Recoupling coefficients and quantum entropies

Introduction

When does there exist a multi-particle quantum state compatible with a given set of reduced states? Long recognized for its importance in many-body quantum physics and quantum chemistry , this quantum marginal problem has seen significant progress in recent years in the context of quantum information theory through the discovery of an underlying group theoretic structure . In particular, a complete solution of its most fundamental version, where the given reduced states are those of the individual particles, has been obtained , and a firm connection to the classification of multiparticle entanglement has been established . Meanwhile, it has been understood that the general problem is computationally hard, even for a quantum computer , and only sufficient conditions are known. The strong subadditivity of the von Neumann entropy is perhaps the most important such condition, and an indispensible tool in quantum statistical physics and quantum information theory . The discovery of any further entropy inequality would be considered a major breakthrough (cf. for recent progress).

In this work we unveil a novel link between the existence of multi-particle quantum states with given marginal eigenvalues and the representation theory of the symmetric group $S_{k}$. For this, we consider the recoupling coefficients of $S_{k}$, where now $k$ plays the role of the semiclassical parameter. We then find that the coefficients’ norm decreases at most polynomially for a sequence of Young diagrams of $k\rightarrow\infty$ boxes converging to the eigenvalues of a given tripartite quantum state $\rho_{ABC}$ and its reduced states $\rho_{A}$, $\rho_{B}$, $\rho_{C}$, $\rho_{AB}$, $\rho_{BC}$; conversely, if there exists no such quantum state then the sequence decreases exponentially in norm.

Our result directly relates to the recent efforts on the quantum marginal problem and the understanding of quantum entropy. In particular, it extends the characterisation of the triples $\rho_{A}$, $\rho_{C}$, $\rho_{AC}$ by the Kronecker coefficient of the symmetric group . The power of this extension is illustrated by the fact that the symmetries of the recoupling coefficients alone imply the strong subadditivity and weak monotonicity of von Neumann entropy (note that entropy is only a function of the eigenvalues). Our result generalizes directly to an arbitrary number of particles and linearly many reduced states, and may thus be regarded as a partial quantum-mechanical version of Chan and Yeung’s description of the set of compatible Shannon entropies in terms of group theory . In fact, our work suggests a new route towards establishing further entropy inequalities by exploiting the symmetries of higher-order representation-theoretic objects.

Our work is inspired by Wigner’s seminal work on the semiclassical behavior of quantum spins, which are described by the representation theory of the group $\operatorname{SU}(2)$ . The recoupling coefficients of $\operatorname{SU}(2)$, known as the Wigner $6j$-symbols in their rescaled, more symmetric form , describe the relation between individual spins $j_{1}$, $j_{2}$, $j_{3}$, their total spin $j_{123}$ and the intermediate spins $j_{12}$ and $j_{23}$ (the Racah W-coefficients are also closely related). As first noted by Wigner, there is a dichotomy similar to our result in the semiclassical limit where all spins are simultaneously large: the $6j$-symbol decays polynomially if there exists a tetrahedron with side lengths $j_{1}$, $j_{2}$, $j_{3}$, $j_{12}$, $j_{23}$, $j_{123}$, and exponentially otherwise . That the asymptotics are in both cases guided by the existence of a geometric object—for Wigner, a tetrahedron with certain side lengths, for us, a quantum state with certain spectral properties—is not accidental. Via Schur-Weyl duality, our limit $k\rightarrow\infty$ can similarly be understood as a semiclassical limit. We furthermore describe how to construct for every tetrahedron a tripartite quantum state in a faithful, i.e. sidelength-encoding way, and for every Wigner $6j$-symbol we construct a corresponding recoupling coefficient of the symmetric group . In this way, Wigner’s problem and its generalization to $\operatorname{SU}(d)$ can be understood as special case of the quantum marginal problem.

We speculate that this surprising connection between tetrahedra and quantum states as well as between Wigner $6j$-symbols and symmetric group recoupling coefficients may help to understand and connect the study of spin foams and spin networks in the context of quantum gravity and condensed matter physics as well as topological quantum computing . Preliminary versions of this work have appeared in .

