Entanglement in the stabilizer formalism
Wouter Borg July 15, 2018
Project Bachelor thesis
Supervisors Michael Walter and Freek Witteveen Study Physics- & Astronomy
Studentnumber 10801081
Abstract
The stabilizer formalism can efficiently describe a subclass of quantum states closed under contraction and subsystems. In this bachelor thesis the stabilizer formalism is worked out and efficient algorithms for the entropy and contraction of stabilizer states are given. The entropy of random stabilizer states is discussed. As a next step, using the algorithms in this bachelor thesis, the entropy of random tensor networks of different configurations can be efficiently calculated. This can lead to a better understanding of holographic duality.
Contents
1 Samenvatting 3 2 Introduction 3 3 Stabilizer formalism 4 3.1 Pauli group . . . 5 3.2 Weyl operator . . . 6 3.3 Stabilizer group . . . 7 3.4 Stabilizer code . . . 7 3.5 Stabilizer subspace . . . 83.6 Example of stabilizer state . . . 12
3.7 Random stabilizer state in SageMath . . . 13
4 Entropy 14 4.1 Entropy of stabilizer states . . . 14
4.2 Example of entropy of qudits . . . 16
4.3 Calculating entropy in SageMath . . . 16
4.4 Random entropy . . . 16
5 Contraction 18 5.1 Contraction of qudits . . . 18
5.2 Contraction in SageMath . . . 21
5.3 Example of contraction . . . 21
5.4 Contraction of tensor networks. . . 22
6 Conclusion 23
1
Samenvatting
Om berekeningen te doen aan grote quantum systemen is zeer veel computerkracht nodig. Dit stijgt exponentieel als het systeem groter wordt. In het stabilizer formalisme kan een interessante groep quantum toestanden veel effici¨enter beschreven worden. Hier stijgt de benodigde computerkracht slechts lineair met de grootte van het systeem. In deze bachelor thesis wordt dit stabilizer formalisme beschreven. Ook worden algoritmes gegeven waarmee enkele interessante eigenschappen van quantum toestanden beschreven met het stabilizer formalisme verkregen kunnen worden.
2
Introduction
Quantum mechanics involves large computations. The source of these large computations lies in the exponential growth of the dimension of the Hilbert space with system size. The conventional way to describe a pure quantum state is with a vector of norm one in Hilbert space. For a state of n qubits the dimension of the Hilbert space is 2n. There are methods which can represent interesting subclasses of quantum states much more efficiently. The sta-bilizer formalism, developed by Daniel Gottesman is such a formalism [1]. This formalism defines quantum states by the set of operators in the Pauli group for which this state is an eigenvector with eigenvalue one. States in this subclass can be represented with only O(n2) bits, which is very few compared to Hilbert spaces of dimension 2n. Although this formalism describes only a small subclass of states, this is an interesting subclass. One reason for this is that this subclass is closed under some interesting operations including the contraction of two states. It is also closed under Clifford unitaries, making the stabilizer formalism very useful in quantum error correction [2-5]. Also, a subsystem of a state described by the stabilizer formalism can also be described by the stabilizer formalism. Using this the entanglement en-tropy of stabilizer states can be efficiently calculated. Another thing that makes the stabilizer formalism interesting is that stabilizer states are almost normally distributed under quantum states so that a random stabilizer state shares much of the properties of a random quantum state. The stabilizer formalism can be used for many purposes, including condensed matter [6] and, as already mentioned, quantum error correction [2-5]. It can also be used for tensor networks. With the stabilizer formalism we can efficiently describe these tensor networks. These tensor networks are useful for the understanding of holographic duality [7]. This is mainly the case because the entropy of a tensor network is bounded by an area law [7]. In this bachelor thesis, we will first describe the stabilizer formalism and give an algorithm which generates a random pure state described by the stabilizer formalism (a stabilizer state). This
will be done in Chapter 2. In Chapter 3 we will describe how to calculate the entropy of a stabilizer state and calculate the entropy of random stabilizer states. In the next chapter we will describe how to contract two stabilizer states and discuss how to make a tensor network with the stabilizer formalism. Finally we summarize and discuss what interesting next steps could be taken following this work.
3
Stabilizer formalism
As mentioned in the introduction, quantum states can be defined by the operators stabilizing them. An operator A is said to stabilize the quantum state |ψi when this state is an eigenvector of A with eigenvalue 1:
A |ψi = |ψi . (1)
If we only look at the operators in a specific group, this restricts the set of states we can describe. This can be very interesting because it can hugely decrease computation time of models and simplify calculations. In the stabilizer formalism the group used is the Pauli group. So in the stabilizer formalism, a quantum state is defined through a subgroup of the Pauli group from which all elements stabilize the state. This formalism can describe highly entangled states and is used in many areas including quantum error correction [2-5] and condensed matter [6]. It was originally proposed for qubits [1], but can be generalized to qudits [8], which are quantum states in a d dimensional Hilbert space. In this bachelor thesis we will only look at odd d, since for odd d all calculations can be done modulo d (instead of sometimes mod d and sometimes mod 2d with even d). So all additions in this chapter will be modulo d. This will simplify the calculation and programming. We also demand d to be a prime. The finite field Fd with elements 0, 1, . . . , d − 1 is used to do calculations modulo d.
In this chapter we will describe the stabilizer formalism. We will first define the Pauli group and introduce the Weyl operator to represent elements in this group. After this, we define the stabilizer group and stabilizer code and use the Weyl operators to describe the stabilizer group with a vector space. Thereafter a formula for the dimension of the stabilizer code is given and proved. As an example the GHZ state is worked out. Finally an algorithm for a random stabilizer state is given.
