Three Dimensional Visualization of Qutrit States
Vinod K. Mishra, Ph.D.
US Army Research Laboratory, Aberdeen, MD 21005
[email protected]
Abstract
The qutrit comes next in complexity after qubit as a resource for quantum information processing. The qubit
density matrix can be easily visualized using Bloch sphere representation of its states. In contrast, this simplicity
is in general unavailable for the 8-dimensional state space of a qutrit. In this work a new way to visualize them in
a 3-dimensional Cartesian space is presented and compared with earlier similar efforts.
1. INTRODUCTION
A qutrit density matrix is of order 3 and so contains 9 parameters constrained by many relations.
One obvious constraint is that the trace of the density matrix is unity and effectively we can
start with the 8 parameter space. It is important to be able to visualize this state space to get a
better understanding of the qutrit properties. So far there have been many attempts at visualizing
this space [1-4] based on different choices of the 3rd order basis matrices. The earliest attempt
used 8 Gell-Mann matrices, which form a complete set for expressing 3 × 3 SU(3) matrices.
The structure of the resulting 8-dimensional parameter space is quite complicated and almost
impossible to visualize in general. The 2- and 3-sections of this space obtained by keeping 2
and 3 nonzero parameters respectively have been described earlier [1,2]. It was found that these
sections have complicated shapes and visualization is not possible for higher order sections.
Recently a new approach [5,6] using Spin-1 matrices has been presented. The 8-dimensional
parameter space was mapped in an ellipsoid and a vector. The 3-dimensional objects make it
easier to visualize the qutrit state space. In this paper we use similar approach and find a new
way to visualize the space in terms of only 2 vectors.
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2. THEORETICAL ANALYSIS
General Qutrit States
The most general qutrit density matrix can be represented using SU (3) invariant form as
1
𝜌𝜌 = 3 𝐼𝐼3 + 𝒏𝒏
��⃗. �𝛌𝛌⃗
(1)
Here 𝐼𝐼3 is 3 × 3 unit matrix, , 𝐧𝐧
�⃗ = (𝑛𝑛 1, n2,…, 𝑛𝑛8) are 8 real parameters, and , �𝛌𝛌⃗ = (𝜆𝜆 1, 𝜆𝜆 2,…, 𝜆𝜆
R
8)
R
R
R
R
are 3 × 3 Gell-Mann matrices. Using them one obtains
1
𝑛𝑛8
⎡3 + 𝑛𝑛3 + √3
⎢
𝜌𝜌 = ⎢ 𝑛𝑛1 + 𝑖𝑖𝑛𝑛2
⎢
⎣ 𝑛𝑛4 + 𝑖𝑖𝑛𝑛5
1
3
𝑛𝑛1 − 𝑖𝑖𝑛𝑛2
− 𝑛𝑛3 +
𝑛𝑛8
√3
𝑛𝑛6 + 𝑖𝑖𝑛𝑛7
𝑛𝑛4 − 𝑖𝑖𝑛𝑛5 ⎤
⎥
𝑛𝑛6 − 𝑖𝑖𝑛𝑛7 ⎥
1
2𝑛𝑛8 ⎥
−
⎦
3
(2)
√3
For this density matrix to represent a physical qutrit, it must satisfy some constraints as given
below.
(i)
Positivity of diagonal elements
(ii)
0 ≤ 3 ± 𝑛𝑛3 +
(3b)
(iii)
𝑇𝑇𝑇𝑇(𝜌𝜌) = 1
(3c)
(iv)
�⃗. 𝒏𝒏
𝒏𝒏
�⃗ ≤ 3
det 𝜌𝜌 ≥ 0
(3d)
1
𝑛𝑛8
√3
Normalization
1
≤ 1, 0 ≤ 3 −
2𝑛𝑛8
√3
≤1
(3a)
Constraint on the length of the state vector
1
Non-negativity of the determinant
Earlier attempts at visualizing the general constraints given above have focused on 8dimensional Cartesian space as given in references [1-4]. The 2- and 3-sections of this 8dimensional objects (only 2 and 3 nonzero elements) have been found, classified, and
2
characterized by these authors. Beyond 3 dimensions it is impossible to visualize the state space
so attempts have been made to find alternative schemes. In this work such a scheme is presented.
