# Towards a resolution of the spin alignment problem

Mohammad A<EMAIL_ADDRESS>National Institute of
Standards and Technology, Boulder, Colorado 80305, USA  JILA, University of
Colorado, 440 UCB, Boulder, CO 80309, USA  Department of Physics, University
of Colorado, Boulder, Colorado, USA  Emanuel Knill  National Institute of
Standards and Technology, Boulder, Colorado 80305, USA  Center for Theory of
Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

Consider minimizing the entropy of a mixture of states by choosing each state
subject to constraints. If the spectrum of each state is fixed, we expect that
in order to reduce the entropy of the mixture, we should make the states less
distinguishable in some sense. Here, we study a class of optimization problems
that are inspired by this situation and shed light on the relevant notions of
distinguishability. The motivation for our study is the spin alignment
conjecture introduced recently in Ref. [11]. In the original version of the
underlying problem, each state in the mixture is constrained to be a freely
chosen state on a subset of $n$ qubits tensored with a fixed state $Q$ on each
of the qubits in the complement. According to the conjecture, the entropy of
the mixture is minimized by choosing the freely chosen state in each term to
be a tensor product of projectors onto a fixed maximal eigenvector of $Q$,
which maximally “aligns” the terms in the mixture. We generalize this problem
in several ways. First, instead of minimizing entropy, we consider maximizing
arbitrary unitarily invariant convex functions such as Fan norms and Schatten
norms. To formalize and generalize the conjectured required alignment, we
define alignment as a preorder on tuples of self-adjoint operators that is
induced by majorization. We prove the generalized conjecture for Schatten
norms of integer order, for the case where the freely chosen states are
constrained to be classical, and for the case where only two states contribute
to the mixture and $Q$ is proportional to a projector. The last case fits into
a more general situation where we give explicit conditions for maximal
alignment. The spin alignment problem has a natural “dual” formulation,
versions of which have further generalizations that we introduce.

###### Contents

  1. I Introduction
  2. II Mathematical preliminaries
  3. III Spin alignment problems and the alignment ordering
  4. IV Results
  5. V “Dual” problems to spin alignment and their generalizations
  6. VI Concluding remarks
  7. A The overlap lemma
  8. B On the relationship between the sum of two projectors and their product

##  I Introduction

Minimizing dispersion in a communication signal is often necessary in both the
theory and practice of information processing. Informally, the term dispersion
is used here so that the more dispersion in a signal the more information
content it carries. How dispersion is quantified depends on context. For
example, in a setting where sources are taken to generate signals in an
independent and identically-distributed manner and processes are assumed to be
memoryless, a useful measure of dispersion is the Shannon entropy for
classical signals associated with probability distributions [20], or
analogously the von Neumann entropy for quantum signals associated with
quantum states [18]. By this point, myriads of measures of dispersion have
been introduced and extensively studied [16, 22, 4]. A sensible requirement
for a measure of dispersion is that it does not decrease under any process
where labels designating outcomes are confused. Such a process is represented
classically by a doubly-stochastic transition matrix; the quantum
generalization of which is a mixed-unitary channel. Put in other words, we
require measures of dispersion to be Schur-concave. In this work, we switch
from concave to convex and primarily focus on convex Schur-convex functions.
To be precise, we study how unitarily invariant convex functions behave under
mixing of signals. The value of these functions at a mixture of signals
depends on the spectra of the individual signals supported in the mixture as
well as the alignment or overlap of the signals. It is our aim to shed light
on the latter kind of dependence.

Maximizing unitarily invariant convex functions is less ubiquitous in quantum
information than minimizing them, and is generally a more difficult task. This
difficulty merits more analysis and research. Quantum channel minimum output
entropy is an information-theoretic quantity that is defined by a minimization
of a Schur-concave function. A reason it is of interest is the equivalence of
additivity (under tensor multiplication) of the minimum output entropy of a
quantum channel to additivity of its Holevo information [21], the
regularization of which is a formula for the channel’s capacity for classical
communication [19]. Despite an existence proof due to Hastings [7], an
explicit example of a quantum channel with a finite-dimensional domain and a
strictly super-additive minimum output entropy is yet to be found. In
addition, such minimizations are relevant in broadcast channel scenarios with
privacy concerns. Ideally, the entropy of the environment or the adversary is
kept at a minimum. Depending on the channels in question, the optimal states
may be mixtures, as in the case of platypus channels [11].

We are interested in situations where we can unambiguously say that a signal
$a$ contains less dispersion than a signal $b$. That is, $f(b)\leq f(a)$ for
all Schur-convex functions $f$. Luckily, we need not argue such an inequality
for every such $f$. It suffices to check that $a$ majorizes $b$. In this
sense, of all the ways of measuring dispersion, majorization is the most
stringent [13, 3]. Majorization theory, which is nearly a century old [6], is
the basis of Nielsen’s celebrated theorem for pure bipartite state conversion
using local operations and classical communication [14]. This is one example
among many of majorization giving insights into information processing.

Here, we elucidate what we mean by alignment and how it is relevant to the
behavior of dispersion under mixing. For a self-adjoint operator $S$, let
$\lambda(S)$ denote the vector whose entries are the eigenvalues of $S$ (with
multiplicity) ordered non-increasingly. Self-adjoint operators
$S_{1},\ldots,S_{\ell}$ acting on $\mathbb{C}^{d}$ are said to be perfectly
aligned if there exists an orthonormal basis
$\\{{|{\phi_{i}}\rangle}\\}_{i=1}^{d}$ such that
$S_{1}=\sum_{i=1}^{d}\lambda_{i}(S_{1}){|{\phi_{i}}\rangle}{\langle{\phi_{i}}|},\>\ldots\>,S_{\ell}=\sum_{i=1}^{d}\lambda_{i}(S_{\ell}){|{\phi_{i}}\rangle}{\langle{\phi_{i}}|}$.
This definition is motivated by Ky Fan’s majorization relation [5], which in
the case of states $\rho_{1},\ldots,\rho_{\ell}$ and a probability measure
$p=(p_{i})_{i=1}^{\ell}$ reads

$\displaystyle\lambda(\sum_{i=1}^{\ell}p_{i}\rho_{i})\preceq\sum_{i=1}^{\ell}p_{i}\lambda(\rho_{i}).$
(I.1)

Here $b\preceq a$ means $b$ is majorized by $a$. These inequalities are
simultaneously saturated if and only if the states are perfectly aligned.
Hence, to minimize the dispersion in a mixture of states subject to spectral
constraints, we should choose the states to be perfectly aligned.

More generally, we may consider a set $G$ whose elements are $\ell$-tuples of
states $(\rho_{i})_{i=1}^{\ell}$ that satisfy constraints including that for
each $i\in[\ell]$, $\lambda(\rho_{i})$ is given. We may then ask if there is a
tuple $(\tilde{\rho}_{i})_{i=1}^{\ell}\in G$ such that for all probability
measures $(p_{i})_{i=1}^{\ell}$,
$\lambda(\sum_{i=1}^{\ell}p_{i}\tilde{\rho}_{i})$ majorizes
$\lambda(\sum_{i=1}^{\ell}p_{i}{\sigma}_{i})$ for all
$(\sigma_{i})_{i=1}^{\ell}\in G$. The previous paragraph shows that the answer
is yes if $G$ contains a tuple of perfectly aligned states. The following
example is of a situation where constraints preclude the existence of such a
tuple.

######  Example I.1.

Let a state of non-maximal rank $\tau$ be given. Let ${|{\alpha}\rangle}$ be a
unit vector in $\text{supp}(\tau):=\ker{\tau}^{\perp}$ and let ${|{e}\rangle}$
be a unit vector in $\text{ker}(\tau)$. Let the set $G$ consists of pairs of
states of the form
$(\rho_{1}=\tau,\rho_{2}=\outerproduct{v_{\gamma}}{v_{\gamma}})$, where
${|{v_{\gamma}}\rangle}=\sqrt{\gamma}{|{\alpha}\rangle}+\sqrt{1-\gamma}{|{e}\rangle}$
for some $\gamma\in[0,1]$. If ${|{\alpha}\rangle}$ is a maximal eigenvector of
$\tau$, then we can see via Eq. I.1 that for all probability distributions
$(p_{1},p_{2})$, $p_{1}\tau+p_{2}\outerproduct{v_{1}}{v_{1}}$ majorizes
$p_{1}\tau+p_{2}\outerproduct{v_{\gamma}}{v_{\gamma}}$ for all
$\gamma\in[0,1]$. In this case, $\tau$ is perfectly aligned with
$\outerproduct{v_{1}}{v_{1}}$. If ${|{\alpha}\rangle}$ is not a maximal
eigenvector of $\tau$, then $\tau$ is not perfectly aligned with
$\outerproduct{v_{\gamma}}{v_{\gamma}}$ for any $\gamma\in[0,1]$. It is
natural to conjecture that the choice $\gamma=1$ minimizes dispersion in the
mixture, because that is where the two states are in a sense as overlapped or
aligned as possible. This conjecture is still open.

We use the spin alignment problem as a case study of these notions of
alignment with the aim of refining Fan’s majorization relation. Introduced by
Leditzky et al. in Ref. [11], the problem arose in the context of proving
additivity of the coherent information of a class of quantum channels with
peculiar communication properties. For a qudit state $Q$ and a probability
measure $\mu$, they considered $n$-qudit states of the form

$\displaystyle\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes I^{c}}.$
(I.2)

Each term in the mixture corresponds to a subset $I\subseteq[n]$ and is a
tensor product of a state $\rho_{I}$ of the qudits in $I$ and a fixed state
$Q$ on each of the qudits in the complement $I^{c}$. Their task was to pick
the tuple of states $(\rho_{I})_{I\subseteq[n]}$ so that the overall operator
in Eq. I.2 has minimum von Neumann entropy. It is not too difficult to see
that an optimal choice has to be one where states in the tuple are pure (see
Lem. III.2 below). However, depending on $\mu$, there may be no such choice
where the summands in Eq. I.2 are perfectly aligned. Leditzky et al.
conjectured that the von Neumann entropy is minimized by choosing the tuple of
states to be $({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$, where ${|{q_{1}}\rangle}$ is a maximal eigenvector of
$Q$. With this tuple, the summands appear to overlap maximally. They showed
that if this conjecture is true, then their channels have additive coherent
information. For more information on these channels, see [11] and the
companion paper [10].

The structure of this paper is as follows. In Sec. II, we introduce notation
and the relevant mathematical background. We formally define and motivate the
class of spin alignment problems in Sec. III. In Sec. IV, we provide our
results, which include resolutions for various instances of spin alignment. In
Sec. V, we introduce a general class of optimization problems that is “dual”
to spin alignment. In Sec. VI, we conclude and discuss open problems and
follow-up lines of inquiry. Appendices App. A and App. B contain proofs of
technical lemmas.