3.1 Pauli group
To define the Pauli group, we first look at the Pauli matrices. We define these Pauli matrices Z, X and Y for qudits which can be in d states:
Z = d−1 X j=0 ωj|ji hj| , X = d−1 X j=0 |j + 1i hj| , Y = τ X†Z† (2)
where ω is the dth root of unity and τ is the square root of ω
ω = e2πi/d, τ = eiπ(d2+1)/d. (3)
For d = 2 these are the well-known Pauli matrices for qubits. However, here we only look at odd prime d. In this case τ is the square root of ω:
τ = ω2−1 (4)
Another property of ω is that the dth power of ω is one:
ωd= 1. (5)
The state |ji represents the different states of one qudit. If we work on such a qudit state |ji with X, Z we get
Z |ji = ωj|ji X |ji = |j + 1i . (6)
We will now give some properties of these Pauli matrices. The Y is defined so that XY Z = τ I. The dth power of both X and Z gives the identity operator and both X and Z are unitary:
Xd= I X−1 = X† Zd= I Z−1= Z†. (7)
The commutation of X and Z is given by
XaZb = ω−abZbXa. (8)
We use these Pauli matrices to define the Pauli group.
Definition 1 (Pauli group for one qudit). The Pauli group for one qudit Pd1 is the group with as group operation matrix multiplication generated by the Pauli matrices X, Y and Z for one qudit: Pd1 =X, Z, Y .
Using that X and Z are unitary (Equation (7)) and the definition of Y (Equation (2)) we see the Pauli group is also generated by X, Z and τ I: P1
d =X, Z, τ I . Before we give the
definition of the Pauli group for n qudits, we will introduce a way to represent elements of the Pauli group. This we do with Weyl operators.
3.2 Weyl operator
The Weyl operator [9] for one qudit is defined as
Definition 2 (Weyl operator). The Weyl operator Wa,b for one qudit is given by Wa,b =
τabZaXb for a, b ∈ Fd and with X and Z the Pauli matrices for one qudit.
Any product of the generators X, Z and τ I of P1
d can be written as a Weyl operator (the
number of X and Z’s) multiplied by a phase ω−φ (the phase accounts for the number of τ I as well as the order of X and Z’s and the phase given by the Weyl operator). This means every element in the Pauli group can be represented by a Weyl operator and a phase. The same can be done for n qudits. We first define the Pauli group for n qudits. This group consists of n fold tensor products of elements of Pd1, each working on a different qudit. The Pauli group for n qudits is the group of all of these tensor products:
Definition 3 (Pauli group for n qudits). The Pauli group for n qudits Pdn is a group with group operation matrix multiplication. It consists of all possible n fould tensor products of elements of the Pauli group for one qudit: Pdn= {A1⊗ A2⊗ · · · ⊗ An, A1, . . . , An∈ Pd1}.
The Weyl operators can also be used to represent an element in the Pauli group for n qudits. A Weyl operators for n qudits is a n-fold tensor poduct of Weyl operators for one qudit. Instead of two numbers a and b in Wa,b we use two vectors a = (a1, a2, . . . , an) ∈ Fnd
and b = (b1, b2, . . . , bn) ∈ Fnd where ak and bk are the a and b for the Weyl operator working
on the kth qudit. We introduce a third vector v made from the first two v ≡ a ⊕ b, so v ∈ F2nd . The Weyl operator for n qudits is now Wv.
Definition 4 (Weyl operator for n qudits). The Weyl operator for n qudits is given by Wv = Wa,b = Wa1,b1⊗· · ·⊗Wan,bnfor the vectors a = (a1. . . an) ∈ F
n
d and b = (b1. . . bn) ∈ Fnd.
Together these vectors form v = a ⊕ b ∈ F2nd .
The multiplication of two Weyl operators gives another Weyl operator and a phase [9]:
WvWw= τ[v,w]Wv+w (9)
where [v, w] is the symplectic inner product, defined as [v, w] ≡ vTΩw with Ω the 2n x 2n
matrix given by Ω ≡ " 0 I −I 0 # . (10)
For odd d, the multiplication of Weyl operators can be done modulo d. However for even d the exponent of τ must be evaluated modulo 2d. But here we only look at odd (and prime) d.
As with one qudit, every element in Pdncan be made by the multiplication of a Weyl operator for n qudits and a phase ω−φ. So every element in Pdn can be represented with a vector v of length 2n and a number φ.
3.3 Stabilizer group
We use the Pauli group to define a quantum state (pure or mixed) using the intersection of elements of the operators stabilizing the state (the state is an eigenstate with eigenvalue 1) and the Pauli group. We will represent this intersection using the stabilizer group. The stabilizer group is defined by
Definition 5 (Stabilizer group). A subgroup G ⊆ Pn
d is called a stabilizer group if ti is an
abelian subgroup of the Pauli group containing no other multiples of I than I.
This definition is useful because for any subgroup of Pdn it holds that if there is some quantum state which is stabilized by all elements in that group, this group is a stabilizer group.
Lemma 3.1. Suppose G ⊆ Pn
d and suppose ∃ |Ψi ∈ (Cd)⊗n such that |Ψi 6= 0 and g |Ψi =
|Ψi ∀g ∈ G, then G is a stabilizer group.
