Session T27: Focus Session: Superselection and Quantum Reference Frames

Quantum frameness for charge-parity-time inversion symmetry

Physical laws are invariant under simultaneous charge-parity-time (CPT) inversion, which is due to relativistic Lorentz covariance and the linearity of quantum mechanics. We show that CPT-superselection can be circumvented by employing a system that possesses CPT frameness, and we construct such resources in two cases: for massive spin-zero particles and for Dirac-spinors. In the case of spin-zero particles, we explicitly construct and quantify all resourceful pure states. Our approach is to treat CPT inversion unitarily by considering the aggregate action of the CPT transformation, rather than sequentially composing a unitary and two anti-unitary transformations, thereby overcoming a major drawback of circumventing time-inversion symmetry alone using an anti-unitary transformation [G. Gour, P. S. Turner and B. C. Sanders, J. Math. Phys. 50, 102105 (2009)]. We discuss an explicit example using pionic communication to overcome CPT superselection.   

The capacity to transmit classical information via black holes

One of the most vexing problems in theoretical physics is the relationship between quantum mechanics and gravity. According to an argument originally by Hawking, a black hole must destroy any information that is incident on it because the only radiation that a black hole releases during its evaporation (the Hawking radiation) is precisely thermal. Surprisingly, this claim has never been investigated within a quantum information-theoretic framework, where the black hole is treated as a quantum channel to transmit classical information. We calculate the capacity of the quantum black hole channel to transmit classical information (the Holevo capacity) within curved-space quantum field theory, and show that the information carried by late-time particles sent into a black hole can be recovered with arbitrary accuracy, from the signature left behind by the stimulated emission of radiation that must accompany any absorption event. We also show that this stimulated emission turns the black hole into an almost-optimal quantum cloning machine, where the violation of the no-cloning theorem is ensured by the noise provided by the Hawking radiation. Thus, rather than threatening the consistency of theoretical physics, Hawking radiation manages to save it instead.   

Constructing holographic spacetimes using entanglement renormalization

We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of the large $N$ limit where on the holographic side the gravity theory becomes classical and a non-fluctuating smooth spacetime description emerges. We show how a number of features of holographic duality in the large $N$ limit emerge naturally from entanglement renormalization, including a classical spacetime generated by entanglement, a sparse spectrum of operator dimensions, and phase transitions in mutual information. We also address questions related to bulk locality below the AdS radius, holographic duals of weakly coupled large $N$ theories, Fermi surfaces in holography, and the holographic interpretation of branching MERA. Some of our considerations are inspired by the idea of quantum expanders which are generalized quantum transformations that add a definite amount of entropy to most states. Since we identify entanglement with geometry, we thus argue that classical spacetime may be built from quantum expanders (or something like them).   

Quantifying asymmetry of quantum states using entanglement

For open systems, symmetric dynamics do not always lead to conservation laws. We show that, for a dynamic symmetry associated with a compact Lie group, one can derive new selection rules from entanglement theory. These selection rules apply to both closed and open systems as well as reversible and irreversible time evolutions. Our approach is based on an embedding of the system's Hilbert space into a tensor product of two Hilbert spaces allowing for the symmetric dynamics to be simulated with local operations. The entanglement of the embedded states determines which transformations are forbidden because of the symmetry. In fact, every bipartite entanglement monotone can be used to quantify the asymmetry of the initial states. Moreover, where the dynamics is reversible, each of these monotones becomes a new conserved quantity.   

How hard is it to decide if a quantum state is separable or entangled?

Suppose that a physical process, described as a sequence of local interactions that can be executed in a reasonable amount of time, generates a quantum state shared between two parties. We might then wonder, does this physical process produce a quantum state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. In order to address this question, we begin by demonstrating a two-message quantum interactive proof system that can decide the answer to a promise version of this problem. We then prove that this promise problem is hard for the class ``quantum statistical zero knowledge'' (QSZK) by demonstrating a polynomial-time reduction from the QSZK-complete promise problem ``quantum state distinguishability'' to our quantum separability problem. Finally, we consider a variant of this question, in which a given physical process accepts a quantum state as input, and the question is to decide if there is an input to this process which makes its output separable across some bipartite cut. We prove that this latter problem is a complete promise problem for the class QIP of problems admitting quantum interactive proof systems.   

Operator extension of strong subadditivity of entropy

We prove an operator inequality that extends strong subadditivity of entropy: after taking a trace, the operator inequality becomes the strong subadditivity of entropy.   

Measuring Entanglement via SICs and 2-designs

We consider measuring entanglement via the classical quadratic R\'{e}nyi entropy of joint probability distributions over the measurement outcomes associated with tensor products of elements of local positive operator valued measures (POVMs). We examine the case of pure $d\times d$ bipartite quantum states and identical local POVMs. In this case, we prove that if the local POVMs are rank 1, then the classical quadratic R\'{e}nyi entropy of such a distribution (denoted by $H$) is independent of the underlying Schmidt bases if and only if the local POVMs are equivalent to spherical 2-designs. We also prove that if the local POVMs admit a cardinality equal to the composite Hilbert space dimension, then $H$ is independent of the underlying Schmidt bases if and only if the local POVMs are symmetric informationally complete POVMs of arbitrary rank. We show that different degrees of entanglement correspond to distinct spheres within the corresponding joint probability simplexes. Furthermore, we derive a separability criterion for mixed isotropic quantum states in terms of probabilities for outcomes of generalized quantum measurements constructed from tensor products of elements of local POVMs formed from spherical 2-designs.   

