Qubit stabilizer states are complex projective 3-designs

I Introduction and main results

In numerical integration, designs are known as cubatures. It follows from the definition that the average of a homogeneous polynomial $p$ of order $2t$ over the complex unit sphere equals $p$’s average over the design. If the design has small order, this realization can be made the basis for fast numerical procedures that compute integrals of smooth functions over high-dimensional spheres.

In quantum information theory, designs are a widely-employed tool for derandomizing probabilistic constructions. Recall that the probabilistic method alon_probabilistic_2004 is a powerful proof technique originally designed to tackle problems in combinatorics. At its core is the observation that the existence of certain extremal combinatorial structures often can be be proved by showing that a suitably chosen random construction would produce an example with high probability. In quantum information, randomized construction often rely on randomly chosen Hilbert space vectors hayden_randomizing_2004 . While this method has brought about spectacular successes (such as the the celebrated proof of strict sub-additivity of entanglement of formation hastings_superadditivity_2009 ), it suffers e.g. from the problem that generic Haar-random states of large quantum systems are unphysical: they cannot be prepared from separable inputs using a polynomial number of operations nielsen_quantum_2010 . Designs, in contrast, can be chosen to consist solely of highly-structured and efficiently preparable vectors, while retaining “generic” properties in a precise sense. Thus considerable efforts have been expended at designing complex projective designs (and their unitary cousins) ambainis_wise_2007 ; dankert_exact_2009 ; gross_evenly_2007 ; low_large_2009 ; brandao_local_2012 .

Lastly, randomized constructions in Hilbert spaces have completely classical applications, e.g. in signal analysis. Take for instance the highly active field of compressed sensing and related topics fora13 : There, one is interested in reconstructing objects that possess some non-trivial structure (e.g. sparsity, or low rank) from a small number of linear measurements. Strong recovery guarantees can be proven for randomly constructed measurement vectors. Once more, this raises the problem of finding sets of structured and well-understood measurements that sufficiently resemble the properties of generic random vectors. The use of designs for this purpose has been proposed in gross_partial_2015 ; ehler_phase_2015 ; kueng_spherical_2015 .

Despite this wealth of applications and non-constructive existence proofs bondarenko_spherical_2010 , explicit constructions for complex designs remain rare. There are varios infinite families of complex projective 2-designs (e.g. maximal sets of mutually unbiased bases klappenecker_mutually_2005 ; bengtsson_geometry_2006 , stabilizer states, or symmetric informationally complete POVMs renes_symmetric_2004 ); sporadic solutions for higher orders conway_sphere_2013 ; bachoc_modular_2001 ; gross_evenly_2007 ; and approximate constructions involving random circuits brandao_local_2012 . To the best of our knowledge, an infinite set of explicit complex projective 3-designs has not been identified before.

Recall that the stabilizer formalism is a ubiqutous tool in quantum information theory gottesman_stabilizer_1997 ; nielsen_quantum_2010 . Stabilizer states (and, slightly more general, stabilizer codes) are joint eigenvectors of generalized Pauli matrices. Constituting the main realization of quantum error correcting codes gottesman_stabilizer_1997 , they can be efficiently prepared hostens_stabilizer_2005 and described in terms of polynomially many parameters nielsen_quantum_2010 . Yet they exhibit non-trivial properties like multi-partite entanglement hein_entanglement_2006 . Stabilizer states were instrumental in the development of measurement-based quantum computation raussendorf_one_2001 ; gross_novel_2007 . In several precise ways, they can be seen as the discrete analogue of Gaussian states gross_hudsons_2006 . Beyond quantum information, stabilizer states have proved to be versatile enough to provide powerful models for one of the most influential recent development in theoretical condensed mater physics: the study of topological order kitaev_fault_2003 ; zeng_quantum_2015 .

Our main result thus identifies yet another aspect according to which stabilizer states capture properties of generic state vectors.