##  II Mathematical preliminaries

In this work, we are concerned with finite-dimensional inner product spaces
over $\mathbb{C}$. We denote such spaces by $\mathcal{H},\mathcal{K}$, etc.
The dimension of $\mathcal{H}$ is denoted by $d_{\mathcal{H}}$. The set of
operators on $\mathcal{H}$ is denoted by $\mathcal{B}(\mathcal{H})$.
$\mathcal{S}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ denotes the real
vector space of self-adjoint operators on $\mathcal{H}$.
$\mathcal{P}(\mathcal{H})\subset\mathcal{S}(\mathcal{H})$ denotes the convex
cone of positive semi-definite operators, and
$\mathcal{D}(\mathcal{H})\subset\mathcal{P}(\mathcal{H})$ denotes the convex
set of states, that is the set of positive semi-definite operators with unit
trace. For $R\in\mathcal{B}(\mathcal{H})$,
$\sigma(R)\in\mathbb{R}^{d_{\mathcal{H}}}$ is the vector whose entries are the
singular values of $R$ ordered non-increasingly. Similarly, for
$S\in\mathcal{S}(\mathcal{H})$, $\lambda(S)\in\mathbb{R}^{d_{\mathcal{H}}}$ is
the vector whose entries are the eigenvalues of $S$ ordered non-increasingly.
We use the word projector to refer to a self-adjoint projection operator and
the word qudit to refer to a physical system that is modeled on a complex
inner product space of dimension $d$. For $n\in\mathbb{N}$, the set
$\\{1,\ldots,n$} is denoted by $[n]$ and its power set is denoted by
$2^{[n]}$. Given a measure $\mu$ on a countable set, its support is denoted by
$\text{supp}(\mu)$. If $x\in\mathbb{R}^{d}$, then $x^{\downarrow}$ and
$x^{\uparrow}$ denote the vectors in $\mathbb{R}^{d}$ with the entries of $x$
ordered non-increasingly and non-decreasingly, respectively. Given
$\mathcal{H}$, we distinguish an ordered orthonormal basis
$\\{{|{q_{i}}\rangle}\\}_{i=1}^{d_{\mathcal{H}}}$. For
$S\in\mathcal{S}(\mathcal{H})$,
$S^{\downarrow}=\sum_{i=1}^{d_{\mathcal{H}}}\lambda_{i}(S){|{q_{i}}\rangle}{\langle{q_{i}}|}$.
Real vectors $v_{1},\ldots,v_{s}$ are said to be similarly ordered if there
exists a permutation matrix $A$ such that
$v_{1}^{\downarrow}=Av_{1},\ldots,v_{s}^{\downarrow}=Av_{s}$. Similar ordering
for vectors is analogous to perfect alignment for self-adjoint operators.
Neither similar ordering nor perfect alignment is transitive. A family of
projectors is perfectly aligned if and only if their supports are nested. That
is, the family is a chain in the lattice of projectors.

Here is a brief review of majorization theory. Given $x,y\in\mathbb{R}^{d}$,
$x$ is said to majorize $y$ if

$\displaystyle\sum_{i=1}^{k}x_{i}^{\downarrow}\geq\sum_{i=1}^{k}y_{i}^{\downarrow}$
(II.1)

for each $k\in[d-1]$ and
$\sum_{i=1}^{d}x_{i}^{\downarrow}=\sum_{i=1}^{d}y_{i}^{\downarrow}$. This
statement is denoted with $x\succeq y$. Majorization is defined on
$\mathcal{S}(\mathcal{H})$ spectrally. For $B,C\in\mathcal{S}(\mathcal{H})$,
if $\lambda(B)\succeq\lambda(C)$, then $B$ is said to majorize $C$, and we
denote that with $B\succeq C$. Majorization induces a preorder on
$\mathbb{R}^{d}$ and $\mathcal{S}(\mathcal{H})$, respectively. Given a set, a
majorant, if it exists, is an element of the set that majorizes all elements
in the set.

There are several useful characterizations of majorization, two of which are
relevant to this work. The first is based on doubly-stochastic processing (see
Theorem II.1.10 on page 33 in Ref. [3]): $x$ majorizes $y$ if and only if
there exist a tuple of permutation matrices $(P_{i})_{i}$ and a probability
measure $(p_{i})_{i}$ such that

$\displaystyle y=\sum_{i}p_{i}P_{i}x.$ (II.2)

In words, $x$ majorizes $y$ if and only if $y$ belongs to the convex hull of
the orbit of $x$ under the action of the symmetric group. That is, $y=Ax$ for
some doubly-stochastic matrix $A$. Analogously, $B\succeq C$ if and only if

$\displaystyle C=\sum_{i}p_{i}U_{i}BU_{i}^{*}$ (II.3)

for a tuple of unitary operators $(U_{i})_{i}$ and a probability measure
$(p_{i})_{i}$. Put differently, $C$ is the value of a mixed-unitary quantum
channel at $B$.

The second characterization is based on unjust transfers, which are related to
the better-known $T$-transforms [13]. For distinct $i,j\in[d]$, and
$t\in[0,1]$, a $T$-transform is a linear map with action

$\displaystyle(x_{1},\ldots,x_{i},\ldots,x_{j},\ldots,x_{d})\mapsto(x_{1},\ldots,(1-t)x_{i}+tx_{j},\ldots,(1-t)x_{j}+tx_{i},\ldots,x_{d}).$
(II.4)

A $T$-transform is a convex combination of the identity and a transposition.
$T$-transforms are also known as Robin Hood operations [1]. The statement
$x\succeq y$ holds if and only if $y$ can be obtained from $x$ by a finite
number of $T$-transforms (see Lemma B.1 on page 32 in Ref. [13]). A direct
consequence of this is that $x\succeq y$ implies $x\oplus z\succeq y\oplus z$
for all $x,y\in\mathbb{R}^{d_{1}}$ and $z\in\mathbb{R}^{d_{2}}$. A transfer on
$\mathbb{R}^{d}$ is an affine transformation determined by a pair of distinct
indices $i,j\in[d]$ and an amount $\varepsilon\geq 0$. It has the following
action

$\displaystyle(x_{1},\ldots,x_{i},\ldots,x_{j},\ldots,x_{d})\mapsto(x_{1},\ldots,x_{i}+\varepsilon,\ldots,x_{j}-\varepsilon,\ldots,x_{d}).$
(II.5)

Here, $i$ is the receiving entry of the transfer and $j$ is the giving entry.
If $x_{i}\geq x_{j}$, then when applied to $x$, the transfer is said to be
unjust.

######  Proposition II.1.

Let $x,y\in\mathbb{R}^{d}$. Then, $y=Tx$ for some $T$-transform $T$ if and
only if $x^{\downarrow}$ can be obtained from $y^{\downarrow}$ with an unjust
transfer.

###### Proof.

Since both $T$-transforms and transfers affect at most a pair of coordinates,
we may assume without loss of generality that $d=2$. Denote $x=(x_{1},x_{2})$
and $y=(y_{1},y_{2})$.

Suppose that there is a $T$-transform $T$ such that $y=Tx$. Then, for some
$t\in[0,1]$

$\displaystyle(y_{1},y_{2})=((1-t)x_{1}+tx_{2},(1-t)x_{2}+tx_{1}).$ (II.6)

Because $y^{\downarrow}_{1}$ is a convex combination of $x_{1}$ and $x_{2}$,
we have $x^{\downarrow}_{1}\geq y^{\downarrow}_{1}$. Let
$\tilde{\varepsilon}:=x^{\downarrow}_{1}-y^{\downarrow}_{1}$, and observe that
$x^{\downarrow}_{1}=y^{\downarrow}_{1}+\tilde{\varepsilon}$ and
$x^{\downarrow}_{2}=x^{\downarrow}_{1}+x^{\downarrow}_{2}-x^{\downarrow}_{1}=y^{\downarrow}_{1}+y^{\downarrow}_{2}-x^{\downarrow}_{1}=y^{\downarrow}_{2}-\tilde{\varepsilon}$,
where we used the fact that $x_{1}+x_{2}=y_{1}+y_{2}$.

Conversely, suppose there exists an unjust transfer of $\varepsilon\geq 0$
that takes $y^{\downarrow}$ to $x^{\downarrow}$. That is,
$x^{\downarrow}_{1}=y^{\downarrow}_{1}+\varepsilon$ and
$x^{\downarrow}_{2}=y^{\downarrow}_{2}-\varepsilon$. Observe that for
$\tilde{t}:=\frac{\varepsilon}{(y^{\downarrow}_{1}-y^{\downarrow}_{2})+2\varepsilon}$,

$\displaystyle
y^{\downarrow}_{1}=(1-\tilde{t})x^{\downarrow}_{1}+\tilde{t}x^{\downarrow}_{2},\quad
y^{\downarrow}_{2}=(1-\tilde{t})x^{\downarrow}_{2}+\tilde{t}x^{\downarrow}_{1}.$
(II.7)

Hence, there exists a $T$-transform $T^{\prime}$ such that
$T^{\prime}x^{\downarrow}=y^{\downarrow}$. Without loss of generality, we may
take $x=x^{\downarrow}$. If $y\neq y^{\downarrow}$, we may compose a
transposition with $T^{\prime}$ to get another $T$-transform $T$ that
satisfies $Tx=y$. ∎

From this proposition, we conclude that $x\succeq y$ if and only if it is
possible to get to $x$ from $y$ with a finite number of transpositions and
unjust transfers. In the particular case of $x,y\in\mathbb{R}^{2}$, $x\succeq
y$ if and only if $x$ can be obtained from $y$ by an unjust transfer, or an
unjust transfer and a transposition. This characterization may be extended to
self-adjoint operators as follows. For $B,C\in\mathcal{S}(\mathcal{H})$,
$B\succeq C$ if and only if there exists a finite number of transpositions and
unjust transfers that take $\lambda(C)$ to $\lambda(B)$.

Let $f$ be a real-valued function such that $\text{dom}(f)$ is a convex set
contained in either $\mathbb{R}^{d}$ or $\mathcal{S}(\mathcal{H})$. Suppose
that $\text{dom}(f)$ is closed under the action of the symmetric group if it
is contained in $\mathbb{R}^{d}$ and closed under the action of the unitary
group if it is contained in $\mathcal{S}(\mathcal{H})$. We say that $f$ is
Schur-convex if for all $a,b\in\text{dom}(f)$, $a\succeq b$ implies $f(a)\geq
f(b)$. If $-f$ is Schur-convex, then $f$ is said to be Schur-concave. Let $F$
be a real-vector-valued function such that $\text{dom}(F)$ is a convex set
contained in either $\mathbb{R}^{d}$ or $\mathcal{S}(\mathcal{H})$. Similarly,
suppose that $\text{dom}(F)$ is closed under the action of the symmetric group
if it is contained in $\mathbb{R}^{d}$ and closed under the action of the
unitary group if it is contained in $\mathcal{S}(\mathcal{H})$. $F$ is called
strictly isotone if for all $a,b\in\text{dom}(F)$ such that $a\succeq b$, it
holds that $F(a)\succeq F(b)$.

An important tool in our analysis is Fan’s maximum principle [5] for self-
adjoint operators. It states that for $T\in\mathcal{S}(\mathbb{C}^{d})$,
$k\in[d]$,

$\displaystyle\sum_{i=1}^{k}\lambda_{i}(T)=\max\\{\tr(TP)\;|\;\text{$P$ is a
rank-$k$ projector}\\}.$ (II.8)

It follows directly from this principle that

$\displaystyle\sum_{i=1}^{s}S_{i}\preceq\sum_{i=1}^{s}S_{i}^{\downarrow}$
(II.9)

for any self-adjoint $S_{1},\ldots,S_{s}$. Moreover, the inequalities above
are simultaneously satisfied with equality if and only if $S_{1},\ldots,S_{s}$
are perfectly aligned.