Do emergent entangled coherent Glauber states violate the no-signaling theorems of quantum theory?

Quantum information theory assumes entanglement cannot be used as a direct stand-alone-communication channel without a light speed limited retarded signal key to unlock the message encrypted in the correlation pattern. This pre-supposes orthogonal base states for the entangled subsystems. Macro-quantum coherent Glauber states emerge as ground/vacuum states in spontaneous broken symmetries that describe the Higgs-Goldstone fields of many real/virtual particles. They are distinguishably non-orthogonal and over-complete. In the bipartite case, Alice's two distinguishable non-orthogonal sender Glauber coherent base states are entangled with Bob's two orthogonal receiver Q-BIT base states. The Born rule for strong von-Neumann projection measurements using the orthodox constant $\sqrt 2 ^{-1}$ normalization gives an entanglement signal \[ \begin{array}{l} S_{Bob} \left( {0/1} \right)=\mathop {\mbox{Tr}}\limits_{Alice} \left\{ {\left| \right\rangle_{Bob} { }_{Bob}\left\langle {(0)1} \right|\left| \right\rangle \left\langle {Bob,Alice} \right|} \right\} \\ =\frac{1}{2}\left( {1+\left| {{ }_{Alice}\left\langle {\sqrt {\left\langle n \right\rangle } e^{\theta }} \mathrel{\left| {\vphantom {{\sqrt {\left\langle n \right\rangle } e^{\theta }} {\sqrt {\left\langle {n'} \right\rangle } e^{\theta '}}}} \right. \kern-\nulldelimiterspace} {\sqrt {\left\langle {n'} \right\rangle } e^{\theta '}} \right\rangle_{Alice} } \right|^{2}} \right) \\ \end{array} \] Emergent spontaneous symmetry breakdown violates the probability interpretation of orthodox quantum theory. It represents an extension of quantum theory in the same way that gravity required an extension of special relativity limiting it to coincident local inertial frames.   

Entanglement witnesses for many qubit systems

Entanglement is one of the essential features of quantum physics and is fundamental to future quantum technologies. The characterization of entanglement has been shown to be equivalent to the characterization of positive, but not completely positive, maps (PnCP) over matrix algebras. In the cases of $2 \times 2$ and $2 \times 3$ dimensional spaces does there exist complete characterization of the separability problem (due to the celebrated Peres-Horodecki criterion). However, for increasingly higher dimensions this task becomes more and more difficult. There has been a considerable effort devoted to constructing PnCP, but a general procedure is still not known. Recently we were able to generalize the Robertson map in a way that naturally meshes with $2N$ qubit systems, i.e., its structure respects the $2^{2N}$ growth of the state space. We proved that this map is positive, but not completely positive, indecomposable and optimal, and as such can be used to detect (bipartite) entanglement. We also determined the relation our maps to entanglement breaking channels. We will discuss these new classes of entanglement witnesses.   

Measurement-Induced Non-locality in an $n$-partite quantum state

We generalize the concept of measurement-induced non-locality (MiN) to $n$-partite quantum states. We get exact analytical expressions for MiN in an $n$-partite pure and $n$-qubit mixed state. We obtain the conditions under which MiN equals geometric quantum discord in an $n$-partite pure state and an $n$-qubit mixed state. We obtain an exact (computable) relation between MiN and entanglement (concurrence) for a bipartite pure state.   

Zitterbewegung in Cold Atoms

In condensed matter systems, the coupling between spatial and spin degrees of freedom through the spin-orbit (SO) interaction offers the possibility of manipulating the electron spin via its orbital motion. The proposal by Datta and Das [1,2] of a `spin transistor' for example, highlights the use of the SO interaction to control the electron spin via electrical means. Recently, arrangements of crossed lasers and magnetic fields have been used to trap and cool atoms in optical lattices and also to create light-induced gauge potentials [3], which mimic the SO interactions in real solids. In this work, we investigate the Zitterbewegung in cold atoms by starting from the effective SO Hamiltonian derived in Ref. [4]. Cross-dressed atoms as effective spins can provide a proper setting in which to observe this effect, as the relevant parameter range of SO strengths may be more easily attainable in this context. We find a variety of peculiar Zitterbewegung orbits in real and pseudo-spin spaces, e.g., cycloids and ellipses - all of which obtained with realistic parameters.\\[4pt] [1] S. Datta and B. Das, Appl. Phys. Lett. 56, 655 (1990);\\[0pt] [2] J. Carlos Egues, et. al., Appl. Phys. Lett. 82, 2658 (2003);\\[0pt] [3] Y. -J. Lin, et. al, Nature 471, 83 (2011);\\[0pt] [4] Jay D. Sau, et. al, PRB 83, 140510(R) (2010).