I.2 Designs and frame potential

In order to state our results more precisely, we need to give a formal definition of complex projective designs and introduce the related notion of frame potential. Following scott_tight_2006 ; Levenshtein_designs_1998 ; koenig_cubature_1999 , we define

where the right-hand-side integration is with respect to the uniform (Haar) measure on the sphere.

In other words, sampling according to $\mu$ should give the same expectation values as sampling according to the uniform measure for any random variable that is a polynomial in $|\langle x,y\rangle|^{2}$ of order at most $t$. From now on, we will only be concerned with the case where $\mu$ is the uniform measure on a finite set of unit vectors.

It is not hard to see that $\mu$ fulfills (1) for all polynomials of order $t$ or less, if equality holds for the specific case of $p(z)=z^{t}$. The resulting value is the $t$-th order frame potential benedetto_tight_2003

It is known that the Haar integral on the r.h.s. of (1) minimizes the frame potential over the set of all measures $\mu$ and that, in fact, its value is given by

This relation is known as Welch bound welch_lower_1974 or Sidelnikov inequality sidelnikov_upper_1975 . In summary, we have:

I.3 Main results

Comparing this explicit characterization of the frame potential to the Sidelnikov inequality (3) allows us to draw the following conclusions:

Let $d^{n}$ be a prime-power dimension. Then the following statements are true

$\operatorname{Stabs}(d,n)$ forms a complex projective 2-design.

$\operatorname{Stabs}(d,n)$ constitutes a complex projective 3-design if and only if $d=2$.

The set $\operatorname{Stabs}(d,n)$ does not constitute a complex projective 4-design.

As indicated before, the first fact was already widely known klappenecker_mutually_2005 ; bengtsson_geometry_2006 ; gross_evenly_2007 . The other results, however, are new to the best of our knowledge. We reemphasize that these assertions follow immediately form the Main Theorem, which may be of independent interest.

I.4 Applications and Outlook

Here, we sketch relations of the result to problems from signal analysis and quantum physics. Elaborating on these connections will be the focus of future work.

i.e. by setting $X=|x\rangle\langle x|$ and $A_{i}=|a_{i}\rangle\langle a_{i}|$. For both problems, strong recovery guarantees for randomly constructed measurements are known. Oftentimes these rely on generic (e.g. Gaussian) measurement ensembles and employing complex projective designs to partially derandomize these result has been proposed in both contexts gross_partial_2015 ; kueng_low_2015 ; ehler_phase_2015 .

Regarding both low rank matrix recovery and phase retrieval, it is known that sampling measurement vectors independently from a 2-design does not do the job gross_partial_2015 , while 4-designs already have an essentially optimal performance kueng_low_2015 ; kabanava_stable_2015 . However, the remaining intermediate case for $t=3$ is not yet fully understood. Numerical studies conducted by Drave and Rauhut drave_bachelor_2015 indicate that random stabilizer-state measurements perform surprisingly well at that task. The combinatorial properties of prime power stabilizer states – e.g. Theorem 2 – may help to clarify this situation. We believe this to be a potentially very insightful open problem.

Finally, we want to point out that one nice structural property of stabilizer states is that they come in bases, i.e. the set of all stabilizer states is a union of different orthonormal bases (see e.g. Theorem 3 below). This allows for a considerably more structured random measurement protocol: Select one such basis at random and iteratively measure the trace inner product of an unknown low rank matrix with all projectors onto the individual basis vectors. After having acquired $D$ data points that way, choose a new stabilizer basis at random and repeat. We refer to kueng_low_2015b for a detailed description of such a protocol. It should be clear that it has immediate applications to quantum state tomography. In the above paper, non-trivial recovery statements have been announced for $t$-designs that admit such a basis structure and have strength $t\geq 3$. Again, stabilizer states obey these criteria and have been used for the numerical experiments conducted there. However the announced recovery statement suffers from a non-optimal sampling rate for 3-designs and the rich combinatorial structure of stabilizer bases might help to amend that situation.

I.5 Relation to previous work and history

The appeal of the question treated here was underscored even more, when we learned a few days prior to submission of this paper to the arxiv e-print server, that yet another researcher – Zak Webb – had independently obtained results related to the ones of Zhu webb2015 .