Recoupling coefficients

The finite-dimensional irreducible representations of the symmetric group $S_{k}$ are labelled by Young diagrams, that is, ordered partitions $\lambda_{1}\geq\ldots\geq\lambda_{l}>0$ of $\sum_{i}\lambda_{i}=k$. We may think of $k$ as the number of boxes of the Young diagram and of $l$ as the number of rows. We write $\lambda\vdash k$ for such a partition, $[\lambda]$ for the associated irreducible unitary representation of $S_{k}$, and denote the dimension of the latter by $d_{\lambda}:=\dim[\lambda]$. Any finite-dimensional representation $V$ of $S_{k}$ can be decomposed into a direct sum of irreducible representations, and if $V$ is a unitary representation then this decompositoni can also be made unitary. Concretely, consider the space of $S_{k}$-linear maps $\operatorname{H}^{V}_{\lambda}:=\operatorname{Hom}_{S_{k}}([\lambda],V)$ and equip $\operatorname{H}^{V}_{\lambda}$ with the re-scaled Hilbert-Schmidt inner product $\braket{\psi,\phi}_{\lambda}:=\operatorname{tr}\psi^{\dagger}\phi/d_{\lambda}$. Then the canonical maps $\Phi^{V}_{\lambda}\colon[\lambda]\otimes\operatorname{H}^{V}_{\lambda}\rightarrow V,v\otimes\phi\mapsto\phi(v)$ are $S_{k}$-linear isometries that can be assembled to a unitary isomorphism $\bigoplus_{\lambda\vdash k}[\lambda]\otimes\operatorname{H}^{V}_{\lambda}\cong V$.

In particular, we may decompose a tensor product $[\alpha]\otimes[\beta]$ of two irreducible representations. This results in the so-called Clebsch-Gordan isomorphism of the symmetric group $S_{k}$,

Its components are $S_{k}$-linear isometries and will be denoted by

The dimension of $\operatorname{H}^{\alpha\beta}_{\lambda}$ is known as the Kronecker coefficient $g_{\alpha\beta\gamma}$; it is fully symmetric in its three indices since the representations of $S_{k}$ are self-dual.

We now consider a triple tensor product. Since the tensor product is associative, we have

Decomposing accordingly using eq. 1, we obtain an isomorphism

By Schur’s lemma, this allows us to identify the multiplicity spaces for each fixed $\lambda$,

The recoupling coefficients of the symmetric group are defined to be the components of the isomorphism (4) for fixed $\mu$ and $\nu$, denoted by

In other words, they are defined by the relation

Here, $V^{d}_{\lambda}$ denotes the irreducible $\operatorname{SU}(d)$-representation with highest weight $\lambda$, and the direct sum runs over all Young diagrams $\lambda$ with $k$ boxes and no more than $d$ rows. In the following we denote by $P^{d}_{\lambda}$ the orthogonal projector onto a direct summand $[\lambda]\otimes V^{d}_{\lambda}$ in (8).

the orthogonal projectors onto the corresponding direct summands in the second and third line of eq. 9, respectively. Observe that each is defined as a product of commuting projectors. The following lemma connects the operator norm of their product to the operator norm of the corresponding recoupling coefficient.

By eq. 6, the recoupling coefficients satisfy the identity

The last equality holds because both $E$ and $F$ are isometries.

Now note that $EE^{\dagger}$ is precisely equal to the orthogonal projector onto

On the other hand, $P$ as defined in eq. 10 is the projector onto

Schur-Weyl duality also leads to an alternative definition of the recoupling coefficients in terms of unitary groups (see discussion at the end of section 3).

Recoupling coefficients and tripartite quantum marginals

If there exists a quantum state $\rho_{ABC}$ with eigenvalues $r_{A}$, $r_{B}$, $r_{C}$, $r_{AB}$, $r_{BC}$, $r_{ABC}$ then there exist Young diagrams $\alpha,\beta,\gamma,\mu,\nu,\lambda\vdash k$ with $k\rightarrow\infty$ boxes and at most $a$, $b$, etc. rows such that

Conversely, if $(r_{A},r_{B},r_{C},r_{AB},r_{BC},r_{ABC})$ is not associated to any tripartite density operator then for every sequence of Young diagrams satisfying eq. 12 we have

where $\varepsilon=\operatorname{poly}(k)\exp(-k\delta^{2}/2)$. Now we use

Using lemma 2, eq. 7 and the triangle inequality, we find that

where the sum extends over Young diagrams with $k$ boxes and at most $a$, $b$, etc. rows, whose normalisation is close to the eigenvalues associated to $\rho_{ABC}$ as specified above. Since the number of terms in the sum is again upper-bounded by $\operatorname{poly}(k)$, we can find sequences of Young diagrams satisfying eq. 12 and eq. 13.