We are interested in maximizing continuous functions over
$\mathcal{D}(\mathcal{H}^{\otimes n})$, specifically those that are convex and
unitarily invariant. Of particular interest are unitarily invariant norms.
Generically denoted by $|||\cdot|||$, these are operator norms that satisfy
$|||\cdot|||=|||U(\cdot)V|||$ for all unitaries $U$ and $V$. For an operator
$R\in\mathcal{B}(\mathbb{C}^{d})$, its Schatten $p$-norm for $p\in[1,\infty)$
is defined as

$\displaystyle||R||_{p}:=\tr(|R|^{p})^{\frac{1}{p}},$ (II.10)

where $|R|$ denotes the absolute value $\sqrt{R^{*}R}$. Schatten norms satisfy
$\lim_{p\rightarrow\infty}||\cdot||_{p}=||\cdot||$, where $||\cdot||$ is the
operator norm. Both Rényi entropies and Tsallis entropies [16, 22] are
monotonic functions of Schatten $p$-norms for $p\in(1,\infty)$. For
$R\in\mathcal{B}(\mathbb{C}^{d})$, $k\in[d]$, the Fan norm of order $k$ is the
sum of the $k$ largest singular values of $R$:

$\displaystyle||R||_{(k)}:=\sum_{i=1}^{k}\sigma_{i}(R).$ (II.11)

For positive semi-definite $R$, $||R||_{(k)}$ is the sum of its $k$ largest
eigenvalues. Both Schatten norms and Fan norms are unitarily invariant norms.
For any $A,B\in\mathcal{B}(\mathbb{C}^{d})$, $||A||_{(k)}\leq||B||_{(k)}$ for
all $k\in[d]$ implies $|||A|||\leq|||B|||$ for every unitarily invariant norm
$|||\cdot|||$. This is Fan’s dominance theorem (see Theorem IV.2.2 on page 93
in Ref. [3]).

For readability, we index factors in tensor products with subsets, each
denoting the domain of the corresponding factor. For example, given
$O\in\mathcal{B}(\mathcal{H})$ and $J\subseteq[n]$, $O^{\otimes J}$ denotes
the fully factorizable operator acting on $|J|$ specific tensor copies of
$\mathcal{H}$—specified to be corresponding to $J$—with each of its factors is
equal to $O$. In general, we do not write the curly brackets of sets in
subscripts.

##  III Spin alignment problems and the alignment ordering

######  Definition III.1.

(Spin alignment problem). Let a local dimension $d$, a number of qudits $n$, a
probability measure $\mu$, and a state
$Q=\sum_{i=1}^{d}\lambda_{i}(Q){|{q_{i}}\rangle}{\langle{q_{i}}|}\in\mathcal{D}(\mathbb{C}^{d})$
be given. If $f$ is a continuous, unitarily invariant, convex function whose
domain includes $\mathcal{D}((\mathbb{C}^{d})^{\otimes n})$, then the spin
alignment problem associated with $(d,n,\mu,Q,f)$ is to maximize the objective
function $f(\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes I^{c}})$
over all state tuples $(\rho_{I})_{I\subseteq[n]}$. The operator argument

$\displaystyle\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes I^{c}}$
(III.1)

is called an alignment operator.

Since the objective function is continuous and the set of alignment operators
is compact, an optimal point exists. For concave functions, such as the von
Neumann entropy, the associated spin alignment problem is a minimization
problem. The orthonormal basis $({|{q_{i}}\rangle})_{i=1}^{d}$ and bases
arising from taking tensor powers of it are called computational bases.
Operators that are diagonal in these bases are called classical. We refer to
assertions that the state tuple $({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$ is optimal as spin alignment conjectures. We mostly focus
on the dependence of the problems on the state $Q$ and the objective function
$f$ and implicitly take the remaining parameters to be arbitrary. For example,
when we speak of the spin alignment problem associated with a state $Q$, we
mean the collection of all spin alignment problems with state $Q$.

We conjecture that the spectrum of an alignment operator with state tuple
$({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I})_{I\subseteq[n]}$ majorizes
the spectra of alignment operators arising with other state tuples for the
same $Q$ and $\mu$. That is, it has the largest possible value of the
order-$k$ Fan norm for each $k\in[d^{n}]$. By the doubly-stochastic processing
characterization of majorization, if this conjecture is true, then every spin
alignment conjecture is true. We call this the strong spin alignment
conjecture. In cases where $Q$ is pure, the conjecture follows directly from
Eq. II.9. So, we focus on the cases where $Q$ is mixed.

Spin alignment problems can be extended to cases where the operators $Q$ are
not the same in each term in the sum in Eq. III.1. For example, we can
consider spin alignment conjectures in the more general case where the
alignment operators are of the form

$\displaystyle\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q_{I^{c}},$ (III.2)

where for $I^{c}\subseteq[n],Q_{I^{c}}=\bigotimes_{j\in I^{c}}Q_{I^{c},j}$
such that for each $j\in[n]$, the operators in the tuple
$(Q_{I^{c},j})_{I^{c}\subseteq[n]}$ are perfectly aligned. In this work, we
consider only alignment operators as in Eq. III.1, though many of our results
apply for these more general alignment operators.

######  Lemma III.2.

For any spin alignment problem, it suffices to restrict to state tuples of
pure states.

###### Proof.

Notice that the alignment operator is a linear function of the state tuple, so
convex combinations of state tuples correspond to convex combinations of
alignment operators. By convexity of the objective function, there exists an
extremal point that is optimal. The extremal points of state tuples are pure-
state tuples. ∎

######  Remark III.3.

If the spin alignment conjecture for a function $g$ is true, then it also
holds that $f=h\circ g$ attains its maximum over alignment operators at
$({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I})_{I\subseteq[n]}$ whenever
$h$ is monotonically increasing. If $h$ is monotonically decreasing, then $f$
attains a minimum there. Examples of this include Schatten norms and the
corresponding Rényi entropies.

In addition to showing additivity of coherent information for platypus
channels, proving the strong spin alignment conjecture would formalize
intuitions we have about how dispersion behaves under mixing. Namely, to
minimize dispersion in a mixture, the optimal procedure ought to be one where
each signal has the lowest dispersion possible and the signals collectively
overlap as much as possible. This can be considered a refinement of Fan’s
majorization relation Eq. I.1 because in the spin alignment problem, while
each summand has minimum dispersion by a pure state choice, we are not always
free to choose the bases so that they are perfectly aligned.

When operators cannot be chosen to be perfectly aligned to achieve minimum
dispersion, we need a way of comparing the alignment of different tuples of
operators. The next definition provides such a way while restricting the
tuples to have matching spectra.

######  Definition III.4.

(Alignment) Let $\mathcal{T}=(T_{1},T_{2},\ldots,T_{\ell})$ and
$\mathcal{R}=(R_{1},R_{2},\ldots,R_{\ell})$ be two tuples of self-adjoint
operators such that for each $i\in[\ell],\lambda(T_{i})=\lambda(R_{i})$. Then,
$\mathcal{T}$ is more aligned than $\mathcal{R}$ if for all probability
measures $(p_{i})_{i=1}^{\ell}$, the following majorization relation holds

$\displaystyle\sum_{i=1}^{\ell}p_{i}T_{i}\succeq\sum_{i=1}^{\ell}p_{i}R_{i}.$
(III.3)

Alignment can be used to construct a preorder on any set of $\ell$-tuples of
operators with matching spectra. We are interested in cases where there exists
a maximal element according to this ordering. The basic example is one where
the set in question contains a tuple of perfectly aligned operators. The
strong spin alignment conjecture may be phrased as follows. The tuple
$({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I}\otimes Q^{\otimes
I^{c}})_{I\subseteq[n]}$ is more aligned than every state tuple of the form
$({|{\psi_{I}}\rangle}{\langle{\psi_{I}}|}_{I}\otimes Q^{\otimes
I^{c}})_{I\subseteq[n]}$.

A key reason to consider alignment as opposed to perfect alignment is that the
latter does not behave well with respect to tensor multiplication.
Specifically, the coordinate-wise tensor product of two perfectly aligned
tuples is not necessarily perfectly aligned. Even worse, the perfect alignment
of $A$ and $B$ does not imply the perfect alignment of $C\otimes A$ and
$C\otimes B$ for self-adjoint $A,B$, and $C$.

######  Example III.5.

Consider
$D={|{q_{1}}\rangle}{\langle{q_{1}}|}+\frac{1}{2}{|{q_{2}}\rangle}{\langle{q_{2}}|}+\frac{1}{3}{|{q_{3}}\rangle}{\langle{q_{3}}|}$.
It is clear that $D$ and ${|{q_{1}}\rangle}{\langle{q_{1}}|}$ are perfectly
aligned. However, $D\otimes D$ and
$D\otimes{|{q_{1}}\rangle}{\langle{q_{1}}|}$ are not perfectly aligned.

In contrast, $(C\otimes A^{\downarrow},C\otimes B^{\downarrow})$ is more
aligned than $(C\otimes A,C\otimes B)$. This is a simple consequence of Eq.
II.9 and the following lemma.

######  Lemma III.6.

Let $A,B\in\mathcal{S}(\mathcal{H})$ and $C\in\mathcal{S}(\mathcal{K})$. If
$A\succeq B$, then $C\otimes A\succeq C\otimes B$.

###### Proof.

This follows from the doubly stochastic characterization of majorization.
$A\succeq B$ is equivalent to the existence of a mixed-unitary quantum channel
$\mathcal{N}$ such that $B=\mathcal{N}(A)$. It is clear that
$\text{id}\otimes\mathcal{N}$ is mixed-unitary and
$\text{id}\otimes\mathcal{N}(C\otimes A)=C\otimes B$.∎

If we restrict the factors on the left to be positive semi-definite, we can
prove the following statement.

######  Proposition III.7.

Let $C_{1},\ldots,C_{\ell}\in\mathcal{P}(\mathcal{K})$ be simultaneously
diagonalizable. For every $A_{1},\ldots,A_{\ell}\in\mathcal{S}(\mathcal{H})$,
$(C_{i}\otimes A^{\downarrow}_{i})_{i=1}^{\ell}$ is more aligned than
$(C_{i}\otimes A_{i})_{i=1}^{\ell}$.

