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Qubit A
UAB
J. Batle, M. Casas
Physics Department, UIB and
IMEDEA-CSIC, Spain
A. R. Plastino
Qubit B
Qubit D
Physics Department, University of
Pretoria, Pretoria, South Africa
Qubit C
A. Plastino
National University La Plata,
La Plata, Argentina
X Int’l Conference on Quantum Optics. Minsk, BELARUS, May 30 - June 3, 2004
I
Optimal Parametrization of Quantum Gates for Two-Qubits Systems
II
Quantum Gates’ Entangling Power: Qubits and Qudits
III
Two-Qubits Space Metrics and the Entangling Power of a Quantum
Gate
IV
Entanglement Distribution in Multiple Qubits Systems
V
Conclusions
References
1
There is a formal compact way to express the general form of any quantum logical
gate U acting on a two-qubits systems. It can be expressed in the form [1]
Local operations
Local operations
Responsible for quantum correlations
(entangling part)
The nuclear part with the exponential depends solely on the vector of values
l = (l1, l2, l3), which are responsible for the non-local features (“entangling part”).
• Thus, we say that gates are equivalent if they associated probability (density)
distributions of finding a state r with a given change in its entanglement DE, once
the gate has been applied to the state, are the same.
• State by state, two gates may “entangle” in a different fashion. But if they are
equivalent, on average they draw the same DE-distribution P(DE).
• The gate (p/4,0,0) is equivalent to the CNOT gate [2]. The study of quantum
gates is greatly simplify to a space of only three parameters: (l1, l2, l3).
2
Pure states of two-qubits
uniformly generated using
the Haar measure
Probability distribution of entanglement changes DE for pure states of two-qubits
generated by several quantum gates: curve 1, (p/4,p/8,0); curve 2, (p/4,p/8,p/16);
curve 3, CNOT; curve 4, (p/4,p/8,-p/8); and curve 5, (p/8,p/8,p/8).
3
• As stated, a quantum gate (QG), represented by a unitary transformation U, acts as
an “entangler”: abstraction of some physical interaction taking place in the system.
• Question arises: Can we quantify this “entanglement capacity”. Relevance in QIP:
a QG beaing robust against the environment is more suitable for a quantum network.
• We use the approach by Zanardi et. al [3]. Entangling power eP(U) of a QG U reads
Average over the set S of
unentangled pure states
Entanglement measure
Quantum evolution U
• Useful in any dimension for bipartite systems. I Qubits. Using the entanglement of
formation E [4], and the previous optimal (l1, l2, l3)-parametrization, we study the
numerics of eP(U) for the paradigmatic CNOT gate. To such an end, we generate a
sample of unentangled states r AB  r A  r B according to the Haar measure, and
compute its eP(U).
• The CNOT gate is optimal (makes eP(U) maximal). If we perturbate around the CNOT
gate through (p/4,x,x), x being continuous, we find and interesting issue: there exists an
infinity of optimal gates in the vicinity of an optimal gate!
4
Unperturbed
CNOT (x=0)
CNOT  (p / 4, 0, 0)
(p/8,0,0) is not
optimal !!!
Entangling power of the perturbed CNOT gate. Small deviations slightly increase
eP(U) around x=0. Large deviations clearly shows that the CNOT gate is optimal.
5
II Qudits
• Two-qudits are systems of dimension N = NA x NA. It is argued in [3] the peculiarity
of the two-qubits (2 x 2) case, as far as eP(U) is concerned. We investigate this point no
longer using the eP(U) definition or any quantum gate in particular.
• We rather introduce a new definition for the entangling power which relies on a
“no gate action”: we look at the probability (density) distribution PR obtained by
ramdomly picking up two pure states according to the Haar measure in NAxNA
dimensions, and determine the relative entanglement change DE in passing from one
of these states to the other. The distribution PR is [2]
Probability (density)
of finding a state with
entanglement E
• E is now S(rA)/ln(NA) (so that ranges between 0 and 1). The ensuing distributions
PR (DE) for 3x3, 4x4, 5x5 and 6x6, as compared to the two-qubits case, make the latter
case rather special.
• This fact suggest a new measure for the entangling capacity of a quantum gate: the
natural width of the corresponding P(DE)-distributions, namely WDE . As a matter of

fact, from numerical inspection, we see a power law decaw WDE  1 / N A for PR (DE)
6
R
NA x NA
pure states
WDE goes as 1 / NA
Probability distributions PR (DE) for 2x2, 3x3, 4x4, 5x5 and 6x6 pure states.
The unique characteristic of the two-qubits case is apparent.
7
Entangling power for bipartite pure states
(generated using Haar invariant measure)
Entangling power for bipartite
mixed states rˆ  Uˆ l Uˆ 
AB
Measure on the simplex
of eigenvalues { l i }
AB
Haar measure on U(N)
•
could be extended to mixed states. Instead, we
make an heuristic approach to the entangling capability of a quantum gate based in
the metrics and special features of the space S of mixed states of two-qubits.
