obtained by straightforward
numerical optimization.
(c) If ξα < π/4 the maximal capacity is achieved when
|ψopt〉 is already entangled.
(d) Unitaries of the CNOT-class achieve their capaci-
ties by acting on optimal states that lie inHA⊗HB. How-
ever, unitaries of the DCNOT and SWAP-classes achieve
their capacities only if the original particles are entangled
with local ancillas. It was conjectured that it is sufficient
to take the size of ancillas equal to the size of original
particles. This conjecture was supported by numerical
simulations for qubits [3].
III. ENTANGLING AND DISENTANGLING
CAPACITIES OF A NON-UNITARY
For non-unitaries we will use a definition similar to Eq.
(2), i.e. we define
E↑(τ) = max
[D(τ(ρ)) − EF (ρ)]
E↓(τ) = max
[EF (ρ)−D(τ(ρ))].
However, in general here we cannot justify reducing
the search to pure states. This is due to the fact that
distillable entanglement is not necessary a convex mea-
sure.
We can argue, nevertheless, that in the case of mix-
tures of unitaries acting on two qubits without ancillas
the distillable entanglement can be regarded as a convex
measure. Indeed, a mixture of optimal states (5) forms a
Bell-diagonal state for which the lower and upper bounds
on distillable entanglement [17]
S(ρA)− S(ρAB) ≤ D(ρ) ≤ ERE(ρAB) (7)
coincide. Here we recall that the relative entropy of en-
tanglement ERE(x) is a convex measure.
If ancillas are used then the situation is more compli-
cated. We leave the question of whether the capacities
are attained on pure states as an open question and cal-
culate the lower bounds on these capacities using pure
states.
IV. MIXTURES OF UNITARIES ACTING ON
TWO QUBITS
The properties of two-qubit unitaries described above
in Section II can help us to generalize that approach to
-0.75-0.5-0.25 0.25 0.5 0.75
FIG. 1: (color online) E↑(τ ) (solid line) and E↓(τ ) (dashed
line) as functions of ∆ for different values of ξ, where p = 0.5.
The highest curve corresponds to ξ = π/4. The lowest curve
corresponds to ξ = 0. Here ∆ is measured in radians.
mixtures of unitaries as in Eq. (1)[18]. Here we use two
methods for calculating E↑(τ) and E↓(τ).
Method I: We make an assumption about the partic-
ular form of the optimal input state, and subsequently
find the optimal values of its parameters.
Method II: We perform a direct numerical optimiza-
tion without making any a priori assumption about the
optimal state (except of its purity).
A. Example I: Discreet CNOT-mixtures
Consider a mixture of unitaries of the CNOT-class,
Uk = e
x . (8)
Here we use Method I. From continuity it follows that
the optimal state is expected to lie on the 2-dimensional
manifold of (superpositions of) states of the form (5) or
their convex mixtures. Moreover, in this special case we
can adopt the argument of [3] (see Sec. II) and claim that
the search can be restricted to pure states only. Thus,
the state optimal for τ is again of the type (5).
As a simplest case let us consider only two unitaries
U1 and U2:
τ(ρ) → pU1ρU †1 + (1 − p)U2ρU
For convenience let us define ∆ = ξ2x − ξ1x, and denote
ξ1x simply by ξ, so ξ
x = ξ +∆. We will fix ξ and analyze
E↑ and E↓ for various ∆. For ∆ = 0 the map reduces
to a unitary (with an appropriate capacity). As smaller
angle means smaller E↑(U), we would expect that if U1 is
mixed with U2, where ∆ < 0, then the entangling capac-
ity of the resulting map will decrease relative to E↑(U).
This intuition is consistent with the results presented on
Fig. 1. Similarly, we might expect that the entangling
capacity of the map will increase with ∆ > 0, and that
-0.4 -0.2 0.2 0.4
FIG. 2: (color online) DCNOT: E↑(τ ) (empty diamonds with
solid line fit) and E↓(τ ) (empty triangles with dashed line fit)
as functions of ∆ for ξ1x = ξ
y = π/8. SWAP: E
↑(τ ) (filled
diamonds with solid line fit) and E↓(τ ) (filled triangles with
dashed line fit) as functions of ∆ for ξ1x = ξ
y = ξ
z = π/8. In
both cases p = 0.5. Here ∆ is measured in radians.
this capacity will reach its maximum for maximal ∆, i.e.
maximal E↑(U2). However, Fig. 1 shows that this is not
the case. E↑(τ) indeed grows while ∆ is positive and
relatively small, reaching its maximum for certain inter-
mediate positive value of ∆ and then starting to decrease.