In comparision to these works, our proof methods are completely different: We rely on counting structures in discrete symplectic vector spaces in order to compute the angle set between stabilizer states, whereas sidelnikov_spherical_1999 is based on algebraic invariant theory and zhu2015 on character theory. As a corollary, we derive an expression for the number of stabilizer states with prescribed inner product to a reference state. This finding might be of independent interest. Also, we show that the set of stabilizer states fails to be a 4-design in dimensions $2^{n}$ and that stabilizer states in dimensions other than powers of two do not even constitute a 3-design. The simultaneously submitted papers seem to have left this possibility open.

II Proof of the main statement

We already mentioned in the introduction that there is a geometric approach to stabilizer states building on the theory of discrete symplectic vector spaces This is connected to the fact that stabilizer states are the natural discrete analogue of Gaussian states of bosonic systems, where the symplectic structure is well-appreciated. For a concise introduction of this point of view, see gross_hudsons_2006 .. This phase space formalism will be introduced in subsection II.2. We formally define stabilizer states and explain how to compute inner products in this language in subsection II.3. We then move on to briefly introducing Grassmannians and some core concepts of discrete symplectic geometry. These tools will be used to establish Theorem 2 in Section III.

II.2 Phase Space Formalism

For $p,q\in Q$, the corresponding Weyl operator (or generalized Pauli operator) is defined as

Again following debeaudrap_linearized_2011 ; appleby_symmetric_2005 , we adopt the convention that any artihmetic expression in the exponent of $\tau$ is not understood to be modulo $d$, but rather as taking place in the integers. This makes a difference for even dimensions (see below). One could argue that it would be slightly cleaner to syntactically distinguish the modular operations appearing in (6) from the non-modular arithmetic in (7). However, the implicit convention does declutter notation and we feel it is ultimately benefitial.

This definition is consistent with established conventions. For example, one recovers the usual Pauli matrices for the qubit case $d=2$. We use the notation $V:=Q\times Q$ and consequently write $w(v):=w(v_{p},v_{q})$ for elements $v=(v_{p},v_{q})\in V$. Furthermore we define the standard symplectic form

and $u=(u_{p},u_{q}),v=(v_{p},v_{q})\in V$. If $d$ is prime, the space $V$ together with the non-degenerate symplectic product (8) forms a symplectic vector space which is called phase space due to its resemblance to the phase space appearing in classical mechanics.

The Weyl operators obey the composition and commutation relations

which can be verified by direct computation.

With elements $(p,q)\in V$, we associate Weyl operators

We conclude this section with two formulas that will be important in what follows and can both be verified immediately. First, the Weyl operators are trace-less, with the exception of the trivial one:

Second, for any vector $v\in V$ and any subspace $W\subseteq V$ one has

II.3 Stabilizer States

Here, we will cast the established theory gottesman_stabilizer_1997 ; nielsen_quantum_2010 of stabilizer states into the language of symplectic geometry required for our proof. For previous similar expositions, see gross_hudsons_2006 ; gross_stabilizer_2013 .

Note that Equation (10) implies that two Weyl operators $w(u)$ and $w(v)$ commute if and only if $\left[u,v\right]=0$. Now consider the image of an entire subspace $M\subseteq V$ under the Weyl representation. We define

and observe that $w(M)$ consists of mutually commuting operators if and only if $\left[m,m^{\prime}\right]=0$ holds for all $m,m^{\prime}\in M$. Spaces having this property are called isotropic. Assume now that $M$ is isotropic.

If $d$ is odd, then the $w(M)$ not only commute, but actually form a group $w(u)w(v)=w(u+v)$. That’s because in (9), the phase factor depends on $[u,v]$ modulo $d$, which is zero by assumption for $u,v\in M$. For even dimensions, however, $[u,v]$ might equal $d$ and in this case, the product $w(u)w(v)=-w(u+v)$ does not lie in $w(M)$ (in other words, $v\mapsto w(v)$ is only a projective representation of the additive group of $M$). This would create problems in our analysis below. Fortunately, it turns out that one can choose phases $c(v)\in\{\pm 1\}$ such that $v\mapsto c(v)w(v)$ does become a true representation of $M$. We will now describe this construction.