###### Proof.

Let $p=(p_{i})_{i=1}^{\ell}$ be an arbitrary probability measure. Let
$\\{{|{j}\rangle}\\}_{j=1}^{d_{\mathcal{K}}}$ be an orthonormal basis for
$\mathcal{K}$ where $C_{1},\ldots.C_{\ell}$ are simultaneously diagonal. For
each $i\in[\ell]$, let
$\sum_{j=1}^{d_{\mathcal{K}}}c_{j|i}{|{j}\rangle}{\langle{j}|}$ be the
corresponding spectral decomposition of $C_{i}$. Observe that

$\displaystyle\sum_{i=1}^{\ell}p_{i}C_{i}\otimes
A^{\downarrow}_{i}=\sum_{i=1}^{\ell}p_{i}\sum_{j=1}^{d_{\mathcal{K}}}c_{j|i}{|{j}\rangle}{\langle{j}|}\otimes
A^{\downarrow}_{i}=\sum_{j=1}^{d_{\mathcal{K}}}{|{j}\rangle}{\langle{j}|}\otimes(\sum_{i=1}^{\ell}p_{i}c_{j|i}A^{\downarrow}_{i}).$
(III.4)

For each $j\in[d_{\mathcal{K}}]$, we know from Eq. II.9 that
$\sum_{i=1}^{\ell}p_{i}c_{j|i}A^{\downarrow}_{i}\succeq\sum_{i=1}^{\ell}p_{i}c_{j|i}A_{i}$.
Hence, there exists a mixed-unitary channel $\mathcal{N}_{j}$ that takes the
left-hand side to the right-hand side. Let $\mathcal{N}$ be the global channel
that conditional on $j$ applies $\mathcal{N}_{j}$ for each
$j\in[d_{\mathcal{K}}]$. It is a mixed-unitary quantum channel and it takes
$\sum_{i=1}^{\ell}p_{i}C_{i}\otimes A^{\downarrow}_{i}$ to
$\sum_{i=1}^{\ell}p_{i}C_{i}\otimes A_{i}$. ∎

The strong spin alignment conjecture subsumes a conjecture about maximizing
unitarily invariant norms at the output of a tensor product of depolarizing
channels. For $q\in[\frac{-1}{d^{2}-1},1]$, the qudit depolarizing channel
$\Delta_{d,q}$ has the action:

$\displaystyle\Delta_{d,q}(\cdot):=q\,\text{id}(\cdot)+(1-q)\frac{\mathds{1}}{d}\tr(\cdot).$
(III.5)

In [9], to show additivity of the Holevo information of depolarizing channels,
King proves that for any $p\in[1,\infty)$, the Schatten $p$-norm of the output
of a tensor product of depolarizing channels is maximized at a pure product
input. It is reasonable to extrapolate and suspect that this holds for all
unitarily invariant norms. The basic idea is that any entanglement in the
input manifests as excess dispersion at the output. For $i\in[n]$, let
$q_{i}\in[0,1]$ and notice that

$\displaystyle\bigotimes_{i=1}^{n}\Delta_{d,q_{i}}(\rho_{[n]})=\sum_{I\subseteq[n]}\nu_{I}\rho_{I}\otimes(\frac{\mathds{1}}{d})^{\otimes
I^{c}},$ (III.6)

where $\nu_{I}=\prod_{i\in I}q_{i}\prod_{j\in I^{c}}(1-q_{j})$ and $\rho_{I}$
denotes the marginal state on $I$ of $\rho_{[n]}$. This is an alignment
operator with $Q=\frac{\mathds{1}}{d}$ arising from a state tuple satisfying
an instance of the quantum marginal problem [17]. Since the conjectured
optimal point $({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$ satisfies an instance of the quantum marginal problem, it
follows that a proof of the strong spin alignment conjecture for
$Q=\frac{\mathds{1}}{d}$ implies a resolution for this problem. This problem
is analogous to the now-solved problem of maximizing unitarily invariant
output norms for single-mode Gaussian channels. Specifically, it was shown in
[12] that coherent states are optimal.

##  IV Results

### A. First observations

First, we show that the spin alignment conjecture is true for the case where
the objective function is $\lambda_{1}$.

######  Proposition IV.1.

$\lambda_{1}(\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes I^{c}})$ is
maximized at
$(\rho_{I})_{I\subseteq[n]}=({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$.

###### Proof.

This follows from Eq. II.8. Specifically,

$\displaystyle\lambda_{1}(\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes
Q^{\otimes I^{c}})\leq\sum_{I\subseteq[n]}\mu_{I}\lambda_{1}(\rho_{I}\otimes
Q^{\otimes I^{c}})\leq\sum_{I\subseteq[n]}\mu_{I}\lambda_{1}(Q^{\otimes
I^{c}}).$ (IV.1)

At $({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I})_{I\subseteq[n]}$, both
inequalities are satisfied with equality. ∎

This proves one of $d^{{n}}-1$ conditions in Eq. II.1 necessary to prove the
strong spin alignment conjecture. The difficulty in proving the remaining ones
lies in the fact that summands of alignment operators with pure-state tuples
cannot be made perfectly aligned in general.

As explained in the previous section, the strong spin alignment conjecture
holds for pure $Q$. Next, for $Q$ of rank $2$ or more, we prove a reduction to
cases where $\lambda_{1}(Q)=\lambda_{2}(Q)$.

######  Proposition IV.2.

Suppose
$Q=\sum_{i=1}^{d}\lambda_{i}(Q){|{q_{i}}\rangle}{\langle{q_{i}}|}\in\mathcal{D}(\mathcal{H})$
has rank $\geq 2$. Define

$\displaystyle\tilde{Q}=\frac{1}{1-(\lambda_{1}(Q)-\lambda_{2}(Q))}\Big{(}\lambda_{2}(Q)({|{q_{1}}\rangle}{\langle{q_{1}}|}+{|{q_{2}}\rangle}{\langle{q_{2}}|})+\sum_{i=3}^{d}\lambda_{i}(Q){|{q_{i}}\rangle}{\langle{q_{i}}|}\Big{)}.$
(IV.2)

If for all $\tilde{\mu}$,
$f(\sum_{I\subseteq[n]}\tilde{\mu}_{I}\rho_{I}\otimes\tilde{Q}^{\otimes
I^{c}})$ is maximized at
$(\rho_{I})_{I\subseteq[n]}=({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$, then for all $\mu$,
$f(\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes I^{c}})$ is maximized
there as well.

###### Proof.

Denote $\varepsilon:=\lambda_{1}(Q)-\lambda_{2}(Q)$. Observe that
$Q=(1-\varepsilon)\tilde{Q}+\varepsilon{|{q_{1}}\rangle}{\langle{q_{1}}|}$.
That is, $Q$ lies on the line segment connecting $\tilde{Q}$ and
${|{q_{1}}\rangle}{\langle{q_{1}}|}$. For any probability measure $\mu$, we
can express

$\displaystyle\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes
I^{c}}=\sum_{K\subseteq[n]}\tilde{\mu}_{K}\tilde{\rho}_{K}\otimes\tilde{Q}^{\otimes
K^{c}},$ (IV.3)

where each factor of $Q$ on the left-hand side was replaced by
$(1-\varepsilon)\tilde{Q}+\varepsilon{|{q_{1}}\rangle}{\langle{q_{1}}|}$ and
$\mu_{I}\rho_{I}\otimes\varepsilon^{|J|}(1-\varepsilon)^{|(I\cup
J)^{c}|}\outerproduct{q_{1}}{q_{1}}^{\otimes J}$ for $I\cup J=K$ were summed
to obtain $\tilde{\mu}_{K}\tilde{\rho}_{K}$. By assumption,

$\displaystyle f(\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes
I^{c}})$
$\displaystyle=f(\sum_{K\subseteq[n]}\tilde{\mu}_{K}\tilde{\rho}_{K}\otimes\tilde{Q}^{\otimes
K^{c}})$ (IV.4) $\displaystyle\leq
f(\sum_{K\subseteq[n]}\tilde{\mu}_{K}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
K}\otimes\tilde{Q}^{\otimes
K^{c}})=f(\sum_{I\subseteq[n]}\mu_{I}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I}\otimes Q^{\otimes I^{c}}),$ (IV.5)

where we reversed the process used to obtain the right-hand side of Eq. IV.3
for $(\rho_{I})_{I\subseteq[n]}=(\outerproduct{q_{1}}{q_{1}}^{\otimes
I})_{I\subseteq[n]}$ to arrive at the last equality. ∎

######  Remark IV.3.

For $d=2$, this reduction implies that $Q=\frac{\mathds{1}}{2}$ may be assumed
without loss of generality. One can therefore apply the argument in Sec. VI.B
of [11] to show that the strong spin alignment conjecture holds for $n=d=2$.

The last result of this subsection relates to problem instances where the
union of the elements of the support of the probability measure $\mu$ is not
$[n]$. That is, those instances where $\mu$ is such that there exists nonempty
$K^{c}\subseteq[n]$ such that $\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes
Q^{\otimes I^{c}}=(\sum_{J\subseteq[n]\setminus K^{c}}\mu_{J}\rho_{J}\otimes
Q^{\otimes J^{c}})\otimes Q^{\otimes K^{c}}$. If the objective function is
multiplicative under tensor multiplication, such as Schatten norms [2], then
we may without loss of generality trace over the qudits in $K^{c}$ and
consider the optimization problem on the rest. Not all Fan norms are
multiplicative under tensor multiplication. However, to prove the strong spin
alignment conjecture, it may be assumed without loss of generality that the
union of the support of $\mu$ is $[n]$.

######  Proposition IV.4.

For every $n_{1},n_{2}\in\mathbb{N}$, if the strong spin alignment conjecture
holds for $\sum_{I\subseteq[n_{1}]}\mu_{I}\rho_{I}\otimes Q^{\otimes I^{c}}$,
then it holds for $\sum_{I\subseteq[n_{1}]}\mu_{I}\rho_{I}\otimes Q^{\otimes
I^{c}}\otimes Q^{\otimes n_{2}}$.

###### Proof.

This a direct consequence of Lem. III.6. ∎

### B. Classical spin alignment

We prove a classical version of the strong spin alignment conjecture. That is,
we prove it with the extra assumption that $(\rho_{I})_{I\subseteq[n]}$ is
such that $\rho_{I}$ is diagonal in the computational basis for each
$I\subseteq[n]$.

######  Proposition IV.5.

For every probability measure $\mu$ and tuple of classical states
$(\rho_{I})_{I\subseteq[n]}$,

$\displaystyle\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes
I^{c}}\preceq\sum_{I\subseteq[n]}\mu_{I}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I}\otimes Q^{\otimes I^{c}},$ (IV.6)

###### Proof.

By the same argument as in Lem. III.2, without loss of generality, the state
tuple $(\rho_{I})_{I\subseteq[n]}$ may be taken to consist of pure classical
states. For $I\subseteq[n]$, we denote
$\rho_{I}={|{t_{I}}\rangle}{\langle{t_{I}}|}$, where $t_{I}$ is a string of
length $|I|$ over the alphabet $[d]$. For example,
${|{123}\rangle}{\langle{123}|}:={|{q_{1}}\rangle}{\langle{q_{1}}|}\otimes{|{q_{2}}\rangle}{\langle{q_{2}}|}\otimes{|{q_{3}}\rangle}{\langle{q_{3}}|}$.