Using the Hölder inequality, the right-hand side can be upper-bounded by the square roots of each of the six traces $\operatorname{tr}(P_{\alpha}^{a}\rho_{A}^{\otimes k})$, $\operatorname{tr}(P_{\beta}^{b}\rho_{B}^{\otimes k})$, etc., which in turn can be upper-bounded via eq. 15. Thus we find

Theorem 3 can be generalized to more than three parties by considering the following quantity: as in eq. 3, successively decompose a tensor product of irreducible representations in two inequivalent ways; the corresponding “generalized recoupling coefficients” then are the components of the resulting isomorphism for fixed intermediate labels $\mu_{j}$ and $\nu_{k}$, and an analogous result can be established for these coefficients, which are symmetric group counterparts of Wigner’s $3nj$-symbols for $\operatorname{SU}(2)$. Just as theorem 3 does not cover the eigenvalues of $\rho_{AC}$, in general only of a linear number of the exponentially many reduced density operators can be controlled in this fashion (e.g., the nearest-neighbor reduced states in a linear chain of particles). What the representation-theoretic quantities involved in controlling all marginal spectra should be is an intriguing question, with possible ramifications to the search for new entropy inequalities of the von Neumann entropy, as we detail in the following sections.

For pure quantum states $\rho_{ABC}$, the Schmidt decomposition implies that necessarily $r_{AB}=r_{C}$ and $r_{A}=r_{BC}$. Therefore, we can discard of the two-body spectra, and the problem reduces to a one-body quantum marginal problem. On the level of representation theory, it suffices to consider single-row Young diagrams $\lambda=(k)$, corresponding to the trivial representation of $S_{k}$; hence, $\mu=\gamma$ and $\alpha=\nu$, and it can be shown easily that

likewise reduces to a Kronecker coefficient of the symmetric group.

In this way, theorem 3 specializes to the well-known relationship between the pure-state one-body quantum marginal problem and the asymptotics of the decomposition of tensor products of irreducible representations of the symmetric group (it also shows that recoupling coefficients can grow with $k$). Theorem 3 generalizes this relationship: It shows that the overlap between two such decompositions – as captured by the recoupling coefficients – similarly characterizes the quantum marginal problem with two overlapping marginals. It would be of great interest to find a geometric explanation of this result in the framework of geometric invariant theory, which might also lead to a more refined understanding of the asymptotics along the lines of for Wigner’s $6j$-symbols. Mathematically, this is related to the “intersection” of moment maps, or to simultaneous Hamiltonian reduction for non-commuting group actions.

We conclude with some remarks on the interpretation of theorem 3 as a semiclassical limit. In , Wigner studied the asymptotics of the recoupling coefficients of $\operatorname{SU}(2)$ which can be defined in complete analogy to definition 1. Given three particles of spin $j_{A}$, $j_{B}$, $j_{C}$ such that the total spin of the first two particles is $j_{AB}$ and of all three particles $j_{ABC}$, the absolute value squared of the $\operatorname{SU}(2)$ recoupling coefficient can be interpreted as the probability of observing that particles two and three have total spin $j_{BC}$. In the semiclassical limit of simultaneously large spins, Wigner showed that this probability oscillates around the inverse volume of the tetrahedron whose edges have length equal to the six spins—if such a tetrahedron exists. In particular, it then decays polynomially with $j$. If no such tetrahedron exists then the recoupling coefficient decays exponentially. This result is understood to mean that “classical” configurations are exponentially more likely than all others in the limit of large quantum numbers. A more precise formula has been given by Ponzano and Regge and only fully proved in .

Symmetries of the recoupling coefficients

In this section we rewrite the Hilbert-Schmidt norm of the recoupling coefficients in a way that makes manifest its symmetries. Both the strong subadditivity property of the von Neumann entropy as well as its weak monotonicity property can then be understood in terms of these symmetries and theorem 3 (see section 5). We first give a diagrammatic argument using the graphical calculus for symmetric monoidal categories . An alternative, purely algebraic proof, is postponed to the end of this section.