To this end, choose a basis $\mathcal{B}=\left\{u_{1},\ldots,u_{\dim M}\right\}$ of $M$. For a given element $m\in M$, let $m=\sum_{i}m_{i}u_{i}$ be the expansion of $m$ with respect to this basis. Define the (basis-dependent) Weyl operators to be:

Using the fact that the $w(u_{i})$ commute, one then obtains for $m,m^{\prime}\in M$

This is the desired representation of $M$.

Stabilizer states turn out to be related to maximal isotropic spaces $M$. We call a subspace $M\subseteq V$ Lagrangian (LAG) – or maximally isotropic – if every vector $v\in V$ that commutes with all elements of $M$ is already contained in $M$. This is precisely the case if

where $M^{\perp}$ denotes the symplectic complement of $M$. A basic result of symplectic geometry (e.g. Satz 9.11 in huppert ) states that this condition is fulfilled if and only if $\dim M=\frac{1}{2}\dim V=n$, or equivalently $|M|=d^{n}$.

We are now ready to state the relation between Lagrangian subspaces and state vectors in Hilbert space:

Let $M\subset V$ be a Lagrangian subspace, let $\mathcal{B}$ be a basis of $M$. Then the following assertions are valid:

Up to a global phase, every $v\in M$ singles out one unit vector $|M,v\rangle\in\mathcal{H}$ – called a stabilizer state that fulfills the eigenvalue equations

Two elements $u,v\in M$ define the same stabilizer state if and only if they belong to the same affine space $\left[v\right]_{M}:=\left\{v+m,\;m\in M\right\}$ modulo $M$. If this is not the case, the resulting stabilizer states are orthogonal, i.e. $\langle M,u|M,v\rangle=0$.

This statement implies that each stabilizer state is uniquely characterized by a Lagrangian subspace $M\subset V$ and one particular affine space $[v]_{M}$ modulo $M$. In the remainder of this article it will be convenient to represent each such affine space by a representative $\zeta\in[v]_{M}\in V$ contained in it. We have opted to denote such representatives of cosets $\zeta,\iota\in V$ by greek letters to notationally underline their origin.

where we have employed (11). The first relation implies that $\rho_{M,v}$ is a projection and the second one that is has rank one. One can check by direct calculation that

holds for every $m\in M$. Consequently, the so that the any vector from the range of $\rho_{M,v}$ fulfills all eigenvalue equations. However, since $\rho_{M,v}$ has rank one, its range corresponds to a single vector that we can associate with $|M,v\rangle\in\mathcal{H}$ up to a global phase. This proves the first claim up to uniqueness which we are going to establish later on.

For the second claim, fix $u,v\in V$ and observe

where we have used (12). But because $M$ is maximally isotropic, $[u-v,m]=0\;\forall m\in M$ implies $u-v\in M$. Thus, there is one $\rho_{M,u}$ for each affine space $u+M\subset V$, and two distinct affine spaces give rise to othogonal states which is just the second claim.

Finally, note that there are $|V/M|=d^{n}=\dim\mathcal{H}$ such affine spaces, which proves that one obtains an ortho-normal basis in this way. Moreover, this establishes the uniqueness part of the first statement and implies, justifying that $|M,v\rangle$ is well-defined up to a global phase. ∎

In the remainder of this section, we will show how to choose consistent bases for two, possibly intersecting, Lagrangian spaces $M,N$ and use these results to come up with formulas for the inner product between two arbitrary stabilizer states.