Proof. G is a stabilizer group iff it is an abelian subgroup of the Pauli group containing no multiples of I other than I. By definition, G is a subgroup of Pdn. A subgroup is abelian
if all its elements commute with each other. From Equation (8) follows that all elements in Pdn commute up to some power α of ω. For any A, B ∈ G using AB = ωαBA we find (AB − BA) |Ψi = (1 − ωα) |Ψi and (AB − BA) |Ψi = |Ψi − |Ψi = 0 Because |Ψi is nonzero,
(1 − ωα) must be zero and hence ωα = 1. So AB = BA and hence A and B commute. This must hold for all A, B ∈ G because A and B were arbitrary and hence G is an abelian subgroup of the Pauli group. If some multiple of I other than I, say βI with β some complex number other than zero or one, is in G the state |Ψi must be stabilized by βI, so βI |Ψi = |Ψi. But since βI |Ψi = β |Ψi and β is not equal to one or zero we find |Ψi = 0 and this is contradictory to our assumptions. So G does not contain multiples of I other than I.
3.4 Stabilizer code
After having defined the stabilizer group we will now look at the quantum states which are described by such a group. We will call the set of these states a stabilizer code. The stabilizer code V is defined as the subspace of (Cd)⊗n in which every element is stabilized by every element of a stabilizer group G.
Definition 6 (Stabilizer code). The stabilizer code V (G) of a stabilizer group G is the subset of quantum states stabilized by all elements in G.
V (G) = {|Ψi | g |Ψi = |Ψi ∀g ∈ G} (11)
If the dimension of the stabilizer code is one, a quantum state is uniquely defined by the corresponding the stabilizer group. States that can be uniquely described in this manner we call stabilizer states.
Definition 7 (Stabilizer state). A stabilizer state is an one dimensional stabilizer code.
3.5 Stabilizer subspace
From Equation (9) it follows that two Weyl operators Wv and Ww commute iff [v, w] =
0. From Definition 5 of the stabilizer group we know all elements in a stabilizer group commute with each other and hence the multiplication of Weyl operators in a stabilizer group corresponds to addition (in F2nd ) of the vectors:
WvWw= τ0Wv+w= Wv+w. (12)
Using a vector v and a phase φvas representation of an element in the stabilizer group G, the
collection of vectors for a stabilizer group G is an isotropic subspace of F2nd . Together with a
stabilizer vector representing the phases such a subspace represents the stabilizer group. We define the stabilizer subspace and vector as:
Definition 8 (Stabilizer subspace and vector). The stabilizer subspace M (G) for a stabilizer group G is M (G) = {v ∈ F2nd | ωφvWv ∈ G} and the stabilizer vector xG ∈ F2nd is defined by
φv = [v, xG]
Here [v, x] is the symplectic inner product as defined before. Because of the nondegeneracy of this symplectic inner product we always can find an xG that meets the requirement φv=
[v, xG]. The vectors in a basis of M correspond to a minimal set of generators for G. Notice
that we recover the stabilizer group from M as
G = {Wvω−[v,x]| v ∈ M }. (13)
We will now give a formula for the dimension of a stabilizer code using its stabilizer subspace. This is interesting because the dimension of a stabilizer code of a pure state one.
Theorem 3.2. The dimension of a stabilizer code V for n qudits with stabilizer subspace M is Dim(V ) = |M |dn
Proof. We can project a state in a stabilizer subspace M onto its stabilizer code with the projection operator ρ∗.
Lemma 3.3. Given a stabilizer code V of the stabilizer group G then
ρ∗ = 1 |G| X g∈G g (14) is a projection onto V (G)
Proof. We first prove ρ∗ is a projection. To prove this, we have to prove (ρ∗)2 = ρ∗ and
(ρ∗)†= ρ∗. We first prove ρ∗ is unitary. Because all elements in G consists of powers of X, Z and τ I, and these are all unitary (see Equation (7)), all elements in G are also unitary and hence ρ∗ is unitary. For (ρ∗)2 we get
(ρ∗)2 = 1 |G|2 X g∈G X g0∈G g · g0 (15)
For some ga, gb, gc ∈ G, we know a product of these is also in G (G is a group with
group operation multiplication). Also, we find ga· gb = ga· gc iff gb = gc. We prove this
by contradiction. So we start with ga· gb = ga· gc and gb 6= gc. Then ga· gb and ga· gc
must have the same inverse. The inverse of a element in an abelian group is unique. Because ga−1· gb−1ga· gb= I we find I = g−1a · g−1c ga· gb = g−1c · gb and we find gb = gc. So ga· gb = ga· gc
iff gb = gc. So we find for a sum over all elements in G:
X
g∈G
g · g0 =X
g∈G
g. (16)
and Equation (15) reduces to
(ρ∗)2= 1 |G|2 X g∈G X g0∈G g = 1 |G| X g∈G g = ρ∗ (17)
This means ρ∗ is a projection operator.
To prove ρ∗ projects onto V (G) we show iff a state is stabilized by ρ∗, it is in V (G). Per definition a state is in V (G) iff it is stabilized by all elements in G. If we work with the operator |G|1 P
g∈G
g onto some state |ψi in V (G) we find:
1 |G| X g∈G g |ψi = 1 |G| X g∈G |ψi = |G| |G||ψi = |ψi . (18)
So ρ∗ stabilizes every element in V (G). If one of the elements in G does not stabilize a state |πi (so |πi is not in V (G)), it either gives a different state or some power of ω (or both).
Because the real part of powers of ω is never bigger than one, the factor in front of |πi will always be smaller than |G| and hence ρ∗ will never stabilize |πi . So ρ∗ is a projection onto V (G)
The density matrix ρ of the system is proportional to ρ∗ according to
ρ = ρ
∗
Dim(V ). (19)
So using Lemma 3.3 we can write ρ∗ as
ρ∗ = 1 |G| X g∈G g = 1 |M | X v∈M Wvω−[v,x]. (20)
From the projection operator we can calculate the dimension of the subspace V by
Dim(V ) = Tr(ρ∗) = Tr 1 |M |
X
v∈M
Wvω−[v,x]. (21)
To calculate this dimension we need to take the trace of a Weyl operator. This we do using the following two lemma’s.
