Background
The authors consider data hiding in the context of general probabilistic theories (GPTs). Data hiding is a concept commonly used in quantum information theory that is concerned with the discrimination of two quantum states that are provided with some known a priori probability distribution \(\{p,1-p\}, 0\leq p\leq1\), by means of a quantum measurement [D. P. DiVincenzo et al., IEEE Trans. Inf. Theory 48, No. 3, 580–598 (2002; Zbl 1071.81511); W. Matthews et al., Commun. Math. Phys. 291, No. 3, 813–843 (2009; Zbl 1179.81020)]. Data hiding occurs if the two states can be easily distinguished by general quantum measurements, but are hard to distinguish when the class of allowed measurements is restricted. In a typical example, the states are shared between two parties, Alice and Bob, and the class of restricted measurement only contains local measurement operations. This class, known as LO, can be relaxed to local operations and (one or two-way) classical communication (LOCC) and further to so-called separable measurement operations (SEP).
Remaining in the bipartite case of Alice and Bob, the quantum states are described by positive semidefinite operators \(\rho_{AB}\) and \(\sigma_{AB}\) with unit trace acting on a tensor product Hilbert space \(\mathcal{H_{AB}}=\mathcal{H}_A\otimes\mathcal{H}_B\). A quantum measurement is described by a finite set of positive semidefinite operators \(\{M^i_{AB}\}\), each corresponding to a specific outcome, that sum up to identity. The probability of getting output \(i\) by measuring state \(\rho\) is given by \(\text{Tr}(\rho M^i)\). In the case where such general measurements are allowed, the distinguishability between two statesman be quantified by their weighted distance \(\|p\rho-(1-p)\sigma\|_1\) in terms of the trace norm. In the case of a restricted class of measurements \(\mathcal{M}\), it has been shown that a corresponding distinguishability norm \(\|\cdot\|_{\mathcal{M}}\) can be defined, such that the distinguishability is quantified by \(\|p\rho-(1-p)\sigma\|_{\mathcal{M}}\) using that norm. A central object of investigation is then given by the ratio of the of the distance in terms of trace norm and the distance in terms of the distinguishability norm, maximised over all pairs of input states and a priori probabilities, \(R_{QM}(\mathcal{M})=\max \|p\rho-(1-p)\sigma\|_1/ \|p\rho-(1-p)\sigma\|_{\mathcal{M}}\).
GPTs can be seen as generalisation of quantum mechanics, as well as classical probability theory, to general probabilistic physical models [J. Barrett, “Information processing in generalized probabilistic theories”, Phys. Rev. A 75, No. 3, Article ID 032304, 21 p. (2007; doi:10.1103/PhysRevA.75.032304)]. A GPT is defined as the triple \((V,C,u)\), where \(V\) is a vector space, \(C\) a closed convex cone, and \(u\) is a functional in the interior of the dual cone \(C^*\), which equals one for all normalised states and is known as unit effect. In the case of quantum mechanics, \(V\) is Hilbert space describing the system, \(C\) the cone of positive semidefinite operators and \(u\) is the trace. Quantum measurements generalise to finite sets of effects \(e_i\) , i.e. functionals in \(C^*\) that sum up to the unit effect, with effect \(e_i\) corresponding to outcome \(i\) of a quantum measurement.
On \(V\), one can define the so-called base-norm as \(\|x\|=\max\{|e(x)|+|(u-e)(x)|\}\), where the maximum is over all effects \(e\). In the case of quantum mechanics the base norm is equal to the trace norm. Going to bipartite systems, the vector space of the joint systems can be identified with the tensor product of the two systems, \(V_{AB}\simeq V_A\otimes V_B\), and also \(u_{AB}=u_A\otimes u_B\). For the cone \(C_{AB}\), it is holds \(C_A\otimes_{\min}C_B\subseteq C_{AB}\subseteq C_A\otimes_{\max}C_B\) with a minimal and maximal tensor product. In the case of quantum mechanics the l.h.s. describes the set of separable operators and the r.h.s. the set of so-called entanglement witnesses. The definitions of classes of locally constraint measurements measurements, i.e. LO, one and two-way LOCC and SEP can also be generalised to general GPTs, and it holds \(LO\subseteq LOCC_\rightarrow\subseteq LOCC \subseteq SEP\).
Contributions
The contributions of this work are multifold. The authors generalise the concept of data hiding to the realm of GPTs. In particular, for a given GPT \(\mathcal{G}\), they define an informationally complete measurement as a finite set of effects, that sum up to the unit effect, and span the dual vector space \(V^*\). For a set \(\mathcal{M}\) of informationally complete measurement they then define a distinguishability norm \(\|\cdot\|_{\mathcal{M}}\) and the data hiding ratio as \(R_\mathcal{G}(\mathcal{M})=\max_{0\neq x\in V}\|x\|/\|x\|_{\mathcal{M}}\).