Let $M,N\subset V$ be two Lagrangian subspaces. Then there exists bases $\mathcal{B}_{M}$ of $M$ and $\mathcal{B}_{N}$ of $N$ such that $w_{\mathcal{B}_{K}}(m)=w_{\mathcal{B}_{M}}(m)=w_{\mathcal{B}_{N}}(m)$ for any $m\in M\cap N$. What is more, for $m\in M$ and $n\in N$, it holds that

Choose a basis $\{u_{1},\dots,u_{\dim{M\cap N}}\}$ of $M\cap N$. By elementary linear algebra, it can be extended both to a basis $\mathcal{B}_{M}$ of $M$ and to a basis $\mathcal{B}_{N}$ of $N$. The first claim follows immediately from (13). For the second claim, note that for from (9), we have that $w_{\mathcal{B}_{M}}(m)w_{\mathcal{B}_{N}}(-n)=\pm w(m-n)$. Thus, by (11), the trace in (15) vanishes unless $m=-n$. In that case, however, $m,n\in K$ and thus, by construction of the bases, $w_{\mathcal{B}_{M}}(m)=w_{\mathcal{B}_{K}}(m)$ and $w_{\mathcal{B}_{N}}(-n)=w_{\mathcal{B}_{K}}(-n)$. Thus

We conclude this subsection with an important observation: The overlap of different stabilizer states is fully characterized by the geometric intersection of their underlying Lagrangian subspaces.

Let $|M,\zeta\rangle,|N,\iota\rangle\in\mathcal{H}$ be two stabilizer states characterized by Lagrangian subspaces $M,N\subset V$ (as well as corresponding bases $\mathcal{B}_{M}$ and $\mathcal{B}_{N}$ if $d$ is even) and representatives $\zeta,\iota\in V$ of cosets $[\zeta]_{M}\in V/M$ and $[\iota]_{N}\in V/N$, respectively. Then, setting $K=M\cap N$, their inner product is given by

The claim follows from direct computation. According to Lemma 1 we can pick bases $\mathcal{B}_{K}$ of $K:=M\cap N$, $\mathcal{B}_{M}$ of $M$ and $\mathcal{B}_{N}$ of $N$ that are compatible with each other. With respect to these bases we can write

where the last equation follows from formula (12). ∎

II.4 Grassmannian subspaces and discrete symplectic geometry

For further reading and proofs of these identities we refer to Chapter 9 in cameron_combinatorics_1994 and move on to introducing some core concepts of symplectic geometry:

of all Lagrangian subspaces transverse to $M$. The set $\mathcal{T}(M)$ appears in various contexts. For instance it labels all graph states (in a sense explaind below) in quantum information theory hein_entanglement_2006

For the purpose of our counting argument, we need to compute the size of $\mathcal{T}(M)\in V$.

Fix $M$ and note that a subset $N\subset V$ has to be both Lagrangian and transverse to $M$ in order to lie in $\mathcal{T}(M)$. These conditions can be made more explicit if we choose a basis $b_{1},\ldots,b_{2n}$ of $V$ which obeys

The name graph state pays tribute to the fact that $A$ can be interpreted as the adjacency matrix of a (possibly weighted) graph. Graph states possess a rich structure and many properties of $|N,\zeta\rangle$ can be deduced from the corresponding graph alone. However, here we content ourselves with pointing out the analogy between graph states and $\mathcal{T}(M)$. For further reading we defer the reader to hein_entanglement_2006 .

Let us now turn to subspaces of the symplectic vector space $V$. It is clear that a proper subspace $W\subset V$ is itself a vector space, however in general it fails to be symplectic. This is due to the fact that the standard symplectic inner product (8) of $V$ becomes degenerate if we restrict it to $W$. Therefore important tools – such as Proposition 1 – cannot be directly applied to the proper subspace $W$. However, this problem can be (partly) circumvent by applying a linear symplectic reduction. For $W\subseteq V$ we define the quotient

This space carries the non-degenerate symplectic form

which is easily seen not to depend on the representatives for $[v]$ and $[w]$. Consequently, the space $\hat{W}$ endowed with $[\cdot,\cdot]_{\hat{W}}$ is a symplectic vector space. We will need such a reduction in the proof of Theorem 4.