The idea of the proof is to start with an arbitrary classical assignment
$(t_{I})_{I\subseteq[n]}$ and flip to 1 sequentially to improve the assignment
making use of the transitivity of the majorization ordering. We start by
considering the first qudit. We write the alignment operator in the form

$\displaystyle\sum_{I\subseteq[n]}\mu_{I}{|{t_{I}}\rangle}{\langle{t_{I}}|}\otimes
Q^{\otimes I^{c}}=Q\otimes B+{|{1}\rangle}{\langle{1}|}\otimes
C+\sum_{j=2}^{d}{|{j}\rangle}{\langle{j}|}\otimes D_{j},$ (IV.7)

where $B$, $C$, and $D_{j}$ can be expressed as

$\displaystyle B$
$\displaystyle=\sum_{I\subseteq[n]\setminus\\{1\\}}\nu_{I}{|{t^{\prime}_{I}}\rangle}{\langle{t^{\prime}_{I}}|}\otimes
Q^{I^{c}},$ (IV.8) $\displaystyle C$
$\displaystyle=\sum_{I\subseteq[n]\setminus\\{1\\}}w_{I}{|{t^{\prime\prime}_{I}}\rangle}{\langle{t^{\prime\prime}_{I}}|}\otimes
Q^{I^{c}},$ (IV.9) $\displaystyle D_{j}$
$\displaystyle=\sum_{I\subseteq[n]\setminus\\{1\\}}\kappa_{j,I}{|{t_{j,I}}\rangle}{\langle{t_{j,I}}|}\otimes
Q^{I^{c}},$ (IV.10)

where the summand $Q\otimes B$ accounts for all $t_{I}$ with $1\not\in I$,
$\outerproduct{1}{1}\otimes C$ accounts for all $t_{I}$ with $1\in I$ and
first letter $1$ (the “good” $t_{I}$) and the $D_{j}$ account for the rest.
The measures $\nu,w,$ and $\kappa_{j}$ are positive, and the operators $B,C,$
and $D_{j}$ are positive semi-definite and classical. We consider each
$j\in[d]\setminus\\{1\\}$ in sequence and flip the first entries of those
$t_{I}$ contributing to $D_{j}$ from $j$ to $1$. This implements the following
change in the classical assignment $(t_{I})_{I\in\subseteq[n]}$: For every
$t_{I}$ of the form
$t_{I}={|{j}\rangle}{\langle{j}|}\otimes{|{t_{j,I^{\prime}}}\rangle}{\langle{t_{j,I^{\prime}}}|}$
contributing to the summand ${|{j}\rangle}{\langle{j}|}\otimes D_{j}$, replace
it with
${|{1}\rangle}{\langle{1}|}\otimes{|{t_{j,I}}\rangle}{\langle{t_{j,I}}|}$. The
representation in Eq. IV.7 changes according to the assignments
$D_{1}\leftarrow D_{1}+D_{j}$ and $D_{j}\leftarrow 0$, so that the new
alignment operator is compatible with $Q$ and $\mu$. For a string $r$ of
length $n-1$ over $[d]$, let $b_{r},c_{r},$ and $d_{j,r}$ be the eigenvalues
for the eigenstate ${|{r}\rangle}$ of $B,C,$ and $D_{j,r}$ before the change.
The change in the spectrum can be implemented by composing transfers of the
form: $d_{j,r}$ is taken from the eigenvalue of
${|{j}\rangle}\otimes{|{r}\rangle}$ and given to the eigenvalue of
${|{1}\rangle}\otimes{|{r}\rangle}$ for each string $r$. These transfers may
be implemented in parallel. For a string $r$, those two entries in the
spectrum before the change are

$\displaystyle(q_{1}b_{r}+c_{r},b_{r}q_{j}+d_{j,r}),$ (IV.11)

while after, they are

$\displaystyle(q_{1}b_{r}+c_{r}+d_{j,r},b_{r}q_{j}).$ (IV.12)

To see that the spectrum after the transfer majorizes the one before it,
observe that

$\displaystyle
q_{1}b_{r}+c_{r}+d_{j,r}\geq\max{(q_{1}b_{r}+c_{r},q_{j}b_{r}+d_{j,r})}.$
(IV.13)

That is, the larger value of the pair did not decrease after the transfer.
Hence, the transfer is either unjust or its composition with a transposition
is unjust. By transitivity of majorization, the same process may now be done
to the remaining qudits. ∎

### C. Schatten norms of integer order

We prove the spin alignment conjecture for Schatten norms of integer order. By
extension, this implies that any monotonic function of these norms, such as
the corresponding Rényi entropies, is optimized at the conjectured optimal
state tuple $({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$. The proof relies on the fact that overlaps between
summands in an alignment operator are simultaneously maximized at the
conjectured optimal state tuple. Here, overlap means the absolute value of the
trace of any product of these summands. In fact, the overlap lemma Lem. A.1 in
App. A establishes the stronger statement that every unitarily invariant norm
of such a product is maximized at
$({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I})_{I\subseteq[n]}$.

######  Corollary IV.6.

Let $I_{1},\ldots,I_{\ell}$ be a family of subsets of $[n]$. For any tuple
$(R_{I_{i}})_{i=1}^{\ell}$ of operators each with trace norm at most 1, the
following inequality holds

$\displaystyle\left|\tr(\;\prod_{i=1}^{\ell}R_{I_{i}}\otimes Q^{\otimes
I_{i}^{c}})\right|\leq\tr(\;\prod_{i=1}^{\ell}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I_{i}}\otimes Q^{\otimes I_{i}^{c}}).$ (IV.14)

###### Proof.

The result follows from an application of Lem. A.1 in the following chain of
inequalities:

$\displaystyle\left|\tr(\;\prod_{i=1}^{\ell}R_{I_{i}}\otimes Q^{\otimes
I_{i}^{c}})\right|$
$\displaystyle\leq\tr(\left|\;\prod_{i=1}^{\ell}R_{I_{i}}\otimes Q^{\otimes
I_{i}^{c}}\right|)$
$\displaystyle\leq\tr(\left|\;\prod_{i=1}^{\ell}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I_{i}}\otimes Q^{\otimes I_{i}^{c}}\right|)\hskip 18.06749pt\textrm{by
Lem.~{}\ref{overlap lemma}}$
$\displaystyle=\tr(\;\prod_{i=1}^{\ell}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I_{i}}\otimes Q^{\otimes I_{i}^{c}}).$ (IV.15)

∎

######  Theorem IV.7.

For arbitrary $m\in\mathbb{N}$, the following inequality holds

$\displaystyle||\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes
I^{c}}||_{m}\leq||\sum_{I\subseteq[n]}\mu_{I}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I}\otimes Q^{\otimes I^{c}}||_{m}$ (IV.16)

###### Proof.

Notice that $||\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes
I^{c}}||_{m}^{m}$ is a linear combination of traces of products of operators
of the form $\rho_{I}\otimes Q^{\otimes I^{c}}$ for some $I\subseteq[n]$. By
Corollary IV.6, the absolute value of each term is maximized at
$({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I})_{I\subseteq[n]}$. By the
triangle inequality for the absolute value and the fact that the terms in the
combination have equal phase at $({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$,

$\displaystyle||\sum_{I\subseteq[n]}\mu_{I}\rho_{I}\otimes Q^{\otimes
I^{c}}||_{m}^{m}\leq||\sum_{I\subseteq[n]}\mu_{I}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I}\otimes Q^{\otimes I^{c}}||_{m}^{m}.$ (IV.17)

The claim follows from the fact that $m$th root function is monotonically non-
decreasing on $[0,\infty)$. ∎

### D. The case where $\text{supp}(\mu)\leq 2$ and $Q$ is proportional to a
projector

We now prove the strong spin alignment conjecture for alignment operators of
the form

$\displaystyle(1-t){|{\psi_{{1}}}\rangle}{\langle{\psi_{{1}}}|}_{I_{1}}\otimes
Q^{\otimes
I_{1}^{c}}+t{|{\psi_{{2}}}\rangle}{\langle{\psi_{{2}}}|}_{I_{2}}\otimes
Q^{\otimes I_{2}^{c}},$ (IV.18)

where $t\in[0,1]$, $I_{1},I_{2}\subseteq[n]$, and $Q$ is proportional to a
projector. Since we have established that we may take $I_{1}\cup I_{2}=[n]$
without loss of generality (see Lem. IV.4), the cases where $I_{1}\subseteq
I_{2}$ or $I_{1}\supseteq I_{2}$ are simply resolved by Eq. II.9. To address
the other cases, we consider the more general problem of the alignment of two
projectors subject to a certain overlap constraint.

Let $\mathcal{K}$ be given. For each tuple $(r_{1},r_{2},c)$, where
$r_{1},r_{2}\in[d_{\mathcal{K}}]\cup\\{0\\}$ and $c\geq 0$, let
$G_{r_{1},r_{2},c}$ denote the set of pairs of projectors $(P_{1},P_{2})$
satisfying the conditions: $\text{rank}(P_{1})=r_{1}$,
$\text{rank}(P_{2})=r_{2}$, and $\tr(|P_{1}P_{2}|)\leq c$.

######  Lemma IV.8.

$G_{r_{1},r_{2},c}$ is nonempty if and only if
$r_{1}+r_{2}-d_{\mathcal{K}}\leq\lfloor c\rfloor$.

###### Proof.

Suppose $r_{1}+r_{2}-d_{\mathcal{K}}>\lfloor c\rfloor$. If $P_{1}$ and $P_{2}$
are projectors satisfying $\text{rank}(P_{1})=r_{1}$ and
$\text{rank}(P_{2})=r_{2}$, then from basic linear algebra

$\displaystyle\dim(\text{supp}(P_{1})\cap\text{supp}(P_{2}))\geq
r_{1}+r_{2}-d_{\mathcal{K}}>\lfloor c\rfloor.$ (IV.19)

Hence, $P_{1}P_{2}$ acts as the identity on a subspace of dimension at least
$\lceil c\rceil$ and so $\tr(|P_{1}P_{2}|)>c$. This implies
$G_{r_{1},r_{2},c}$ is empty.

If on the other hand $r_{1}+r_{2}-d_{\mathcal{K}}\leq\lfloor c\rfloor$, let
$P$ be a projector of rank equal to $\max(r_{1}+r_{2}-d_{\mathcal{K}},0)$.
Consider the two projectors $P_{1}=P+R_{1}$ and $P_{2}=P+R_{2}$, where
$R_{1},R_{2}$ are projectors satisfying $R_{1}\perp P,R_{2}\perp P,R_{1}\perp
R_{2}$, and
$\text{rank}(R_{1})=r_{1}-\text{rank}(P),\text{rank}(R_{2})=r_{2}-\text{rank}(P)$.
Then $P_{1}P_{2}=P$, $\tr(|P_{1}P_{2}|)=\tr(P)\leq\lfloor c\rfloor$, and so
$(P_{1},P_{2})\in G_{r_{1},r_{2},c}$. ∎

From now on, we take for granted that $r_{1}+r_{2}-d_{\mathcal{K}}\leq\lfloor
c\rfloor$. Moreover, since $\tr(|P_{1}P_{2}|)\leq\min(\tr(P_{1}),\tr(P_{2}))$
for any two projectors $P_{1}$ and $P_{2}$, we take $c\leq\min(r_{1},r_{2})$.
Otherwise, the constraint on the product may as well not be considered.

The crucial ingredient of our analysis is a characterization of the angle
between two subspaces due to Camille Jordan [8]. This characterization,
commonly referred to as Jordan’s lemma, allows us to reason about the
relationship between the eigenvalues of a sum of two projectors and the
singular values of their product.

######  Theorem IV.9.

There exists $(P^{\text{opt}}_{1},P^{\text{opt}}_{2})\in G_{r_{1},r_{2},c}$
that is maximal in the alignment ordering on $G_{r_{1},r_{2},c}$, and it
satisfies $\tr(|P^{\text{opt}}_{1}P^{\text{opt}}_{2}|)=c$. Moreover, if
$c\in\mathbb{N}$, then $[P^{\text{opt}}_{1},P^{\text{opt}}_{2}]=0$.