Recall from section 2 that the multiplicity spaces $\operatorname{H}^{\alpha\beta}_{\lambda}$ are given by the space of $S_{k}$-linear maps from $[\lambda]$ to $[\alpha]\otimes[\beta]$. In each multiplicity space, let us choose maps $\Phi^{\alpha\beta}_{\lambda,i}$ that form an orthonormal basis with respect to the inner product $\braket{\psi,\phi}_{\lambda}=\operatorname{tr}\psi^{\dagger}\phi/d_{\lambda}$ introduced in section 2. We will represent these maps graphically by

in the graphical calculus. The maps $\Phi^{\alpha\beta}_{\lambda,i}$ are nothing but components of the Clebsch–Gordan isometries $\Phi_{\lambda}^{\alpha\beta}$ defined in eq. 2. We thus obtain the following graphical expression for the matrix elements of the recoupling coefficients with respect to the bases fixed above from (6) by taking a trace over $[\lambda]$:

Our goal is to transform the right-hand side expression in (17) into a form that renders its symmetries apparent. For this, we recall that the irreducible representations of the symmetric group are self-dual, i.e., $[\lambda]\cong[\lambda]^{*}$, because they can be defined over the reals. It follows that there exists a single copy of the trivial representation $\mathbf{1}$ in each tensor product $[\lambda]\otimes[\lambda]$, i.e., $\operatorname{H}^{\lambda=\lambda}_{\mathbf{1}}$ is one-dimensional. We shall denote the corresponding basis vector by

omitting the leg corresponding to the identity object $\mathbf{1}$ as is usual in the graphical calculus. It can be concretely written as a maximally entangled state $\sum_{i}\ket{\lambda,i}\otimes\ket{\lambda,i}/{\sqrt{d_{\lambda}}}$ in any real orthonormal basis $\ket{\lambda,i}$ of $[\lambda]$ (i.e., in a basis such that $S_{k}$ acts by real orthogonal matrices). We denote the adjoint of (18) by reversing arrows. It is then easy to see that we have the “teleportation identity”

We can use (18) and its adjoint to raise and lower indices, i.e., to reverse the direction of arrows. We thus obtain the following important property of the Clebsch–Gordan isometries (cf. [47, (7-205a)]):

form orthonormal bases of the space $([\alpha]\otimes[\beta]\otimes[\lambda])^{S_{k}}$ of $S_{k}$-invariant vectors in the triple tensor product.

Since the dimensions of $([\alpha]\otimes[\beta]\otimes[\lambda])^{S_{k}}$ and of $\operatorname{H}^{\alpha\beta}_{\lambda}$ agree by self-duality of $[\lambda]$, it suffices to show that both sets of vectors are orthonormal. For the first set, observe that it follows from the teleportation identity (19) that

since the $\Phi^{\alpha\beta}_{\lambda,i}$ form an orthonormal basis.

For the second set, we find similarly that

We finally introduce the symmetric notation:

We note that the vectors (20) depend on the choice of arrow that was reversed. However, by lemma 4 any such choice gives rise to unitarily equivalent bases of the space of $S_{k}$-invariants! We thus obtain the following result:

By inserting the teleportation identity (19) once for each of the six arrows, we obtain

By first applying the unitary transformation that relates the second orthonormal basis in lemma 4 to the first (at the vertices $k$ and $l$) and then using definition (20) (at all four vertices), this is in turn equal to

The right-hand side of (21) is the symmetric group analogue of a Wigner $6j$-symbols, which can be similarly from the recoupling coefficients of $\operatorname{SU}(2)$.However, for our purposes it was important to use the recoupling coefficients in theorem 3, since the dimensions of irreducible $S_{k}$-representations grow exponentially with $k$ and thus affect the asymptotics. It is immediately apparent from the graphical expression that it has the symmetries of a tetrahedron. We record the following consequence of this symmetry, which has a well-known counterpart for $\operatorname{SU}(2)$; cf. [31, (B4)]:

are invariant under exchanging the columns $(\beta,\lambda)\leftrightarrow(\mu,\nu)$ and also under exchanging the columns $(\alpha,\gamma)\leftrightarrow(\mu,\nu)$.

