We present experimentally and numerically accessible quantities that can be used to differentiate among various families of random entangled states. To this end, we analyze the entanglement properties of bipartite reduced states of a tripartite pure state. We introduce a ratio of simple polynomials of low-order moments of the partially transposed reduced density matrix, and we show that this ratio takes well-defined values in the thermodynamic limit for various families of entangled states. This allows us to sharply distinguish entanglement phases in a way that can be understood from a quantum information perspective based on the spectrum of the partially transposed density matrix. We analyze in particular the entanglement phase diagram of Haar-random states, states resulting from the evolution of chaotic Hamiltonians, stabilizer states (which are outputs of Clifford circuits), matrix-product states, and fermionic Gaussian states. We show that for Haar-random states, the resulting phase diagram resembles the one obtained via the negativity, and that for all the cases mentioned above, a very distinctive behavior is observed. Our results can be used to experimentally test necessary conditions for different types of mixed-state randomness in quantum states formed in quantum computers and programmable quantum simulators.
We present experimentally and numerically accessible quantities that can be used to differentiate among various families of random entangled states. To this end, we analyze the entanglement properties of bipartite reduced states of a tripartite pure state. We introduce a ratio of simple polynomials of low-order moments of the partially transposed reduced density matrix, and we show that this ratio takes well-defined values in the thermodynamic limit for various families of entangled states. This allows us to sharply distinguish entanglement phases in a way that can be understood from a quantum information perspective based on the spectrum of the partially transposed density matrix. We analyze in particular the entanglement phase diagram of Haar-random states, states resulting from the evolution of chaotic Hamiltonians, stabilizer states (which are outputs of Clifford circuits), matrix-product states, and fermionic Gaussian states. We show that for Haar-random states, the resulting phase diagram resembles the one obtained via the negativity, and that for all the cases mentioned above, a very distinctive behavior is observed. Our results can be used to experimentally test necessary conditions for different types of mixed-state randomness in quantum states formed in quantum computers and programmable quantum simulators.
Entanglement plays a crucial role in quantum information science and many-body physics, yet quantifying it in mixed quantum many-body systems has remained a notoriously difficult problem. Here, we introduce families of quantitative entanglement witnesses, constructed from partial transpose and realignment moments, which provide rigorous bounds on entanglement monotones as well as entanglement dimensionality. Our witnesses can be efficiently measured using SWAP tests or variants of Bell measurements, thus making them directly implementable on current hardware. Leveraging our witnesses, we present several novel results on entanglement properties of mixed states, both in quantum information and many-body physics. We develop efficient algorithms to test whether mixed states with bounded entropy have low or high entanglement, which previously was only possible for pure states. We also provide an efficient algorithm to test the Schmidt rank using only two-copy measurements, and the operator Schmidt rank using four-copy measurements. Further, our witnesses robustly certify the quantum circuit depth in the presence of noise, as well as the Schmidt rank of mixed states. Finally, we show that the entanglement phase diagram of Haar random states, quantified by the partial transpose negativity, can be fully established solely by computing our witness, a result that also applies to any state 4 -design. Our witnesses can also be efficiently computed for matrix product states, thus enabling the characterization of entanglement in extensive many-body systems. Finally, we make progress on the entanglement required for quantum cryptography, establishing rigorous limits on pseudoentanglement and pseudorandom density matrices with bounded entropy. Our work opens new avenues for quantifying entanglement in large and noisy quantum systems.
The projected ensemble -- an ensemble of pure states on a subsystem conditioned on projective measurement outcomes on its complement -- provides a finer probe of ergodicity and information structure than the reduced density matrix of the subsystem in bipartite quantum states. This framework can be generalised to partial projected ensembles in tripartite settings, where outcomes from part of the measured subsystem are discarded, leading to ensembles of mixed states. We show that information measures defined for such ensembles, in particular the Holevo information, yield a more detailed characterisation of how quantum information is distributed between subsystems compared to conventional entanglement measures. Using exact analytical results supported by numerical results, we uncover a qualitative change in the scaling of the Holevo information with system size in partial projected ensembles generated by Haar-random states, as the relative sizes of the subsystem are varied. In one phase, the Holevo information decays exponentially with system size, while in the other it grows linearly, thereby defining distinct information phases separated by sharp transitions signalled by non-analyticities in the Holevo information. The exponentially decaying phase rigorously establishes the existence of a measurement-invisible quantum-correlated phase -- a manifestation of many-body information scrambling with no bipartite analogue. We contrast this information-phase diagram with the entanglement-phase structure of tripartite Haar-random states obtained from logarithmic negativity, and show that the Holevo information reveals additional fine structure beyond conventional entanglement measures. Finally, we show that these information phases, as characterised by the Holevo information, emerge in the dynamics of chaotic quantum circuits and discuss the associated timescales.
The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics.The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve.In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties.We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension.We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity.As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned.This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point.We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement.Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes.We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.
The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics. The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve. In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties. We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension. We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity. As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned. This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point. We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement. Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes. We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.
Quantum many-body systems display an extraordinary degree of complexity, yet many of their features are universal: they depend not on microscopic details, but on a few fundamental physical aspects such as symmetries. A central challenge is to distill these universal characteristics from model-specific ones. Random quantum states sampled from a uniform distribution, the Haar measure, provide a powerful framework for capturing this typicality. Here, we experimentally study the entanglement and symmetries of random many-body quantum states generated by evolving simple product states under ergodic Floquet models. We find excellent agreement with the predictions from the Haar-random state ensemble. First, we measure the Rényi-2 entanglement entropy as a function of the subsystem size, observing the Page curve. Second, we probe the subsystem symmetries using entanglement asymmetry. Finally, we measure the moments of partially transposed reduced density matrices obtained by tracing out part of the system in the generated ensembles, thereby revealing distinct entanglement phases. Our results offer an experimental perspective on the typical entanglement and symmetries of many-body quantum systems.
Entanglement features of the ground state of disordered quantum matter are often captured by an infinite-randomness fixed point that, for a variety of models, is the random singlet phase. Although a copious number of studies covers bipartite entanglement in pure states, at present, less is known for mixed states and tripartite settings. Our goal is to gain insights in this direction by studying the negativity spectrum in the random singlet phase. Through the strong disorder renormalization group technique, we derive analytic formulas for the universal scaling of the disorder-averaged moments of the partially transposed reduced density matrix. Our analytic predictions are checked against a numerical implementation of the strong disorder renormalization group and against exact computations for the XX spin chain (a model in which free fermion techniques apply). Importantly, our results show that the negativity and logarithmic negativity are not trivially related after the average over the disorder.