A complex conjugation of unitary quantum map is a second-order map (supermap) that maps a unitary operator $U$ to its complex conjugate $U^*$. First, we present a deterministic quantum protocol that universally implements the complex conjugation supermap when we are given a blackbox quantum circuit, guaranteed to implement some unitary operation, whose only known description is its dimension. We then discuss the complex conjugation supermap in the context of entanglement theory and derive a conjugation-based expression of the $G$-concurrence. Finally, we present a physical process involving identical fermions from which the complex conjugation protocol is derived as a simulation of the process using qudits.
Polycategorical Constructions for Unitary Supermaps of Arbitrary Dimension
University of Oxford（牛津大学）；HKU-Oxford Joint Laboratory for Quantum Information and Computation（香港大学-牛津大学量子信息与计算联合实验室）；University College London（伦敦大学学院）；Universite Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；Inria（法国国家数字创新研究院）；CentraleSupélec（中央理工-高等电力学院）；Perimeter Institute for Theoretical Physics（理论物理珀利马研究所）
We consider unitary conjugation channels with continuous random phases. The spectral properties of the channel average are examined, thereby the asymptotic behaviors of the repeated quantum interactions of the motion are derived. We then study the channels with uniformly distributed continuous phases on an interval. In this context, it is shown that discrete phases are sufficient to achieve the channel average.
We present a method to engineer the unitary charge conjugation operator, as given by quantum field theory, in the highly controlled context of quantum optics, thus allowing one to simulate the creation of charged particles with well-defined momenta simultaneously with their respective antiparticles. Our method relies on trapped ions driven by a laser field and interacting with a single mode of a light field in a high Q cavity.
In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a K…
This paper addresses the problem of designing universal quantum circuits to transform $k$ uses of a $d$-dimensional unitary input-operation into a unitary output-operation in a probabilistic heralded manner. Three classes of protocols are considered, parallel circuits, where the input-operations can be simultaneously, adaptive circuits, where sequential uses of the input-operations are allowed, and general protocols, where the use of the input-operations may be performed without a definite causal order. For these three classes, we develop a systematic semidefinite programming approach that fin…
We consider generalizations of depolarizing channels to maps in which the identity channel is replaced by a convex combinations of unitary conjugations. We show that one can construct unital channels of this type for which the input which achieves maximal output purity is unique. We give conditions under which multiplicativity of the maximal p-norm and additivity of the minimal output entropy. We also show that the Holevo capacity need not equal log d - the minimal entropy as one might expect for a convex combination of unitary conjugations. Conversely, we give examples for which this conditio…
Many quantum computational tasks have inherent symmetries, suggesting a path to enhancing their efficiency and performance. Exploiting this observation, we propose representation matching, a generic probabilistic protocol for reducing the cost of quantum computation in a quantum network. We show that the representation matching protocol is capable of reducing the communication or memory cost to almost minimum in various tasks, including remote execution of unitary gate arrays, permutation gates and unitary conjugation, as well as the storage and retrieval of unitary gates.
The understanding of symmetry operations has brought enormous advancements in physics, ranging from elementary particle to condensed matter systems. In quantum mechanics, symmetry operations are described by either unitary or antiunitary operators, where the latter are unphysical transformations that cannot be realized in physical systems. So far, quantum simulators of unitary and dissipative processes, the only allowed physical dynamics, have been realized in key experiments. Here, we present an embedding quantum simulator able to encode unphysical operations in a multilevel single trapped io…
No quantum circuit can turn a completely unknown unitary gate into its coherently controlled version. Yet, coherent control of unknown gates has been realised in experiments, making use of a different type of initial resources. Here, we formalise the task achieved by these experiments, extending it to the control of arbitrary noisy channels, and to more general types of control involving higher dimensional control systems. For the standard notion of coherent control, we identify the information-theoretic resource for controlling an arbitrary quantum channel on a $d$-dimensional system: specifi…
We present a new scheme for teleporting a quantum state between two parties whose local reference frames are misaligned by the action of a finite symmetry group. Unlike other proposals, our scheme requires the same amount of classical communication and entangled resources as conventional teleportation, does not reveal any reference frame information, and is robust against changes in reference frame alignment while the protocol is underway. The mathematical foundation of our scheme is a unitary error basis which is permuted up to a phase by the conjugation action of the group. We completely cla…
In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present a proof of this theorem which is both elementary and economical. Central to our proof is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis involves a judicious i…
The application of open quantum systems in biological processes such as photosynthetic complexes has recently received renewed attention. In this paper, we introduce a quantum algorithm for simulation of Markovian dynamics of the Fenna-Matthew-Olson (FMO) complex that exists in photosynthesis using a "universal set" of a one-parameter semigroup of generators. We investigate the details of each generator that has been obtained from spectral decomposition of the Gorini-Kossakowski-Sudarshan (GKS) matrix by using linear combination and unitary conjugation. Also, we present a simple quantum circui…
We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, PPT channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we p…
Generalising in the sense of Hahn's spin echo, we completely characterise those unitary propagators of effective multi-qubit interactions that can be inverted solely by {\em local} unitary operations on $n$ qubits (spins-$\tfrac{1}{2}$). The subset of $U\in \mathbf{SU}(2^n)$ satisfying $U^{-1}=K_1 U K_2$ with pairs of local unitaries $K_1, K_2\in\mathbf{SU}(2)^{\otimes n}$ comprises two classes: in type-I, $K_1$ and $K_2$ are inverse to one another, while in type-II they are not. {Type-I} consists of one-parameter groups that can jointly be inverted for all times $t\in\R{}$ because their Hamil…
Isometry operations encode the quantum information of the input system to a larger output system, while the corresponding decoding operation would be an inverse operation of the encoding isometry operation. Given an encoding operation as a black box from a $d$-dimensional system to a $D$-dimensional system, we propose a universal protocol for isometry inversion that constructs a decoder from multiple calls of the encoding operation. This is a probabilistic but exact protocol whose success probability is independent of $D$. For a qubit ($d=2$) encoded in $n$ qubits, our protocol achieves an exp…
Quantum Wasserstein isometries of the $n$-qubit state space: a Wigner-type result
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；ELTE Eötvös Loránd University（罗兰大学）
We consider the simulation of interacting high-dimensional systems using pairwise interacting qubits. The main tool in this context is the generation of effective many-body interactions, and we examine a number of different protocols for obtaining them. These methods include the usage of higher-order processes (commutator method), unitary conjugation or graph state encoding, as well as teleportation based approaches. We illustrate and compare these methods in detail and analyze the time cost for simulation. In the second part of the article, we investigate the influence of noise on the simulat…