This is an immediate consequence of proposition 5, since the right-hand side norm in (21) is invariant under reflection of the diagram by the axes through the edges labeled by $\alpha$ and $\beta$, respectively. ∎

We now give an alternative, algebraic proof of proposition 5 and corollary 6 that follows along the same lines as the graphical proof. In quantum information theory, maximally entangled states on a Hilbert space $\mathcal{H}\otimes\mathcal{H}$ are defined by the formula

with respect to an orthonormal basis $\ket{i}$. They satisfy the fundamental identity

for any operator $X$ on $\mathcal{H}$, where $X^{T}$ denotes the transpose in the basis $\ket{i}$. Thus they are invariant under operations of the form $U\otimes\overline{U}$, where $U\in\operatorname{U}(\mathcal{H})$ is a unitary and where $\overline{U}$ denotes its complex conjugate with respect to the basis $\ket{i}$ . In particular, this implies that for any basis $\ket{\lambda,i}$ of $[\lambda]$ in which $S_{k}$ acts by orthogonal transformations,

is the (unique up to phase) invariant vector in $[\lambda]\otimes[\lambda]$—as we had asserted before above eq. 18. By using (24) it is straightforward to verify that the following two well-known properties hold:

This is the algebraic version of eq. 19. It follows that for any two operators $X\colon\mathcal{K}\rightarrow\mathcal{K}^{\prime}\otimes\mathcal{H}$ and $Y\colon\mathcal{H}\otimes\mathcal{L}\rightarrow\mathcal{L}^{\prime}$ we have the relation

The normalized trace of any operator $X$ can be written as

For any $\alpha$, $\beta$ and $\lambda$, we shall consider the following sets of vectors in $([\alpha]\otimes[\beta]\otimes[\lambda])^{S_{k}}$,

constructed as in lemma 4. We now prove algebraically that each set forms an orthonormal basis. For the first,

by (27) and the definition of the inner product. For the second set of vectors,

We now consider the recoupling coefficients. First, (6) and (27) give

We may now apply (26) to $X=\ket{\mu\gamma\lambda,i}$ and $Y=\Phi^{\alpha\beta}_{\mu,j}$ in order to rewrite

Continuing in this way and using definition (28), we obtain the following expression for the matrix elements of the recoupling coefficient:

The sum of their absolute values squared over all indices $i$, $j$, $k$ and $l$ is equal to

where $P^{\alpha\beta\mu}$ denotes the orthogonal projection onto $([\alpha]\otimes[\beta]\otimes[\mu])^{S_{k}}$, $P^{\alpha\alpha}=\lvert\Psi^{+}_{\alpha}\rangle\langle\Psi^{+}_{\alpha}\rvert$, etc. Equation (29) is the algebraic analogue of proposition 5. As before, corollary 6 is a direct consequence of its symmetries.

Entropy inequalities from symmetries: strong subadditivity

We now prove the strong subadditivity and weak monotonicity of the von Neumann entropy as a direct consequence of theorem 3 and the symmetry properties in corollary 6.

To start, we note that it follows from the first invariance asserted in corollary 6 and the polynomial upper bound (7) that

for sequences of normalized Young diagrams that converge to the respective spectra of the reduced density operators. Since for large $k$, $\frac{1}{k}\log_{2}\dim[\lambda]\rightarrow H(\bar{\lambda})=\sum_{i}-\bar{\lambda}_{i}\log_{2}\bar{\lambda}_{i}$ , we conclude that the von Neumann entropy is strongly subadditive:

where $S(\rho):=-\operatorname{tr}\rho\log\rho$ denotes the von Neumann entropy of a density operator $\rho$.

For $[\beta]$ the trivial representation, this proof of strong subadditivity reduces to the proof of subadditivity given in (cf. the discussion at the end of section 3). The weak monotonicity,

follows similarly from the second invariance in corollary 6.

As we have mentioned in the introduction, it is an important open question to decide whether the von Neumann entropy satisfies any other linear entropy inequalities beyond strong subadditivity and weak monotonicity. Our proofs of the latter are markedly different from previous proofs in the literature, which are built on operator convexity or asymptotic equipartition (cf. the review ). In our approach, we interpret an entropy inequality as the asymptotic shadow of a dimensional relation such as (31). We establish the latter by exploiting the symmetries of a corresponding representation-theoretic object – the recoupling coefficients – together with a lower bound from spectrum estimation. This hints towards an intriguing route towards establishing new entropy inequalities—namely, by constructing novel representation-theoretic objects (e.g., by composing Clebsch–Gordan maps) and uncovering their symmetries (as can conveniently be done using the graphical calculus).