###### Proof.

It suffices to show that there exists
$(P^{\text{opt}}_{1},P^{\text{opt}}_{2})\in G_{r_{1},r_{2},c}$ such that

$\displaystyle s_{1}P_{1}^{\text{opt}}+s_{2}P_{2}^{\text{opt}}\succeq
s_{1}P_{1}+s_{2}P_{2}$ (IV.20)

holds for all $s_{1},s_{2}\geq 0$ and $(P_{1},P_{2})\in G_{r_{1},r_{2},c}$. We
show existence by explicit construction. Let arbitrary $s_{1},s_{2}\geq 0$ be
given and consider an arbitrary pair $(P_{1},P_{2})\in G_{r_{1},r_{2},c}$. We
modify the pair such that it remains in $G_{r_{1},r_{2},c}$ and the
corresponding Fan norms do not decrease until we reach an optimal pair.

We can decompose $\mathcal{K}$ into minimal subspaces invariant under both
$P_{1}$ and $P_{2}$. According to Jordan’s lemma, these subspaces are either
one- or two-dimensional. Thus, we may write
$\mathcal{K}=\mathcal{K}^{(1)}\oplus\mathcal{K}^{(2)}$, where
$\mathcal{K}^{(1)}=\bigoplus_{i_{1}=1}^{m_{1}}\mathcal{K}_{i_{1}}^{(1)}$ is
the direct sum of invariant subspaces of dimension 1, and
$\mathcal{K}^{(2)}=\bigoplus_{i_{2}=1}^{m_{2}}\mathcal{K}_{i_{2}}^{(2)}$ is
the direct sum of invariant subspaces of dimension 2. Moreover, for each
$i_{2}\in[m_{2}]$, $P_{1}|_{\mathcal{K}_{i_{2}}}$ and
$P_{2}|_{\mathcal{K}_{i_{2}}}$ are projectors of rank $1$. Denote the overlap
$\tr(|P_{1}P_{2}|)=\tilde{c}_{1}+\tilde{c}_{2}$ where
$\tilde{c}_{1}=\tr(|P_{1}P_{2}|_{\mathcal{K}^{(1)}}|)\;\text{and}\;\tilde{c}_{2}=\tr(|P_{1}P_{2}|_{\mathcal{K}^{(2)}}|)$.
Observe that $\tilde{c}_{1}$ is an integer as it equals the dimension of
$\text{supp}(P_{1}|_{\mathcal{K}^{(1)}})\cap\text{supp}(P_{2}|_{\mathcal{K}^{(1)}})$.

First, we reduce to the case where $m_{2}\in\\{0,1\\}$. For each
$i_{2}\in[m_{2}]$, we notate

$\displaystyle
P_{1}|_{\mathcal{K}_{i_{2}}^{(2)}}=:{|{\alpha_{i_{2}}}\rangle}{\langle{\alpha_{i_{2}}}|},P_{2}|_{\mathcal{K}_{i_{2}}^{(2)}}=:{|{\beta_{i_{2}}}\rangle}{\langle{\beta_{i_{2}}}|}.$
(IV.21)

Notice that
$\tilde{c}_{2}=\sum_{i_{2}=1}^{m_{2}}|\braket{\alpha_{i_{2}}}{\beta_{i_{2}}}|$.
By Lem. B.2, the vector
$(|\braket{\alpha_{i_{2}}}{\beta_{i_{2}}}|)_{i_{2}=1}^{m_{2}}$ is majorized by
the vector
$(\underbrace{1,\ldots,1}_{\lfloor\tilde{c}_{2}\rfloor\;\text{times}},\tilde{c}_{2}-\lfloor\tilde{c}_{2}\rfloor,0,\ldots,0)$.
For each $i_{2}\in[m_{2}]$, there are
${|{\alpha_{i_{2}}^{\prime}}\rangle},{|{\beta_{i_{2}}^{\prime}}\rangle}\in\mathcal{K}_{i_{2}}^{(2)}$,
such that

$\displaystyle(|\braket{\alpha_{i_{2}}^{\prime}}{\beta_{i_{2}}^{\prime}}|)_{i_{2}=1}^{m_{2}}=(\underbrace{1,\ldots,1}_{\lfloor\tilde{c}_{2}\rfloor\;\text{times}},\tilde{c}_{2}-\lfloor\tilde{c}_{2}\rfloor,0,\ldots,0).$
(IV.22)

Specifically, for each $1$ on the right-hand side we choose the two vectors to
be equal and for each $0$ we choose them to be orthogonal. For the remaining
entry, we pick the two such that they have overlap
$\tilde{c}_{2}-\lfloor\tilde{c}_{2}\rfloor$. By Lem. B.1, the eigenvalues of
$(s_{1}P_{1}+s_{2}P_{2})|_{\mathcal{K}^{(2)}}$ are a strictly isotone function
of the nonzero singular values of $P_{1}P_{2}|_{\mathcal{K}^{(2)}}$, and so we
may make the reassignments

$\displaystyle{|{\alpha_{i_{2}}}\rangle}\leftarrow{|{\alpha_{i_{2}}^{\prime}}\rangle}\;,\;{|{\beta_{i_{2}}}\rangle}\leftarrow{|{\beta_{i_{2}}^{\prime}}\rangle}$
(IV.23)

for each $i_{2}\in[m_{2}]$. Save for at most one, this replaces each
$\mathcal{K}_{i_{2}}^{(2)}$ with two $1$-dimensional invariant subspaces. That
is, we may assume without loss of generality that $m_{2}\in\\{0,1\\}$ and that
$\tilde{c}_{2}<1$.

Now, suppose $c$ is an integer.
$\lambda((s_{1}P_{1}+s_{2}P_{2})|_{\mathcal{K}^{(1)}})$ has $s_{1}+s_{2}$ with
multiplicity $\tilde{c}_{1}$ along with the smaller values $s_{1},s_{2}$ and
$0$. If $\tilde{c}_{2}>0$, then we may adjust the action of the projectors on
the sole invariant $2$-dimensional subspace so that they are equal when
restricted to it. This does not cause the two projectors to exit
$G_{r_{1},r_{2},c}$ as $c-\tilde{c}_{1}$ is a positive integer. Hence, we may
assume without loss of generality that $\tilde{c}_{2}=0$. Then, the number of
one-dimensional invariant subspaces with $s_{1}$ is $r_{1}-\tilde{c}_{1}\geq
c-\tilde{c}_{1}$ and the number of one-dimensional invariant subspaces with
$s_{2}$ is $r_{2}-\tilde{c}_{1}\geq c-\tilde{c}_{1}$. We pair up
$c-\tilde{c}_{1}$ many of the two kinds of invariant one-dimensional subspaces
to get $c-\tilde{c}_{1}$ two-dimensional subspaces. We adjust the projectors
so that they are equal when restricted to these $c-\tilde{c}_{1}$ subspaces.
For each such subspace, this causes a transfer on the spectrum of
$s_{1}P_{1}+s_{2}P_{2}$ of the form:

$\displaystyle s_{1}\leftarrow 0\;,\;s_{2}\leftarrow s_{1}+s_{2}.$ (IV.24)

Since this is either an unjust transfer or an unjust transfer composed with a
transposition, the Fan norms of $s_{1}P_{1}+s_{2}P_{2}$ do not decrease
afterwards. Denote the pair arising after all the transfers are done with
$(P_{1}^{\text{opt}},P_{2}^{\text{opt}})$. Notice that
$\tr(|P_{1}^{\text{opt}}P_{2}^{\text{opt}}|)=c$ and
$[P_{1}^{\text{opt}},P_{2}^{\text{opt}}]=0$.

On the other hand, consider the case where $c$ is not an integer. We may take
$\tilde{c}_{1}=\lfloor c\rfloor$ by the arguments in the previous paragraph.
If $m_{2}=0$, then consider two invariant $1$-dimensional subspaces where the
two projectors are orthogonal. Such a pair exists because $c<\lceil
c\rceil\leq\min(r_{1},r_{2})$. Then, we adjust the action of the two
projectors on the direct sum of these two subspaces so that $m_{2}=1$ and
$\tilde{c}_{2}=c-\lfloor c\rfloor$. That is, we make the projectors align
there as much as possible. From Eq. B.3, this is either an unjust transfer or
an unjust transfer composed with a transposition on the spectrum of
$s_{1}P_{1}+s_{2}P_{2}$. If $m_{2}=1$, then we adjust the overlap in the
$2$-dimensional invariant subspace until $\tilde{c}_{2}=c-\lfloor c\rfloor$.
Denote the pair arising after these adjustments with
$(P^{\text{opt}}_{1},P^{\text{opt}}_{2})$. ∎

By Thm. IV.9, if $c\in\mathbb{N}$, then any pair of commuting projectors in
$G_{r_{1},r_{2},c}$ whose product has rank $c$ is optimal. By Lem. A.1, the
projectors in the alignment operator in Eq. IV.18 have overlap at most
$(\text{rank}(Q))^{|(I_{1}\cup I_{2})^{c}|}$.

######  Corollary IV.10.

For $t\in[0,1],I_{1},I_{2}\subseteq[n]$ and $Q$ proportional to a projector,

$\displaystyle(1-t){|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes{I_{1}}}\otimes
Q^{\otimes
I_{1}^{c}}+t{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes{I_{2}}}\otimes
Q^{\otimes
I_{2}^{c}}\succeq(1-t){|{\psi_{{1}}}\rangle}{\langle{\psi_{{1}}}|}_{I_{1}}\otimes
Q^{\otimes
I_{1}^{c}}+t{|{\psi_{{2}}}\rangle}{\langle{\psi_{{2}}}|}_{I_{2}}\otimes
Q^{\otimes I_{2}^{c}}.$

##  V “Dual” problems to spin alignment and their generalizations

We introduce a general class of conjectures that articulates the intuition
that under a fixed global spectrum constraint, the least locally dispersed
states are classical. The inspiration came from considering spin alignment
problems with $Q\propto\mathds{1}$ and using Fan’s maximum principle Eq. II.8.
Specifically, for a positive measure $\nu$ and a state tuple
$({|{\psi_{I}}\rangle}{\langle{\psi_{I}}|})_{I\subseteq[n]}$, we consider

$\displaystyle\max_{P:\text{rank}(P)=k}\frac{1}{k}\tr(P_{[n]}\sum_{I\subseteq[n]}\nu_{I}{|{\psi_{I}}\rangle}{\langle{\psi_{I}}|}\otimes\mathds{1}^{\otimes
I^{c}})$
$\displaystyle=\max_{P:\text{rank}(P)=k}\frac{1}{k}\sum_{I\subseteq[n]}\nu_{I}\tr(P_{I}{|{\psi_{I}}\rangle}{\langle{\psi_{I}}|})$
(V.1)
$\displaystyle\leq\max_{P:\text{rank}(P)=k}\frac{1}{k}\sum_{I\subseteq[n]}\nu_{I}\lambda_{1}(P_{I}).$
(V.2)

When
$({|{\psi_{I}}\rangle}{\langle{\psi_{I}}|})_{I\subseteq[n]}=({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I})_{I\subseteq[n]}$, the inequality is satisfied with equality. This
maximization problem may be generalized by letting the optimization be over
quantum states with a prescribed global spectrum, and by replacing
$\lambda_{1}$ with any continuous, convex, unitarily invariant function.
Moreover, since the objective function is convex, we may take the convex
closure of the feasible set without loss of generality. We formalize this
below.

