# A simple solution to the problem of self-assembling cubic diamond crystals

Lorenzo Rovigatti  Dipartimento di Fisica, Sapienza Università di Roma, P.le
Aldo Moro 5, 00185 Rome, Italy  CNR-ISC Uos Sapienza, Piazzale A. Moro 2,
IT-00185 Roma, Italy  John Russo  Dipartimento di Fisica, Sapienza Università
di Roma, P.le Aldo Moro 5, 00185 Rome, Italy  Flavio Romano  Dipartimento di
Scienze Molecolari e Nanosistemi, Università Ca’ Foscari di Venezia Campus
Scientifico, Edificio Alfa, via Torino 155, 30170 Venezia Mestre, Italy
European Centre for Living Technology (ECLT) Ca’ Bottacin, 3911 Dorsoduro
Calle Crosera, 30123 Venice, Italy  Michael Matthies  School of Molecular
Sciences and Center for Molecular Design and Biomimetics, The Biodesign
Institute, Arizona State University, 1001 South McAllister Avenue, Tempe,
Arizona 85281, USA  Lukáš Kroc  School of Molecular Sciences and Center for
Molecular Design and Biomimetics, The Biodesign Institute, Arizona State
University, 1001 South McAllister Avenue, Tempe, Arizona 85281, USA  Petr Šulc
School of Molecular Sciences and Center for Molecular Design and Biomimetics,
The Biodesign Institute, Arizona State University, 1001 South McAllister
Avenue, Tempe, Arizona 85281, USA

###### Abstract

The self-assembly of colloidal diamond (CD) crystals is considered as one of
the most coveted goals of nanotechnology, both from the technological and
fundamental points of view. For applications, colloidal diamond is a photonic
crystal which can open new possibilities of manipulating light for information
processing. From a fundamental point of view, its unique symmetry exacerbates
a series of problems that are commonly faced during the self-assembly of
target structures, such as the presence of kinetic traps and the formation of
crystalline defects and alternative structures (polymorphs). Here we
demonstrate that all these problems can be systematically addressed via SAT-
assembly, a design framework that converts self-assembly into a satisfiability
problem. Contrary to previous solutions (requiring four or more components),
we prove that the assembly of the CD crystal only requires a binary mixture.
Moreover, we use molecular dynamics simulations of a system composed by nearly
a million nucleotides to test a DNA nanotechnology design that constitutes a
promising candidate for experimental realization.

The bottom-up realization of complex structures from elementary components has
been the main goal of nanotechnology since Feynman first imagined ”What would
happen if we could arrange the atoms one by one the way we want them” [1]. In
the last decade, one of the most sought-after arrangements is that of a cubic
diamond (CD) crystal. Realizing a diamond structure on the colloidal scale
would in fact open the doors to the realization of optical metamaterials [2,
3]. In analogy to semiconductors, which have an electronic band gap, a
metamaterial lattice can be designed to possess an optical band gap, which
prevents the transport of photons within a specific range of wavelengths
through the material. If such a lattice can be assembled with cavity sizes
comparable to the wavelength of visible or infrared light, it would open new
possibilities of manipulating such radiation in the same way semiconductors
manipulate electrons.

Efforts to reliably produce diamond structures have spanned many decades of
research and have employed a variety of techniques, from traditional top-down
approaches like lithography and holography [4], to bottom-up strategies like
the self-assembly of liquid crystals, nanoparticles and colloidal particles
[5, 6]. To reach the lengthscales that are required for photonic materials,
self-assembly has emerged as one of the most promising routes for the
realization of diamond cubic structure. Notwithstanding recent successes [6,
7, 8], realizing a diamond lattice remains notoriously difficult, and the
reasons are manyfold.

The first major problem is represented by the low volume fraction of the CD
crystal. The CD crystal is an open crystalline structure which is mechanically
stable at volume fractions that are almost half those of ordinary closed
packed structures, like fcc and bcc. This makes the assembly particularly
difficult for the isotropic and short-range potentials that typically govern
colloidal particles, and which tend to favour close-packed structures. To
overcome this problem, one of the most promising approaches has been to use
anisotropic interactions [9], and in particular colloidal particles with
tetrahedrally arranged bonding sites. These particles are termed _Patchy
Particles_ (Fig. 1a) and they acquire anisotropic interactions either via
their shape [10] or via chemical patterning of their surface [11, 12, 13, 14,
15, 16, 17, 18].

The second common problem with the assembly process is that during the
nucleation process different crystalline phases (called _polymorphs_) can
nucleate [19, 20]. Systems that can assemble a CD lattice are often found to
be able to assemble into the hexagonal diamond (HD) lattice, resulting in
imperfect crystals with defects and stacking faults. Avoiding or minimizing
the formation of polymorphs requires specific solutions: to avoid the
formation of stacking faults in CD formations usually requires either
torsional interactions [21, 22] or hierarchical assembly [23, 24, 22, 25].
Importantly, some of these solutions lack a convincing experimental
counterpart.

The final obstacle to a successful self-assembly strategy is the presence of
_kinetic traps_. Despite the reversible nature of the bonds between different
particles, the thermodynamic equilibrium of self-assembling systems can be
easily derailed by locally stable minima that force the system to stay in a
kinetically trapped state. Multiple strategies have emerged to avoid kinetic
traps, from suppressing local motifs found in kinetic traps but not in the
desired structures (such as odd-numbered rings [26]) to designing specific
protocols that steer the non-equilibrium assembly in the correct direction
[27, 28, 29, 30].

Recently we have introduced a novel framework, named SAT-assembly [31, 32], to
design patchy particle systems that can assemble into any desired structure.
Through a mapping to a Boolean satisfiability problem (SAT), the structure is
encoded into the interaction map describing the bonds that can be formed
between the different particles. Crucially, the interaction map can also
encode the avoidance of alternative structures, which can be both competing
polymorphs or commonly occurring kinetic traps. We can then use highly
optimized SAT-solver algorithms to find solution in terms of interaction
matrix between patchy particles that assemble target lattice and avoid
specified alternative structures. Here we systematically explore the space of
solutions that can form the CD crystal while avoiding the HD polymorph, and
demonstrate that the minimum number of particle species required for the
defect-free assembly of CD is only 2. We thus present the binary mixture
solution that encodes this design, and propose a possible realization as a DNA
origami design, which we study with oxDNA, a coarse-grained model of DNA [33,
34, 35, 36] (Fig. 1b). Via molecular simulations, we explore the nucleation
pathway of the solution and reveal that it occurs through a two-step
nucleation process aided by gas-liquid demixing. Requiring the minimal number
of components (a binary mixture), together with its facile crystallizability,
the solution is the most promising candidate for experimental realization of a
colloidal diamond crystal yet using patchy particles without torsional
modulation of their interacting patches, which allows for realization e.g.
with gold nanoparticles or colloids with patches realized as sptially
localized DNA brushes [37] or with DNA origamis with patches as single-
stranded overhangs [38].

Figure 1:  Proposed realization of the CD lattice crystal. a Finite size
cluster of 128 Patchy Particles (the 16-particle cubic diamond unit cell is
multiplied 2x2x2), which interact anisotropically through colored patches
according to the interaction matrix provided by SAT-assembly. b Mean structure
of the CD lattice assembled from two distinct types of DNA origamis, as
represented by the oxDNA model.

##  I The minimal solution

PP species | Patch Coloring
---|---
1: | (1,A) | (2,B) | (3,C) | (4,D)
2: | (1,E) | (2,F) | (3,G) | (4,H)
Color interactions
(A,H),(B,B),(C,C),(D,F),(E,G)
Table 1: Designed patchy particles for self-assembly into a cubic diamond
crystal lattice. It consists of 2 patchy particle species and 8 colors. The
patch coloring is in format (patch number, patch color), where patches are
numbered from 1 to 4, and colors are assigned a letter from A to H. The color
interaction then lists all pairs of interacting colors, where colors B and C
are self-complementary.

Since it is desirable to keep the number of distinct particle types as low as
possible for ease of manufacturing and experimental preparation, we seek to
find the minimum set of particle species that can assemble into a CD lattice
without any defects or alternative assemblies. We formulate the design problem
as a SAT problem [31, 32], which is a collection of Boolean clauses (see
Methods and Supp. Table S1) that encode the topology as given by the
16-particle unit cell for CD lattice (Supp. Table S2). We then gradually
enumerate all possible solutions by augmenting the set of conditions by
additional clauses that contain negations of already discovered solutions, so
that the SAT problem finds new previously unknown solutions. We then test each
discovered solution for its ability to assemble the hexagonal diamond lattice
with 32 sites in the unit cell (Supp. Table S3).

The SAT procedure proves that it is not possible to obtain a solution that
would avoid hexagonal diamond formation with a single particle type ($N_{s}=1$
and $1\leq N_{c}\leq 4$ ). Therefore, we next scanned all patchy particle
systems with $N_{s}=2$ and $N_{c}=8$ and found that there exists a solution to
the SAT-assembly problem that is able to form a CD cubic lattice $16$-particle
unit cell and avoids at the same time formation of the HD lattice with
$32$-particle unit cell. The solution is listed in Table 1. Each of the two
particle types have the same stoichiometry in the CD lattice, so for the full
crystal assembly they are mixed in equal concentrations.

##  II DNA Lattice Simulation

Figure 2:  Projection of the mean structure of Fig.1b on the x-y plane. The z
coordinate of each particle is encoded in the coloring. Red markers represent
the centre of mass of every DNA origami particle computed using the mean
structure. The blue point cloud around the red markers corresponds to the
centre of mass positions of the structures during the production run, aligned
to the centre of mass position of the mean structure (red markers).

We start by mapping the minimal solution in the oxDNA model to simulate a
proposed tetrahedral design realization (Fig. 1b). The DNA tetrahedron has
single-stranded overhangs in each of its vertices that correspond to the DNA
patches. As strands in each vertex are identical, they do not impose any
torsional restraint in terms of orientations at which two particles can bind
to each other, as opposed to e.g. shape complementary DNA origami designs
[39], which presents larger experimental challenge to assemble and correctly
optimize the structure design to ensure pairing of the shape complementary
regions [40]. Patches of complementary colors would correspond to Watson-Crick
base pairing complementary single-stranded DNA overhangs, and self-
complementary color would correspond to a palindromic DNA sequence. After
assembling the CD lattice cluster out of individual DNA origamis as described
in Methods, we ran a sampling molecular dynamics simulations at $T=293$ K. The
system consists of 951040 bases, making it the largest nucleotide-level
coarse-grained DNA simulation that has been realized to date. The mean
structure is the assembled cluster obtained from the MD simulation and it is
shown in Fig. 1b. Its projection in Fig. 2 shows how the centre of mass of
each DNA origami moves relative to its mean structure throughout the
simulation. Our oxDNA model shows that the target lattice is an energy minima
of the proposed design (i.e. it is mechanically stable), and is a candidate
for experimental realization.

We note that despite oxDNA model being highly coarse-grained, representing
each nucleotide as a single rigid body, sampling the kinetics of assembly of
individual origamis into the target lattice is still out of reach. In the
following section we address these limitations by adopting a more efficient
(and more coarse-grained) Patchy Particle model.

##  III Patchy Particle Simulation Results

Figure 3: Nucleation run at $k_{B}T/\epsilon=0.095$ and $\rho\sigma^{3}=0.2$.
a Potential energy as a function of simulation time steps. Parts of the
trajectory are highlighted in red for the metastable liquid phase and in cyan
for the crystalline phase. (inset) Structure factor corresponding to the
highlighted parts of the trajectory. $\mathbf{b}$ Snapshot of a configuration
in the metastable liquid phase. Particles in crystalline environments are
depicted in blue for the CD crystal, and orange for the HD crystal. The color
shade represents the distance from the surface. Particles in disordered
environments are hidden and only bonds between them are represented with black
lines. $\mathbf{c}$ Snapshot of a configuration in the crystal phase. A single
CD crystal is formed, for which we plot a surface mesh to highlight the growth
of crystalline planes.

To demonstrate the correct assembly of a CD crystal from our design, in Fig. 3
we show results from molecular dynamics simulations at temperature
$k_{B}T/\epsilon=0.095$ and number density $\rho\sigma^{3}=0.2$ for $N=10000$
particles using a generalization of the Kern-Frenkel Patchy Particle model
with differentiable interaction potentials (dKF model, introduced in Methods).
Fig. 3a plots the potential energy as a function of the simulated time for a
significant trajectory. The trajectory quickly reaches a transient state
(partially colored in red), where the energy relaxation is very slow, followed
by an abrupt change towards a low-energy state (partially colored in cyan). A
comparison of the structure factor $S(q)$ for these two states (inset in Fig.
3a) shows that the first one is characterized by an amorphous structure at
high q-vectors (short distance), but a large inhomogeneity at small q-vectors
(with a peak appearing at $q\sigma\to 0$), characteristic of a phase-separated
system. The low-energy state shows a similar density inhomogeneity, but it is
also characterized by sharp peaks at finite $q$, typical of a crystalline
state. These observations are confirmed by a visual inspection of these states
(Fig. 3 b and c). Fig. 3b is a snapshot taken in the phase-separated state.
Particles in crystalline environments are depicted in blue for the CD crystal,
and in orange for the HD crystal. Particles in disordered environments are
represented with bond lines. The snapshot shows that the system has phase-
separated into a liquid phase (top part of the simulation box) and a gas phase
(bottom half of the simulation box). Inside the liquid phase there is an
equilibrium population of pre-critical crystalline clusters that constantly
form and dissolve. Note that for these small clusters the hexagonal symmetry
(orange particles) is still present, albeit only on the surface of the nuclei.
Fig. 3c shows the final crystalline state into which the system self-
assembles. A single CD crystal is formed, and an enveloping surface is drawn
to highlight the different crystalline planes. The snapshot shows the lateral
growth of the (1,1,1) plane.