Lemma 3.4.
X
k∈Fd
ωkp = 0 (p 6= 0) (22)
for ω = e2πi/d the dth root of unity and some nonzero number p ∈ Fd.
Proof. ωp X k∈Fd ωkp = ωp+ ω2p+ ω3p+ · · · + ωdp = 1 + ωp+ ω2p+ · · · + ω(d−1)p= X k∈Fd ωkp (23)
Since ωp is not equal to one we find P
k∈Fd
ωkp= 0
Lemma 3.5. The trace of a Weyl operator is nonzero if it is a multiple of the identity operator.
Proof. First, we use that the trace of a tensor product is the product of the traces of the factors in the tensor product: Tr(A1⊗ A2⊗ · · · ⊗ An) = Tr(A1)Tr(A2) . . . Tr(An). It is thus
sufficient to prove that the trace of a Weyl operator for one qudit is nonzero if and only if it is a multiple of the identity operator. We divide in four cases, one where the power of both the Z and X Pauli matrices is zero , one where the power of only the X Pauli matrix is zero, one where the power of only the Z Pauli matrix is zero and one where the power of both the Z and X Pauli matrices is nonzero. In the first case we get the identity matrix and the trace
is d. In the second case, using Lemma 3.4, we get: Tr(Zα) = P
k∈Fd
ωkα= 0. In the third case we only have powers of X. Since X maps every state to the next, it has no diagonal elements. If we apply X multiple times, it keeps on mapping to different states (and so the power of X stays zero on the diagonal unless Xk is the identity).
Xk=
d−1
X
j=0
|j + ki hj| (24)
The trace of Xk becomes
Tr Xk = d−1 X i=0 d−1 X j=0 hi|j + ki hj|ii (25)
This is only nonzero if k = 0 (mod d), and in that case Xk is equal to the identity operator. For the last case we use a that power of Z is always a diagonal matrix and a power of X has no diagonal elements. So a product of powers of X and Z will also have no diagonal elements and hence the trace of this will be zero. So the trace of a Weyl operator is always zero, unless the Weyl operator is the identity operator.
From Lemma 3.5 we get that the trace of a Weyl operator is 0 unless the Weyl operator is the identity operator. Hence only the I1⊗ I2· · · ⊗ In term in Equation (21) contributes
which gives a factor dn. We also see that [v, x] = 0 if v=0 (which it always is if Wv= I). So
we get for the dimension of the stabilizer code
Dim(V ) = 1 |M |Tr X v∈M Wvω−[v,x] = dn |M |. (26)
From Theorem 3.2 we find for a pure state, where the dimension of the stabilizer code V is 1, dn= |M |. This means we only need n generators for M and thus only n + 1 (including
the stabilizer vector x) vectors of length 2n to define a pure stabilizer state. This can be done using a stabilizer tableau T , which is a matrix of the generator vectors vi and their phases
φi.
Definition 9 (Stabilizer tableau and Weyl block). A Weyl block B(M ) for a stabilizer sub-space M and vector x is a matrix of the generators v1, . . . , vk of M :
B(M ) = v1 .. . vk (27)
A stabilizer tableau T (M, x) is a Weyl block with an extra colomn consisting of the phases φ1, . . . , φk given by φi= [vi, x] : T (M, x) = v1 φ1 .. . ... vk φk (28)
Each row in this stabilizer tableau stands for a generator Wviω
−φ of the stabilizer group.
These generators are independent iff the vectors v1. . . vk are linearly independent, and
com-mute iff [vi, vj] = 0 (∀i, j) as is clear from Equation 9. The number of bits needed to describe
a stabilizer state in this manner is proportional to 2n2+ 2n and thus grows quadratic with n. This is, for big n, a huge difference with the normal way to describe a quantum state, which grows exponentially with n.
3.6 Example of stabilizer state
As an example, we will look at the maximally entangled stabilizer state of three qudits with Hilbert space of dimension 3 |ψi = |0 0 0i + |1 1 1i + |2 2 2i. This is the Greenberger-Horne-Zeilinger state GHZ3 for three qudits. We want to represent this state |ψi as a stabilizer
tableau T , so that T only represents the state |ψi. Following Theorem 3.2 we need three independent Weyl operators with |ψi as eigenvector to form a Weyl block. The eigenvalues of these operators become the phase factors which we can use to make a stabilizer tableau. In the computational basis for d = 3 the X and Z matrices working on one qudit are:
X = 0 0 1 1 0 0 0 1 0 Z = 1 0 0 0 ω 0 0 0 ω2 .
By inspection, we find three independent operators stabilizing |ψi.