III Proof of the main Theorem

Let $D=d^{n}$ be a prime power. The $t$-th frame potential of the set of all stabilizer states in dimension $D$ is given by

where $\kappa_{M}(d,n,k)$ is the number of Lagrangian subspaces $N$ whose intersection with an arbitrary fixed Lagrangian subspace $M$ is $k$-dimensional.

Stabilizer states constitute an orbit of a particular finite unitary group – the Clifford group. Due to this symmetry, the second summation in $\mathcal{F}_{t}(\operatorname{Stabs}(d,n))$ is superfluous and we can write

where $x_{k}\in\operatorname{Stabs}(d,n)$ is an arbitrary fixed stabilizer state. Theorem 3 assures that any such $x_{k}$ is unambiguously specified by a Lagrangian subspace $M$ of $V$ and coset $[\zeta]_{M}\in M/V$. Since the choice of $x_{k}$ in (22) was arbitrary, we can choose $x_{k}=|M,0\rangle$ – i.e. it is specified by $M$ and the particularly simple representative $0\in V$ of the coset $_{M}$. Such a choice of $x_{k}$ together with Theorem 3 allows us to rewrite (22) as

because instead of summing over stabilizer states, we may as well sum over their characterizing Lagrangian subspaces and cosets instead. Such a reformulation allows us to employ Lemma 2 which implies

where $K=M\cap N$ denotes the intersection. If this intersection is $k$-dimensional, $|K|=d^{k}$ and consequently $\left|\langle N,\zeta|M,0\rangle\right|^{2t}=d^{-t(n-k)}$, provided that $[\zeta,m]=0$ for all elements $m\in K$. This requirement for a non-vanishing overlap is met if and only if $\zeta\in K^{\perp}$. The number of representatives $\zeta$ which obey this property (and single out different stabilizer states) is given by the order of the quotient space $|K^{\perp}/N|$. Since $N\subseteq K^{\perp}$ (which follows from $K\subseteq N$ and $N^{\perp}=N$), such a quotient space is well defined and its order amounts to

Consequently, for each pair of Lagrangians $M,N$ with $k$-dimensional intersection, $d^{n-k}$ out of a total of $d^{n}$ stabilizer states specified by $N$ give rise to a non-vanishing overlap $\left|\langle N,\zeta|M,0\rangle\right|^{2t}=d^{-t(n-k)}$ with the fixed stabilizer state $x_{k}=|M,0\rangle$. Inserting this insight into (23) reveals

where we have replaced the summation over the different Lagrangian subspaces with an equivalent summation over the dimension $k$ of the intersections $M\cap N$. ∎

Lemma 3 shows that we can compute the stabilizer frame potential $\mathcal{F}_{t}(\operatorname{Stabs}(d,n))$ provided that the number $\kappa_{M}(d,n,k)$ is known for any Lagrangian subspace $M$ and any intersection space dimesion $k\in\left\{0,\ldots,n\right\}$. The following two statements characterize that number.

The fact that each Lagrangian $M$ admits $|\mathcal{G}(d,n,k)|=\binom{n}{k}_{d}$ different $k$-dimensional subspaces $K$ (formula (17)) immediately yields the following corollary.

We need to count in how many ways one can choose a Lagrangian space $N\subset V$ that intersects $M$ exactly in $K$. Our strategy will be to relate the set of such extensions $N$ of $K$ to a set $\mathcal{T}$ as in Proposition 1. To that end, set $\hat{W}:=K^{\perp}/K$. Note that $K\subseteq K^{\perp}$ (because $K\subseteq M$ and $M$ is Lagrangian) implies

Therefore $\hat{W}$ is the linear symplectic reduction of $K^{\perp}$ as defined in (19). The space $\hat{W}$ endowed with the induced symplectic product $[\cdot,\cdot]_{\hat{W}}$ defined in (20) forms a symplectic vector space with dimension

Note that any isotropic space $N$ containing $K$ is in particular contained in $K^{\perp}$. The canonical projection $N\mapsto N/K$ sets up a one-to-one correspondence between $n$-dimensional subspaces of $K^{\perp}$ containing $K$ and $(n-k)$-dimensional subspaces of $\hat{W}$. We need two properties of this correspondence:

(i) $N/K\subset\hat{W}$ is isotropic if and only if $N\subset V$ is. Proof: This follows immediately from (20).