The nucleation pathway described so far is akin to a two-step nucleation
process in which nucleation is preceded by the condensation of dense liquid
droplets. To better analyse the nucleation pathway we map the gas-liquid phase
diagram of the system via direct-coexistence simulations. The system is first
equilibrated in the liquid state at a density $\rho$ and then put in an
elongated box ($L_{x}=3L_{y}=3L_{z}$), where the slab of liquid is bounded by
two interfaces with the gas parallel to the $yz$ plane. Simulations are then
run at constant density and constant temperature until liquid-gas equilibrium
is reached. A snapshot from an equilibrated configuration is shown in Fig. 4a.
We run direct coexistence simulations at densities
$\rho\sigma^{3}=0.5,0.55,0.6$ and temperature ranging from
$k_{B}T/\epsilon=0.08$ and $k_{B}T/\epsilon=0.1$. At the highest temperature,
$k_{B}T/\epsilon=0.1$, the final equilibrium state consist of a single fluid
phase (supercritical state) while the other temperatures maintain the liquid-
slab configuration. From these we can measure equilibrium concentration and
densities of the two phases. In Fig. 4b we plot the isoplethic phase diagram
at constant concentration $x=0.5$ between the two species: the black line
indicates the coexistence density of the gas (left black branch) and liquid
(right black branch) phases. In order to nucleate, the system is quenched
inside the coexistence region, where the formation of liquid droplets favours
the nucleation process. This is tested with several independent simulations of
$N=10000$ patchy particles prepared in a homogeneous state and then quenched
inside at $k_{B}T/\epsilon=0.095$ at different densities. Each simulation is
represented as a point in Fig. 4b joined by the horizontal dashed line. Open
red symbols represent simulations which did not crystallize within the
simulation time, while blue full symbols represent systems that spontaneously
nucleated. Fig. 4b shows that the nucleation rate has a maximum inside the
coexistence region, located approximately under the critical point. We also
quench several systems at $\rho\sigma^{3}=0.1$ at different temperatures
(joined by the vertical dashed line). These simulation show that the
nucleation rate also has a maximum in temperature. A snapshot of a slab of the
system at ($\rho\sigma^{3}=0.1$, $k_{B}T/\epsilon=0.092$) is shown in Fig. 4c,
which highlights the typical formation of a cubic crystal. While lowering the
temperature guarantees a smaller nucleation barrier for nucleation, at low
temperature ($k_{B}T/\epsilon\leq 0.09$) the dynamics of the system becomes
arrested and instead of compact liquid droplets the system is frozen in a gel-
state. A snapshot of this state at ($\rho\sigma^{3}=0.1$,
$k_{B}T/\epsilon=0.08$) is shown in Fig. 4d.

##  IV Discussion and Conclusions

We have used an inverse design framework based on formulating the design
problem as a Boolean Satisfiability Problem to obtain the minimum set of
patchy particles that, without torsional potential, can assemble into a Cubic
Diamond (CD) lattice. We have demonstrated that at least two different types
of patchy particles are required to obtain a system that self-assembles into
the desired lattice and avoid the competing hexagonal diamond geometry, and
have provided an explicit solution with two species and eight colours. We have
observed that in the patchy particle simulations the system first forms a
metastable liquid phase, from which the cubic diamond then crystallizes.
Finally, we have showed a possible realization of the CD lattice made out of
DNA tetrahedrons designed and verified using tools for DNA nanotechnology and
coarse-grained simulations.

Figure 4: a-b Isoplethic phase diagram at equimolar concentration obtained via
direct coexistence simulations (see snapshot a). Black lines are the liquid-
gas coexistence densities. Error bars represent the dispersion over three
independent trajectories at total density $\rho\sigma^{3}=0.5,0.55,0.6$. The
horizontal dashed line connect simulations runs at $k_{B}T/\epsilon=0.095$
starting from homogeneous conditions, while the vertical dashed line connects
the points at $\rho\sigma^{3}=0.1$. Full (blue) symbols represent trajectories
that crystalline spontaneously, while open (red) symbols represent states that
remain in a disordered phase. Slabs of width $10\sigma$ for
$\rho\sigma^{3}=0.1$ and c $k_{B}T/\epsilon=0.092$ and d
$k_{B}T/\epsilon=0.08$.

Our methods can also be used to design a system that avoids certain kinetic
intermediates, thus possibly using SAT-assembly approach to optimize not only
the yield of the target structure, but also the assembly kinetics. The
rearrangement from a metastable liquid phase into the crystal lattice is a
slow process, and our future work will explore if the SAT solver can be used
to design interactions that further destabilize the liquid phase, thus
enhancing the rate of formation of crystal lattice, as well as the effects of
changing the number of different species and distinct patch colors onto the
overall phase diagram of the system.

Our inverse design approach can be further generalized to other geometries,
including finite-size 2D or 3D multicomponent nanostructures, with
applications to biotemplated self-assembly and optical metamaterial
manufacturing.

A more efficient model would be needed to capture the assembly kinetics of the
DNA tetrahedrons, such as an intermediate multiscale model between Patchy
Particle and oxDNA representation, which might represent part of the structure
at nucleotide level (such as ssDNA overhangs) and part of the structure as a
rigid body. Studies and development of such models will also be the focus of
future work.

##  V Acknowledgments

JR acknowledges support from the European Research Council Grant DLV-759187.
PŠ acknowledges support from the ONR Grant N000142012094. JR and PŠ
acknowledge support from the Università Ca’ Foscari for a Visiting Scholarship
and from NSF 1931487 - ERC DLV-759187 research collaboration grant. PŠ further
acknowledges the use of the Extreme Science and Engineering Discovery
Environment (XSEDE), which is supported by National Science Foundation grant
number TG-BIO210009.

##  VI Methods

###  VI.1 SAT-assembly

We use the SAT-assembly framework to find the colorings of patches of the
patchy particles which are compatible with the desired target structure. The
procedure has been described in detail in Refs. [32, 31]. Briefly, we
formulate the inverse design problem as a Boolean Satisfiability Problem. The
parameters of the SAT-assembly are the number of species of patchy particles
($N_{s}=2$ in our case), and the number of colors $N_{c}$ that the patchy
particles will use. The interaction between particles is then determined by
the color interaction matrix, which sets which patch colors can bind to each
other (also allowing for self-complementarity). In the case of realization of
patchy particles with DNA nanostructures such as DNA origami [41, 38],
complementary coloring of patches would corresponds to e.g. a single-stranded
DNA overhangs with complementary sequences that can hybridize into a duplex.
Self-complementary colors would then correspond to palindromic DNA sequences.

To formulate the Boolean Satisfiability problem, each possible coloring and
color interaction is assigned a Boolean variable which can be either true or
false. Furthermore, we define Boolean variables that correspond to which
positions in the target CD lattice are occupied by what type of particle, and
how each respective patchy particle needs to be placed in each of the lattice
positions as specified by its assigned orientation (out of the possible
$N_{o}=12$ orientations, as specified in Supp. Table S1 and Supp. Table S4).
The Boolean variables and clauses are listed in Supp. Table S1 and described
in Section SI in Supplementary Information.

We developed a modified version of the MiniSAT SAT-solver tool [42] to
enumerate all possible solutions for a given combination of $N_{s}$ and
$N_{c}$ that can form a DC lattice, and then tests all generated solutions for
their ability to assemble a specified undesired lattice (HD 32-unit cell in
our case) to quickly filter out the desired solution that can form the DC unit
cell but is unable to be arranged into the HD unit cell in a way that
satisfies all bonds.

###  VI.2 DNA System Simulations

We setup the oxDNA simulations starting from a finite cluster of 128 particles
(Fig. 1), where the unit cell of 16 elements is repeated two times along each
axis. The Athena [43] and oxView [44] design tools were used to construct a
plausible tetrahedral DNA origami monomer which is aligned to the idealized
lattice positions of the cluster using the Biopython SVDSuperimposer package
[45]. Prior to molecular dynamics simulation, the presassembled system needs
to be relaxed to correct for possible overlaps or overstretched bonds that
were created during the placing of the DNA origamis onto the patchy particle
representation. For each of the paired bases in the system an additional
pairwise attractive potential was generated using the generate_force script of
the oxDNA_analysis_tools package [44] with the stiffness parameter of the trap
set to stiff = 3. This potential is active during the entire relaxation
process. For simulations we chose the oxDNA2 model with averaged sequence
interaction strength and a salt concentration of 1 M. The system was first
relaxed using a short Monte Carlo simulation in the canonical NVT ensemble
with T = 300K. The relaxation then continued using MD simulation for 7 400 000
steps with time step $dt=0.001$ using the Bussi Thermostat [46] with the
$\tau$ value set to 1, T = 293K and oxDNA’s options max_backbone_force = 10,
max_density_multiplier = 4. Next, we equilibrated the system without the
max_backbone_force parameter and the pairwise attractive potential for
$175\,400\,000$ simulation steps using the Langevin thermostat. The final
production run performed $142\,400\,000$ MD steps which, taking into account
the value of the diffusion constant set in the simulation ($D_{0}=1$ in oxDNA
units), corresponds to about 0.6 ms [33].

###  VI.3 Mean Structure Calculation

The oxDNA_analysis_tools [44] scripts were used to obtain a mean structure of
the production run: every configuration in the production trajectory is
aligned to a randomly chosen configuration from the production run using the
Biopython SVDSuperimposer package [45]. This is done to maximize overlap
between the sampled states. Next we compute an average of all coordinates
describing the system. The final result is in an artificial structure that is
a good global descriptor of the analysed trajectory.

###  VI.4 Patchy Particle Simulations

In this work, we employ a generalization of the Kern-Frenkel potential [47] to
describe the directional interaction between patches. While the Kern-Frenkel
potential is discontinous and designed for use in Monte Carlo simulations, we
introduce here its differentiable version (dKF potential) that makes it
possible to implement it with a Molecular Dynamics (MD) code for simulation of
the assembly kinetics, and also allows for efficient implementation on GPUs,
thus allowing us to study much longer timescales and larger systems sizes of
the nucleating system.

Patchy particles feel a mutual repulsion modelled through a WCA interaction
[48]:

$U_{ij}(r)=\left\\{\begin{array}[]{l
l}4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}+\frac{1}{4}\right]&r\leq
2^{\frac{1}{6}}\sigma\\\ 0&r>2^{\frac{1}{6}}\sigma\end{array}\right.$ (1)

where $r$ is the centre-to-centre particle distance, $\epsilon$ is the energy
scale and $\sigma$ is the particle diameter. We take the latter two quantities
as units of measurements of energy and length, respectively.

The patch-patch interaction is a square-well-like attractive potential
modulated by an orientation-dependent function. The range of the interaction
is set by the parameter $\delta$, while its (half) angular width is controlled
by $\cos\theta_{\rm max}$. The interaction between patch $i$ on particle
$\alpha$, identified by the unit vector $\hat{\alpha}_{i}$, and patch $j$ on
particle $\beta$, identified by the versor $\hat{\beta}_{j}$, is given by

$V_{\rm pp}(\vec{r_{\rm pp}},\hat{\alpha}_{i},\hat{\beta}_{j})=\\\
-\epsilon\exp\left(-\frac{1}{2}\left(\frac{r_{\rm
pp}-\sigma_{c}}{\delta}\right)^{10}\right)\Omega(-\hat{r},\hat{\alpha_{i}})\Omega(\hat{r}_{\rm
pp},\hat{\beta_{j}})$ (2)

where $\vec{r}_{\rm pp}=\vec{r}_{\alpha}-\vec{r}_{\beta}$, $r_{\rm
pp}=|\vec{r}_{\rm pp}|$, $\hat{r}_{\rm pp}=\vec{r}_{\rm pp}/r_{\rm pp}$ and
$\Omega$ is a steep modulating function that takes into account the
orientation of a patch with respect to the unit vector connecting the
particles’ centres and takes the following form:

$\Omega(\hat{r},\hat{\gamma_{k}})=\exp\left(-\frac{1}{2}\left(\frac{1-\hat{r}\cdot\hat{\gamma}_{k}}{1-\cos\theta_{\rm
max}}\right)^{20}\right).$ (3)

In this work we set $\delta=0.2$ and $\cos\theta_{\rm max}=0.97$ so that
patches can be involved in no more than one bond at a time.