• We generate mixed unetangled states according to a given measure [5-8] and apply
the quantum gate under consideration (CNOT in our case). The corresponding difference in entanglement is viewed through the “distance” between final an initial states,
using an appropiate metric. Averages are taken and distribution of distances obtained.
• The metric we use is the Bures [9] distance
Simple idea: the further the gate sends an unentangled state r , the stronger it is.
It is a semi-quantitative argument. The entanglement capability WDE can be computed
proportional to the average distance < d>, as taken from the distribution P(d).
8
zone I
WDE  < d >
S of two-qubits
Û CNOT Spacemixed
states
d1, 2
zone II
zone II
zone I + zone II
d1, 2
zone I
Probability (density) distributions P(dBures)
of finding a state r being set a
distance dBures away from set S, on average, after the action of the CNOT gate UCNOT
9
• Bipartite pure states
Multipartite pure states rˆ     in1 i
• Physically-motivated situation: in the two-qubits system, the environment can be
regarded as a third interacting party.
• Interesting to study i) the general case of multipartite networks of qubits on one side,
while on the other hand discussing ii) how the dimension (number n of qubits) affects
the distribution of entanglement pairs when we apply, locally, a certain quantum gate.
i) Following the work of Coffman et. al [10], the pairwise entanglements present
between pairs of qubits in a tripartite system, in the form of a rABC pure state, as
measured by the concurrence CXY, related to the entanglement of formation E, and
defining the reduced matrices rA=TrBC [rABC], rAB=TrC [rABC], rAC=TrB [rABC] reads
New definition
ii) Suppose that we apply the CNOT gate to a given pair AB of qubits. From the
entanglement distributions obtained, we see that the existence of a third or fourth
qubit in the system somehow dilutes the pairwise entanglement CAB available to
AB, prescribed by, just like the action of a rough “thermal bath”.
10
A
<d>1/3
C
B
2
2
dW  CA2( BC)  CAB
 CAC
Legitimate multipartite
entanglement measure
Probability distribution of dW, after a Monte Carlo generation, of the three-qubit
pure states according to the invariant Haar measure. Remarkable bias.
11
AB
A
B
Û
AB
CNOT
P(DEAB )
A' B'
DEAB
A
C
ABC
B
AB
ˆ
U CNOT  IˆC
A' B' C ABCD
A
B
D
C
AB
ˆ
U CNOT  IˆC  IˆD
A' B' CD
12
AB
ˆ
U CNOT  IˆC
AC,BC qubit pairs
AB qubit pair
random AB,AC,BC pairs
Û
AB
CNOT
13
AB qubit pair from
n=2, 3 and 4 qubits
AB
ˆ
U CNOT  IˆC  IˆD
AB
ˆ
U CNOT  IˆC
Û
AB
CNOT
14
In the present work we have exposed the results of a systematic Monte Carlo survey
of the action of quantum gates as applied to multipartite quantum systems.
I
After reducing the ultimate non-local aspects of any quantum gate (two-qubits)
to three parameters (l1, l2, l3), we have focused our attention to the definition of the
“entangling power” eP(U) of a quantum evolution from:
i) Pure bipartite states. We compute eP(U) for the CNOT gate. A continuous
perturbation around the CNOT concludes that around an optimal gate
(makes eP(U) maximal) there are infinitely many other maximal gates.
Introduce new definition of entanglement capability: natural width WDE
ii) Mixed bipartite states. Heuristic approach to the entangling power based
on the metrics of the 15-dimensional space of two-qubits mixed states.
II
We have studied some basic properties of the distribution of entanglement in
multipartite systems (network of qubits) and the effects produced by two-qubits gates
acting upon them. The fact that the entanglement between pairs becomes diluted by the
presence of third or fourth parties is apparent from the concomitant P(DE)-distrib.
Their natural WDE decreases almost exponentially fast with the number of parties n.
15
[1] G. Vidal, K. Hammeter and J.I. Cirac, Phys. Rev. Lett. 88, 237902 (2002).
[2] J. Batle, A.R. Plastino, M. Casas and A. Plastino. To be published.
[3] P. Zanardi, C. Zalka and L. Faoro, Phys. Rev A 62, 030301 (2000).
[4] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[5] K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein, Phys. Rev. A 58,
883 (1998).
[6] K. Zyczkowski, Phys. Rev. A 60, 3496 (1999).
[7] J. Batle, M. Casas, A. Plastino and A. R. Plastino, Phys. Lett. A 318, 506 (2003).
[8] J. Batle, A.R. Plastino, M. Casas and A. Plastino, Eur. Phys. J. B 35, 391 (2003).
[9] D. J. C. Bures, Trans. Am. Math. Soc. 135, 199 (1969).
[10] V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev. A 61, 052306 (2000).