A *quantum state* is a mathematical tool – a Hermitian, positive semi-definite
density matrix $\rho$ – used to compute probability of a given outcome of any
measurement. It is compelling to investigate
geometry of the set of quantum
states of order $N$. Composite quantum states are described in a Hilbert space
with a tensor product structure, e.g. $\mathcal{H}_A \otimes \mathcal{H}_B$.
A state with classical correlations can be represented as a convex combination
of product states, $\rho^{AB}_{\texttt{sep}}= \sum_{j=1}^k q_j \; \rho_j^A \otimes \rho_j^B$,
and is called *separable*. All other states display quantum correlations and are called
*entangled* [1]. Key questions one may ask concern
the volume and
the structure
of the set of
entangled states.

The convex set $\Omega_N$ of quantum states of order $N$ has $N^2-1$ dimensions.
In the case of a single qubit it forms the *Bloch ball*, $\Omega_2=B_3$, with the pure states comprising the Bloch sphere. To investigate properties of such multidimensional sets
arising for $N\ge 3$ we use their projection onto a $3$-space and apply the notion
of joint
numerical range,
$W(X_1,X_2,X_3)$, which represents the
set of expectation
values of three Hermitian observables, $x_i=\texttt{Tr}\rho X_i$, taken over entire
set $\Omega_N$.

Figure 1. The numerical range $W(X_1,X_2,X_3)$ of two selected
triples of operators of order four listed here that gives a projection of the
set $\Omega_4$ of $2$ qubit states onto $\mathbb{R}^3$, is shown in blue.
Subsets in green represent the
separable numerical range $\color{green}W_{\color{green}{sep}}$, which forms
a $3D$ projection of the set $\Omega^{\texttt{sep}}_N$ of separable states. The remaining
blue states are entangled. Such an approach allows one to study the
dynamics of
quantum entanglement. |

Figure 2. a) $3D$ model of the difference of two sets
${\color{blue}W} \setminus {\color{green}W_{\color{green}{sep}}}$ shown in Fig. 1.a – registering
three expectation values $(x_1,x_2,x_3)$, which represent a point in this set, confirms entanglement of
$\rho$. b) Model of separable join numerical range of
three operators of order four generated by a 3D printer at
Complexity Garage. |

Consider a quantum system in an $N$-dimensional Hilbert space. If no specific
constraints are imposed on the system, the Hilbert space is isotropic, hence
*all quantum states are equal*. The situation changes, if one can identify
physical subsystems. In a bipartite scenario one can distinguish, for instance, product states
and maximally entangled, generalized Bell states. For a particular multipartite
system, it is legitimate to ask, which state is the most entangled with respect
to a given measure of entanglement. A state is called
absolutely maximally entangled
(AME), if it is maximally entangled with respect to any bipartite
partition of the system. Such states, useful for quantum teleportation and quantum
error correction, are known for the
four-qutrit system,
$\mathcal{H}=\mathcal{H}_3^{\otimes 4}$, and
several larger systems.
They do not exist for
seven
qubits [5], and a focus of our current research is whether such states do exist for
four parties with six levels each.

A set of $3$ mutually unbiased bases (MUBs) leads an informationally complete measurement of a single qubit. Any two such orthogonal bases of are related by a unitary complex Hadamard matrix, so the uncertainty related to this joint measurement is minimal. It is simple to show that in dimension $N$ there are no more than $N+1$ MUBs. For $N=3$, $4$ and $5$ this bound is also saturated, as there exist $4$, $5$ and $6$ MUBs, respectively. For $N=6$ only $3$ MUBs are found up till now, and we would like to know how many MUBs exist in dimension six to optimize the quantum measurement scheme.

The standard construction of $5$ MUBs for a two-qubit system consists of three orthogonal bases consisting of separable states and two bases with all states maximally entangled. In [2] we found a constellation of $20$ iso-entangled states giving the full set of MUBs. Partial traces (into any subsystem) of four orthogonal states of any basis forms a regular tetrahedron inside the Bloch ball, while the corners of five such tetrahedrons yield a regular dodecahedron.