## References

  * Feynman [2018] R. Feynman, There’s plenty of room at the bottom, in  _Feynman and computation_ (CRC Press, 2018) pp. 63–76.
  * Soukoulis and Wegener [2010] C. M. Soukoulis and M. Wegener, Optical metamaterials—more bulky and less lossy, Science 330, 1633 (2010).
  * Ngo  _et al._ [2006] T. Ngo, C. Liddell, M. Ghebrebrhan, and J. Joannopoulos, Tetrastack: Colloidal diamond-inspired structure with omnidirectional photonic band gap for low refractive index contrast, Applied physics letters 88, 241920 (2006).
  * Werts  _et al._ [2002] M. H. Werts, M. Lambert, J.-P. Bourgoin, and M. Brust, Nanometer scale patterning of langmuir- blodgett films of gold nanoparticles by electron beam lithography, Nano Letters 2, 43 (2002).
  * Ducrot  _et al._ [2017] É. Ducrot, M. He, G.-R. Yi, and D. J. Pine, Colloidal alloys with preassembled clusters and spheres, Nature materials 16, 652 (2017).
  * Liu  _et al._ [2016] W. Liu, M. Tagawa, H. L. Xin, T. Wang, H. Emamy, H. Li, K. G. Yager, F. W. Starr, A. V. Tkachenko, and O. Gang, Diamond family of nanoparticle superlattices, Science 351, 582 (2016).
  * Michelson  _et al._ [2022] A. Michelson, B. Minevich, H. Emamy, X. Huang, Y. S. Chu, H. Yan, and O. Gang, Three-dimensional visualization of nanoparticle lattices and multimaterial frameworks, Science 376, 203 (2022).
  * He  _et al._ [2020] M. He, J. P. Gales, É. Ducrot, Z. Gong, G.-R. Yi, S. Sacanna, and D. J. Pine, Colloidal diamond, Nature 585, 524 (2020).
  * Russo  _et al._ [2021] J. Russo, F. Leoni, F. Martelli, and F. Sciortino, The physics of empty liquids: from patchy particles to water, Reports on Progress in Physics (2021).
  * van Anders  _et al._ [2013] G. van Anders, N. K. Ahmed, R. Smith, M. Engel, and S. C. Glotzer, Entropically patchy particles: engineering valence through shape entropy, Acs Nano 8, 931 (2013).
  * Zhang and Glotzer [2004] Z. Zhang and S. C. Glotzer, Self-assembly of patchy particles, Nano Letters 4, 1407 (2004).
  * Pawar and Kretzschmar [2010] A. B. Pawar and I. Kretzschmar, Fabrication, assembly, and application of patchy particles, Macromolecular rapid communications 31, 150 (2010).
  * Bianchi  _et al._ [2011] E. Bianchi, R. Blaak, and C. N. Likos, Patchy colloids: state of the art and perspectives, Physical Chemistry Chemical Physics 13, 6397 (2011).
  * Romano and Sciortino [2011] F. Romano and F. Sciortino, Colloidal self-assembly: patchy from the bottom up, Nature materials 10, 171 (2011).
  * Suzuki  _et al._ [2009] K. Suzuki, K. Hosokawa, and M. Maeda, Controlling the number and positions of oligonucleotides on gold nanoparticle surfaces, Journal of the American Chemical Society 131, 7518 (2009).
  * Kim  _et al._ [2011] J.-W. Kim, J.-H. Kim, and R. Deaton, Dna-linked nanoparticle building blocks for programmable matter, Angewandte Chemie International Edition 50, 9185 (2011).
  * Wang  _et al._ [2012] Y. Wang, Y. Wang, D. R. Breed, V. N. Manoharan, L. Feng, A. D. Hollingsworth, M. Weck, and D. J. Pine, Colloids with valence and specific directional bonding, Nature 491, 51 (2012).
  * Feng  _et al._ [2013] L. Feng, R. Dreyfus, R. Sha, N. C. Seeman, and P. M. Chaikin, Dna patchy particles, Advanced Materials 25, 2779 (2013).
  * Mahynski  _et al._ [2016] N. A. Mahynski, L. Rovigatti, C. N. Likos, and A. Z. Panagiotopoulos, Bottom-up colloidal crystal assembly with a twist, ACS nano 10, 5459 (2016).
  * Leoni and Russo [2021] F. Leoni and J. Russo, Non-classical nucleation pathways in stacking-disordered crystals, arXiv preprint arXiv:2105.05506 (2021).
  * Romano and Sciortino [2012] F. Romano and F. Sciortino, Patterning symmetry in the rational design of colloidal crystals, Nature communications 3, 975 (2012).
  * Tracey  _et al._ [2019] D. F. Tracey, E. G. Noya, and J. P. K. Doye, Programming patchy particles to form complex periodic structures, The Journal of Chemical Physics 151, 224506 (2019).
  * Morphew  _et al._ [2018] D. Morphew, J. Shaw, C. Avins, and D. Chakrabarti, Programming hierarchical self-assembly of patchy particles into colloidal crystals via colloidal molecules, ACS nano 12, 2355 (2018).
  * Patra and Tkachenko [2018] N. Patra and A. V. Tkachenko, Programmable self-assembly of diamond polymorphs from chromatic patchy particles, Physical Review E 98, 032611 (2018).
  * Ma and Ferguson [2019] Y. Ma and A. L. Ferguson, Inverse design of self-assembling colloidal crystals with omnidirectional photonic bandgaps, Soft matter 15, 8808 (2019).
  * Neophytou  _et al._ [2021] A. Neophytou, D. Chakrabarti, and F. Sciortino, Facile self-assembly of colloidal diamond from tetrahedral patchy particles via ring selection, Proceedings of the National Academy of Sciences 118 (2021).
  * Bisker and England [2018] G. Bisker and J. L. England, Nonequilibrium associative retrieval of multiple stored self-assembly targets, Proceedings of the National Academy of Sciences 115, E10531 (2018).
  * Whitelam and Tamblyn [2020] S. Whitelam and I. Tamblyn, Learning to grow: Control of material self-assembly using evolutionary reinforcement learning, Physical Review E 101, 052604 (2020).
  * Whitelam and Tamblyn [2021] S. Whitelam and I. Tamblyn, Neuroevolutionary learning of particles and protocols for self-assembly, Physical review letters 127, 018003 (2021).
  * Bupathy  _et al._ [2021] A. Bupathy, D. Frenkel, and S. Sastry, Temperature protocols to guide selective self-assembly of competing structures, arXiv preprint arXiv:2110.11274 (2021).
  * Romano  _et al._ [2020] F. Romano, J. Russo, L. Kroc, and P. Šulc, Designing patchy interactions to self-assemble arbitrary structures, Physical Review Letters 125, 118003 (2020).
  * Russo  _et al._ [2022] J. Russo, F. Romano, L. Kroc, F. Sciortino, L. Rovigatti, and P. Šulc, SAT-assembly: A new approach for designing self-assembling systems, Journal of Physics: Condensed Matter (2022).
  * Snodin  _et al._ [2016] B. E. Snodin, F. Romano, L. Rovigatti, T. E. Ouldridge, A. A. Louis, and J. P. Doye, Direct simulation of the self-assembly of a small DNA origami, ACS nano 10, 1724 (2016).
  * Rovigatti  _et al._ [2015] L. Rovigatti, P. Šulc, I. Z. Reguly, and F. Romano, A comparison between parallelization approaches in molecular dynamics simulations on gpus, Journal of computational chemistry 36, 1 (2015).
  * Ouldridge  _et al._ [2011] T. E. Ouldridge, A. A. Louis, and J. P. Doye, Structural, mechanical, and thermodynamic properties of a coarse-grained DNA model, The Journal of chemical physics 134, 02B627 (2011).
  * Šulc  _et al._ [2012] P. Šulc, F. Romano, T. E. Ouldridge, L. Rovigatti, J. P. K. Doye, and A. A. Louis, Sequence-dependent thermodynamics of a coarse-grained DNA model, Journal of Chemical Physics 137, 5101 (2012).
  * Xiong  _et al._ [2020] Y. Xiong, S. Yang, Y. Tian, A. Michelson, S. Xiang, H. Xin, and O. Gang, Three-dimensional patterning of nanoparticles by molecular stamping, ACS nano 14, 6823 (2020).
  * Tian  _et al._ [2020] Y. Tian, J. R. Lhermitte, L. Bai, T. Vo, H. L. Xin, H. Li, R. Li, M. Fukuto, K. G. Yager, J. S. Kahn, _et al._ , Ordered three-dimensional nanomaterials using dna-prescribed and valence-controlled material voxels, Nature materials 19, 789 (2020).
  * Gerling  _et al._ [2015] T. Gerling, K. F. Wagenbauer, A. M. Neuner, and H. Dietz, Dynamic DNA devices and assemblies formed by shape-complementary, non–base pairing 3D components, Science 347, 1446 (2015).
  * Park  _et al._ [2019] S. H. Park, H. Park, K. Hur, and S. Lee, Design of dna origami diamond photonic crystals, ACS Applied Bio Materials 3, 747 (2019).
  * Rothemund [2006] P. W. K. Rothemund, Folding DNA to create nanoscale shapes and patterns, Nature 440, 297 (2006).
  * Een [2005] N. Een, Minisat: A sat solver with conflict-clause minimization, in  _Proc. SAT-05: 8th Int. Conf. on Theory and Applications of Satisfiability Testing_ (2005) pp. 502–518.
  * Jun  _et al._ [2021] H. Jun, X. Wang, M. Parsons, W. Bricker, T. John, S. Li, S. Jackson, W. Chiu, and M. Bathe, Rapid prototyping of arbitrary 2D and 3D wireframe DNA origami, Nucleic Acids Research 49, 10265 (2021).
  * Poppleton  _et al._ [2020] E. Poppleton, J. Bohlin, M. Matthies, S. Sharma, F. Zhang, and P. Šulc, Design, optimization and analysis of large DNA and RNA nanostructures through interactive visualization, editing and molecular simulation, Nucleic Acids Research 48, e72 (2020).
  * Cock  _et al._ [2009] P. J. A. Cock, T. Antao, J. T. Chang, B. A. Chapman, C. J. Cox, A. Dalke, I. Friedberg, T. Hamelryck, F. Kauff, B. Wilczynski, and M. J. L. de Hoon, Biopython: freely available Python tools for computational molecular biology and bioinformatics, Bioinformatics 25, 1422 (2009).
  * Bussi  _et al._ [2007] G. Bussi, D. Donadio, and M. Parrinello, Canonical sampling through velocity rescaling, The Journal of chemical physics 126, 014101 (2007).
  * Kern and Frenkel [2003] N. Kern and D. Frenkel, Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction, The Journal of chemical physics 118, 9882 (2003).
  * Weeks  _et al._ [1971] J. D. Weeks, D. Chandler, and H. C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, Journal of Chemical Physics 54, 5237 (1971).

##  Appendix A SUPPLEMENTARY INFORMATION

###  S.1 SI SAT-assembly clauses

Id | Clauses | Boolean expression
---|---|---
(i) | $C^{\rm int}_{c_{i},c_{j},c_{k}}$ | $\neg x^{\rm int}_{c_{i},c_{j}}\lor\neg x^{\rm int}_{c_{i},c_{k}}$
(ii) | $C^{\rm pcol}_{s,p,c_{k},c_{l}}$ | $\neg x^{pcol}_{s,p,c_{k}}\lor\neg x^{\rm pcol}_{s,p,c_{l}}$
(iii) | $C^{L}_{l,s_{i},o_{i},s_{j},o_{j}}$ | $\neg x^{L}_{l,s_{i},o_{i}}\lor\neg x^{L}_{l,s_{j},o_{j}}$
(iv) | $C^{\rm lint}_{l_{i},k_{i},l_{j},k_{j},c_{i},c_{j}}$ | $(x^{A}_{l_{i},k_{i},c_{i}}\land x^{A}_{l_{j},k_{j},c_{j}})\implies x^{\rm int}_{c_{i},c_{j}}$
(v) | $C^{\rm LS}_{l,s,o,c,k}$ | $x^{L}_{l,s,o}\implies\left(x^{A}_{l,k,c}\iff x^{\rm pcol}_{s,\phi_{o}(k),c}\right)$
(vi) | $C^{\rm all\,s.}_{s}$ | $\bigvee_{\forall l,o}x^{L}_{l,s,o}$
(vii) | $C^{\rm all\,c.}_{c}$ | $\bigvee_{\forall s,p}x^{\rm pcol}_{s,p,c}$
Table S1: SAT clauses and variables.

The color interaction is given by binary variables $x^{\rm int}_{c_{i},c_{j}}$
which are 1 if color $c_{i}$ is compatible with color $c_{j}$ and 0 otherwise.
The patch coloring for each PP species is described by binary variables
$x^{\rm pcol}_{s,p,c}$ which are 1 if patch $p$ of species $s$ has color $c$
and 0 otherwise. The arrangement of the particle species in the lattice is
described by $x^{L}_{l,s,o}$ which is 1 if the position $l$ is occupied by a
PP of species $s$ in the specific orientation $o$. The mapping $\phi_{o}(k)=p$
for a given orientation $o$ means that PP’s patch $p$ overlaps with slot $k$
in a given lattice position. The variable $x^{A}_{l,k,c}$ is 1 if slot $k$ of
lattice position $l$ is occupied by a patch with color $c$ and 0 otherwise.
The clauses and variables are defined for all possible combinations of colors
$c\in[1,N_{c}]$, patches $p\in[1,N_{p}]$, slots $k\in[1,N_{p}]$, PP species
$s\in[1,N_{s}]$, orientations $o\in[1,N_{o}]$, and lattice positions
$l\in[1,L]$. Clauses $C^{\rm lint}$ are defined only for slots $k_{i},k_{j}$
that are in contact in neighboring lattice positions $l_{i},l_{j}$. For a
given $s$, clause $C^{\rm all\,s.}_{s}$ is defined as a list of
$x^{L}_{l,s,o}$ for all possible values of $l$ and $o$, joined by
disjunctions. Clause $C^{\rm all\,c.}_{c}$ is defined analogously. The SAT
problem is a conjunction of all clauses. The color interaction matrix is given
by the indices of the $x^{\rm int}_{c_{i},c_{j}}$ that are true in the found
solution, and the coloring (which particle species is assigned which color) is
given by the indices of variables $x^{\rm pcol}_{s,p,c}$ that are true. We
first define the SAT problem using the topology of CD lattice as given in
Table S2. For each identified solution, we formulate a new SAT problem with
additional new clauses that require $x^{\rm int}_{c_{i},c_{j}}$ and $x^{\rm
pcol}_{s,p,c}$ variables that correspond to the identified solution are true.
We then reformulate clauses $C^{L}$ to correspond to the HD topology (Table
S3). If such a SAT problem is satisfiable, we discard the solution and repeat
the process, until we find a solution that satisifies CD topology but does not
satisify the HD topology.