Let complex inner product spaces $\mathcal{H}_{1},...,\mathcal{H}_{n}$ be
given and denote $\mathcal{H}_{I}=\bigotimes_{i\in I}\mathcal{H}_{i}$ for
$I\subseteq[n]$. For a probability measure
$p\in\mathbb{R}^{\Pi_{i\in[n]}d_{\mathcal{H}_{i}}}$, we consider the
maximization problem:

$\displaystyle\max_{\tau_{[n]}\in\mathcal{D}(\mathcal{H}_{[n]})}\quad\quad\sum_{I\subseteq[n]}f_{I}(\tau_{I}),$
(V.3) $\displaystyle\text{subject to:}\quad\quad\;\lambda(\tau_{[n]})\preceq
p,$ (V.4)

where for all $I\subseteq[n]$, $f_{I}$ is continuous, convex and unitarily
invariant on a domain containing $\mathcal{D}(\mathcal{H}_{I})$. We conjecture
that there exists a classical optimal point. This is self-evident in the case
where $p$ has only one nonzero value, as all the functions in the sum can be
maximized simultaneously. More interesting are the cases where the
optimization is over strictly mixed states. An example of such a problem in
the quantum information literature appears in Ref. [24]. Namely, Eq. 39 there
gives an upper bound on the fidelity of entanglement generating pure codes in
terms of a sum as above where for each $I\subseteq[n]$,
$f_{I}(\cdot)\propto||\cdot||_{\alpha}^{\alpha}$ for a fixed $\alpha\in[1,2]$.

Below, we show that this “dual” conjecture holds in the special case where
non-constant functions in $(f_{I})_{I\subseteq[n]}$ correspond to members of a
partition of $[n]$.

######  Proposition V.1.

Let $\mathcal{W}\subseteq 2^{[n]}$ be a partition of $[n]$. The problem

$\displaystyle\max_{\tau_{[n]}\in\mathcal{D}(\mathcal{H}_{[n]})}\quad\quad\sum_{I\in\mathcal{W}}f_{I}(\tau_{I}),$
(V.5) $\displaystyle\text{subject to:}\quad\;\;\lambda(\tau_{[n]})\preceq p.$
(V.6)

has a classical solution.

###### Proof.

Without loss of generality, let $\tau_{[n]}\in\mathcal{D}(\mathcal{H}_{[n]})$
have spectrum $p$. For each $I\in\mathcal{W}$, if necessary, a unitary quantum
channel $\mathcal{U}_{I}\otimes\text{id}_{I^{c}}$ may be applied so that
$\tau_{I}$ is classical. This does not affect the objective function. For each
$I\in\mathcal{W}$, let $\mathcal{N}_{\tau,I}$ denote the quantum channel that
fully decoheres in a basis of $\tau_{I}$. That is, it has the pinching action
$\mathcal{N}_{\tau,I}(\cdot)=\text{diag}(\cdot)$. It is clear that
$\tau^{\prime}_{[n]}=\bigotimes_{I\in\mathcal{W}}\mathcal{N}_{\tau,I}(\tau_{[n]})$
is classical, and for each $I\in\mathcal{W}$, $\tau^{\prime}_{I}=\tau_{I}$.
Since pinching is mixed-unitary (see chapter 4 of Ref. [23] ), then
$\lambda(\tau^{\prime}_{[n]})\preceq p$. We may optimize further by noticing
that $\tau^{\prime}_{[n]}$ is a convex combination of classical states each
with spectrum equal to $p$. One of them has a higher objective value. ∎

##  VI Concluding remarks

We generalized and systematically studied spin alignment problems. We gave
non-trivial examples of cases where tuples of spectrally constrained self-
adjoint operators are more aligned than others. These can be considered
refinements of Ky Fan’s relation Eq. I.1. We conclude with a non-exhaustive
list of problems in this area that we hope will help guide future research.

  1. 1.

(Schatten norms of non-integer order) We used Cor. IV.6 to resolve the spin
alignment problem for Schatten norms of integer order. It is of interest to
know if a similar approach can be used in cases where the order of the norm is
not an integer. Suppose that $A_{0},A_{1},B_{0},B_{1}\geq 0$ satisfy
$\lambda(A_{0})=\lambda(B_{0})$, $\lambda(A_{1})=\lambda(B_{1})$. For a string
$s\in\\{0,1\\}^{*}$, define the ordered products
$\Pi_{s}(A_{0},A_{1}):=\Pi_{i\in[|s|]}A_{s(i)}$ and
$\Pi_{s}(B_{0},B_{1}):=\Pi_{i\in[|s|]}B_{s(i)}$. If for all strings $s$,
$\tr(\Pi_{s}(A_{0},A_{1}))\geq|\tr(\Pi_{s}(B_{0},B_{1}))|$, does it hold that
$||A_{0}+A_{1}||_{p}\geq||B_{0}+B_{1}||_{p}$ for all $p\in[1,\infty)$? If so,
then the overlap lemma may be used to prove the spin alignment conjecture for
all Schatten norms, and by extension for the von Neumann entropy as well.

  2. 2.

(Algebraic approximations of Fan norms) The Schatten norms of integer order
may be said to be algebraic norms, in that they can be defined using
polynomials of the absolute value of the operator in question. Since
$\lim_{m\rightarrow\infty}||\cdot||_{m}=||\cdot||$, they approximate the Fan
norm of order $1$ arbitrarily well. This allows for a resolution of the
question posed in Ex. I.1 in the case where $\text{supp}(\tau)$ has dimension
$2$. Clearly, the trace norm is algebraic as well. Do there exist algebraic
operator norms that approximate other Fan norms? If so, is there any
interesting structure to the corresponding polynomials, especially with regard
to spin alignment?

  3. 3.

(Separable refinement of Ky Fan’s relation) Given $A_{0},A_{1},B_{0},B_{1}\geq
0$, does the following majorization relation

$\displaystyle\lambda(A_{0}\otimes B_{0}+A_{1}\otimes
B_{1})\preceq\lambda(A^{\downarrow}_{0}\otimes
B^{\downarrow}_{0}+A^{\downarrow}_{1}\otimes B^{\downarrow}_{1})$ (VI.1)

hold? In the special case where $\text{rank}(B_{0})=\text{rank}(B_{1})=1$, we
know the answer to be yes by way of a characterization of separable states due
to Nielsen and Kempe [15]. Specifically, they showed that separable states
contain more dispersion globally than locally. If the relation Eq. VI.1 holds
more generally, then it may be used to prove a separable version of the strong
spin alignment conjecture.

#### Acknowledgements

The authors thank Eli Halperin, Graeme Smith, and Mark Wilde for fruitful
discussions regarding the manuscript, and Felix Leditzky and Scott Glancy for
comments that helped improve the presentation of the material. At the time
this work was performed, M. A. Alhejji was supported as an Associate in the
Professional Research Experience Program (PREP) operated jointly by NIST and
the University of Colorado Boulder. This is a contribution of the National
Institute of Standards and Technology, not subject to U.S. copyright.

##  Appendix A The overlap lemma

######  Lemma A.1.

(The overlap lemma) Let $I_{1},\ldots,I_{\ell}$ be a family of subsets of
$[n]$. For $i\in[\ell]$, let $Q_{I_{i}^{c}}=\bigotimes_{j\in
I_{i}^{c}}Q_{I_{i}^{c},j}$, where for each $j\in I_{i}^{c}$,
$Q_{I_{i}^{c},j}\in\mathcal{B}(\mathcal{H}_{j})$ has largest singular value
$\alpha_{i,j}>0$ and is of the form
$Q_{I_{i}^{c},j}=\alpha_{i,j}{|{q_{1}}\rangle}{\langle{q_{1}}|}\oplus
W_{i,j}$. For any unitarily invariant norm $|||\cdot|||$, the maximization
problem

$\displaystyle\max_{(R_{I_{i}})_{i=1}^{\ell}}|||\;\;\prod_{i=1}^{\ell}R_{I_{i}}\otimes
Q_{I_{i}^{c}}\;|||,$ (A.1)

where the variable tuple $(R_{I_{i}})_{i=1}^{\ell}$ is of operators each with
trace norm at most $1$, has $({|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I_{i}})_{i=1}^{\ell}$ as a solution.

We argue by induction on the number of sets $\ell$. For that, we need the
following statement.

######  Lemma A.2.

Let $\mathcal{K}_{1},\mathcal{K}_{2},$ and $\mathcal{K}_{3}$ be given complex
inner product spaces. Let $A_{1}\in\mathcal{B}(\mathcal{K}_{1})$ and
$A_{3}\in\mathcal{B}(\mathcal{K}_{3})$ be such that $||A_{1}||\leq 1$ and
$||A_{3}||\leq 1$. For all
$T_{2,3}\in\mathcal{B}(\mathcal{K}_{2}\otimes\mathcal{K}_{3})$ and
$S_{1,2}\in\mathcal{B}(\mathcal{K}_{1}\otimes\mathcal{K}_{2})$ of trace norm
at most $1$, it holds that $||(A_{1}\otimes T_{2,3})(S_{1,2}\otimes
A_{3})||_{1}\leq 1$.

###### Proof.

Because $||\cdot||_{1}$ is homogeneous and convex, it suffices to prove the
inequality holds in the cases where
$S_{1,2}={|{s}\rangle}{\langle{\tilde{s}}|}_{1,2}$ and
$T_{2,3}={|{t}\rangle}{\langle{\tilde{t}}|}_{2,3}$ for arbitrary unit vectors
${|{s}\rangle},{|{\tilde{s}}\rangle}\in\mathcal{K}_{1}\otimes\mathcal{K}_{2}$
and
${|{t}\rangle},{|{\tilde{t}}\rangle}\in\mathcal{K}_{2}\otimes\mathcal{K}_{3}$.
Since $||A_{1}||\leq 1$ and $||A_{3}||\leq 1$, the norms of
${|{v}\rangle}_{1,2}=A_{1}\otimes\mathds{1}_{2}{|{s}\rangle}_{1,2}$ and
${|{u}\rangle}_{2,3}=\mathds{1}_{2}\otimes
A_{3}^{*}{|{\tilde{t}}\rangle}_{2,3}$ are bounded from above by $1$. Hence, it
suffices to prove that

$\displaystyle||(\mathds{1}_{1}\otimes{|{t}\rangle}{\langle{u}|}_{2,3})({|{v}\rangle}{\langle{\tilde{s}}|}_{1,2}\otimes\mathds{1}_{3}||_{1}\leq
1$ (A.2)

for arbitrary unit vectors
${|{v}\rangle},{|{\tilde{s}}\rangle}\in\mathcal{K}_{1}\otimes\mathcal{K}_{2}$
and ${|{t}\rangle},{|{u}\rangle}\in\mathcal{K}_{2}\otimes\mathcal{K}_{3}$.

