The Quantum Formalization Drive

A quantum computer stores information in a system that cannot be perfectly isolated: it is coupled to an environment whose degrees of freedom are neither controlled nor observed. The coupling turns pure states of the system into mixed ones (decoherence), and this is the main obstacle to building reliable quantum memories. One response, developed in the late…

A decoherence-free subsystem (DFS) is a part of a quantum system that the coupling to its environment cannot touch: when the system operators of the system-environment interaction all respect a common block structure, information stored in the degeneracy of one block evolves trivially, however strong the coupling is (Zanardi–Rasetti 1997; Lidar–Chuang–Whaley…

Every experiment that manipulates a quantum system does so through some apparatus: a laser mode driving an atom, a gate electrode biasing a quantum dot, a coil producing a magnetic field. The apparatus is itself a quantum system, and the system and apparatus interact through a fixed Hamiltonian. Whether this interaction lets the apparatus act as a classical…

State redistribution asks how cheaply one party can hand part of a shared quantum state to another. Alice and Bob share many copies of a state; Alice holds two shares, $A$ and $C$, and wants to transfer $A$ to Bob, who already holds $B$, using quantum communication and pre-shared entanglement, without disturbing the correlations with anything else. Devetak and Yard…

A private state is the quantum analogue of a shared secret key: two parties, Alice and Bob, hold key systems $A$ and $B$ of dimension $K$ whose standard-basis measurement outcomes are perfectly correlated, uniformly random and independent of any eavesdropper holding a purification. Distilling such states from many copies of a noisy state $\rho_{AB}$ is the task of q…

In classical statistics the Fisher information of a parametric family of probability distributions bounds how well its parameter can be estimated (the Cramér–Rao inequality), and it can only decrease when the data are processed by a Markov kernel. Quantum estimation theory, quantum metrology and information geometry need the analogue for families of density…

Quantum discord was introduced by Ollivier and Zurek (quant-ph/0105072) and by Henderson and Vedral (quant-ph/0105028) as the part of the correlations of a bipartite quantum state that no local measurement can reveal. It is nonzero for some unentangled states, and it has been proposed as a resource behind mixed-state quantum computation with little…

In quantum tomography of a large system one rarely measures an informationally complete set of observables. For an $n$-qubit system with $n$ large, one typically knows only the expectation values of a few local observables, and many states are compatible with that data. The principle of maximum entropy (Jaynes, 1957; Wichmann, 1963) selects among them the state…

We extend the concept of the relative error to mixed-state cloning and related physical operations, in which the ancilla contains some a priori information about the input state. The lower bound on the relative error is obtained. It is shown that this result contributes to the stronger no-cloning theorem.

Previous work on symmetric group equivariant neural networks generally only considered the case where the group acts by permuting the elements of a single vector. In this paper we derive formulae for general permutation equivariant layers, including the case where the layer acts on matrices by permuting their rows and columns simultaneously. This case…

The Peres-Horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states. The partial transpose operation admits, in the continuous case, a geometric interpretation as mirror reflection in phase space. This recognition leads to uncertainty principles, stronger than the traditional…

An optical transmitter irradiates a target region containing a bright thermal-noise bath in which a low-reflectivity object might be embedded. The light received from this region is used to decide whether the object is present or absent. The performance achieved using a coherent-state transmitter is compared with that of a quantum illumination transmitter,…

The quantum internet holds promise for performing quantum communication, such as quantum teleportation and quantum key distribution, freely between any parties all over the globe. For such a quantum internet protocol, a general fundamental upper bound on the performance has been derived [K. Azuma, A. Mizutani, and H.-K. Lo, arXiv:1601.02933]. Here we…

We prove the security of quantum key distribution against the most general attacks which can be performed on the channel, by an eavesdropper who has unlimited computation abilities, and the full power allowed by the rules of classical and quantum physics. A key created that way can then be used to transmit secure messages in a way that their security is…

In this paper we use the AdS/CFT correspondence to refine and then establish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT.…

A pure-loss bosonic channel is a simple model for communication over free-space or fiber-optic links. More generally, phase-insensitive bosonic channels model other kinds of noise, such as thermalizing or amplifying processes. Recent work has established the classical capacity of all of these channels, and furthermore, it is now known that a strong converse…

The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce new codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by…

We show how adaptive protocols of quantum and private communication through bosonic Gaussian channels can be simplified into much easier block versions that involve resource states with finite energy. This is achieved by combining the adaptive-to-block reduction technique devised earlier [S. Pirandola et al., Nat. Commun. 8, 15043 (2017)], based on…

Almost all modern communication systems rely on electromagnetic fields as a means of information transmission, and finding the capacities of these systems is a problem of significant practical importance. The Additive White Gaussian Noise (AWGN) channel is often a good approximate description of such systems, and its capacity is given by a simple formula.…

Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: previous…

We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the specified error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff…

We give an operational definition of the quantum, classical and total amount of correlations in a bipartite quantum state. We argue that these quantities can be defined via the amount of work (noise) that is required to erase (destroy) the correlations: for the total correlation, we have to erase completely, for the quantum correlation one has to erase…

The importance of symmetries has recently been recognized in quantum machine learning from the simple motto: if a task exhibits a symmetry (given by a group $\mathfrak{G}$), the learning model should respect said symmetry. This can be instantiated via $\mathfrak{G}$-equivariant Quantum Neural Networks (QNNs), i.e., parametrized quantum circuits whose gates are…

Fusion of positive energy representations is defined using Connes' tensor product for bimodules over a von Neumann algebra. Fusion is computed using the analytic theory of primary fields and explicit solutions of the Knizhnik-Zamolodchikov equation.

One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information…