###  S.2 SII Cubic and Hexagonal diamond topologies

We include here the topology for a unit cell of cubic diamond crystal lattice
16-unit cell, and for hexagonal diamond lattice 32-unit cell. These cells are
created by merging smaller unit cells of cubic diamond and hexagonal diamond
respectively.

Position $l_{i}$ | Slot $s_{i}$ | Position $l_{j}$ | Slot $s_{j}$
---|---|---|---
2 | 4 | 15 | 4
10 | 2 | 14 | 4
4 | 1 | 8 | 3
12 | 3 | 13 | 2
3 | 2 | 6 | 3
1 | 4 | 5 | 4
11 | 4 | 16 | 4
2 | 3 | 5 | 3
4 | 4 | 7 | 2
7 | 4 | 10 | 4
12 | 2 | 14 | 2
3 | 3 | 7 | 3
11 | 1 | 13 | 1
2 | 2 | 6 | 4
4 | 3 | 5 | 2
1 | 1 | 16 | 1
9 | 3 | 14 | 1
8 | 2 | 10 | 1
7 | 1 | 9 | 2
12 | 1 | 16 | 3
3 | 4 | 8 | 4
11 | 2 | 14 | 3
2 | 1 | 16 | 2
1 | 2 | 15 | 1
9 | 4 | 13 | 4
8 | 1 | 9 | 1
10 | 3 | 13 | 3
4 | 2 | 6 | 2
12 | 4 | 15 | 2
3 | 1 | 5 | 1
1 | 3 | 6 | 1
11 | 3 | 15 | 3
Table S2:  Cubic diamond 16-unit cell topology: List of lattice positions $l_{i}$ and $l_{j}$ that are neighbors in the unit cell of the lattice and their respective slot numbers $s_{i}$, $s_{j}$ through which they are bound. The unit cell of size 16 is obtained by pasting together two unit cells of size 8. Position $l_{i}$ |  Slot $s_{i}$ |  Position $l_{j}$ |  Slot $s_{j}$
---|---|---|---
8 | 4 | 17 | 1
2 | 4 | 23 | 1
10 | 2 | 14 | 2
4 | 1 | 6 | 2
12 | 3 | 29 | 1
22 | 3 | 26 | 2
17 | 3 | 29 | 2
3 | 2 | 7 | 2
27 | 3 | 30 | 4
7 | 3 | 11 | 1
18 | 3 | 30 | 2
1 | 4 | 24 | 1
11 | 4 | 15 | 4
28 | 3 | 29 | 4
8 | 3 | 12 | 1
2 | 3 | 14 | 1
13 | 3 | 28 | 1
23 | 3 | 27 | 2
4 | 4 | 22 | 1
9 | 2 | 13 | 2
17 | 4 | 24 | 2
27 | 4 | 31 | 4
24 | 3 | 28 | 2
7 | 4 | 18 | 1
12 | 2 | 13 | 4
19 | 2 | 21 | 4
26 | 4 | 32 | 3
3 | 3 | 15 | 1
2 | 2 | 7 | 1
20 | 4 | 24 | 4
4 | 3 | 16 | 1
1 | 1 | 5 | 1
14 | 3 | 27 | 1
9 | 3 | 31 | 1
19 | 3 | 31 | 2
10 | 4 | 16 | 2
18 | 2 | 22 | 2
20 | 3 | 32 | 2
3 | 4 | 21 | 1
25 | 3 | 29 | 3
11 | 2 | 14 | 4
5 | 3 | 9 | 1
2 | 1 | 6 | 1
15 | 3 | 25 | 1
6 | 4 | 20 | 1
1 | 2 | 8 | 1
9 | 4 | 15 | 2
19 | 4 | 23 | 4
16 | 3 | 26 | 1
10 | 3 | 32 | 1
21 | 3 | 25 | 2
4 | 2 | 8 | 2
12 | 4 | 16 | 4
28 | 4 | 32 | 4
17 | 2 | 21 | 2
3 | 1 | 5 | 2
25 | 4 | 31 | 3
18 | 4 | 23 | 2
5 | 4 | 19 | 1
26 | 3 | 30 | 3
20 | 2 | 22 | 4
6 | 3 | 10 | 1
1 | 3 | 13 | 1
11 | 3 | 30 | 1
Table S3:  Hexagonal 32-unit cell topology, obtained by joining 4 unit cell lattices. List of lattice positions $l_{i}$ and $l_{j}$ that are neighbors in the unit cell and their respective slot numbers $s_{i}$, $s_{j}$ through which they are bound  Orientation $o$ | Mapping $\phi_{o}$
---|---
1 | (1,2,3,4)
2 | (1,4,2,3)
3 | (1,3,4,2)
4 | (2,4,3,1)
5 | (2,1,4,3)
6 | (2,3,1,4)
7 | (4,1,3,2)
8 | (4,2,1,3)
9 | (4,3,2,1)
10 | (3,1,2,4)
11 | (3,4,1,2)
12 | (3,2,4,1)
Table S4:  List of orientations $o$ for a PP with a tetrahedral symmetry of
patch positions. Each orientation determines a mapping $\phi_{o}$, which
specifies which patches overlap with patches in the original position before
applying the rotation. The first mapping for $o=1$ corresponds to no rotation.


# Quantum nonlocality without entanglement depending on nonzero prior
probabilities in optimal unambiguous discrimination

Donghoon Ha  Department of Applied Mathematics and Institute of Natural
Sciences, Kyung Hee University, Yongin 17104, Republic of Korea  Jeong San Kim
<EMAIL_ADDRESS>Department of Applied Mathematics and Institute of Natural
Sciences, Kyung Hee University, Yongin 17104, Republic of Korea

###### Abstract

Nonlocality without entanglement(NLWE) is a nonlocal phenomenon that occurs in
quantum state discrimination of multipartite separable states. In the
discrimination of orthogonal separable states, the term NLWE is used when the
quantum states cannot be discriminated perfectly by local operations and
classical communication. In this case, the occurrence of NLWE is independent
of nonzero prior probabilities of quantum states being prepared. Recently, it
has been found that the occurrence of NLWE can depend on nonzero prior
probabilities in minimum-error discrimination of nonorthogonal separable
states. Here, we show that even in optimal unambiguous discrimination, the
occurrence of NLWE can depend on nonzero prior probabilities. We further show
that NLWE can occur regardless of nonzero prior probabilities, even if only
one state can be locally discriminated without error. Our results provide new
insights into classifying sets of multipartite quantum states in terms of
quantum state discrimination.

## Introduction

Although orthogonal quantum states can be perfectly discriminated in quantum
mechanics, there also exists a set of orthogonal, non-entangled(separable)
states in multipartite quantum systems that cannot be perfectly discriminated
only by local operations and classical communications(LOCC)chef2000 ; barn2009
; berg2010 ; bae2015 ; benn19991 . This set of multipartite quantum states
reveals the possible existence of separable measurement(SEP) that cannot be
realized by LOCC. In discriminating separable states, a phenomenon that can be
achieved by global measurements but cannot be achieved only by LOCC is called
nonlocality without entanglement(NLWE).
The first NLWE phenomenon was shown through nine $3\otimes 3$ orthogonal
product statesbenn19991 . On the other hand, Walgate et al.walg2000  showed
that any two multipartite orthogonal pure states can be perfectly
discriminiated within finite-round using LOCC. Since then, there have been
many studies focused on local indistinguishability or local distinguishability
of multipartite orthogonal states benn19992 ; ghos2001 ; walg2002 ; divi2003 ;
ghos2004 ; fan2004 ; watr2005 ; nise2006 ; cohe2007 ; duan2009 ; klei2011 ;
band2012 ; yang2013 ; zhan20142 ; chit20141 ; wang2015 ; yang2016 ; zhan2016 ;
zhan2017 ; hald2019 ; jian2020 . Recently, there has also been an effort to
find NLWE in generalized probabilistic theories bhat2020 .
The NLWE phenomenon occurs in the perfect discrimination of orthogonal
separable states as well as minimum-error discrimination (MD)hels1976 ;
hole1979 ; yuen1975  of nonorthogonal separable states. MD of two multipartite
nonorthogonal pure states can always be achieved using LOCCvirm2001 ,
regardless of the prior probabilities. However, MD of more than two states is
not (e.g., double trine ensemblepere1991 ; chit2013 ); the minimum error
probability cannot be obtained using asymptotic LOCCchit20142 ; alocc .
Recently, Ha and Kwonha2021  found a set of three nonorthogonal $2\otimes 2$
product states where NLWE in terms of MD occurs depending on the nonzero prior
probabilities. In perfect discrimination of orthogonal states, the nonzero
prior probabilities have no effect on the local indistinguishability or the
local distinguishability. That is, NLWE in terms of perfect discrimination
does not depend on nonzero prior probabilities.
In unambiguous discrimination(UD)ivan1987 ; pere1988 ; diek1988 ; jaeg1995 ,
quantum states are discriminated without error (i.e., unambiguously), instead,
allowing inconclusive results. In order for pure states to be unambiguously
discriminated, they must be linearly independentchef19981 . Moreover, in order
to unambiguously discriminate a multipartite pure state using LOCC, the
existence of a product state that detects only the state is additionally
requiredchef2004 ; duan2007 ; band2009 .
The optimization of UD is to minimize the probability of obtaining
inconclusive results. Whereas optimal UD of two multipartite pure states can
always be achieved using LOCCvirm2001 ; chen2001 ; chen2002 ; ji2005 , optimal
UD of more than two multipartite states is not in general. A typical example
is “indistinguishable product basis” presented by Duan et al.duan2007 , in
which all states cannot be unambiguously discriminated using LOCC. We also
note that NLWE in terms of optimal UD may occur regardless of nonzero prior
probabilities. However, it is still a natural question whether there is a set
of separable states that cause NLWE in terms of optimal UD depending on
nonzero prior probabilities.
In this work, we consider a complete product basis, in which only one state
can be discriminated unambiguously using LOCC, and establish a necessary and
sufficient condition that the occurrence of NLWE in terms of optimal UD
depends on nonzero prior probabilities (Corollary 2). We provide an example of
six $2\otimes 3$ product states where NLWE in terms of optimal UD occurs
depending on nonzero prior probabilities (Example 1). We also provide an
example that is not indistinguishable product basis but causes NLWE in terms
of optimal UD regardless of nonzero prior probabilities (Example 2). We
further investigate the case where optimal UD measurement detects only one
state unambiguously (Theorem 1), and obtain a necessary and sufficient
condition for optimal UD measurement to detect two or more states
unambiguously, regardless of nonzero prior probabilities (Corollary 1).

## Results

### Optimal Unambiguous Discrimination

Let $\\{|\psi_{1}\rangle,\ldots,|\psi_{n}\rangle\\}$ be a set of $n$ linearly
independent pure states and $\eta_{i}$ be the corresponding non-zero
probability of $|\psi_{i}\rangle$ being prepared for $i=1,\ldots,n$. The UD of
$\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ is to discriminate the prepared
state without error using a positive operator-valued measure(POVM) that
consists of $n+1$ positive semidefinite operators $M_{0},M_{1},\ldots,M_{n}$
on $\mathcal{H}_{\rm S}$ satisfying $\sum_{i=0}^{n}M_{i}=\mathbbm{1}$. Here
$\mathcal{H}_{\rm S}$ is the subspace spanned by
$\\{|\psi_{1}\rangle,\ldots,|\psi_{n}\rangle\\}$ and $\mathbbm{1}$ is the
identity operator on $\mathcal{H}_{\rm S}$. $M_{0}$ provides inconclusive
results for the prepared state and $M_{i}(i=1,\ldots,n)$ gives conclusive
results for $|\psi_{i}\rangle$ unambiguously.
The no-error condition for $\\{M_{i}\\}_{i=0}^{N}$ to discriminate
$\\{|\psi_{i}\rangle\\}_{i=1}^{n}$ unambiguouslypang2009 , that is,
$\langle\psi_{i}|M_{j}|\psi_{i}\rangle=0$ $\forall i\neq j$, requires

$M_{i}=p_{i}|\tilde{\psi}_{i}\rangle\langle\tilde{\psi}_{i}|,\ i=1,\ldots,n,$
(1)

where $p_{i}$ is the probability of success in unambiguously discriminating
$|\psi_{i}\rangle$ and $|\tilde{\psi}_{i}\rangle$ is the reciprocal vector
defined by

$|\tilde{\psi}_{i}\rangle=\sum_{j=1}^{n}(\Omega^{-1})_{ji}|\psi_{j}\rangle,$
(2)

where $\Omega^{-1}$ is the inverse matrix of $n$-by-$n$ Gram matrix $\Omega$
with $(\Omega)_{ij}=\langle\psi_{i}|\psi_{j}\rangle$. We note that
$\langle\tilde{\psi}_{i}|\tilde{\psi}_{j}\rangle=(\Omega^{-1})_{ij}$. Because
$\\{|\tilde{\psi}_{1}\rangle,\ldots,|\tilde{\psi}_{n}\rangle\\}$ is a set of
linearly independent vectors in $\mathcal{H}_{\rm S}$ satisfying
$\langle\tilde{\psi}_{i}|\psi_{j}\rangle=\delta_{ij}$, selecting a UD
measurement can be understood as selecting a point
$\vec{p}=(p_{1},\ldots,p_{n})^{\sf T}\in\mathbb{R}^{n}$ such that