Consider the inner factor operator
${\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}:\mathcal{K}_{3}\rightarrow\mathcal{K}_{1}$.
To see that its trace norm is at most 1, let
${|{v}\rangle}_{1,2}=\sum_{i}\sqrt{\lambda_{i}}{|{\alpha_{i}}\rangle}_{1}\otimes{|{\beta_{i}}\rangle}_{2}$
and
${|{u}\rangle}_{2,3}=\sum_{j}\sqrt{\mu_{j}}{|{\gamma_{j}}\rangle}_{2}\otimes{|{\delta_{j}}\rangle}_{3}$
be Schmidt decompositions. Then,

$\displaystyle{\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}=\sum_{i,j}\sqrt{\mu_{j}}\sqrt{\lambda_{i}}\braket{\gamma_{j}}{\beta_{i}}{|{\alpha_{i}}\rangle}_{1}{\langle{\delta_{j}}|}_{3}$
(A.3)

and

$\displaystyle|{\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}|^{2}=\sum_{i,j,j^{\prime}}\sqrt{\mu_{j}\mu_{j^{\prime}}}\lambda_{i}\braket{\gamma_{j}}{\beta_{i}}\braket{\beta_{i}}{\gamma_{j^{\prime}}}{|{\delta_{j^{\prime}}}\rangle}{\langle{\delta_{j}}|}_{3}.$
(A.4)

Denote
$b_{j}={\langle{\gamma_{j}}|}(\sum_{i}\lambda_{i}{|{\beta_{i}}\rangle}{\langle{\beta_{i}}|}){|{\gamma_{j}}\rangle}$
and notice that $\sum_{j}b_{j}\leq 1$. It follows from the Schur-concavity of
$\tr(\sqrt{\cdot})$ and the Schur-Horn theorem that

$\displaystyle||{\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}||_{1}$
$\displaystyle=\tr(\sqrt{|{\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}|^{2}})$ (A.5)
$\displaystyle\leq\tr(\sqrt{\text{diag}(|{\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}|^{2})})$
(A.6) $\displaystyle=\sum_{j}\sqrt{\mu_{j}b_{j}}\leq 1.$ (A.7)

Finally,
$||{\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}\otimes{|{t}\rangle}_{2,3}{\langle{\tilde{s}}|}_{1,2}||_{1}=||{\langle{u}|}_{2,3}{|{v}\rangle}_{1,2}||_{1}\;||{|{t}\rangle}_{2,3}{\langle{\tilde{s}}|}_{1,2}||_{1}\leq
1.$ ∎

###### Proof of Lem. A.1.

Since $|||\cdot|||$ is homogeneous, it may be assumed without loss of
generality for each $i\in[\ell]$ that $||Q_{I_{i}^{c},j}||=1$ for all $j\in
I_{i}^{c}$. We proceed via induction on $\ell$. If $\ell=1$, the statement of
the lemma follows from the facts that
$||{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I_{1}}||_{1}=1$ and that
$|||\cdot|||$ is convex and unitarily invariant. Suppose that the statement
holds in cases where the family of sets has $\ell>1$ elements. Consider
$\prod_{i=1}^{\ell+1}R_{I_{i}}\otimes Q_{I_{i}^{c}}$ and notice that

$\displaystyle R_{I_{1}}\otimes Q_{I_{1}^{c}}R_{I_{2}}\otimes
Q_{I_{2}^{c}}=\underbrace{Q_{I_{1}^{c}\cap
I_{2}}R_{I_{1}}R_{I_{2}}Q_{I_{2}^{c}\cap I_{1}}}_{\tilde{R}_{I_{1}\cup
I_{2}}}\otimes\;\underbrace{Q_{I_{1}^{c}\setminus I_{2}}Q_{I_{2}^{c}\setminus
I_{1}}}_{\tilde{Q}_{(I_{1}\cup I_{2})^{c}}}.$ (A.8)

The operator $\tilde{Q}_{(I_{1}\cup I_{2})^{c}}$ is completely factorizable on
$I_{1}\cup I_{2}$ and has maximal singular value $1$. Moreover, each of its
factors can be written as ${|{q_{1}}\rangle}{\langle{q_{1}}|}\oplus W$ for
some operator $W$. By Lem. A.2, $||\tilde{R}_{I_{1}\cup I_{2}}||_{1}\leq 1$
and so we may estimate

$\displaystyle|||\prod_{i=1}^{\ell+1}R_{I_{i}}\otimes Q_{I_{i}^{c}}|||$
$\displaystyle=|||\tilde{R}_{I_{1}\cup I_{2}}\otimes\;\tilde{Q}_{(I_{1}\cup
I_{2})^{c}}\prod_{i=3}^{\ell+1}R_{I_{i}}\otimes Q_{I_{i}^{c}}|||$ (A.9)
$\displaystyle\leq|||{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes I_{1}\cup
I_{2}}\otimes\;\tilde{Q}_{(I_{1}\cup
I_{2})^{c}}\prod_{i=3}^{\ell+1}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I_{i}}\otimes Q_{I_{i}^{c}}|||$ (A.10)
$\displaystyle=|||\prod_{i=1}^{\ell+1}{|{q_{1}}\rangle}{\langle{q_{1}}|}^{\otimes
I_{i}}\otimes Q_{I_{i}^{c}}|||,$ (A.11)

where the inequality is by the induction hypothesis. ∎

##  Appendix B On the relationship between the sum of two projectors and their
product

In this appendix, we prove lemmas necessary to elucidate the relationship
between a non-negative linear combination $s_{1}P_{1}+s_{2}P_{2}$ of two
projectors $P_{1},P_{2}\in\mathcal{S}(\mathcal{K})$ and their product
$P_{1}P_{2}$. By [8], $\mathcal{K}$ may be decomposed into a direct sum of
subspaces, each of dimension at most $2$, that are invariant under the action
of both $P_{1}$ and $P_{2}$. Moreover, when restricted to each invariant
subspace, the two projectors have rank at most 1. So, we may write
$\mathcal{K}$ as an orthogonal direct sum of one- and two-dimensional minimal
invariant subspaces

$\displaystyle\mathcal{K}=\bigoplus_{i_{1}=1}^{m_{1}}\mathcal{K}_{i_{1}}^{(1)}\oplus\bigoplus_{i_{2}=1}^{m_{2}}\mathcal{K}_{i_{2}}^{(2)},$
(B.1)

where the $\mathcal{K}_{i}^{(1)}$ are one-dimensional and the
$\mathcal{K}_{i}^{(2)}$ are two dimensional. By minimal, we mean that the
subspaces contain no proper nonzero invariant subspace. For each
$i_{2}\in[m_{2}]$, we notate

$\displaystyle
P_{1}|_{\mathcal{K}_{i_{2}}^{(2)}}=:{|{\alpha_{i_{2}}}\rangle}{\langle{\alpha_{i_{2}}}|}\;,\;P_{2}|_{\mathcal{K}_{i_{2}}^{(2)}}=:{|{\beta_{i_{2}}}\rangle}{\langle{\beta_{i_{2}}}|}.$
(B.2)

$P_{1}$ and $P_{2}$ commute if and only if $m_{2}=0$. The difficulty in
reasoning about the eigenvalues of a linear combination of two projectors lies
in these subspaces where they do not commute.

When restricted to $\bigoplus_{i_{2}=1}^{m_{2}}\mathcal{K}_{i_{2}}^{(2)}$, the
nonzero singular values of $P_{1}P_{2}$ are
$(|\braket{\alpha_{i_{2}}}{\beta_{i_{2}}}|)_{i_{2}=1}^{m_{2}}$. The
eigenvalues of the restriction
$(s_{1}P_{1}+s_{2}P_{2})|_{\mathcal{K}_{i_{2}}^{(2)}}$ may be computed as

$\displaystyle\frac{1}{2}((s_{1}+s_{2})\pm\sqrt{(s_{1}-s_{2})^{2}+4s_{1}s_{2}|\braket{\alpha_{i_{2}}}{\beta_{i_{2}}}|^{2}}).$
(B.3)

Hence, the eigenvalues of $s_{1}P_{1}+s_{2}P_{2}$ are a function of the
singular values of the product $P_{1}P_{2}$. We show next that it is in fact a
strictly isotone function (see page 41 of Ref. [3]). A strictly isotone
function $G$ is one that preserves the majorization ordering in the sense that
if $v\succeq w$, then $G(v)\succeq G(w)$. That is, the less dispersed the
singular values of $P_{1}P_{2}$, the less dispersed the eigenvalues of
$s_{1}P_{1}+s_{2}P_{2}$. This is a consequence of the fact that for $a,b\geq
0$, the map $x\mapsto\sqrt{a+bx^{2}}$ is convex on $\mathbb{R}_{\geq 0}$.

######  Lemma B.1.

Let $A\subseteq\mathbb{R}$ be convex and $g:A\rightarrow\mathbb{R}_{\geq 0}$
be a convex function. For $t\in\mathbb{R}$, define the mapping
$G:A^{m}\rightarrow\mathbb{R}^{2m}$ with action

$\displaystyle(v_{1},\ldots,v_{m})\mapsto(t+g(v_{1}),\ldots,t+g(v_{m}))\oplus(t-g(v_{1}),\ldots,t-g(v_{m})).$
(B.4)

Then, $G$ is strictly isotone.

###### Proof.

Let $v,w\in A^{m}$ be such that $v\succeq w$. Observe that for $k\in[m]$,

$\displaystyle\sum_{j=1}^{k}G(v)^{\downarrow}_{j}=kt+\sum_{j=1}^{k}g(v)_{j}^{\downarrow}\geq
kt+\sum_{j=1}^{k}g(w)_{j}^{\downarrow}=\sum_{j=1}^{k}G(w)^{\downarrow}_{j},$
(B.5)

where the inequality follows from the convexity of $g$ and the doubly-
stochastic characterization of majorization (see, for example, Theorem II.3.3
on page 41 of Ref. [3]). If $k>m$, then

$\displaystyle\sum_{j=1}^{k}G(v)^{\downarrow}_{j}$
$\displaystyle=kt+\sum_{j=1}^{m}g(v)_{j}^{\downarrow}-\sum_{j=1}^{k-m}g(v)_{j}^{\uparrow}=kt+\sum_{j=1}^{2m-k}g(v)_{j}^{\downarrow}$
(B.6) $\displaystyle\geq
kt+\sum_{j=1}^{2m-k}g(w)_{j}^{\downarrow}=\sum_{j=1}^{k}G(w)^{\downarrow}_{j}.$
(B.7)

Since $\sum_{j=1}^{2m}G(\cdot)_{j}=2mt$, $G(v)\succeq G(w)$. ∎

The following lemma exhibits the maximal elements in the majorization ordering
in a superset of the possible tuples of nonzero singular values of
$P_{1}P_{2}|_{\bigoplus_{i_{2}=1}^{m_{2}}\mathcal{K}_{i_{2}}^{(2)}}$.

######  Lemma B.2.

Given $m\in\mathbb{N},e\geq 0$, define the set

$\displaystyle S_{e}:=\\{x\in\mathbb{R}^{m}\;|\;\forall
i\in[m],x_{i}\in[0,1],\sum_{i=1}^{m}x_{i}=e\\}.$ (B.8)

The element $(\underbrace{1,\ldots,1}_{\lfloor
e\rfloor\;\text{times}},e-\lfloor e\rfloor,0,\ldots,0)$ is a majorant of
$S_{e}$.

###### Proof.

Let $x\in S_{e}$ be arbitrary. For $k\in[m]$, $k\leq\lfloor e\rfloor$, observe
that $\sum_{i=1}^{k}x^{\downarrow}_{i}\leq\sum_{i=1}^{k}1=k$. And for
$k>\lfloor e\rfloor$,
$\sum_{i=1}^{k}x^{\downarrow}_{i}\leq\sum_{i=1}^{m}x^{\downarrow}_{i}\leq e$.
∎

## References

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