Sums of matrices and quantum marginals

Do there exist Hermitian $d\times d$-matrices $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ with given prescribed eigenvalues for $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{A}+\mathcal{B}$, $\mathcal{B}+\mathcal{C}$ and $\mathcal{A}+\mathcal{B}+\mathcal{C}$?

This is a natural generalization of the problem of determining the relation between the eigenvalues of $A$, $B$ and $A+B$, posed by Weyl, whose solution conjectured by Horn was proved in the celebrated works .

In , it was shown how the one-body quantum marginal problem degenerates to Weyl’s problem in an appropriate limit. We will now show that problem 8 can likewise be considered as a special case of the quantum marginals problem for overlapping subsystems characterized by theorem 3—both on the level of geometry and on the level of representation theory.

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be Hermitian $d\times d$-matrices. Without loss of generality, we may assume that $\mathcal{A},\mathcal{B},\mathcal{C}\geq 0$ and that $1-\operatorname{tr}(\mathcal{A}+\mathcal{B}+\mathcal{C})\geq\lVert\mathcal{A}+\mathcal{B}+\mathcal{C}\rVert_{\infty}$ (else, we may add suitable multiples of the identity and rescale). Generalizing a construction from , we define a tripartite density operator $\rho_{ABC}$ as the reduced density operator of the four-party pure state

Let $\rho_{ABC}$ be the quantum state with purification (33). Then the non-zero eigenvalues of $\rho_{ABC}$ and all its reduced density operators are given by

Observe that $\ket{\psi_{ABCD}}$ is built from a sum of (unnormalized) maximally entangled states (23) on $AD$, $BD$ and $CD$, respectively. By using (24) and the orthogonality properties of the construction (33), we thus find that

If we only trace out the first two systems, then we instead get a block decomposition of the form

where $\ket{\phi}_{CD}=\sum_{k=1}^{d}\sqrt{\mathcal{C}}\ket{k}_{C}\otimes\ket{k}_{D}+\sqrt{1-\operatorname{tr}(\mathcal{A}+\mathcal{B}+\mathcal{C})}\ket{00}_{CD}$. Using (27), we find that $\braket{\phi_{CD}}{\phi_{CD}}=1-\operatorname{tr}(\mathcal{A}+\mathcal{B})$, so that the second claim follows as above.

which is established similarly. All other marginal spectra can be computed in the same way. ∎

We have thus obtained an embedding of triples of matrices into the space of tripartite density operators that preserves the eigenvalue information. We remark that lemma 9 can in particular be used to obtain entropy inequalities for convex combinations of Hermitian matrices from entropy inequalities for multipartite quantum states. E.g., strong subadditivity (32) yields after some manipulations the following inequality

where we have abbreviated $a=\operatorname{tr}\mathcal{A}$, $b=\operatorname{tr}\mathcal{B}$, and $c=\operatorname{tr}\mathcal{C}$, and where $h(x):=-x\log x-(1-x)\log(1-x)$ denotes the binary entropy function.

The state $\rho_{ABC}$ has rank at most $d+1$ and it satisfies the equality

where $r_{I,1}$ denotes the maximal eigenvalue of the reduced density operator $\rho_{I}$.

In other words, the states $\rho_{ABC}$ constructed above saturate the “polygonal inequality”

which holds for arbitrary quantum states, with equality. We now show the following converse to corollary 10.

The first inequality is obtained by omitting the terms with negative signs, and the second by using the variational principle for the maximal eigenvalue of $\rho_{ABC}$. It is thus immediate that we have equality if and only if $\ket{000}_{ABC}$ is a maximal eigenvector of $\rho_{ABC}$ and

A priori, the right-hand side can run over all indices $(i,j,k)\neq(0,0,0)$ and $l\neq 0$ by orthogonality of the bases in the Schmidt decomposition. But (36) implies that in fact precisely two out of the three indices $(i,j,k)$ have to be zero, so that we obtain