$\sum_{i=1}^{n}p_{i}|\tilde{\psi}_{i}\rangle\langle\tilde{\psi}_{i}|\geqslant
0,\quad\mathbbm{1}-\sum_{i=1}^{n}p_{i}|\tilde{\psi}_{i}\rangle\langle\tilde{\psi}_{i}|\geqslant
0.$ (3)

The optimization task of UD is to maximize the average probability of success,

$P=\sum_{i=1}^{n}\eta_{i}p_{i}.$ (4)

whose maximum is denoted by $P_{\max}$. The optimal $\vec{p}$ that gives
$P_{\max}$ is uniquesugi2010 , which can also be verified from optimality
condition including Lagrangian stability; see Lemma 1 in the Methods Section.
Some components of optimal $\vec{p}$ may be zero. In other words, the optimal
UD measurement of $\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ may not detect
all given quantum states unambiguously. If $p_{i}$ is zero, the UD measurement
does not detect $|\psi_{i}\rangle$ unambiguously. The following theorem shows
a necessary and sufficient condition for the optimal UD measurement of
$\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ to detect only $|\psi_{1}\rangle$
unambiguously. The proof of Theorem 1 is provided in the Methods Section.
Theorem 1. For the optimal UD of $\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$,
the optimal measurement detects only $|\psi_{1}\rangle$ unambiguously if and
only if

$\frac{\eta_{i}}{\eta_{1}}\leqslant\frac{|\langle\tilde{\psi}_{i}|\tilde{\psi}_{1}\rangle|^{2}}{|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{2}}\
\forall i.$ (5)

In the case of $n=2$jaeg1995 , it is easy to see that the condition
$\eta_{2}/\eta_{1}\leqslant|\langle\psi_{1}|\psi_{2}\rangle|^{2}$ is
equivalent that the optimal $p_{2}$ is zero. As we can check in the proof of
Theorem 1 in Method Section, the choice of $|\psi_{1}\rangle$ in Theorem 1 can
be arbitrary. That is, any of $\\{|\psi_{1}\rangle,\ldots,|\psi_{n}\rangle\\}$
can be used to play the role of $|\psi_{1}\rangle$ in Theorem 1.
Corollary 1. For any set of nonzero prior probabilities
$\\{\eta_{i}\\}_{i=1}^{n}$, the optimal UD measurement of
$\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ detects more than one state
unambiguously if and only if, for any $j$, there is $k$ satisfying
$\langle\tilde{\psi}_{j}|\tilde{\psi}_{k}\rangle=0$.

###### Proof.

For fixed $j$, let us first suppose that there exists $k$ satisfying
$\langle\tilde{\psi}_{j}|\tilde{\psi}_{k}\rangle=0$, which implies

$\frac{\eta_{k}}{\eta_{j}}>\frac{|\langle\tilde{\psi}_{k}|\tilde{\psi}_{j}\rangle|^{2}}{|\langle\tilde{\psi}_{j}|\tilde{\psi}_{j}\rangle|^{2}}$
(6)

because the left-hand side is positive and the right-hand side is zero. Thus,
from Theorem 1, the optimal UD measurement of
$\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ cannot be achieved by detecting
only $|\psi_{j}\rangle$ unambiguously. Since this argument is true for any
$|\psi_{j}\rangle$, the optimal UD measurement must detect more than one state
unambiguously.
Now, let us suppose that $\langle\tilde{\psi}_{j}|\tilde{\psi}_{k}\rangle\neq
0$ for all $k$. We can see that

$\frac{\eta_{k}}{\eta_{j}}=\frac{|\langle\tilde{\psi}_{k}|\tilde{\psi}_{j}\rangle|^{2}}{|\langle\tilde{\psi}_{j}|\tilde{\psi}_{j}\rangle|^{2}}\
\forall k$ (7)

with nonzero prior probabilities defined as

$\eta_{i}=\frac{|\langle\tilde{\psi}_{i}|\tilde{\psi}_{j}\rangle|^{2}}{\sum_{i^{\prime}=1}^{n}|\langle\tilde{\psi}_{i^{\prime}}|\tilde{\psi}_{j}\rangle|^{2}}.$
(8)

Thus, from Theorem 1, the optimal UD measurement of
$\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ detects only $|\psi_{j}\rangle$
unambiguously. ∎

For example, let us consider four pure states

$\begin{array}[]{rcl}|\psi_{1}\rangle&=&\frac{1}{\sqrt{3}}(|0\rangle-|2\rangle-|3\rangle),\\\\[2.84526pt]
|\psi_{2}\rangle&=&\frac{1}{\sqrt{3}}(|1\rangle-|2\rangle+|3\rangle),\\\\[2.84526pt]
|\psi_{3}\rangle&=&|2\rangle,\\\\[2.84526pt]
|\psi_{4}\rangle&=&|3\rangle,\end{array}$ (9)

with the reciprocal vectors

$\begin{array}[]{rcl}|\tilde{\psi}_{1}\rangle&=&\sqrt{3}|0\rangle,\\\\[2.84526pt]
|\tilde{\psi}_{2}\rangle&=&\sqrt{3}|1\rangle,\\\\[2.84526pt]
|\tilde{\psi}_{3}\rangle&=&|0\rangle+|1\rangle+|2\rangle,\\\\[2.84526pt]
|\tilde{\psi}_{4}\rangle&=&|0\rangle-|1\rangle+|3\rangle.\end{array}$ (10)

Since $\langle\tilde{\psi}_{1}|\tilde{\psi}_{2}\rangle$ and
$\langle\tilde{\psi}_{3}|\tilde{\psi}_{4}\rangle$ are zero, for any $i$, there
exists $j$ satisfying $\langle\tilde{\psi}_{i}|\tilde{\psi}_{j}\rangle=0$.
Therefore, for any set of nonzero prior probabilities, the optimal UD
measurement detects more than one state unambiguously.

### Multipartite State Discrimination

Let $\mathcal{H}$ be a multipartite Hilbert space and
$\\{|\Psi_{i}\rangle\\}_{i=1}^{n}$ a set of $n$ linearly independent pure
states in $\mathcal{H}$, where each $|\Psi_{i}\rangle$ is prepared with
nonzero prior probability $\xi_{i}$. We use $|\Psi_{i}\rangle$,
$|\tilde{\Psi}_{i}\rangle$, and $P_{\max}^{\mbox{\tiny Global}}$ instead of
$|\psi_{i}\rangle$, $|\tilde{\psi}_{i}\rangle$, and $P_{\max}$ to highlight
the multipartite quantum system to be considered. Also, we use $P_{\rm
max}^{\overline{\mbox{\tiny LOCC}}}$ to denote the maximum of average success
probability that can be obtained in UD restricted to asymptotic LOCC, whereas
$P_{\max}^{\mbox{\tiny Global}}$ is the maximum of average success probability
that can be obtained using global measurement. If there exists a product state
$|\Phi\rangle$ that satisfies $\langle\Phi|\Psi_{1}\rangle\neq 0$ and
$\langle\Phi|\Psi_{i}\rangle=0\,\forall i\neq 1$, $|\Psi_{1}\rangle$ can be
unambiguously discriminated using finite-round LOCC, otherwise the state
cannot be unambiguously discriminated even using SEP chef2004 ; duan2007 ;
band2009 . In particular, when $\\{|\Psi_{i}\rangle\\}_{i=1}^{n}$ is a basis
of $\mathcal{H}$ (that is, $\mathcal{H}_{\rm S}=\mathcal{H}$), the existence
of a product state $|\Phi\rangle$ satisfying $\langle\Phi|\Psi_{1}\rangle\neq
0$ and $\langle\Phi|\Psi_{i}\rangle=0\,\forall i\neq 1$ is determined by the
corresponding reciprocal vector $|\tilde{\Psi}_{1}\rangle$duan2007 . To be
precise, $|\Psi_{1}\rangle$ can be unambiguously discriminated using LOCC if
and only if $|\tilde{\Psi}_{1}\rangle$ is a product vector. Note that when
$\mathcal{H}_{\rm S}\neq\mathcal{H}$, even if $|\tilde{\Psi}_{1}\rangle$ is
entangled, a product state that detects only $|\Psi_{1}\rangle$ may exist in
$\mathcal{H}$.
In the case of $n=2$ virm2001 ; chen2001 ; chen2002 ; ji2005 , the globally
optimal UD is always possible only with finite-round LOCC no matter what two
pure states with arbitrary prior probabilities are given, i.e., $P_{\rm
max}^{\overline{\mbox{\tiny LOCC}}}=P_{\max}^{\mbox{\tiny Global}}$. However,
in the case of $n>2$, $P_{\rm max}^{\overline{\mbox{\tiny LOCC}}}$ can be less
than $P_{\rm max}^{\mbox{\tiny Global}}$. An example is the indistinguishable
product basisduan2007  in which no state can be unambiguously discriminated
using asymptotic LOCC, i.e., $P_{\max}^{\overline{\mbox{\tiny LOCC}}}=0$. It
should be noted that $P_{\max}^{\overline{\mbox{\tiny LOCC}}}=0$ means
$P_{\max}^{\overline{\mbox{\tiny LOCC}}}<P_{\max}^{\mbox{\tiny Global}}$
because $P_{\max}^{\mbox{\tiny Global}}$ is always nonzero for any linearly
independent pure states. Moreover, this example illustrates the case that NLWE
in terms of optimal UD occurs regardless of nonzero prior probabilities.
The following theorem shows that if $\\{|\Psi_{i}\rangle\\}_{i=1}^{n}$ is a
basis of $\mathcal{H}$ and only $|\Psi_{1}\rangle$ can be unambiguously
discriminated using LOCC, $P_{\max}^{\overline{\mbox{\tiny
LOCC}}}=P_{\max}^{\mbox{\tiny Global}}$ is determined by whether the optimal
UD measurement requires unambiguous detection other than $|\Psi_{1}\rangle$.
The proof of Theorem 2 is provided in the Methods Section.
Theorem 2. When $\mathcal{H}_{\sf S}=\mathcal{H}$ and only
$|\tilde{\Psi}_{1}\rangle$ is a product vector, the globally optimal UD of
$\\{\xi_{i},|\Psi_{i}\rangle\\}_{i=1}^{n}$ can be achieved using LOCC if and
only if

$\frac{\xi_{i}}{\xi_{1}}\leqslant\frac{|\langle\tilde{\Psi}_{i}|\tilde{\Psi}_{1}\rangle|^{2}}{|\langle\tilde{\Psi}_{1}|\tilde{\Psi}_{1}\rangle|^{2}}\
\ \forall i.$ (11)

Theorem 2 tells us that the possibility of globally optimal UD using LOCC can
depend on nonzero prior probabilities. In other words, when
$|\Psi_{1}\rangle,\ldots,|\Psi_{n}\rangle$ are fixed, both
$P_{\max}^{\overline{\mbox{\tiny LOCC}}}=P_{\max}^{\mbox{\tiny Global}}$ and
$P_{\max}^{\overline{\mbox{\tiny LOCC}}}<P_{\max}^{\mbox{\tiny Global}}$ can
occur according to $\xi_{1},\ldots,\xi_{n}$. As we can check in the proof of
Theorem 2 in Method Section, the choice of $|\tilde{\Psi}_{1}\rangle$ in
Theorem 2 can be arbitrary. That is, any of
$\\{|\tilde{\Psi}_{1}\rangle,\ldots,|\tilde{\Psi}_{n}\rangle\\}$ can be used
to play the role of $|\tilde{\Psi}_{1}\rangle$ in Theorem 2.
Corollary 2. For the optimal UD of $n$ product states
$|\Psi_{1}\rangle,\ldots,|\Psi_{n}\rangle$ such that $\mathcal{H}_{\sf
S}=\mathcal{H}$ and only $|\tilde{\Psi}_{1}\rangle$ is a product vector, if
$\langle\tilde{\Psi}_{i}|\tilde{\Psi}_{1}\rangle$ is nonzero for all $i$, NLWE
occurs depending on the nonzero prior probabilities. Otherwise NLWE occurs for
any nonzero prior probabilities.