Reduction of qutrit constraints to be applicable in 3-dimensions
Following the formalism of reference [5,6], an alternative representation of the qutrit density
matrix based on the symmetric part of the 2 qubit Bloch matrix representing a spin-1 state is
given by
𝜔𝜔1
(𝑞𝑞3 + 𝑖𝑖𝑎𝑎3 )/2
(𝑞𝑞2 − 𝑖𝑖𝑎𝑎2 )/2
𝜔𝜔2
−(𝑞𝑞1 + 𝑖𝑖𝑎𝑎1 )/2�
𝜌𝜌 = �(𝑞𝑞3 − 𝑖𝑖𝑎𝑎3 )/2
𝜔𝜔3
(𝑞𝑞2 + 𝑖𝑖𝑎𝑎2 )/2 −(𝑞𝑞1 − 𝑖𝑖𝑎𝑎1 )/2
(4)
The parameters in this representation are connected to spin-1 observables.
𝜔𝜔𝑖𝑖 =< 𝑆𝑆𝑖𝑖2 >= 𝑇𝑇𝑇𝑇(𝜌𝜌𝑆𝑆𝑖𝑖2 ),
(5a)
𝑞𝑞𝑘𝑘 =< 𝑆𝑆𝑖𝑖 𝑆𝑆𝑗𝑗 + 𝑆𝑆𝑗𝑗 𝑆𝑆𝑖𝑖 >= 𝑇𝑇𝑇𝑇�𝜌𝜌(𝑆𝑆𝑖𝑖 𝑆𝑆𝑗𝑗 + 𝑆𝑆𝑗𝑗 𝑆𝑆𝑖𝑖 )�, 𝑘𝑘 ≠ 𝑖𝑖, 𝑗𝑗
(5c)
𝑎𝑎𝑖𝑖 =< 𝑆𝑆𝑖𝑖 >= 𝑇𝑇𝑇𝑇(𝜌𝜌𝑆𝑆𝑖𝑖 ),
(5b)
Comparison of this parametrization of qutrit density matrix with the earlier one based on the
Gell-Mann matrices gives the following identities.
1
1
1
1
1
1
𝑛𝑛1 = 2 𝑞𝑞3 ,𝑛𝑛2 = 2 𝑎𝑎3 , 𝑛𝑛4 = 2 𝑞𝑞2 , 𝑛𝑛5 = 2 𝑎𝑎2 , 𝑛𝑛6 = − 2 𝑞𝑞1 , 𝑛𝑛7 = 2 𝑎𝑎1
1
1
1
(6)
𝑛𝑛3 = 2 (𝜔𝜔1 − 𝜔𝜔2 ), 𝑛𝑛8 = √3 �− 3 + 2 (𝜔𝜔1 + 𝜔𝜔2 )�
We use the characteristic polynomial approach as given in reference [7] to discuss the
constraints on the qutrit density matrix𝜌𝜌. For it to be a physical quantity it must be Hermitian,
of unit trace, and positive semidefinite. The last condition implies that all 3 eigenvalues of 𝜌𝜌
are non-negative. These constraints translate into the following 3 relations.
(i)
(ii)
(iii)
𝑇𝑇𝑇𝑇(𝜌𝜌) = 1
𝑇𝑇𝑇𝑇(𝜌𝜌2 ) ≤ 1
3𝑇𝑇𝑇𝑇(𝜌𝜌2 ) − 2𝑇𝑇𝑇𝑇(𝜌𝜌3 ) ≤ 1
(7a)
(7b)
(7c)
Applying them to the qutrit density matrix leads to the following relations.
(i)
𝜔𝜔1 + 𝜔𝜔2 + 𝜔𝜔3 = 1
(8a)
3
This is the normalization condition.
(ii)
(iii)
1
∑3𝑖𝑖=1 �𝜔𝜔2𝑖𝑖 + ( 𝑎𝑎2𝑖𝑖 + 𝑞𝑞𝑖𝑖2 )� ≤ 1
(8b)
2
1
1
∑3 𝜔𝜔 (𝑎𝑎2𝑖𝑖 + 𝑞𝑞𝑖𝑖2 ) ≤ 𝜔𝜔1 𝜔𝜔2 𝜔𝜔3 + (𝑎𝑎2 𝑎𝑎3 𝑞𝑞1 + 𝑎𝑎3 𝑎𝑎1 𝑞𝑞2 + 𝑎𝑎1 𝑎𝑎2 𝑞𝑞3 − 𝑞𝑞1 𝑞𝑞2 𝑞𝑞3 ) (8c)
4 𝑖𝑖=1 𝑖𝑖
4
In reference [5, 6], these constraints were given in a basis in which all 𝑞𝑞𝑖𝑖 are zero and were
further used to define an ellipsoid and a vector. The unitary dynamics of qutrit was also
presented in terms of geometric transformations of these two entities.