Thus we may define $d\times d$-matrices $X_{A}$, $X_{B}$ and $X_{C}$ such that

Finally, we use the polar decomposition to write $X_{A}=U_{A}\lvert X_{A}\rvert$, etc., and set $\sqrt{\mathcal{A}}:=\lvert X_{A}\rvert$, etc. Then (33) is indeed a purification of the quantum state $(U^{\dagger}_{A}\otimes U^{\dagger}_{B}\otimes U^{\dagger}_{C})\rho_{ABC}(U_{A}\otimes U_{B}\otimes U_{C})$, which is locally unitarily equivalent to $\rho_{ABC}$. ∎

The following theorem shows that problem 8 – and, in particular, the existence of Wigner’s tetrahedra – is in a precise mathematical sense a special case of the quantum marginal problem with overlapping marginals covered by theorem 3. This generalizes the corresponding result for the one-body quantum marginal problem mentioned above and in particular gives a geometric proof of the latter.

There exist Hermitian $d\times d$-matrices $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ with $\operatorname{spec}(\mathcal{A}+\mathcal{B}+\mathcal{C})=s_{\mathcal{A}+\mathcal{B}+\mathcal{C}}$, $\operatorname{spec}(\mathcal{A}+\mathcal{B})=s_{\mathcal{A}+\mathcal{B}}$, $\operatorname{spec}\mathcal{A}=s_{A}$, etc. as their partial sums.

$(1)\Rightarrow(2)$ is the content of lemma 9. For $(2)\Rightarrow(1)$, we use proposition 11 to obtain Hermitian $d\times d$-matrices $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ such that $\rho_{ABC}$ is locally unitarily equivalent to the state $\rho^{\prime}_{ABC}$ with purification (33). Since the spectra of $\rho_{ABC}$ and its reduced density operators are left invariant by local unitaries, lemma 9 implies that the partial sums of these matrices $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ have the desired spectra. ∎

Similar statements can be proved for all marginal spectra (i.e., including $s_{\mathcal{A}+\mathcal{C}}$, since lemma 9 holds for all reduced density operators) as well as for an arbitrary number of summands. Thus the quantum marginal problem with overlaps is a precise generalization of the problem of characterizing the eigenvalues of partial sums of Hermitian matrices.

We now show an analogous statement to theorem 12 on the level of representation theory—namely, that the recoupling coefficients of the unitary group can be obtained as special recoupling coefficients of the symmetric group.

To see this, let $\lambda$ be a Young diagram. In , the restriction of an irreducible $\operatorname{U}(k)$-representation $V^{k}_{\lambda}$ to the subgroup of permutation matrices $S_{k}\subseteq U(k)$ has been computed:

In the right-hand side of (37), $\operatorname{Ind}$ denotes an induced representation and $\alpha$ is the unique Young diagram with number of boxes $\lvert\alpha\rvert$ equal to the number of rows of $\mu$ such that

where $S_{\mu}:=S_{\mu_{1}}\times S_{\mu_{2}}\times\dots\subseteq S_{\lvert\mu\rvert}$ is the Young subgroup corresponding to $\mu$, $N_{S_{\lvert\mu\rvert}}(S_{\mu})$ its normalizer in $S_{\lvert\mu\rvert}$, and $S_{\alpha}\subseteq S_{\lvert\alpha\rvert}$ the Young subgroup of $\alpha$. Note that $S_{\alpha}$ indeed acts on the subspace $[\lambda]^{S_{\mu}}$.

If $k-\lvert\lambda\rvert\geq\lambda_{1}$, then $\lambda^{\prime}:=(k-\lvert\lambda\rvert,\lambda)$ is again a Young diagram, and

Here and in the following, we write “…” for a sum of irreducible $S_{k}$-representations whose Young diagrams have longer first rows than all the preceding ones.

Otherwise, if $k-\lvert\lambda\rvert<\lambda_{1}$ then the first row of any Young diagram that appears in the restriction of $V^{k}_{\lambda}$ is longer than $k-\lvert\lambda\rvert$.

Since induction is transitive, we can rewrite (37) as

The Pieri formula asserts that the $S_{k}$-representation induced from a tensor product of an irreducible $S_{\lvert\alpha\rvert}$-representation $[\nu]$ with the trivial $S_{k-\lvert\alpha\rvert}$-representation $\mathbf{1}$ is given by the sum over all irreducible $S_{k}$-representations with a Young diagram that can be obtained by adding $k-\lvert\alpha\rvert$ boxes to $\nu$, with no two in the same column (see, e.g., [62, §2.2, (4)]). The first row of any such Young diagram is of length at least $k-\lvert\alpha\rvert$. As $\lvert\alpha\rvert$ is equal to the number of rows of $\mu$, we obtain the lower bound

on the length of the first row of any irreducible $S_{k}$-representation that occurs in the restriction of $V^{k}_{\lambda}$.