###### Proof.

If $\langle\tilde{\Psi}_{i}|\tilde{\Psi}_{1}\rangle\neq 0$ for all $i$,
satisfying Inequality (11) for all $i$ depends on nonzero prior probabilities;
Inequality (11) for each $i$ holds by the nonzero prior probabilities defined
as

$\xi_{i}=\frac{|\langle\tilde{\Psi}_{i}|\tilde{\Psi}_{1}\rangle|^{2}}{\sum_{j=1}^{n}|\langle\tilde{\Psi}_{j}|\tilde{\Psi}_{1}\rangle|^{2}},$
(12)

whereas Inequality (11) for $i=2$ does not hold by the nonzero prior
probabilities defined as

$\xi_{i}=\left\\{\begin{array}[]{lll}\frac{1+|\langle\tilde{\Psi}_{2}|\tilde{\Psi}_{1}\rangle|^{2}}{1+\sum_{j=1}^{n}|\langle\tilde{\Psi}_{j}|\tilde{\Psi}_{1}\rangle|^{2}}&\mbox{for}&i=2,\\\\[5.69054pt]
\frac{|\langle\tilde{\Psi}_{i}|\tilde{\Psi}_{1}\rangle|^{2}}{1+\sum_{j=1}^{n}|\langle\tilde{\Psi}_{j}|\tilde{\Psi}_{1}\rangle|^{2}}&\mbox{for}&i\neq
2.\end{array}\right.$ (13)

It follows from Theorem 2 that the possibility of the globally optimal UD
using LOCC depends on nonzero prior probabilities. Thus, NLWE occurs depending
on the nonzero prior probabilities.
However, if $\langle\tilde{\Psi}_{j}|\tilde{\Psi}_{1}\rangle=0$ for some $j$,
no set of nonzero prior probabilities satisfies Inequality (11) for all $i$.
It follows from Theorem 2 that for any set of nonzero prior probabilities
$\\{\xi_{i}\\}_{i=1}^{n}$, the globally optimal UD of
$\\{\xi_{i},|\Psi_{i}\rangle\\}_{i=1}^{n}$ cannot be achieved using LOCC.
Therefore, NLWE occurs regardless of nonzero prior probabilities. ∎

Example 1. Let us consider six $2\otimes 3$ product states,

$\begin{array}[]{rclrcl}|\Psi_{1}\rangle&=&\frac{1}{6}(|0\rangle+|1\rangle)\otimes(4|0\rangle+|1\rangle+|2\rangle),\\\\[2.84526pt]
|\Psi_{2}\rangle&=&\frac{1}{3}|1\rangle\otimes(2|0\rangle+2|1\rangle+|2\rangle),\\\\[2.84526pt]
|\Psi_{3}\rangle&=&\frac{1}{3}|1\rangle\otimes(2|0\rangle+|1\rangle+2|2\rangle),\\\\[2.84526pt]
|\Psi_{4}\rangle&=&\frac{1}{5}(2|0\rangle+|1\rangle)\otimes(2|1\rangle+|2\rangle),\\\\[2.84526pt]
|\Psi_{5}\rangle&=&\frac{1}{5}(|0\rangle+2|1\rangle)\otimes(|1\rangle+2|2\rangle),\\\\[2.84526pt]
|\Psi_{6}\rangle&=&\frac{1}{2}(|0\rangle+|1\rangle)\otimes(|1\rangle+|2\rangle).\end{array}$
(14)

Example 1 satisfies $\mathcal{H}_{\rm S}=\mathcal{H}$ because

$\displaystyle\begin{array}[]{l}{\rm
span}\\{|\Psi_{1}\rangle,|01\rangle,|02\rangle,|10\rangle,|11\rangle,|12\rangle\\}=\mathcal{H},\\\\[2.84526pt]
{\rm span}\\{|\Psi_{2}\rangle,|\Psi_{3}\rangle,|12\rangle\,\\}={\rm
span}\\{|10\rangle,|11\rangle,|12\rangle\\},\\\\[2.84526pt] {\rm
span}\\{|\Psi_{4}\rangle,|\Psi_{5}\rangle,|\Psi_{6}\rangle\\}={\rm
span}\\{|01\rangle,|02\rangle+|11\rangle,|12\rangle\\}.\end{array}$ (18)

The reciprocal vectors are

$\begin{array}[]{rcl}|\tilde{\Psi}_{1}\rangle&=&\frac{3}{2}|00\rangle,\\\\[2.84526pt]
|\tilde{\Psi}_{2}\rangle&=&\frac{3}{2}(|00\rangle-2|02\rangle-|10\rangle+2|11\rangle),\\\\[2.84526pt]
|\tilde{\Psi}_{3}\rangle&=&-3(|00\rangle-|02\rangle-|10\rangle+|11\rangle),\\\\[2.84526pt]
|\tilde{\Psi}_{4}\rangle&=&\frac{5}{6}(3|00\rangle+4|01\rangle-8|02\rangle-3|10\rangle+2|11\rangle+2|12\rangle),\\\\[2.84526pt]
|\tilde{\Psi}_{5}\rangle&=&\frac{5}{3}(3|00\rangle+|01\rangle-5|02\rangle-3|10\rangle+2|11\rangle+2|12\rangle),\\\\[2.84526pt]
|\tilde{\Psi}_{6}\rangle&=&-\frac{1}{2}(13|00\rangle+8|01\rangle-28|02\rangle-12|10\rangle+8|11\rangle+8|12\rangle).\end{array}$
(19)

Only $|\tilde{\Psi}_{1}\rangle$ is a product vector because the reduced non-
negative operators of
$|\tilde{\Psi}_{2}\rangle\langle\tilde{\Psi}_{2}|,\ldots,|\tilde{\Psi}_{6}\rangle\langle\tilde{\Psi}_{6}|$
for the qubit are not all rank one.

Figure 1: The occurrence of NLWE according to $a,b\in(0,1)$. NLWE occurs in
the green region where $a$ is smaller than $b$ or $\frac{125}{169}(1-b)$.
However, NLWE does not occur in the yellow area where $a$ is equal to or
greater than $b$ and $\frac{125}{169}(1-b)$.

According to Theorem 2, we can achieve the globally optimal UD between six
states of (14) using LOCC if and only if

$\frac{\xi_{1}}{9}\geqslant\frac{\xi_{2}}{9},\frac{\xi_{3}}{36},\frac{\xi_{4}}{25},\frac{\xi_{5}}{100},\frac{\xi_{6}}{169}.$
(20)

Furthermore, since the given six states are product vectors, we can see from
Corollary 2 that the occurrence of NLWE in terms of optimal UD depends on
nonzero prior probabilities. Figure 1 illustrates two cases classified by the
occurrence of NLWE when considering nonzero prior probabilities determined by
two variables $a\in(0,1)$ and $b\in(0,1)$ as follows:

$\xi_{1}=\xi_{2}=\frac{9a}{143},\ \xi_{3}=\frac{18(1-a)}{143},\
\xi_{4}=\frac{25b}{143},\ \xi_{5}=\frac{100b}{143},\
\xi_{6}=\frac{125(1-b)}{143}.$ (21)

For this case, NLWE does not occur if and only if $a$ is equal to or greater
than $b$ and $\frac{125}{169}(1-b)$.
Example 2. Let us consider the case where the first state of Example 1 is
replaced with

$\begin{array}[]{rcl}|\Psi_{1}\rangle&=&\frac{1}{15}(3|0\rangle+4|1\rangle)\otimes(|0\rangle+2|1\rangle+2|2\rangle).\end{array}$
(22)

In this case as well, Eq. (18) is satisfied, so the given six states form a
basis for $\mathcal{H}$. The reciprocal set corresponding to
$\\{|\Psi_{i}\rangle\\}_{i=1}^{6}$ consists of

$\begin{array}[]{rcl}|\tilde{\Psi}_{1}\rangle&=&5|00\rangle,\\\\[2.84526pt]
|\tilde{\Psi}_{2}\rangle&=&-\frac{3}{2}(2|02\rangle+|10\rangle-2|11\rangle),\\\\[2.84526pt]
|\tilde{\Psi}_{3}\rangle&=&-2|00\rangle+3|02\rangle+3|10\rangle-3|11\rangle,\\\\[2.84526pt]
|\tilde{\Psi}_{4}\rangle&=&\frac{5}{18}(4|00\rangle+12|01\rangle-24|02\rangle-9|10\rangle+6|11\rangle+6|12\rangle),\\\\[2.84526pt]
|\tilde{\Psi}_{5}\rangle&=&\frac{5}{9}(4|00\rangle+3|01\rangle-15|02\rangle-9|10\rangle+6|11\rangle+6|12\rangle),\\\\[2.84526pt]
|\tilde{\Psi}_{6}\rangle&=&-\frac{2}{3}(10|00\rangle+6|01\rangle-21|02\rangle-9|10\rangle+6|11\rangle+6|12\rangle).\end{array}$
(23)

Also, only $|\tilde{\Psi}_{1}\rangle$ is a product vector because the reduced
nonnegative operator of $|\tilde{\Psi}_{i}\rangle\langle\tilde{\Psi}_{i}|$ is
not rank one for $i\neq 1$. Since
$\langle\tilde{\Psi}_{1}|\tilde{\Psi}_{2}\rangle$ is zero, according to
Corollary 2, no matter what nonzero prior probabilities
$\xi_{1},\ldots,\xi_{6}$ are given, the globally optimal UD of
$\\{\xi_{i},|\Psi_{i}\rangle\\}_{i=1}^{6}$ cannot be achieved using LOCC. This
example that is not an indistinguishable product basisduan2007  shows that
NLWE can occur regardless of nonzero prior probabilities. We also note that,
in two qubits, there is no basis consisting of four product states that
satisfy the assumption of Theorem 2; see Methods Section.

## Discussion

We have shown that NLWE in terms of optimal UD can depend on the nonzero prior
probabilities of quantum states being prepared. We have first established a
necessary and sufficient condition for optimal UD measurement of linearly
independent pure states to detect only one specific state unambiguously
(Theorem 1). We have also provided a necessary and sufficient condition for
optimal UD measurement to detect two or more states unambiguously, regardless
of nonzero prior probabilities (Corollary 1). By generalizing Theorem 1 to
multipartite state discrimination, we have established a necessary and
sufficient condition for globally optimal UD of linearly independent pure
states using LOCC provided that only one state can be discriminated
unambiguously (Theorem 2). From Theorem 2, we have provided a sufficient
condition that NLWE occurs depending on nonzero prior probabilities, as well
as a sufficient condition that NLWE occurs regardless of nonzero prior
probabilities (Corollary 2). Finally, we have illustrated the occurrence of
NLWE in Corollary 2 by providing Examples 1 and 2.
Our result means that sets of linearly independent product states can be
classified into three types in terms of optimal UD: Type I where NLWE does not
occur regardless of nonzero prior probabilities (e.g., two multipartite pure
stateswalg2000 ; virm2001 ; chen2001 ; chen2002 ; ji2005 ), Type II where NLWE
occurs regardless of nonzero prior probabilities (e.g., domino statesbenn19991
, indistinguishable product basisduan2007 , and Example 2), and Type III where
NLWE occurs depending on nonzero prior probabilities (e.g. Example 1).
Surprisingly, Type III is a new type that only occurs in nonorthogonal states.
The quantum states of Type III can be useful for possible construction of
cryptographical schemes such as quantum data hidingterh2001 ; divi2002 ;
egge2002 ; matt2009  or quantum secret sharingraha2015 ; wang2017  with
nonorthogonal states. In most quantum data-hiding schemes, classical bits are
hidden by orthogonal quantum states of Type II which cannot be perfectly
discriminated using LOCC regardless of nonzero prior probabilities. In many
quantum secret sharing schemes, classical bits are shared among parties using
orthogonal quantum states of Type I that can be perfectly discriminated using
LOCC regardless of nonzero prior probabilities. By using nonorthogonal quantum
states of Type III, an unified cryptographic scheme can be constructed that
can be both data-hiding and secret-sharing scheme depending on nonzero prior
probabilities. We believe this is an interesting challenge as a future work.
Also, we have provided a new example of Type II (Example 2). Interestingly,
this example is different from the domino statesbenn19991 , which can be
unambiguously discriminated using LOCC. Moreover, this example is also
different from the indistinguishable product basisduan2007 , which cannot be
unambiguously discriminated using LOCC. In Example 2, one state can be
unambiguously discriminated using LOCC, but all other states cannot. This
suggests that Type II can be subdivided into more inequivalent classes.

## Methods

### Semidefinite programming to optimal UD of linearly independent pure states

The optimal UD of $\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ can be expressed
as

$\begin{array}[]{lll}\mbox{maximize}&{\rm Tr}(\rho M_{\vec{p}})\\\\[2.84526pt]
\mbox{subject to}&M_{\vec{p}}\geqslant 0\ \mbox{and}\
\mathbbm{1}-M_{\vec{p}}\geqslant 0,\end{array}$ (24)

where

$\rho=\sum_{i=1}^{n}\eta_{i}|\psi_{i}\rangle\langle\psi_{i}|,\ \
M_{\vec{p}}=\sum_{i=1}^{n}p_{i}|\tilde{\psi}_{i}\rangle\langle\tilde{\psi}_{i}|.$
(25)

Eq. (24) is called a primal problem. $M_{\vec{p}}$ is said to be feasible if
it safisfies the two constraints of (24), and strictly feasible if both
$M_{\vec{p}}$ and $\mathbbm{1}-M_{\vec{p}}$ are positive-definite.
When constructing the Lagrangianboyd2004  as

$\displaystyle\mathcal{L}(M_{\vec{p}},E,K)={\rm Tr}(\rho M_{\vec{p}})+{\rm
Tr}(M_{\vec{p}}E)+{\rm Tr}[(\mathbbm{1}-M_{\vec{p}})K]={\rm Tr}K+{\rm
Tr}[M_{\vec{p}}(\rho+E-K)],$ (26)

we can derive the dual problemelda2003 ; elda2004  of (24):

$\begin{array}[]{lll}\mbox{minimize}&{\rm Tr}K\\\\[2.84526pt] \mbox{subject
to}&\langle\tilde{\psi}_{i}|E|\tilde{\psi}_{i}\rangle\geqslant 0\ \forall i,\
K\geqslant 0,\ \mbox{and}\ K=\rho+E,\end{array}$ (27)

where $E$ and $K$ are two Hermitian operators on $\mathcal{H}_{\rm S}$
corresponding to the Lagrange multipliers associated with
$M_{\vec{p}}\geqslant 0$ and $\mathbbm{1}-M_{\vec{p}}\geqslant 0$. As in the
primal problem, a pair $(E,K)$ is said to be feasible if they satisfy the dual
constraints of (27).
The first and second constraints of (27) are the positivity conditions of
Lagrange multipliers and the last condition means the Lagrangian stability,
i.e., $\partial\mathcal{L}/\partial M_{\vec{p}}=0$. The first constraint of
(27) does not mean that $E$ is positive semidefinite. The first and last
constraints of (27) can be expressed as