(ii) $N/K\subset\hat{W}$ is transverse to $M/K$ if and only if $M\cap N=K$. Proof: Basic linear algebra shows

with equality if and only if $M\cap N=K$. Hence $\dim(M+N)/K\leq 2(n-k)$ with the same condition for equality. For the right hand side:

with equality if and only if the two spaces are transverse.

It follows that $M/K$ is a Lagrangian subspace of $\hat{W}$ and there is a one-to-one correspondence between Lagrangian spaces $N$ intersecting $M$ in $K$ and Lagrangian subspaces of $\hat{W}$ transverse to $M/K$. Employing Proposition 1 then yields the desired result. ∎

Finally, we are going to require an explicit characterization of the number $S(d,n)$ of stabilizer states. We borrow it from (gross_hudsons_2006, , Corollary 21):

Formula (25) combined with Corollary 2 allows us to write down the frame potential (Lemma 3) explicitly:

with $S(d,n)$ defined in (25). Note that this is a purely combinatorical expression that depends solely on $d$ and $n$. Analyzing its recursive dependence on $n$ allows us to establish the main result of this work – Theorem 2.

Let us start with the base case (4) which is readily established. Indeed, setting $n=1$ and evaluating formula (27) reveals that for any $d$ and $t$ $\mathcal{F}_{t}(\operatorname{Stabs}(d,n))$ amounts to

where we have used $\binom{n}{0}_{d}=\binom{n}{n}_{d}=1$. Let us now move on to establishing the recursive behavior. Replacing $n$ by $(n+1)$ in formula (27) and employing Pascal’s identity (18) as well as trivial coefficients for Gaussian binomials yields

where we have encorporated the first and last terms in the first and second summation, respectively. Note that the second summation just corresponds to $\sum_{k=0}^{n}\binom{n}{k}_{d}d^{\frac{1}{2}(n-k)(n-k+3-2t)}$ – which in that very form also appears in (27). Importantly, a similar equivalence is true for the first sum appearing in (28). Taking a closer look at the overall exponent of $d$ in that summation reveals

and the first term is independent of the summation index. Consequently the first sum in (28) actually corresponds to $d^{n-(t-2)}\sum_{k=0}^{n}\binom{n}{k}_{d}d^{\frac{1}{2}(n-k)(n-k+3-2t)}$ and we can conclude

We conclude this article with presenting a proof of Corollary 1 which establishes some substantial insights into the structure of stabilizer states.

Start with the case $t=2$. Then the result of Theorem 2 reads

But the Welch Bound (3) satisfies identical relations:

The 3-design case can be proved along similar lines. We have

The two base values (31) and (33) coincide for $d\leq 2$. Otherwise, the former is strictly larger than the latter. Comparing the recursion factors yields

Finally, let us move on the the 4-design case, where we have

Comparing (37) to (40) reveals $\mathcal{F}_{4}\left(\operatorname{Stabs}(d,1)\right)\geq\mathcal{W}_{4}\left(d\right)$ with equality if and only if $d=1$. An analogous relation holds for (38) and (41) which assures that $\mathcal{F}_{4}(\operatorname{Stabs}(d,n))$ and $\mathcal{W}_{4}(d^{n})$ only ever coincide in the trivial case $d=1$.

Acknowledgements: The authors want to thank P. Turner for insightful discussions and H. Zhu, as well as Z. Webb for informing us of their impeding work zhu2015 ; webb2015 .

The work of DG and RK is supported by the Excellence Initiative of the German Federal and State Governments (Grants ZUK 43 & 81), the ARO under contract W911NF-14-1-0098 (Quantum Characterization, Verification, and Validation), the DFG projects GRO 4334/1,2 (SPP1798 CoSIP), and the State Graduate Funding Program of Baden-Württemberg.

References