(X ⊗ X ⊗ X) |ψi = |ψi (29)
(Z1⊗ Z2⊗ Z3) |ψi = |ψi (30)
(Z3⊗ Z2⊗ Z1) |ψi = |ψi (31)
Because the eigenvalues are all 1, the stabilizer vector is zero. The stabilizer tableau for |ψi is T (|ψi) = 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 2 1 0 0 0 0 . (32)
We can check whether the rows are linearly independent, as they should be and whether the operators commute by checking if [B1, B2] = 0, [B1, B3] = 0 and [B2, B3] = 0 (where Bi is the
ith row of the Weyl block B of |ψi) and find that this is indeed the case. The Weyl block B forms a basis of the stabilizer subspace of the state. So the stabilizer subspace M for |ψi has as basis vectors [0,0,0,1,1,1], [1,2,0,0,0,0] and [0,2,1,0,0,0] and stabilizer vector x = (0, 0, 0, 0, 0, 0) This example can be easily generalized to higher (prime) dimensions, thus for the stabilizer state ψn = |0 0 0i + |1 1 1i + · · · + |n − 1 n − 1 n − 1i we will find as generators for
the stabilizer group X ⊗ X ⊗ · · · ⊗ X, Z1⊗ Z2⊗ · · · ⊗ Zn and Zn⊗ Zn−1⊗ · · · ⊗ Z1.
3.7 Random stabilizer state in SageMath
It is interesting to look at the properties of random stabilizer states because a random stabi-lizer state acts as a random quantum state. Here we will choose a random stabistabi-lizer state by generating a random stabilizer tableau. A random Weyl block is a basis of a random stabilizer subspace. To make a pure stabilizer state of n qudits we use Theorem 3.2 and find we only need n linearly independent vectors as generators for the stabilizer subspace. So a basis of a stabilizer subspace must consist of n nonzero linear independent vectors between which the symplectic inner product is zero (so all operators commute). To make a random stabilizer tableau we add a colomn of random numbers to the Weyl block. In Algorithm 1 we give the pseudocode of the program used to generate a random stabilizer tableau. In this algorithm we start with a random nonzero matrix H with one row representing one generator, generate a random nonzero vector v from the kernel of ΩHT so to make sure the generators commute (Ω is the matrix used in the symplectic inner product). If the rows of H and v are linearly independent we add v as a row to H. We repeat this until H has rank n. Next, we choose a random vector φ ∈ F2nd representing the phases. Now [H|φ] is a random stabilizer tableau.
Algorithm 1 Random stabilizer state input: Number of qudits n
Dimension of Hilbert space d
output: Tableau for a random stabilizer state of n qudits initialize: Random nonzero 1 x 2n matrix H over Fd
while rank H < n:
Generate random vector v from nullspace of ΩHT
if v is not in the rowspace of H: add v as a row to H
φ ← Random vector of length 2n over Fd
4
Entropy
4.1 Entropy of stabilizer states
The entanglement entropy is an important property of a quantum state. An algorithm calcu-lating the entanglement entropy of a stabilizer state can be derived and turns out to be very efficient [10]. After deriving such an algorithm we will discuss an example and look at the entropy of random stabilizer states. First we give the definition of the entropy of a quantum state.
Definition 10 (Von Neumann entropy). For a quantum state with density matrix ρ the entropy S(ρ) is given by
S(ρ) = −Tr ρ log2(ρ). (33)
We can rewrite this in terms of the eigenvalues λ1, . . . , λk of ρ and the dimension of the
stabilizer code V of the system using ρ has Dim(V ) nonzero eigenvalues:
S(ρV) = −Tr ρ log2(ρ) = − Dim(V )
X
i=1
λi log2(λi). (34)
The terms for zero eigenvalues give zero. The density matrix ρ is the projection operator ρ∗ multiplied with a normalization factor of Dim(V )1 , as in Equation (19). Because the eigenvalues of the projection operator are all equal to one or zero, the nonzero eigenvalues λi of ρ are all
equal to Dim(V )1 . Using this the expression for the entropy becomes
S(ρV) = − Dim(V ) X i 1 Dim(V )log2 1 Dim(V ) = log2 Dim(V ). (35)
For a pure state the dimension of the stabilizer code is one. This gives a entropy of zero for a pure state, as expected. Given a pure state we can divide this in two subsystems A, B ⊂ {1, . . . , n} of respectively a and b qudits. We demand these subsystems to have only different qudits, so A ∩ B = 0. Together A and B have to give the whole system, so A ∪ B = {1, . . . , n}. So A, B are a partition of the system. The Hilbert space of the whole system is the tensor product of the Hilbert spaces of A and B.
H = HA⊗ HB (36)
The density matrix ρA of such a subspace is given by
ρA= TrB ρAB
For the entropy, it doesn’t matter which subspace we use since S(ρA) = S(ρB). S(ρA) is
a measure for the entanglement between the subsystems A and B [2]. We will now give a formula to calculate the stabilizer subspace and entanglement entropy of a subsystem of a stabilizer state.
Theorem 4.1. Given a pure stabilizer state |ΨABi with stabilizer subspace MAB of n qudits
divided in two subsystems A, B ⊂ {1, . . . , n} with respectively a and b qubits so that A ∪ B = {1, . . . , n} and A ∩ B = 0 (and n = a + b). Then ρA is a (mixed) stabilizer state with stabilizer subspace MA given by
MA= {vA| vAB∈ MAB, vB= 0}. (38)
The entanglement entropy of this subspace is given by
SA(|ΨABi) = a − Dim(MA)log2(d). (39)
Proof. To prove this, we first look at the density matrix ρAB of the whole system. If we write
out the projection on the stabilizer code of |ΨABi as given in Lemma 3.3 and use Theorem
3.2 this density matrix becomes
ρAB = ρ∗ Dim(V ) = 1 Dim(V ) 1 |MAB| X v∈M Wvω−[v,x]= d−n X v∈M Wvω−[v,x] (40)
From this we calculate the density matrix ρA of the subsystem A using Equation 37.
ρA= TrB ρAB = TrB d−n X v∈M Wvω−[v,x] = d−n X v∈M ω−[v,x]TrB Wv. (41)
We again use Lemma 3.5 for the trace of Weyl operators.