Here we show that an alternative scheme containing only 3 dimensional vectors is another
possibility. For that we define (i) two scalars𝑓𝑓 , 𝑔𝑔, and (ii) components of a vector R in terms
of the density matrix parameters.
𝑓𝑓 = 𝜔𝜔2 𝜔𝜔3 + 𝜔𝜔3 𝜔𝜔1 + 𝜔𝜔1 𝜔𝜔2,
(9a)
1
𝑔𝑔 = 𝜔𝜔1 𝜔𝜔2 𝜔𝜔3 + 4 (𝑎𝑎2 𝑎𝑎3 𝑞𝑞1 + 𝑎𝑎3 𝑎𝑎1 𝑞𝑞2 + 𝑎𝑎1 𝑎𝑎2 𝑞𝑞3 − 𝑞𝑞1 𝑞𝑞2 𝑞𝑞3 )
𝑅𝑅𝑖𝑖 =
1
4
(𝑎𝑎2𝑖𝑖 + 𝑞𝑞𝑖𝑖2 ),
(9b)
(9c)
It is to be noted that the scalar quantity 𝑔𝑔 must be non-zero and positive and so the values of
(𝑎𝑎𝑖𝑖 , 𝑞𝑞𝑖𝑖 ) which make it negative are non-physical. Then inequalities in eqns. (8b) and (8c) can
be rewritten using eqns. (9a-c) as
1
𝑓𝑓
1
𝑔𝑔
∑3𝑖𝑖=1 𝑅𝑅𝑖𝑖 ≤ 1
(10a)
∑3𝑖𝑖=1 𝜔𝜔𝑖𝑖 𝑅𝑅𝑖𝑖 ≤ 1
(10b)
Define the 3-dimensional vectors
��⃗
𝛀𝛀 = �√𝜔𝜔1 , √𝜔𝜔2 , �𝜔𝜔3 �
𝑅𝑅
𝑅𝑅
𝑅𝑅
𝐮𝐮
�⃗ = {𝑢𝑢1 , 𝑢𝑢2 , 𝑢𝑢3 } = �� 𝑓𝑓1 , � 𝑓𝑓2 , � 𝑓𝑓3 �
𝜔𝜔1 𝑅𝑅1
𝐯𝐯�⃗ = {𝑣𝑣1 ,𝑣𝑣2 , 𝑣𝑣3 } = ��
𝑔𝑔
(11a)
𝜔𝜔2 𝑅𝑅2
,�
𝑔𝑔
(11b)
𝜔𝜔3 𝑅𝑅3
,�
𝑔𝑔
�
(11c)
4
These vectors are always in the positive octant of a sphere by construction. The two inequalities
given in eqn. (11a-b) assume very simple forms as constraints on the lengths of these 3dimensional vectors.
��⃗ | = 1
|𝛀𝛀
(12a)
�⃗. 𝒗𝒗
𝒗𝒗
�⃗ ≤ 1
(12c)
�⃗. 𝒖𝒖
𝒖𝒖
�⃗ ≤ 1
(12b)
A Useful Combination: Expressing two 3-dimensional constraints by a single vector
It should be noted that eqns. (12b-c) lead to 5 possible combinations.
(i)
(ii)
(iii)
𝑢𝑢 = |𝒖𝒖
��⃗| = 1, 𝑣𝑣 = |𝒗𝒗
�⃗| = 1: Pure states.
|𝒖𝒖
�⃗| < 1, |𝒗𝒗
�⃗| = 1 and |𝒖𝒖
�⃗| = 1, |𝒗𝒗
�⃗| < 1: Mixed States
|𝒖𝒖
�⃗| < 1, |𝒗𝒗
�⃗| < 1, | 𝒖𝒖
�⃗| < |𝒗𝒗
�⃗| or | 𝒖𝒖
�⃗| > |�𝒗𝒗⃗|: Mixed States
For the purpose of expressing the two length constraints by constraints on a single vector, we
define a new vector as a simple combination of the two.
�𝑲𝑲
�⃗ = 𝟏𝟏 𝜶𝜶 (𝒖𝒖
�⃗ + 𝒗𝒗
�⃗)
𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
(14a)
𝟐𝟐
Its length is given by
��⃗| =
𝐾𝐾 = |𝑲𝑲
𝟏𝟏
𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
𝜶𝜶
𝟐𝟐
√𝑢𝑢2 + 𝑣𝑣 2 + 2𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢,
(16)
Here 𝛼𝛼 is the angle between the two vectors �𝒖𝒖⃗ and 𝒗𝒗
�⃗. The maximum and minimum values of
the new vector are obtained as
(i)
(ii)
𝐾𝐾𝑚𝑚𝑚𝑚𝑚𝑚 = 1 corresponds to
𝑢𝑢 = |��⃗
𝒖𝒖| = 1, 𝑣𝑣 = |𝒗𝒗
�⃗| = 1: Pure states.