In other words, if U1 and p are fixed, then maximal E
is achieved for some intermediate U2 with ξ
x > ξ
x, but
not for ξ2x = π/4. This result might seem counterintuitive
from the first sight, but it has a clear explanation. Let α1,
α2 be the corresponding optimal values of α in Eq. (5)
for U1 and U2 respectively, then the optimal value of ατ
will lie somewhere in between, i.e. satisfy α1 > ατ > α2.
For ξ2x = π/4, U2 can realize its entangling capacity of 1
ebit if α2 = 0. However, when U2 acts on a state with
ατ > α2, then it creates less than 1−H [(cosατ )2] ebit.
Disentangling capacity, E↓(τ), behaves differently. It
is monotonic with ∆. It equals E↑ for ∆ = 0 as expected,
and it is strictly larger than E↑ for all other values of ∆.
The last observation shows behavior completely opposite
to that of unitaries. In a sense, it is easier for non-local
map to destroy entanglement rather than create it, while
for unitary operations the opposite holds [7].
This approach can be similarly applied to any finite
number of unitaries and to continuous distribution of uni-
taries, which will be discussed in Section IVC.
B. Example II: Discreet DCNOT and
SWAP-mixtures
For mixtures of unitaries of DCNOT and SWAP-class
Method II was used. The details of numerical calcula-
tions are presented in Appendix. We conjectured that
it is sufficient if local ancillas are qubits. Selected re-
sults are shown in Fig. 2. We can see that for DCNOT-
mixture the behavior of E↑(τ) and E↓(τ) is qualitatively
similar to CNOT-mixture (Fig. 1). However, for SWAP-
mixture slightly different behavior is obtained. In partic-
ular E↓(τ) exhibits maximum at an intermediate value
of ∆. It is also noticeable that for D > 0.357, E↑(τ) is
smaller for SWAP-mixture than for DCNOT-mixture. It
is a counterintuitive result that SWAP-mixture which is
naturally considered as “stronger” that DCNOT-mixture
has lower entangling capacity. However, again similar to
the line of thought in Example I we can argue that for
(relatively) large ∆ the second unitary, U2 is too strong,
and therefore when it acts on the optimal state (optimal
for the mixture, not for itself) it causes more destruc-
tion that corresponding U2 of DCNOT-class would have
caused.
C. Example III: Entangling capacity of noisy
unitary with Gaussian fluctuations
So far we analyzed discrete mixtures of unitaries. In
this section we analyze a continuous distribution, which is
usually what experimentalists deal with. These distribu-
tions arise due to uncertainty in one or more parameters.
Such uncertainties may be caused by the limits of cali-
bration precision of the devices and by high sensitivity of
systems used to generate desired interactions. For exam-
ple, the strength of exchange coupling between donors in
silicon based solid-state architectures for quantum com-
puting exhibit significant uncertainty resulting in error
in gate operation [19].
In particular, we consider the case when a non-unitary
map arises if a unitary from CNOT-class is subject to a
Gaussian noise.
Recent work [20] analyzed the capability of noisy
Hamiltonians to create entanglement. In particular, in-
teractions of the form Eq. (8), where ξ is Gaussian dis-
tributed with the mean ξ = π/4 and standard deviation
Ω, were considered. Without noise this operation (CNOT
operation) is able to create a maximally entangled state
if it acts on a disentangled pure state. The authors an-
alyzed the situation when the noisy operation acts on
initially disentangled state, which is by itself subject to
a Gaussian noise. Its capability to create entanglement
was characterized in terms of the condition for insepara-
bility of the resulting mixed state (via Peres-Horodecci
separability criterion).
The aim of our analysis is different. We consider noisy
interactions with ξ ∈ [0, π/4] and calculate their entan-
gling and disentangling capacities in terms of ξ and Ω.
Thus, we give a comprehensive quantitative characteriza-
tion of the non-local content of these noisy maps in terms
of their entangling and disentangling capacities. Unlike
[20] we do not test the resulting state on inseparability,
rather calculate its distillable entanglement explicitly.
The action of a unitary U = exp[iξσAx σ
x ], where ξ is
Gaussian distributed with the mean ξ and the standard
deviation Ω, on the state ρAB can be seen as a non-
0.2 0.4 0.6 0.8
FIG. 3: (color online) E↑(τG) (solid line) and E
↓(τG) (dashed
line) as a function of ξ for several values of Ω: 0.01, 0.18, 0.35,
0.52, 0.69, and 0.86. The dotted line corresponds to Ω = 0, i.e.
a unitary transformation. ξ and Ω are measured in radians.
unitary CPTP map
τG(ρ) →
(ξ−ξ)2
2Ω2 UρU †dξ, (10)
which is a continuous mixture of unitaries of the CNOT-
class. Similarly to the Sec. IVA we consider pure initial
state, i.e. ρ = |ψ〉〈ψ|, where ψ takes the form (5), cal-
culate the distillable entanglement of the output mixed
state, and maximize it over α. Figure 3 presents nu-
merical results for E↑(τ) and E↓(τ) as functions of ξ
for several values of Ω. We can see that already for
Ω ≈ 0.01 rad we obtain only very small deviation from
the (dis)entangling capacity of the unitary, i.e. τG with-
out noise - Ω = 0. As Ω increases the disentangling
capacity increases and the entangling capacity decreases.
