We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement $M$ and preparation $\ensuremath{\rho}$ the experimenter can determine, given enough time, the probability of a given outcome $a$: $p(a\ensuremath{\mid}M,\ensuremath{\rho})$. How large does the Hilbert space of the quantum system have to be in order to allow us to find density matrices and measurement operators that will reproduce the given probability distribution? In this paper, we prove a simple lower bound for the dimension of the Hilbert space. The main insight is to relate this problem to the construction of quantum random access codes, for which interesting bounds on the Hilbert space dimension already exist. We discuss several applications of our result to hidden-variable or ontological models, to Bell inequalities, and to properties of the smooth min-entropy.
We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement M and preparation \ensuremathρ the experimenter can determine, given enough time, the probability of a given outcome a: p(a\ensuremath∣M,\ensuremathρ). How large does the Hilbert space of the quantum system have to be in order to allow us to find density matrices and measurement operators that will reproduce the given probability distribution? In this paper, we prove a simple lower bound for the dimension of the Hilbert space. The main insight is to relate this problem to the construction of quantum random access codes, for which interesting bounds on the Hilbert space dimension already exist. We discuss several applications of our result to hidden-variable or ontological models, to Bell inequalities, and to properties of the smooth min-entropy.
Abstract The prepare-and-measure scenario offers the possibility to infer the dimension of an unknown physical system in a device-independent way, i.e. using only raw measurement data with apparatuses regarded as black boxes. We provide here a general lower bound on the dimension necessary to observe arbitrary quantum behaviours in this scenario based on simple matrix analysis. This bound holds even if the preparer and measurer share randomness. This is relevant in scenarios where the parties are free to access this resource or it is not safe to assume that the devices are not correlated. We further use this result to bound the success probability of random access codes in general as a function of the dimension of the quantum systems sent from one party to another and we provide constructions of dimension witnesses.
Classical probabilistic models of (noisy) quantum systems are not only relevant for understanding the non-classical features of quantum mechanics, but they are also useful for determining the possible advantage of using quantum resources for information processing tasks. A common feature of these models is the presence of inaccessible information, as captured by the concept of preparation contextuality: There are ensembles of quantum states described by the same density operator, and hence operationally indistinguishable, and yet in any probabilistic (ontological) model, they should be described by distinct probability distributions. In this work, we quantify the inaccessible information of a model in terms of the maximum distinguishability of probability distributions associated to any pair of ensembles with identical density operators, as quantified by the total variation distance of the distributions. We obtain a family of lower bounds on this maximum distinguishability in terms of experimentally measurable quantities. In the case of an ideal qubit this leads to a lower bound of, approximately, 0.07. These bounds can also be interpreted as a new class of robust preparation non-contextuality inequalities. Our non-contextuality inequalities are phrased in terms of generalizations of max-relative entropy and trace distance for general operational theories, which could be of independent interest. Under sufficiently strong noise any quantum system becomes preparation non-contextual, i.e., can be described by models with zero inaccessible information. Using our non-contextuality inequalities, we show that this can happen only if the noise channel has the average gate fidelity less than or equal to 1/D(1+1/2+...+1/D), where D is the dimension of the Hilbert space.
Introduction The use of parameters to describe an experimenter's control over the devices used in an experiment is familiar in quantum physics, for example in connection with Bell inequalities. Parameters are also interesting in a different but related context, as we noticed when we proved a formal separation in quantum mechanics between linear operators and the probabilities that these operators generate. In comparing an experiment against its description by a density operator and detection operators, one compares tallies of experimental outcomes against the probabilities generated by the operators but not directly against the operators. Recognizing that the accessibility of operators to experimental tests is only indirect, via probabilities, motivates us to ask what probabilities tell us about operators, or, put more precisely, “what combinations of a parameterized density operator and parameterized detection operators generate any given set of parametrized probabilities?” Here, we review and augment recent proofs that any given parameterized probabilities can be generated in very diverse ways, so that a parameterized probability measure, detached from any of the (infinitely many) parameterized operators that generate it, becomes an interesting object in its own right. By detaching a parameterized probability measure from the operators that may have led us to it, we (1) strengthen Holevo's bound on a quantum communication channel and (2) clarify a role for multiple levels of modeling in an example based on quantum key distribution. We then inquire into some parameterized probability measures generated by entangled states and into the topology of the associated parameter spaces; in particular we display some previously overlooked topological features of level sets of these probability measures.
I explore the possibility that a quantum system S may be described completely by the combination of its standard quantum state ∣ψ⟩ and a (hidden) quantum state ∣ϕ⟩ (that lives in the same Hilbert space), such that the outcome of any standard projective measurement on the system S is determined once the two quantum states are specified. I construct an algorithm that retrieves the standard quantum-mechanical probabilities, which depend only on ∣ψ⟩, by assuming that the (hidden) quantum state ∣ϕ⟩ is drawn at random from some fixed probability distribution Pr(.) and by averaging over Pr(.). Contextuality and Bell nonlocality turn out to emerge automatically from this algorithm as soon as the dimension of the Hilbert space of S is larger than 2. If ∣ϕ⟩ is not completely random, subtle testable deviations from standard quantum mechanics may arise in sequential measurements on single systems.
A natural measure for the amount of quantum information that a physical system E holds about another system A = A 1 ,..., A n is given by the min-entropy H min ( A | E ). Specifically, the min-entropy measures the amount of entanglement between E and A , and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography, decoupling used in channel coding, to physical processes such as thermalization or the thermodynamic work cost (or gain) of erasing a quantum system. As such, it is a central question to determine the behaviour of the min-entropy after some process M is applied to the system A . Here we introduce a new generic tool relating the resulting min-entropy to the original one, and apply it to several settings of interest, including sampling of subsystems and measuring in a randomly chosen basis. The results on random measurements yield new high-order entropic uncertainty relations with which we prove the optimality of cryptographic schemes in the bounded quantum storage model. This is an abridged version of the paper; the full version containing all proofs and further applications can be found in [13]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A natural measure for the amount of quantum information that a physical system E holds about another system A = A1, .. . , Anis given by the min-entropy Hmin(A|E). In particular, the min-entropy measures the amount of entanglement between E and A, and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography, decoupling used in channel coding, to physical processes such as thermalization or the thermodynamic work cost (or gain) of erasing a quantum system. As such, it is a central question to determine the behavior of the minentropy after some process M. is applied to the system A. Here, we introduce a new generic tool relating the resulting min-entropy to the original one, and apply it to several settings of interest. A simple example of such a process is the one of sampling, where a subset S of the systems A1, ... , An is selected at random. Our tool allows us to quantify the entanglement that E has with the selected systems AS, i.e., Hmin(AS|ES) as a function of the original Hmin(A|E). We give two applications of this result. First, it directly provides the first local quantum-to-classical randomness extractors for use in quantum cryptography, as well as decoupling operations acting on only a small fraction AS of the input A. Moreover, it gives lower bounds on the dimension of k-out-of-n fully quantum random access encodings. Another natural example of such a process is a measurement in, e.g., BB84 bases commonly used in quantum cryptography. We establish the first entropic uncertainty relations with quantum side information that are nontrivial whenever E is not maximally entangled with A. As a consequence, we are able to prove optimality of quantum cryptographic schemes in the noisy-storage model. This model allows for the secure implementation of two-party cryptographic primitives under the assumption that the adversary cannot store quantum information perfectly. A special case is the bounded-quantum-storage model (BQSM), which assumes that the adversary's quantum memory device is noise free but limited in size. Ever since the inception of the BQSM, it has been a vexing open question to determine whether the security is possible as long as the adversary can only.