ρA= d−a X vAB∈M s. th. vB=0 ω−[vA,xA]W vA (42)
This means ρA is the density matrix of a mixed stabilizer state. We can also write ρA in
terms of its stabilizer subspace MA of subsystem A. The density matrix ρA is a projection
on the subsystem A divided by the dimension of the stabilizer code VA of this subsystem. So
we can write for ρA:
ρA= 1 Dim(VA) 1 |MA| X vA∈MA ω−[vA,xA]W vA. (43)
By comparing this with Equation (42) we find Dim(VA) = d
a
|MA|. We also see that MAconsists
of the vectors vA such that vAB ∈ MAB and vB= 0:
The entropy SA(|Ψi) becomes
SA(|Ψi) = log2(Dim(VA)) = a − Dim(MA)log2(d) (45)
where we have used |MA| = dDim(MA).
4.2 Example of entropy of qudits
As an example, we calculate the entanglement entropy of one qudit in the GHZ3 state |ψi =
|0 0 0i+|1 1 1i+|2 2 2i from Example 3.6. We choose the partition as A = {1} and B = {2, 3} and calculate the entropy using Equation (39). Because the GHZ3 state is pure, this entropy
is a measure for the entanglement between the first qudit and the other two. The stabilizer subspace MAB of |ψi is given in example 3.6 (here it was called M and was the span of
the vectors [0,0,0,1,1,1], [1,2,0,0,0,0] and [0,2,1,0,0,0]). The stabilizer subspace MA of the
subsystem A consists of vectors vA such that vAB ∈ MAB and vB= 0, as given in Theorem
4.1. In this example there is only one vector in MAB for which the second, third, fifth and
sixth entries are zero, and this is the zero vector (0, 0, 0, 0, 0, 0). So in this example MA is
the subspace with only the zero vector and hence its dimension is zero. Filling in a = 1 and d = 3 in Equation (39) gives:
SA(MAB) = (1 − 0)log2(d) = log2(d). (46)
We can also calculate the entropy of subsystem B, expecting the same value as we got for subsystem A. Again using Equation (38) we see MB consists of the vectors for which the
first and fourth entries is zero. This is the case for the vectors (0, 1, 2, 0, 0, 0), (0, 2, 1, 0, 0, 0) and (0, 0, 0, 0, 0, 0). So the dimension of MB is one and the entropy (with d = 3 and b = 2)
becomes
SB(MAB) = (2 − 1)log2(d) = log2(d) = SA (47)
as expected.
4.3 Calculating entropy in SageMath
We calculate the entropy of a subsystem A of a stabilizer state in SageMath from the density matrix and a partition list. We do this using Equation (39). The pseudocode for this is given in Algorithm 2.
4.4 Random entropy
It is interesting to look at the entropy of a random stabilizer state. Using Algorithm 1 we generate random stabilizer states. We calculated the entropy of such a random stabilizer
Algorithm 2 Entropy of stabilizer state
input: Tableau of pure stabilizer state M of n qudits List of qudits in partition P
Dimension of Hilbert space d
output: Entanglement entropy of a subsystem of the stabilizer state initialize: 2n x 2n zero matrix K
for each qudit q in partition:
Set the qth and q + nth entrys on the diagonal of K to one. S ← (rank of K · M - number of qudits in P )log2(d)
return S
Figure 1: The mean entropy of random stabilizer states of n qudits with dimension of Hilbert space d. The partition is two subsystems of half of the qudits each. The blue dots are the average entropy of fifteen random stabilizer states with the same n and d. The entropy has been evaluated for random stabilizer states with all combinations of n ≤ 20 and a prime d ≤ 20.
state of n qudits with as partition half of the state using Algorithm 2. The mean entropy of a random stabilizer state has a lower bound of n2log2(d) − 1 [11]. Because a maximally entangled state has an entropy of n2log2(d) we expect to find values between these bounds. In Figure 1 the results are shown. We see that the results match the expectations. So most stabilizer states are highly entangled, which, among others, makes the stabilizer formalism an interesting formalism.
5
Contraction
5.1 Contraction of qudits
An important operation on a quantum state is the contraction with another state [7]. We divide a pure state |ΨiAB of n = a + b qudits in two subsystems A, B ⊂ {1, . . . , n} of respectively a and b qudits in the same way as in Chapter 3 (So with A ∪ B = {1, . . . , n} and A ∩ B = 0). The Hilbert space of this state is the tensor product of the Hilbert spaces of A and B, as in Equation 36. We describe this state with the vector space MAB ⊂ F2nd and
phase vector xAB ∈ F2nd as in Definition 8. We can contract this state with another pure state
|φiA, of a qudits which is described with vector space NA⊂ F2ad and phase vector yA∈ F2ad .
The Hilbert space of this state is the same as the Hilbert space of subsystem A of the state |ΨiAB. This contraction can be seen as a projection of |ΨiAB onto |φiAand can be written as hφA|ΨABi. We will show here the state resulting form this contraction |Ψ0i
B ≡ α hφA|ΨABi
(with α some normalization constant) is a stabilizer state as well with as Hilbert space the same as that of subsystem B of |ΨiAB, or zero.