The former fact should not be surprizing as it is known
that entanglement can be destroyed even by local CPTP
unital maps [21]. Thus, the more dispersed the distribu-
tion of ξ becomes, the easier for τG to destroy entangle-
ment and the harder to create it. Nevertheless, we see
that even when Ω is relatively large τG is still able to
create considerable amount of entanglement.
V. DISCUSSION AND CONCLUSION
We have discussed the entangling and disentangled ca-
pacities of nonlocal CPTP unital maps, i.e. maps that
can be represented as probabilistic mixtures of unitaries,
and have calculated these capacities in some particular
cases for two qubits. Three classes of unitaries were con-
sidered, namely the CNOT, DCNOT, and SWAP classes.
We have observed that the disentangling capacity was
always larger than corresponding entangling capacity,
which contrasts with the unitary case where the both
capacities are equal for qubits and for higher dimensions
disentangling capacity cannot be greater than entangling
capacity [7].
In the case of the CNOT-class our results were
obtained via straightforward generalization of the
method for CNOT-class unitaries. We argue that the
(dis)entangling capacity is achieved when a map acts on
the optimal pure state from the same family as in the uni-
tary case. Both discrete and continuous mixtures were
analyzed. In the case of the DCNOT and SWAP-class
direct numerical optimization was performed. We have
conjectured that dimensions of the local ancillas are equal
to the dimensions of the original particles, i.e. the ancil-
las were taken to be qubits.
A number of open question can be addressed in a future
research.
It will be interesting and useful to prove (or disprove)
the general conjecture that the sizes of local ancillas can
be taken equal to the sizes of original particles.
Here we have calculated single-shot capacities. In the
case of unitaries it had been shown that in the asymptotic
regime one cannot do better [5, 6]. It is important to
check whether this result holds in the non-unitary case.
In the case of DCNOT and SWAP-mixtures we per-
formed maximization over pure states only thereby ob-
taining lover bounds on E↑(τ) and E↓(τ), but not their
actual values.
In our future research we will address the question of
whether these bounds are tight. It might be the case
that optimal states for these operations are mixed and,
consequently, the capacities are higher than we have cal-
culated.
Acknowledgments
This work was funded by the U.K. Engineering
and Physical Sciences Research Council, Grant No.
EP/C528042/1, and supported by the European Union
through the Integrated Project QAP (IST-3-015848) and
SECOQC.
APPENDIX A
We have used two-dimensional ancilla on each side.
Consider a general state of four qubits in the tensor-
product of the computational bases of the original parti-
cles A, B and the ancillary particles A′, B′
|Ψ〉AA′BB′ =
i,j,k,l
ci,j,k,l|i〉A|j〉A′ |k〉B |l〉B′ . (A1)
There are 16 terms in the above superposition with 16
complex amplitudes ci,j,k,l, therefore |Ψ〉 can be parame-
terized using 30 real numbers (if we take into account the
global phase and normalization). We will parameterize it
in the following way [22]. First, to facilitate our analysis
it is easier to incorporate four indexes i, j, k and l, each of
which runs from 0 to 1, into a single index, x, that runs
from 1 to 16. This can be done by using the formula
x = 8i + 4j + 2k + l + 1, which is essentially a formula
for converting a number from the Boolean representation
to the decimal. Second, we present amplitudes cx in the
cx = |cx|eiθx , (A2)
where
|cx|2 = 1 and θ1 = 0. Third, we introduce
new parameters φx such that
cx = sinφx−1
cosφy, (A3)
where φ0 = π/2. Thus the state |Ψ〉 is parameterized by
30 angles. The advantage of this parametrization is that
we restrict their values only to the interval [0, 2π] that
simplifies numerics.
We proceed as follows. A program generates a vec-
tor of 30 random numbers in the interval [0, 2π]. This is
the initial state. We then apply the non-local map and
obtain a final state. We calculate the value of the gain
in entanglement ∆S = S(τ(Ψ)BB′ ) − S(τ(Ψ)AA′BB′) −
S(TrAA′ |Ψ〉〈Ψ|). After that we vary the values of the ran-
dom vector by a small amount and repeat these calcula-
tions again, thereby obtaining a gradient of the change in
entanglement in that point. We move along the gradient
to obtain the next |Ψ〉, and the procedure is repeated.
Eventually, the program reaches the maximum where it
stops.
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