Within the GPT of quantum mechanics, for subsystems of dimensions \(d_A\) and \(d_B\), the authors use an argument based on quantum teleportation [C. H. Bennett et al., Phys. Rev. Lett. 70, No. 13, 1895–1899 (1993; Zbl 1051.81505)] to show that in the case of (one or two-way) LOCC, and separable measurements the data hiding ratio is of order \(\min\{\sqrt{d_A},\sqrt{d_B}\}\), which is an improvement of previously known upper bounds [W. Matthews and A. Winter, Commun. Math. Phys. 285, No. 1, 161–174 (2009; Zbl 1228.81117); W. Matthews et al., Commun. Math. Phys. 291, No. 3, 813–843 (2009; Zbl 1179.81020); F. G. S. L. Brandão and M. Horodecki, Commun. Math. Phys. 333, No. 2, 761–798 (2015; Zbl 1317.81021)] that are of order \((d_Ad_B)^\frac{1}{4}\) and \(\min\{d_A,d_B\}\), respectively. The same scaling is observed in a modified version of the GPT of quantum mechanics in which composite systems are obtained via the minimal tensor product.
The authors also consider a GPT describing a hypothetical class of physical models in which the state space is a Euclidian ball of arbitrary dimension. This is known as a spherical model. For local dimensions \(d_A,d_B\), it is shown that \(R_{Sph}(SEP)\geq \min\{d_A,d_B\}-1\). Hence, for the same local dimensions, the spherical model can provide a higher data hiding ratio against all locally constrained sets of measurement. This means that quantum mechanics, even in it’s modified version using using the minimal tensor product is not optimal in terms of data hiding.
This motivates the main focus of this study, namely the determination of constraints that must be obeyed in any possible GPT. To this end the authors introduce a so-called ultimate data hiding ratio \(R_{\mathcal{M}}(d_A,d_B)\), which is defined, for given local dimensions \(d_A,d_B\) and a locally constraint set of measurements \(\mathcal{M}\), as the supremum over all data ratios \(R_{\mathcal{M}}\) achieved by composite GPTs and local dimensions \(d_A,d_B\). It is shown that the lower bound provided by the spherical model is optimal up to an additive constant. The main result of this work is an upper bound on the ultimate data hiding ratio, \(R_{SEP}(d_Ad_B)\leq R_{LOCC}(d_Ad_B)\leq R_{LOCC_\rightarrow}(d_Ad_B)\leq R_{LO}(d_Ad_B)\leq\min\{d_A,d_B\}\).
The upper bound is obtained by showing that for the composite GPTs in question, the hiding ratio against LO can be expressed in terms of a ratio of the projective norm and the injective norm. Such norms can be defined on the tensor product \(V_A\otimes V_B\) for any two finite dimensional Banach spaces \(V_A\) and \(V_B\). The injective norm is defined as \(\|X\|_\epsilon=\max\{(f\otimes g)(X):f\in V^*_A, \|f\|_*\leq1,g\in V^*_B,\|g\|_*\leq1\}\) and the projective norm as \(\|X\|_\pi=\inf\{\sum_i\|x_i\|\|y_i\|,X=\sum_ix_i\otimes y_i\}\), where the sum is finite. Relating those norms to the distinguishability norms on composite systems, it is then shown that \(R_{LO}(d_Ad_B)\leq\max_{0\neq X\in V_A\otimes V_B}\|X\|_\pi/\|X\|_\epsilon\). The results then follows from showing that \(\|\cdot\|_\pi\leq\min\{dim(V_A),dim(V_B)\}\|\cdot\|_\epsilon\).
Finally, the authors consider a number of examples of special classes of GPTs. In particular they consider GPTs with a centrally symmetric state space, generalising the spherical model, as well as a class of GPTs the vector spaces of which is endowed with a representation of a compact group, which are a generalisation to quantum mechanics and classical probability theory. The authors construct a generalisation of the Werner states in quantum mechanics, and use them for data hiding. By using such states, the authors obtain a general lower bound on the data hiding ratio that only depends on a few geometrically meaningful parameters.
Significance
This work is significant in a number of ways. By improving the upper bound on the data hiding ratios in the GPT of quantum mechanics, the authors provide a valuable tool in the study of quantum data hiding, which can serve as a primitive in a number of other quantum information protocols. By providing an example of a GPT that allows for better data hiding than quantum mechanics, showing that, as the authors put it, ‘nature is non-classical but not as non-classical as it could have been’, this work is also of interest from a more foundational point of view. Further, by generalising the concept of data hiding to GPTs and establishing a connection between the base and distinguishability norms, which have a clear operational meaning in the context of data hiding, and the projective and injective norms on tensor products in Banach spaces, the authors uncover a previously unknown link between data hiding and the theory of Banach spaces, possibly opening up a new direction of research.