Figure 3. a) Eigenbases of three Pauli operators $\sigma_x$, $\sigma_y$ and $\sigma_z$
provide the complete set of $3$ MUBs for a single qubit;
b) partial traces of $20$ two-qubit pure states, which form a complete set of
$5$ iso-entangled MUBs
in $N=4$, yield a regular dodecahedron. Any state
of this unique constellation is obtained by local unitary transformations of the fiducial state
\[|\phi\rangle = \frac{1}{20}\Big(a_+|00\rangle - 10i |01\rangle + (8i-6)|10\rangle + a_- |11\rangle\Big),\]
where $a_\pm = -7 \pm 3\sqrt{5} + i\left(1 \pm\sqrt{5}\right)$. |

To identify distinguished multipartite states, we can analyze properties of reduced density matrices. An entangled quantum state of $N$ subsystems is called $m$-resistant, if it remains entangled after losing an arbitrary subset of $m$ particles, but becomes separable after losing any number of particles larger than $m$. One can establish an analogy to the problem of designing a topological link consisting of $N$ rings such that after cutting any $(m + 1)$ of them, the remaining rings become disconnected.

Figure 4. a) Borromean link forms an example
of a $0$-resistant
link of three rings corresponding to the polynomial
${\color{red}a}{\color{green}b}{\color{blue}c}$ and the GHZ state
$$|\psi_0\rangle=|GHZ_3\rangle=\frac{1}{\sqrt{2}}\Big(|000\rangle+|111\rangle\Big).$$
b) An example of a $1$-resistant link of four rings, represented by the polynomial
${\color{red}a}{\color{green}b}{\color{blue}c}+
{\color{red}a}{\color{green}b}{\color{orange}d}+
{\color{red}a}{\color{blue}c}{\color{orange}d}+
{\color{green}b}{\color{blue}c}{\color{orange}d}$, corresponds to the $1$-resistant state of four qubits,
$$|\psi_1\rangle=\frac{1}{\sqrt{20}}\Big(4|0000\rangle+|0111\rangle+|1011\rangle+|1101\rangle+|1110\rangle\Big).$$
After cutting any single ring one arrives at the Borromean link of three rings shown in panel a),
while after tracing out any of four qubits, one becomes an entangled mixed state of three qubits,
the entanglement of which is $0$-resistant. |

Several quantum states and operations are distinguished by attaining extremal values of quantities relevant for information processing. To provide a universal reference point we are interested in their mean values averaged over of the entire set of (pure or mixed) quantum states, computed with respect to physically motivated probability measures. These tasks can be achieved with help of the theory of random matrices. Such generic quantum states and typical quantum operations describe physical systems corresponding to classically chaotic dynamics.

Spectral properties of a quantum operation, including its
*spectral gap*,
defined as the difference between moduli of two largest eigenvalues of the
superoperator, describes the convergence to the equilibrium. In the case of
a discrete random operation acting on a set of states of size $N$ the spectrum
consists of the leading eigenvalue, $\lambda_1=1$, and the
bulk of eigenvalues
in the disk of radius $R=1/N$ in the complex plane. The spectrum of a generic
Lindblad operator, which describes a continuous dynamics of an open quantum
system, displays a universal,
lemon-like shape,
discovered in [3].

Figure 5. Spectral density of the bulk of rescaled eigenvalues of random
Lindblad operators generating continuous dynamics for $N = 50$ and $N=100$. Solid lines
represent analytic boundaries of the lemon-like support of the spectrum derived in [3]. |

The first step to devise new quantum technologies able to overcome limitations
of currently used classical computing machines is to identify, which components
of quantum theory can provide such an advantage - we need to recognise what actually
constitutes quantum resources. Once they are identified, the next step is to
characterise them. Informally this corresponds to translating qualitative
statements like *entanglement is a useful cryptographic resource* to
quantitative ones like *entangled state $A$ is five times better than $B$, because
it allows one to securely communicate five times more bits*.