Theorem 5.1. Given a pure stabilizer state |ΨiAB , of n = a+b qudits with vector space MAB
and phase vector xAB devided in two subsystems A, B ⊂ {1, . . . , n} so that A ∪ B = {1, . . . , n}
and A ∩ B = 0 and the pure stabilizer state |φiA, of a qudits with vector space NA and phase
vector yA, the contraction of these stabilizer states hφA|ΨABi gives a stabilizer state iff
[mA, xA− yA] = 0 ∀mA∈ NAs. th. (mA, 0) ∈ MAB. (48)
Otherwise it gives zero. Given QAB = MAB ∩ (NA⊕ F2bd) this stabilizer state has a stabilizer
subspace QB given by
QB= {mB |mAB ∈ QAB} (49)
and phase vector zB such that
Proof. We will proof this using density matrices. The density matrices for |ΨiAB and |φAi
are given by (using Equation (40)): |ΨABi hΨAB| = d−n X vAB∈MAB WvABω −[vAB,xAB] (51) |φAi hφA| = d−a X wA∈NA WwAω −[wA,yA]. (52)
These we use to write out the density matrix ρ0B of the state |Ψ0Bi.
ρ0B≡ Ψ0B Ψ0B = TrA |ΨABi hΨAB|φAi hφA| = d−a−n X vAB∈MAB X wA∈NA TrA (WwAω −[wA,yA]⊗ I B)WvABω −[vAB,xAB] (53)
We can use Equation (9) to multiply the Weyl operators, and get everything independent of A out of the trace:
ρ0B = d−a−n X
vAB∈MAB
wA∈NA
ω−[wA,yA]−[vAB,xAB]τ[wA,vA]W
vBTrA WwA+vA. (54)
Using Lemma 3.5 we find the trace is only nonzero if WwA+vA = I and in that case, the
trace gives a factor da. The Weyl operator W
wA+vA is only equal to the identity operator if
wA= −vA. Hence only the terms where [vA, wA] = [vA, −vA] = 0 are nonzero and we find:
ρ0B = d−n X vAB∈MAB wA∈NA s. th.wA=−vA ω−[wA,yA]−[vAB,xAB]W vB. (55)
To simplify the sum we introduce the map Π which maps a vector in the intersection between MAB and NA⊕ F2bd to vB:
Π : MAB∩ (NA⊕ F2bd ) → F2bd. (56)
The kernel of this map consists of all vectors vAB ∈ MAB for which vAis in NA and vB is 0:
Ker(Π) = MAB ∩ (NA⊕ {0}) ≡ ˜NA⊕ {0} (57)
where we have introduces ˜NA as the A part of the kernel of Π. When −wA is in the vector
space NA, wA is also in NA (everything is mod d, and thus −wA = (d − 1)wA). Hence we
can sum over vA∈ NA instead of wA∈ NA and replace the wA in the sum with −vA. For
the B part of vAB, we sum over the image of Π.
ρ0B= d−n X vB∈Im(Π) WvB ω −[xB,vB] X vAs. th. vA∈NA (vA⊕vB)∈MAB ω[yA−xA,vA] (58)
For every vB ∈ Im(Π) we choose a ˜vA such that ˜vA ∈ NA and (˜vA⊕ vB) ∈ MAB. Now we
can replace the second sum with a sum over vA∈ ˜vA+ ˜NA. Recall ˜NAconsists of the A part
of the vectors in the kernel of Π. ρ0B = d−n X vB∈Im(Π) WvB ω −[xB,vB] X vA∈˜vA+ ˜NA ω[yA−xA,vA] (59)
The second sum is now equal to the sum over v∗A∈ ˜NAwhere we replace vA by ˜vA+ v∗A.
= d−n X vB∈Im(Π) WvB ω −[xB,vB]+[yA−xA,˜vA] X v∗A∈ ˜NA ω[yA−xA,v∗A] (60)
The last sum is independent of vB, so we calculate it apart from the first sum. We divide
in two cases: one where [yA− xA, ˜NA] is zero and one where it is nonzero. In the first case,
the sum gives a factor of | ˜NA|. In this case [yA− xA, ˜vA] is independent of the choice of ˜vA
because [yA− xA, ˜vA] = [yA− xA, ˜vA+ v∗A] ∀v∗A∈ ˜NA and this covers all choices of ˜vA.
For the second case, we use Lemma 3.4. For every v∗A ∈ ˜NA, we know all multiples of v∗A
are in ˜NA: (αv∗A) ∈ ˜NA ∀α ∈ Fd. So the sum over v∗A∈ ˜NA consists of several sums over
α ∈ Fd of α times some vector ˆvA in ˜NA. Hence by taking p = [yA− xA, ˆvA] in Lemma
3.4 and using that [a, bc] = b[a, c] for vectors a, c and number b, we get for the second case (where [yA− xA, ˜NA] 6= 0):
X
v∗A∈ ˜NA
ω[yA−xA,v∗A]= 0 (61)
and hence |Ψ0Bi = 0.
When [yA− xA, ˜NA] = 0 and hence the last sum of Equation (60) gives | ˜NA|, we get for
(the unnormalized) ρ0B: ρ0B= d−n| ˜NA| X vB∈Im(Π) WvB ω −[xB,vB]+[yA−xA,˜vA]. (62)
|Ψ0Bi is now a stabilizer state, with the image of Π as the isotropic subspace of |Ψ0Bi. To find the phase vector zB such that
ρ0B = d−n X
vB∈Im(Π)
WvB ω
−[zB,vB] (63)
we can solve the equation
−zBΩBvB = −xBΩB vB+ (yA− xA) ΩA˜vA (64)
5.2 Contraction in SageMath
Algorithm 3 Contraction of stabilizer states input: Tableau of first stabilizer state M
Stabilizer vector of first stabilizer state x Tableau of second stabilizer state N
Stabilizer vector of second stabilizer state y Dimension of Hilbert space d
output: Stabilizer vector and tableau of the state resulting from the contraction of the two stabilizer states.