𝐾𝐾 < 1 corresponds to
|𝒖𝒖
�⃗| < 1, |𝒗𝒗
�⃗| = 1, and|𝒖𝒖
��⃗| = 1, |𝒗𝒗
�⃗| < 1: Mixed States
(iii)
|𝒖𝒖
�⃗| < 1, |𝒗𝒗
�⃗| < 1, | 𝒖𝒖
�⃗| < |𝒗𝒗
�⃗| , and|𝒖𝒖
�⃗| > | 𝒗𝒗
�⃗|: Mixed States
𝐾𝐾𝑚𝑚𝑚𝑚𝑚𝑚 = 0 corresponds to
𝑢𝑢 = 0 𝑎𝑎𝑎𝑎𝑎𝑎 𝑣𝑣 = 0: Maximum mixed states
5
It should be noted that this vector does not capture the full 6 degrees of freedom represented by
vectors ��⃗
𝒖𝒖 and �𝒗𝒗⃗. Still it is a useful combination for deciding the purity of a state.
Visualization of the qutrit states in 3-dimensions
The way to visualize the qutrit states is as follows.
(i)
The states are confined inside the positive octant of a 3 dimensional sphere of unit
radius. This is due to the fact that the components of the vectors are always positive
by construction.
(ii)
The pure qutrit states are characterized by points on the surface whose distance from
the center is given by 𝐾𝐾 = 1. It is bounded by 0 ≤ 𝛼𝛼 ≤
𝜋𝜋
2
𝜋𝜋
and 0 ≤ 𝜒𝜒 ≤ 2 . It should
be noted that the “pure qutrit state surface” thus obtained is the surface of the positive
octant of the sphere and not the whole sphere. It is somewhat similar to the case with
the Bloch ball of a qubit.
(iii)
The mixed states reside inside the volume interior to the “pure qutrit state surface”.
This is similar to the Bloch ball of a qubit.
(iv)
The maximum mixed state resides at the center.
4. CONCLUSIONS
In this work we have shown that after applying the positivity constraints to the qutrit density
matrix one is left with 6 degrees of freedom. The resulting 6-dimensional qutrit state space
can be alternatively visualized in three dimensions using 2 three-dimensional vectors. This is
yet another approach to this problem alongside a similar one in [5,6] and also to the idea of
hyper-entanglement [7,8]. In future, the effect of unitary transformations on these vectors and
relation to Heisenberg-Weyl algebra will be investigated. The problem of applying these ideas
to the higher-dimensional Hilbert spaces of qudits and effect of quantum error correcting
codes will be explored as well.
5. ACKNOWLEDGEMENT
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I thank Jacob Taylor (NIST and QuICS, University of Maryland) for encouragement and
discussions.
6. REFERENCES
[1] S. K. Goyal, B. N. Simon, R. Singh, and S. Simon, “Geometry of the generalized Bloch
sphere for qutrits “, arXiv: 1111.4427
[2] Istok P. Mendas, “The classification of three-parameter density matrices for a qutrit”, J.Phys.
A: Math. Gen. 39 (2006) 11313-11324
[3] G. Sarbicki and I. Bengtsson, “Dissecting the Qutrit “, J. Phys. A: Math. Theor. 46, 035306
(2013)
[4] Ingemar Bengtsson, Stephan Weis and Karol Zyczkowski, “Geometry of the set of mixed
quantum states: An apophatic approach“, arXiv: 1111.4427v2
[5] Pawel Kurzynski, Adrian Kolodziejski, Wieslaw Laskowski, and Marein Markiewicz,
“Three-dimensional visualization of a qutrit”, arXiv:1601.07361v1
[6] Pawel Kurzynski, “Multi-Bloch Vector Representation of the Qutrit”, arXiv: 0912.3155v1
[7] Omar Gamel, “Entangled Bloch Spheres: Bloch Matrix and Two Qubit State Space”,
arXiv:1602.01548v1
[8] Zhenda Xie, Tian Zhong, Sajan Shrestha, XinAn Xu, Junlin Liang, Yan-Xiao
Gong, Joshua C. Bienfang, Alessandro Restelli, Jeffrey H. Shapiro, Franco N. C. Wong,
and Chee Wei Wong “Harnessing high-dimensional hyperentanglement through a biphoton
frequency comb”,Nature Photonics 9, 536–542 (2015)
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