$\langle\tilde{\psi}_{i}|K|\tilde{\psi}_{i}\rangle\geqslant\eta_{i}\ \ \forall
i,$ (28)

therefore $E$ can be omitted. When $E$ is omitted, $K$ is called feasbile if
it is positive-semidefinite and satisfies Inequality (28) for all $i$. We use
the superscript ⋆ to express the optimality of primal and dual variables.
The primal optimal value $P_{\max}$ is an lower bound of the dual optimal
value ${\rm Tr}K^{\star}$ because

$\displaystyle{\rm Tr}(\rho
M_{\vec{p}})\leqslant\mathcal{L}(M_{\vec{p}},E,K)={\rm Tr}K$ (29)

for any feasible $M_{\vec{p}}$ and $(E,K)$. From Inequality (29), it can be
easily seen that feasible $M_{\vec{p}}$ and $(E,K)$ with ${\rm Tr}(\rho
M_{\vec{p}})={\rm Tr}K$ satisfy

${\rm Tr}(M_{\vec{p}}E)={\rm Tr}[(\mathbbm{1}-M_{\vec{p}})K]=0,$ (30)

which is called the complementary slackness condition. Applying $K=\rho+E$ to
${\rm Tr}(M_{\vec{p}}E)=0$, we can have

${\rm Tr}[M_{\vec{p}}(K-\rho)]=0$ (31)

or equivalently,

$p_{i}(\langle\tilde{\psi}_{i}|K|\tilde{\psi}_{i}\rangle-\eta_{i})=0\ \
\forall i$ (32)

which are the complementary slackness condition that replaces ${\rm
Tr}(M_{\vec{p}}E)=0$ when $E$ is omitted. Due to the weak duality ${\rm
Tr}M_{\vec{p}}^{\star}\leqslant{\rm Tr}K^{\star}$, if feasible $M_{\vec{p}}$
and $(E,K)$ satisfying Eq. (30) exist, they are optimal and the strong duality
${\rm Tr}M_{\vec{p}}^{\star}={\rm Tr}K^{\star}$ holds. In addition, when
strong duality holds, the optimal primal and dual variables satisfy Eq. (30).
The convex optimization problem given in (24) satisfies the well-known
Slater’s conditionboyd2004  (i.e., the existence of strictly feasible
$M_{\vec{p}}$) as a sufficient condition for strong duality, so primal and
dual problems provide the same optimal value, i.e., ${\rm
Tr}K^{\star}=P_{\max}$. This implies that Eq. (30) is a necessary and
sufficient condition for feasible $M_{\vec{p}}$ and $(E,K)$ to be optimal.
Therefore, $M_{\vec{p}}$ and $(E,K)$ are optimal if and only if they satisfy
the so-called Karush-Kuhn-Tucker (KKT) conditionelda2003 ; elda2004 ; boyd2004
; naka2015 :

$\begin{array}[]{llll}\mbox{(i)}&M_{\vec{p}}\geqslant 0,\
\mathbbm{1}-M_{\vec{p}}\geqslant 0;\\\\[5.69054pt]
\mbox{(ii)}&\langle\tilde{\psi}_{i}|E|\tilde{\psi}_{i}\rangle\geqslant 0\
\forall i,\ K\geqslant 0,\ K=\rho+E;\\\\[5.69054pt] \mbox{(iii)}&{\rm
Tr}(M_{\vec{p}}E)=0,\ {\rm Tr}[(\mathbbm{1}-M_{\vec{p}})K]=0.\end{array}$ (33)

The approach to finding an optimal solution using KKT condition is called the
linear complementarity problem (LCP) approachbae2013 ; bae2015 . The
characteristic of LCP is that it considers both primal and dual variables,
that it is neither the minimum nor the maximum problem [i.e., there is no
objective function], and that the constraints are KKT condition (33).
From KKT condition (33), we can see that both
$\mathbbm{1}-M_{\vec{p}}^{\star}$ and $K^{\star}$ are positive-semidefinite,
nonzero, and not full rank. $M_{\vec{p}}^{\star}$ is also positive-
semidefinite and nonzero, but it may be full rank. Note that if
$(M_{\vec{p}},E,K)$ and $(\bar{M}_{\vec{p}},\bar{E},\bar{K})$ satisfy KKT
condition, so do $(M_{\vec{p}},\bar{E},\bar{K})$ and $(\bar{M}_{\vec{p}},E,K)$
because both $M_{\vec{p}}$ and $\bar{M}_{\vec{p}}$ are primal optimal and both
$(E,K)$ and $(\bar{E},\bar{K})$ are dual optimal. Furthermore, the optimality
condition for $(M_{\vec{p}},E,K)$ can be converted to one for
$(M_{\vec{p}},K)$ by replacing the constraints for $E$ with (28) and (31).
Thus, $(M_{\vec{p}},K)$ is optimal if and only if

$\begin{array}[]{llll}\mbox{(i)}&p_{i}\geqslant 0\ \forall i,\
\mathbbm{1}-M_{\vec{p}}\geqslant 0;\\\\[5.69054pt] \mbox{(ii)}&K\geqslant 0,\
{\rm Tr}[(\mathbbm{1}-M_{\vec{p}})K]=0;\\\\[5.69054pt]
\mbox{(iii)}&p_{i}(\langle\tilde{\psi}_{i}|K|\tilde{\psi}_{i}\rangle-\eta_{i})=0\
\forall i;\\\\[5.69054pt]
\mbox{(iv)}&\langle\tilde{\psi}_{i}|K|\tilde{\psi}_{i}\rangle\geqslant\eta_{i}\
\forall i.\end{array}$ (34)

If Condition (33) holds for $(M_{\vec{p}},K,E)$, then Condition (34) holds for
$(M_{\vec{p}},K)$. Conversely, if $(M_{\vec{p}},K)$ satisfies Condition (34),
$(M_{\vec{p}},K,E)$ with $E=K-\rho$ satisfies Condition (33).
We will selectively use Conditions (33) and (34) because both have their
respective advantages. Eliminating the dual variable $E$ simplifies the form
of the dual problem and KKT condition, so Condition (34) is useful for
demonstrating the optimality of given variables or for deriving the condition
that fixed variables must be optimal. On the other hand, leaving $E$ as in
Condition (33) allows a useful condition, Lagrangian stability, to be used.
The Lagrangian stability condition is useful for demonstrating the uniqueness
of optimal variables.
Lemma 1. $\vec{p}^{\star}$ is unique.
Proof. Assume that both $M_{\vec{p}}$ and $\bar{M}_{\vec{p}}$ are primal
optimal and $(K,E)$ is dual optimal. Then, both $(M_{\vec{p}},E,K)$ and
$(\bar{M}_{\vec{p}},E,K)$ satisfy Condition (33). Since ${\rm
Tr}[(\mathbbm{1}-M_{\vec{p}})K]=0$, it follows that

$M_{\vec{p}}K=KM_{\vec{p}}=K.$

In a similar way,

$\bar{M}_{\vec{p}}K=K\bar{M}_{\vec{p}}=K.$

From the above equation and $K=\rho+E$, we obtain

$\begin{array}[]{ll}\rho
M_{\vec{p}}+EM_{\vec{p}}=K=\rho\bar{M}_{\vec{p}}+E\bar{M}_{\vec{p}}.\end{array}$

From Eq. (25), we have

$\rho M_{\vec{p}}=\sum_{i=1}^{n}|\psi_{i}\rangle\langle\tilde{\psi}_{i}|(\rho
M_{\vec{p}}+EM_{\vec{p}})|\psi_{i}\rangle\langle\tilde{\psi}_{i}|=\sum_{i=1}^{n}|\psi_{i}\rangle\langle\tilde{\psi}_{i}|(\rho\bar{M}_{\vec{p}}+E\bar{M}_{\vec{p}})|\psi_{i}\rangle\langle\tilde{\psi}_{i}|=\rho\bar{M}_{\vec{p}}$

in which the first and last equalities hold due to

${\rm Tr}(M_{\vec{p}}E)={\rm Tr}(\bar{M}_{\vec{p}}E)=0.$

Since $\rho$ is invertible, $M_{\vec{p}}=\bar{M}_{\vec{p}}$. This means that
$M_{\vec{p}}^{\star}$ is unique. Therefore, $p^{\star}$ is unique.∎
The uniqueness of optimal primal variables tells us that the optimal UD of $n$
linearly independent pure states is a problem of finding only one point
$\vec{p}^{\star}$ in $\mathbb{R}^{n}$.
The optimal dual variable $K^{\star}$ is generally not unique, but it is
uniquely determined when the zero eigenvalue of
$\mathbbm{1}-M_{\vec{p}}^{\star}$ is not degeneratepang2009 . In this case, it
is easy to find the detail relation between
$\\{\eta_{i},|\psi_{i}\rangle\\}_{i=1}^{n}$ and the optimal UD measurement(or
$\vec{p}^{\star}$). The simplest example is the case where only
$p_{1}^{\star}$ is nonzero. In this case, the optimal UD measurement detects
only $|\psi_{1}\rangle$ unambiguously.

### Proof of Theorem 1

In this section, we prove Theorem 1 using the optimality condition in (34).
For the necessity, suppose that

$p_{2}^{\star}=\cdots=p_{n}^{\star}=0.$

Because of primal constraints (i),

$0\leqslant
p_{1}^{\star}\leqslant|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-1}.$

Since the kernel of $\mathbbm{1}-M_{\vec{p}}^{\star}$ must have a nonzero
vector,

$p_{1}^{\star}=|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-1}.$

That is,

$M_{\vec{p}}^{\star}=|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-1}|\tilde{\psi}_{1}\rangle\langle\tilde{\psi}_{1}|.$

Because $K^{\star}$ is a positive-semidefinite operator on the kernel of
$\mathbbm{1}-M_{\vec{p}}^{\star}$,

$K^{\star}=a|\tilde{\psi}_{1}\rangle\langle\tilde{\psi}_{1}|$

for some nonnegative number $a$. By (iii) of Condition (34), $a$ is determined
as $\eta_{1}|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-2}$, that is,

$K^{\star}=\eta_{1}|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-2}|\tilde{\psi}_{1}\rangle\langle\tilde{\psi}_{1}|.$

Due to (iv) of Condition (34), the following inequality holds,

$\frac{\eta_{i}}{\eta_{1}}\leqslant\frac{\langle\tilde{\psi}_{i}|K^{\star}|\tilde{\psi}_{i}\rangle}{\eta_{1}}=\frac{|\langle\tilde{\psi}_{i}|\tilde{\psi}_{1}\rangle|^{2}}{|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{2}}\
\ \forall i,$

which proves the necessity of our theorem.
To prove sufficiency, we assume that

$\frac{\eta_{i}}{\eta_{1}}\leqslant\frac{|\langle\tilde{\psi}_{i}|\tilde{\psi}_{1}\rangle|^{2}}{|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{2}}\
\ \forall i.$

It is straightforward to verify that the following $(M_{\vec{p}},K)$ satisfies
Condition (34);

$M_{\vec{p}}=|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-1}|\tilde{\psi}_{1}\rangle\langle\tilde{\psi}_{1}|$

and

$K=\eta_{1}|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-2}|\tilde{\psi}_{1}\rangle\langle\tilde{\psi}_{1}|.$

Thus, $(M_{\vec{p}},K)$ is optimal, so

$p_{i}^{\star}=\delta_{i1}|\langle\tilde{\psi}_{1}|\tilde{\psi}_{1}\rangle|^{-1},$

and $p_{2}^{\star}=\cdots=p_{n}^{\star}=0$. This completes our proof.

### Proof of Theorem 2

In this section, we suppose that only $|\tilde{\Psi}_{1}\rangle$ is a product
vector, which implies only $|\Psi_{1}\rangle$ can be unambiguously
discriminated using LOCC.
To prove the necessity, we assume that the globally optimal UD of
$\\{\xi_{i},|\Psi_{i}\rangle\\}_{i=1}^{n}$ is possible using LOCC. Thus, the
globally optimal UD of $\\{\xi_{i},|\Psi_{i}\rangle\\}_{i=1}^{n}$ is the
optimally unambiguous detection of $|\Psi_{1}\rangle$. From Theorem 1,
Inequality (11) holds for each $i$, which proves the necessity of our theorem.
To prove sufficiency, suppose Inequality (11) holds for each $i$. According to
Theorem 1, the globally optimal UD measurement of
$\\{\xi_{i},|\Psi_{i}\rangle\\}_{i=1}^{n}$ detects only one state
$|\Psi_{1}\rangle$ unambiguously. Thus, the optimally unambiguous detection of
$|\Psi_{1}\rangle$ means the projective measurement

$\\{|\langle\tilde{\Psi}_{1}|\tilde{\Psi}_{1}\rangle|^{-1}|\tilde{\Psi}_{1}\rangle\langle\tilde{\Psi}_{1}|,\mathbbm{1}-|\langle\tilde{\Psi}_{1}|\tilde{\Psi}_{1}\rangle|^{-1}|\tilde{\Psi}_{1}\rangle\langle\tilde{\Psi}_{1}|\\}.$

Moreover, this projective measurement can be done using finite-round LOCC. To
show this, let $\mathcal{H}_{k}$ be each party’s Hilbert space, i.e.,
$\mathcal{H}=\bigotimes_{k=1}^{m}\mathcal{H}_{k}$. Since
$|\tilde{\Psi}_{1}\rangle$ is a product vector, its normalized form can be
written as

$\begin{array}[]{c}|\langle\tilde{\Psi}_{1}|\tilde{\Psi}_{1}\rangle|^{-1/2}|\tilde{\Psi}_{1}\rangle=\bigotimes_{k=1}^{m}|\phi_{k}\rangle,\end{array}$

in which $|\phi_{k}\rangle$ is a pure state in $\mathcal{H}_{k}$. Each party
performs a projective measurement
$\\{|\phi_{k}\rangle\langle\phi_{k}|,\mathbbm{1}_{k}-|\phi_{k}\rangle\langle\phi_{k}|\\}$,
and all parties share their local measurement results through classical
communication. Here $\mathbbm{1}_{k}$ is the identity operator on
$\mathcal{H}_{k}$. If the measurement result is
$|\phi_{k}\rangle\langle\phi_{k}|$ for all party $k$, they unambiguously
discriminate $|\Psi_{1}\rangle$, otherwise it is considered as inconclusive
results. Thus, we can detect only $|\Psi_{1}\rangle$ unambiguously and
optimally using only finite-round LOCC. This implies that the globally optimal
UD of $\\{\xi_{i},|\Psi_{i}\rangle\\}_{i=1}^{n}$ can be achieved using LOCC,
which proves the sufficiency of our theorem.