Equality in (40) can occur only if each row of $\mu$ contains a single box, i.e., for $\mu=(1,\ldots,1,0,\ldots,0)$, such that $\lvert\alpha\rvert=\lvert\mu\rvert=\lvert\lambda\rvert$. Then $S_{\mu}$ is the trivial group, $S_{\alpha}=S_{\lvert\alpha\rvert}=S_{\lvert\lambda\rvert}$, and the corresponding summand in (39) is equal to

By the Pieri formula, (41) contains an irreducible $S_{k}$-representation with first row of length $k-\lvert\lambda\rvert$ if and only if $\lambda_{1}\leq k-\lvert\lambda\rvert$ (since we only add boxes to $\lambda$). Moreover, if this condition is satisfied then there is only a single option, namely to place one box in each of the $k-\lvert\lambda\rvert$ leftmost columns, resulting in the Young diagram $\lambda^{\prime}=(k-\lvert\lambda\rvert,\lambda)$. ∎

We now consider the decomposition of a tensor product of irreducible $\operatorname{U}(k)$-representations,

where we assume that $k-\lvert\alpha\rvert\geq\alpha_{1}$ and $k-\lvert\beta\rvert\geq\beta_{1}$. The multiplicities $c^{\alpha,\beta}_{\lambda}$ are known as the Littlewood–Richardson coefficients, and they are independent of the choice of $k$ (if $k$ is at least as large as the number of rows in the Young diagrams involved) . Moreover, $c^{\alpha,\beta}_{\lambda}$ is non-zero only if $\lvert\alpha\rvert+\lvert\beta\rvert=\lvert\lambda\rvert$. It follows from the points above that

On the other hand, by applying (38) to the individual tensor factors we find that

where $g_{\alpha^{\prime},\beta^{\prime},\lambda^{\prime}}$ are the Kronecker coefficients. In the last inequality, we have used that $g_{\alpha^{\prime},\beta^{\prime},\lambda^{\prime}}>0$ only if $\lvert\lambda\rvert\leq\lvert\alpha\rvert+\lvert\beta\rvert$ [63, Theorem 2.9.22]. By comparing coefficients we find that $c^{\alpha,\beta}_{\lambda}=g_{\alpha^{\prime},\beta^{\prime},\lambda^{\prime}}$ for all triples of Young diagrams with $\lvert\alpha\rvert+\lvert\beta\rvert=\lvert\gamma\rvert$ and $k$ large enough. We thus recover a well-known result due to Littlewood and Murnaghan that states that the Littlewood–Richardson coefficients are a special case of the Kronecker coefficients .

What is more, the argument shows that the Clebsch–Gordan embeddings $\Phi^{\alpha^{\prime}\beta^{\prime}}_{\lambda^{\prime}}$ for $S_{k}$ can be obtained by restricting the ones of $\operatorname{U}(k)$. In view of (6), this implies directly that the recoupling coefficients are the same, since they are built solely from the action on the multiplicity spaces. Again, the recoupling coefficients for $\operatorname{U}(k)$ do not depend on the choice of $k$ (if $k$ is at least as large as the number of rows in the Young diagrams involved).

Acknowledgements

We thank M. Backens, M. Gromov, D. Gross, H. Haggard, F. Hellmann, W. Kamiński, A. Knutson, G. Mitchison, M.B. Ruskai, L. Vinet and R. Werner for valuable discussions.

We acknowledge financial support by the German Science Foundation (grant CH 843/2-1), the Swiss National Science Foundation (grants PP00P2-128455, 20CH21-138799 (CHIST-ERA project CQC)), the Swiss National Center of Competence in Research ‘Quantum Science and Technology (QSIT)’, the Swiss State Secretariat for Education and Research supporting COST action MP1006, the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 337603, the Simons Foundation, FQXi, and a Sapere Aude: DFF-Starting Grant. M.B. Şahinoğlu acknowledges support of the Excellence Scholarship and Opportunity Programme of ETH Zürich.

References