More formally, characterisation requires one to understand when different resources can be interconverted and how efficiently this can be done, which is captured by the mathematical framework of resource theories. Finally, there is also the third essential step: finding optimal ways to experimentally implement protocols exhibiting quantum advantage, while taking into account realistic constraints. Our goal is to develop a theoretical framework underpinning quantum technologies, with a particular focus on quantum computing and quantum thermodynamics, by investigating all three aspects of quantum resource theories.

The specific research tasks include: investigating possibilities for quantum advantage within thermodynamic scenarios, developing quantitative methods to characterise resource dissipation, [4], operationally motivating coherence resources through communication scenarios, constructing experimentally feasible protocols for probing quantum thermodynamic phenomena and devising classical simulation algorithms for the certification and verification of quantum devices.

Figure 6. a)
The resource resonance effect: by carefully engineering
the interconversion process (controlled by irreversibility parameter $\nu$) the
dissipation of resources (measured by transformation error $\varepsilon$) arising
from finite-size effects can be greatly suppressed; b)
Markovian cooling of a
two-level system: quantum memoryless dynamics (dotted arrow) allows one to cool
the system below the lowest temperature achievable by classical memoryless
dynamics (solid arrow). |

Similarly to a phone line that has a "bad connection" or a corrupted computer memory, quantum information can be affected by environmental noise and imperfect information processing. Quantum error correction allows the packaging quantum information in interesting ways, such that it can be safely stored and transmitted, forming a back-end to practically all types of quantum communication and computation. The existence and performance of quantum codes is related to many interesting topics in quantum information theory, such as entanglement and the sharing of quantum correlations, ground states of local Hamiltonians, and the quantum marginal problem.

Figure 7. Experimental implementation of quantum error correction in a
ultracold atomic mixture:
(a) The phonons of the bosonic bath couple a tweezer with $N$ atoms to a tweezer with one atom.
(b) Preparation of the logical state using a control qubit and quantum gates.
(c) Error correction on the encoded state. |

>Together with mathematicians from the Jagiellonian University and other places in Poland and abroad we also analyze various mathematical aspects of quantum information. Our joint weekly seminar has been running since 1995: Quantum Chaos and Quantum Information.

Figure 8. In search for optimal bounds for
operation entropy
of
complementary quantum channels. |

^{[1]} I. Bengtsson and K. Życzkowski, Geometry of Quantum States: *An Introduction to Quantum Entanglement*, Cambridge University Press, 2005; II ed. 2017.

^{[2]} J. Czartowski, D. Goyeneche, M. Grassl, K. Życzkowski, Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs, Phys. Rev. Lett. **124**, 090503 (2020).

^{[3]} S. Denisov, T. Laptyeva, W. Tarnowski, D. Chruściński and K. Życzkowski, Universal spectra of random Lindblad operators, Phys. Rev. Lett. **123**, 140403 (2019).

^{[4]} K. Korzekwa, C. Chubb and M. Tomamichel, Avoiding irreversibility: engineering resonant conversions of quantum resources, Phys. Rev. Lett. **122**, 110403 (2019).

^{[5]} F. Huber, O. Gühne, and J. Siewert, Absolutely Maximally Entangled States of Seven Qubits Do Not Exist, Phys. Rev. Lett. **118**, 200502 (2017).

We are grateful for the support by the following research projects:

- Polish National Science Centre project Maestro 7, Uncertainty Relations and Quantum Entanglement number DEC-2015/18/A/ST2/00274, run in 2016-2021;
- Team-Net project Near Term Quantum Computing financed by the Foundation for Polish Science (FNP), pursued during 2019-2023;
- Polish National Science Centre project Preludium Bis1, Optimized Generalized Quantum Measurements, number 2019/35/O/ST2/01049, run in 2020-2024.