### Nonexistence of complete $2\otimes 2$ product basis where only one state
can be unambiguously discriminated

Suppose that $\\{|\Psi_{i}\rangle\\}_{i=1}^{4}$ is a normalized basis of
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ and consists of four product states
$|\Psi_{i}\rangle=|\alpha_{i}\rangle\otimes|\beta_{i}\rangle$. Also, assume
that the reciprocal vector $|\tilde{\Psi}_{1}\rangle$ is a product vector,
i.e.,
$|\tilde{\Psi}_{1}\rangle=|\tilde{\alpha}\rangle\otimes|\tilde{\beta}\rangle$.
This assumption implies that $|\tilde{\alpha}\rangle$ is orthogonal to two of
$\\{|\alpha_{i}\rangle\\}_{i=2}^{4}$ or $|\tilde{\beta}\rangle$ is orthogonal
to two of $\\{|\beta_{i}\rangle\\}_{i=2}^{4}$.
When $|\alpha_{2}\rangle$ and $|\alpha_{3}\rangle$ are orthogonal to
$|\tilde{\alpha}\rangle$, the two vectors are linearly dependent because they
are in $\mathbb{C}^{2}$. Since $|\Psi_{2}\rangle$ and $|\Psi_{3}\rangle$ are
linearly independent, $|\beta_{2}\rangle$ and $|\beta_{3}\rangle$ span
$\mathbb{C}^{2}$, and thus all vectors orthogonal to both $|\Psi_{2}\rangle$
and $|\Psi_{3}\rangle$ are in
$\\{|\tilde{\alpha}\rangle\otimes|\varphi\rangle:|\varphi\rangle\in\mathbb{C}^{2}\\}$.
This means that $|\tilde{\Psi}_{4}\rangle$ is a product vector because
$\langle\tilde{\Psi}_{4}|\Psi_{i}\rangle=0$ for $i\neq 4$. In any case, we can
similarly show that at least one of $\\{|\tilde{\Psi}_{i}\rangle\\}_{i=2}^{4}$
is a product vector. Therefore, there is no set of four linearly independent
$2\otimes 2$ product states where only one state can be unambiguously
discriminated.

## Acknowledgements

This work was supported by Quantum Computing Technology Development
Program(NRF-2020M3E4A1080088) through the National Research Foundation of
Korea(NRF) grant funded by the Korea government(Ministry of Science and ICT).

## References

  * (1) Chefles, A. Quantum state discrimination. Contemporary Physics 41, 401 (2000).
  * (2) Barnett, S. M. & Croke, S. Quantum state discrimination. Adv. Opt. Photon. 1, 238 (2009).
  * (3) Bergou, J. A. Discrimination of quantum states. J. Mod. Opt. 57, 160 (2010).
  * (4) Bae, J. & Kwek, L.-C. Quantum state discrimination and its applications. J. Phys. A: Math. Theor. 48, 083001 (2015).
  * (5) Bennett, C. H., DiVincenzo, D. P., Fuchs, C. A., Mor, T., Rains, E., Shor, P. W., Smolin, J. A. & Wootters, W. K. Quantum nonlocality without entanglement. Phys. Rev. A 59, 1070 (1999).
  * (6) Walgate, J., Short, A. J., Hardy, L. & Vedral, V. Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett. 85, 4972 (2000).
  * (7) Bennett, C. H., DiVincenzo, D. P., Mor, T., Shor, P. W., Smolin, J. A. & Terhal, B. M. Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385 (1999).
  * (8) Ghosh, S., Kar, G., Roy, A., Sen(De), A. & Sen, U. Distinguishability of Bell states. Phys. Rev. Lett. 87, 277902 (2001).
  * (9) Walgate, J. & Hardy, L. Nonlocality, asymmetry, and distinguishing bipartite states. Phys. Rev. Lett. 89, 147901 (2002).
  * (10) Divincenzo, D. P., Mor, T., Shor, P. W., Smolin, J. A. & Terhal, B. M. Unextendible product bases, uncompletable product bases and bound entanglement. Commun. Math. Phys. 238, 379 (2003).
  * (11) Ghosh, S., Kar, G., Roy, A. & Sarkar, D. Distinguishability of maximally entangled states. Phys. Rev. A 70, 022304 (2004).
  * (12) Fan, H. Distinguishability and indistinguishability by local operations and classical communication. Phys. Rev. Lett. 92, 177905 (2004).
  * (13) Watrous, J. Bipartite subspaces having no bases distinguishable by local operations and classical communication. Phys. Rev. Lett. 95, 080505 (2005).
  * (14) Niset, J. & Cerf, N. J. Multipartite nonlocality without entanglement in many dimensions. Phys. Rev. A 74, 052103 (2006).
  * (15) Cohen, S. M. Local distinguishability with preservation of entanglement. Phys. Rev. A 75, 052313 (2007).
  * (16) Duan, R., Feng, Y., Xin, Y. & Ying, M. Distinguishability of quantum states by separable operations. IEEE Trans. Inf. Theory 55, 1320 (2009).
  * (17) Kleinmann, M., Kampermann, H. & Bruß, D. Asymptotically perfect discrimination in the local-operation-and-classical-communication paradigm. Phys. Rev. A 84, 042326 (2011).
  * (18) Bandyopadhyay, S. Entanglement, mixedness, and perfect local discrimination of orthogonal quantum states. Phys. Rev. A 85, 042319 (2012).
  * (19) Yang, Y.-H., Gao, F., Tian, G.-J., Cao, T.-Q. & Wen, Q.-Y. Local distinguishability of orthogonal quantum states in a $2\bigotimes 2\bigotimes 2$ system. Phys. Rev. A 88, 024301 (2013).
  * (20) Zhang, Z.-C., Gao, F., Tian, G.-J., Cao, T.-Q. & Wen, Q.-Y. Nonlocality of orthogonal product basis quantum states. Phys. Rev. A 90, 022313 (2014).
  * (21) Chitambar, E., Duan, R. & Hsieh, M.-H. When do local operations and classical communication suffice for two-qubit state discrimination? IEEE Trans. Inf. Theory 60, 1549 (2014).
  * (22) Wang, Y.-L., Li, M.-S., Zheng, Z.-J. & Fei, S.-M. Nonlocality of orthogonal product-basis quantum states. Phys. Rev. A 92, 032313 (2015).
  * (23) Xu, G.-B., Yang, Y.-H., Wen, Q.-Y., Qin, S.-J. & Gao, F. Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system. Sci. Rep. 6, 31048 (2016).
  * (24) Zhang, Z.-C., Gao, F., Cao, Y., Qin, S.-J. & Wen, Q.-Y. Local indistinguishability of orthogonal product states. Phys. Rev. A 93, 012314 (2016).
  * (25) Zhang, X., Weng, J., Tan, X. & Luo, W. Indistinguishability of pure orthogonal product states by LOCC. Quantum Inf. Process. 16, 168 (2017).
  * (26) Halder, S., Banik, M., Agrawal, S. & Bandyopadhyay, S. Strong quantum nonlocality without entanglement. Phys. Rev. Lett. 122, 040403 (2019).
  * (27) Jiang, D.-H. & Xu, G.-B. Nonlocal sets of orthogonal product states in an arbitrary multipartite quantum system. Phys. Rev. A 102, 032211 (2020).
  * (28) Bhattacharya, S. S., Saha, S., Guha, T. & Banik, M. Nonlocality without entanglement: Quantum theory and beyond. Phys. Rev. Research. 2, 012068 (2020).
  * (29) Helstrom, C. W. Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
  * (30) Holevo, A. S. Probabilistic and Statistical Aspects of Quantum Thoery (North-Holland, 1979).
  * (31) Yuen, H., Kennedy, R. & Lax, M. Optimum testing of multiple hypotheses in quantum detection theory. IEEE Trans. Inf. Theory 21, 125 (1975).
  * (32) Virmani, S., Sacchi, M. F., Plenio, M. B. & Markham, D. Optimal local discrimination of two multipartite pure states. Phys. Lett. A 288, 62 (2001).
  * (33) Peres, A. & Wootters, W. K. Optimal detection of quantum information. Phys. Rev. Lett. 66, 1119 (1991).
  * (34) Chitambar, E. & Hsieh, M.-H. Revisiting the optimal detection of quantum information. Phys. Rev. A 88, 020302 (2013).
  * (35) Chitambar, E., Leung, D., Mančinska, L., Ozols, M. & Winter, A. Everything you always wanted to know about LOCC (but were afraid to ask). Commun. Math. Phys. 328, 303 (2014).
  * (36) LOCC consists of finite- and infinite-round LOCC. Asymptotic LOCC is the closure of finite-round LOCC. It contains LOCC. We also note that SEP contains asymptotic LOCC, but not vice versa.
  * (37) Ha, D. & Kwon, Y. Quantum nonlocality without entanglement: explicit dependence on prior probabilities of nonorthogonal mirror-symmetric states. npj Quantum Inf. 7, 81 (2021).
  * (38) Ivanovic, I. D. How to differentiate between non-orthogonal states. Phys. Lett. A 123, 257 (1987).
  * (39) Peres, A. How to differentiate between non-orthogonal states. Phys. Lett. A 128, 19 (1988).
  * (40) Dieks, D. Overlap and distinguishability of quantum states. Phys. Lett. A 126, 303 (1988).
  * (41) Jaeger, G. & Shimony, A. Optimal distinction between two non-orthogonal quantum states. Phys. Lett. A 197, 83 (1995).
  * (42) Chefles, A. Unambiguous discrimination between linearly independent quantum states. Phys. Lett. A 239, 339 (1998).
  * (43) Chefles, A. Condition for unambiguous state discrimination using local operations and classical communication. Phys. Rev. A 69, 050307(R) (2004).
  * (44) Duan, R., Feng, Y., Ji, Z. & Ying, M. Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication. Phys. Rev. Lett. 98, 230502 (2007).
  * (45) Bandyopadhyay, S. & Walgate, J. Local distinguishability of any three quantum states. J. Phys. A: Math. Theor. 42, 072002 (2009).
  * (46) Chen, Y.-X. & Yang, D. Optimal conclusive discrimination of two nonorthogonal pure product multipartite states through local operations. Phys. Rev. A 64, 064303 (2001).
  * (47) Chen, Y.-X. & Yang, D. Optimally conclusive discrimination of nonorthogonal entangled states by local operations and classical communications. Phys. Rev. A 65, 022320 (2002).
  * (48) Ji, Z., Cao, H. & Ying, M. Optimal conclusive discrimination of two states can be achieved locally. Phys. Rev. A 71, 032323 (2005).
  * (49) Pang, S. & Wu, S. Optimum unambiguous discrimination of linearly independent pure states. Phys. Rev. A 80, 052320 (2009).
  * (50) Sugimoto, H., Hashimoto, T., Horibe, M. & Hayashi, A. Complete solution for unambiguous discrimination of three pure states with real inner products. Phys. Rev. A 82, 032338 (2010).
  * (51) Terhal, B. M., DiVincenzo, D. P. & Leung, D. W. Hiding bits in Bell states. Phys. Rev. Lett. 86, 5807 (2001).
  * (52) Divincenzo, D. P., Leung, D. W. & Terhal, B. M. Quantum data hiding. IEEE Trans. Inf. Theory 48, 580 (2002).
  * (53) Eggeling, T. & Werner, R. F. Hiding classical data in multipartite quantum states. Phys. Rev. Lett. 89, 097905 (2002).
  * (54) Matthews, W., Wehner, S. & Winter, A. Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding. Commun. Math. Phys. 291, 813 (2009).
  * (55) Rahaman, R. & Parker, M. G. Quantum scheme for secret sharing based on local distinguishability. Phys. Rev. A 91, 022330 (2015).
  * (56) Wang, J., Li, L., Peng, H. & Yang, Y. Quantum-secret-sharing scheme based on local distinguishability of orthogonal multiqudit entangled states. Phys. Rev. A 95, 022320 (2017).
  * (57) Boyd, S. & Vandenberghe, L. Convex Optimization (Cambridge University Press, Cambridge, 2004).
  * (58) Eldar, Y. C. A semidefinite programming approach to optimal unambiguous discrimination of quantum states. IEEE Trans. Inf. Theory 49, 446 (2003).
  * (59) Eldar, Y. C., Stojnic, M. & Hassibi, B. Optimal quantum detectors for unambiguous detection of mixed states. Phys. Rev. A 69, 062318 (2004).
  * (60) Nakahira, K., Kato, K. & Usuda, T. S. Generalized quantum state discrimination problems. Phys. Rev. A 91, 052304 (2015).
  * (61) Bae, J. Structure of minimum-error quantum state discrimination. New J. Phys. 15, 073037 (2013).