MAB ← the direct sum of the vector space spanned by M and Fbd (b is the rank of N ) with
as NAB ← the direct sum of Fad (a is the rank of M ) and the vector space spanned by N
˜
MA ← A basis of all the nonzero vectors in the intersection of MAB and NAB
If [YTΩXA] = 0:
return 0
Z ← the solution for z of −[z, vB] = −[xB, vB] + [(y − xA), vA] ∀v = vA⊕ vB∈ ˜MA(with
x = xA⊕ xB)
return ˜MA, z
We calculate the contraction of two states in SageMath using the vectors x,y and isotropic subspaces MAB and NA, representing two stabilizer groups using Theorem 5.1. The
pseu-docode for this algorithm is given in Algorithm 3.
5.3 Example of contraction
As an example we contract some states with the GHZ3 state which we used in the Examples
3.6 and 4.2. This is also used as a test for the program in SageMath. First we calculate the contraction of the GHZ3 state with the state |ζi = |0i + |1i, which also consist of qudits with
dimension of Hilbert space d = 3 and the qudit in this state is the same as the first qudit in the GHZ3 state. This contraction gives:
|ψi = h0| + h1|
|0 0 0i + |1 1 1i + |2 2 2i = |0 0i + |1 1i . (65) We can also calculate this with the stabilizer subspaces. Using Theorem 5.1 we find the sta-bilizer subset of the new state |ψi is the image of the map Π as defined in Equation (56). The stabilizes group of |ψi consist of the B part of the vectors in the stabilizer group of the GHZ3
state ,as given in the previous examples, for which the A part is in the stabilizer group of |ζi. The stabilizer group of |ζi consists of (0, 0), (0, 1), (0, 2) and hence the vectors in the stabi-lizer group of |ψi are (2, 1, 1, 1), (0, 0, 2, 2), (2, 1, 2, 2), (0, 0, 1, 1), (2, 1, 1, 1), (0, 0, 0, 0), (2, 1, 0, 0)
and (1, 2, 0, 0) which has a basis of (0, 0, 1, 1) and (1, 2, 0, 0). The phase vector of the GHZ3
state as well as the one from the state |0i + |1i are zero, so solving Equation (64) with xA= xB = yA= 0 gives zB = 0. A basis for the stabilizer subspace of |ψi is
"
0 0 1 1
1 2 0 0
#
and together with the phase vector (0, 0, 0, 0) this indeed uniquely gives the state |0 0i + |1 1i (again d=3 and the same two qudits as the first two in the GHZ3state). As a second example,
to involve the phase vectors more, we contract the GHZ3 state with the state |1 1i. This
contraction gives
|πi = h1 1| |0 0 0i + |1 1 1i + |2 2 2i = |1i . (66) The stabilizer subspace NA of |1 1i has a basis of (1, 0, 0, 0), (0, 1, 0, 0) and phase vector
(0, 0, 2, 2). Hence a basis for MAB ∩ (NA⊕ F2bd ) (with MAB the stabilizer subspace of GHZ3)
is (1,0,2,0,0,0), (0,2,1,0,0,0). The stabilizer group of |πi thus has a basis of (1, 0) and the phase vector is the solution of (filling in Equation (64))
−zB " 0 1 −1 0 # " 2 2 0 0 # = [0, 0, 2, 2] 0 0 1 0 0 0 0 1 −1 0 0 0 0 −1 0 0 1 0 0 1 0 0 0 0 (67)
which gives zB = (0, 2). This phase vector together with the basis of the stabilizer subspace
(1.0) give |πi = |1i as expected. The program in SageMath gives the same results.
5.4 Contraction of tensor networks.
The contraction discussed above is only a special case of a contraction where one state is fully contracted. More generally, the contraction of states is defined as
Definition 11 (Contraction). Given two quantum states |ΨABi and |ΦBCi their contraction
along the B system is given by the quantum state |φi with as components hac|φACi =X
b
hab|ΨABi hbc|ΦABi (68)
for all |ai and |ci quantum states working on respectively the subsystems of A and C. We can write this in terms of a full contraction (the contraction discussed in Section 5.1) using a maximally entangled state:
where
|χBB0i =
X
b
|bbi (70)
For stabilizer states |ΨABi and |ΦBCi the state |ΨABi ⊗ |ΦB0Ci is a stabilizer state and since
the maximally entangled state |χBB0i is a stabilizer state as well, we can use the special
contraction to calculate any contraction of stabilizer states. A tensor network can be made using stabilizer states and their contraction. In Figure 2 a simple example of a tensor network with two tensors (Ψ and Φ) is shown. The tensors are quantum states. The line connecting the two tensors stands for a contraction of the B subsystems of the tensors. If the tensors are stabilizer states, we can calculate the contraction using Theorem 5.1 in the way described above. In this way we can also calculate the entanglement entropy of a subsystem of the tensor network. For large tensor networks this works completely analogously.
Figure 2: A simple tensor network for stabilizer states |ΨABi and |ΦB0Ci
6
Conclusion
In this bachelor thesis the stabilizer formalism is described. Using this formalism we derived an algorithm calculating the entanglement entropy in a stabilizer state and the contraction between two stabilizer states. We also gave an algorithm generating random stabilizer states and saw the entropy of these random states behaved as expected. These algorithms can be used for many purposes. A next step could be to generate random tensor networks on different geometries by choosing random stabilizer states and contracting these. It would be interesting to see how the entropy behaves in these tensor networks, for example under which conditions the area bound for the entropy saturates. This can lead to better understanding of holographic duality.
7
References
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