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arXiv:1708.07846 · EPJ Web of Conferences

On the Generation of Random Ensembles of Qubits and Qutrits Computing Separability Probabilities for Fixed Rank States

A. Khvedelidze, I. Rogojin

Published Jan 1, 2018• 2 citations
arXivHTMLPDF

Abstract

The generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. In particular, we consider the generation of random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank.

DOI
10.1051/epjconf/201817302010

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We detect a certain pattern of behavior of separability probabilities [Formula: see text] for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where [Formula: see text] and [Formula: see text] are the Bloch radii ([Formula: see text]) of the qubit reduced states ([Formula: see text]). We observe a relative repulsion of radii effect, that is [Formula: see text], except for rather narrow “crossover” intervals [Formula: see text]. Among the seven specific cases we study are, firstly, the “toy” seven-dimensional [Formula: see text]-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in [Formula: see text]-dimensions, for [Formula: see text], with [Formula: see text] corresponding to HS (flat/Euclidean) measure. We also examine the real (two-rebit) [Formula: see text], the [Formula: see text]-states [Formula: see text], and Bures (minimal monotone)–for which no nontrivial crossover behavior is observed–instances. In the two [Formula: see text]-states cases, we derive analytical results; for [Formula: see text], we propose formulas that well-fit our numerical results; and for the other scenarios, rely presently upon large numerical analyses. The separability probability crossover regions found expand in length (lower [Formula: see text]) as [Formula: see text] increases. This report continues our efforts [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A 48 (2015) 035306.] from a univariate ([Formula: see text]) framework — in which they found separability probabilities to hold constant with [Formula: see text] — to a bivariate ([Formula: see text]) one. We also analyze the two-qutrit and qubit–qutrit counterparts reported in Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form [Formula: see text]. A physics.stack.exchange link to a contribution by Mark Fischler addressing, in considerable detail, the construction of suitable bivariate distributions is indicated at the end of the paper.

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Paul B. Slater

We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and (the 9-dimensional) two-rebits of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of $\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$ and a rebit-retrit one of $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$. These four values for $2 \times m$ systems ($m=2,3$) suggest certain numerator/denominator sequences involving powers of $m$, which we further investigate for $m>3$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 π^2} \approx 0.54038$, as well as more generally, that the probabilities based on induced measures are equal across these three sets of $X$-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is $\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$, yielding $\frac{8}{33}$, its $k=1$ induced measure counterpart is $\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yielding $\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$, where $\varepsilon$ is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".

arXiv:1803.10680•Mar 28, 2018•1 citation
The generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. In particular, we consider the generation of random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank.
We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by {Ż}yczkowski and Khvedelidze, Rogojin and Abgaryan, for the generation of random (complex or real) N×NN \times NN×N density matrices of rank k≤Nk \leq Nk≤N with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix AAA of dimensions k×k+2(N−k)k \times k+ 2 (N-k)k×k+2(N−k), while for a real scenario, one employs a Ginibre matrix BBB of dimensions k×k+1+2(N−k)k \times k+1+ 2 (N-k)k×k+1+2(N−k). Then, the k×kk \times kk×k product AA†A A^{\dagger}AA† or BBTB B^TBBT is diagonalized-padded with zeros to size N×NN \times NN×N-and rotated, obtaining a random density matrix. Implementing the procedure for rank-4 rebit-retrit states, for 800 million Ginibre-matrix realizations, 6,192,047 were found separable, for a sample probability of .00774006-suggestive of an exact value 3875000=32⋅4323⋅54=.0774\frac{387}{5000} =\frac{3^2 \cdot 43}{2^3 \cdot 5^4}=.07745000387​=23⋅5432⋅43​=.0774. A conjecture for the HS separability probability of rebit-retrit systems of full rank is 8606561=22⋅5⋅4338≈0.1310775\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.13107756561860​=3822⋅5⋅43​≈0.1310775 (the two-rebit counterpart has been proven to be 2964=2926\frac{29}{64}=\frac{29}{2^6}6429​=2629​). Subject to these conjectures, the ratio of the rank-4 to rank-6 probabilities would be 590491000000=31026⋅56≈0.059049\frac{59049}{1000000}=\frac{3^{10}}{2^6 \cdot 5^6} \approx 0.059049100000059049​=26⋅56310​≈0.059049, with the common factor 43 cancelling. As to the intermediate rank-5 probability, a 2006 theorem of Szarek, Bengtsson and {Ż}ycskowski informs us that it must be one-half the rank-6 probability-itself conjectured to be 271000=3323⋅53\frac{27}{1000} =\frac{3^3}{2^3 \cdot 5^3}100027​=23⋅5333​, while for rank 3 or less, the associated probabilities must be 0 by a 2009 result of Ruskai and Werner. We are led to re-examine a 2005 qubit-qutrit analysis of ours, in these regards, and now find evidence for a 702673=2⋅5⋅735⋅11≈0.0261878\frac{70}{2673}=\frac{2 \cdot 5 \cdot 7}{ 3^5 \cdot 11} \approx 0.0261878267370​=35⋅112⋅5⋅7​≈0.0261878 rank-4 to rank-6 probability ratio.
Mixed state ensembles such as the Bures-Hall and Hilbert-Schmidt measure are probability distributions that characterise the statistical properties of random density matrices and can be used to determine the typical features of mixed quantum states. Here we extend this framework by considering the properties of random states with fixed average energy, and the ensemble-averaged density matrix is derived under this additional physical constraint. This gives rise to a type of microcanonical ensemble for random mixed states and we connect its properties to a statistical mechanical entropy and temperature. Our results are illustrated using a variety of simple spin systems, and we find that they can exhibit exotic features such as phase transitions in the absence of energetic interactions and finite relative energy fluctuations in the thermodynamic limit.
We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
Abstract The geometric separability probability of composite quantum systems has been extensively studied in the recent decades. One of the simplest but strikingly difficult problem is to compute the separability probability of qubit–qubit and rebit–rebit quantum states with respect to the Hilbert–Schmidt measure. A lot of numerical simulations confirm the P r e b i t − r e b i t = 29 64 and P q u b i t − q u b i t = 8 33 conjectured probabilities. We provide a rigorous proof for the separability probability in the real case and we give explicit integral formulas for the complex and quaternionic case. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit–qubit (and qubit-qutrit) states of the form of ( D 1 C C ∗ D 2 ) depends on D = D 1 + D 2 (on single qubit subsystems), moreover it depends only on the Bloch radii ( r ) of D and it is constant in r . Using the Peres–Horodecki criterion for separability we give a mathematical proof for the 29 64 probability and we present an integral formula for the complex case which hopefully will help to prove the 8 33 probability, too. We prove Milz and Strunz’s conjecture for rebit–rebit and qubit–qubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function f ( x ) =
We detect a certain pattern of behavior of separability probabilities [Formula: see text] for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where [Formula: see text] and [Formula: see text] are the Bloch radii ([Formula: see text]) of the qubit reduced states ([Formula: see text]). We observe a relative repulsion of radii effect, that is [Formula: see text], except for rather narrow “crossover” intervals [Formula: see text]. Among the seven specific cases we study are, firstly, the “toy” seven-dimensional [Formula: see text]-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in [Formula: see text]-dimensions, for [Formula: see text], with [Formula: see text] corresponding to HS (flat/Euclidean) measure. We also examine the real (two-rebit) [Formula: see text], the [Formula: see text]-states [Formula: see text], and Bures (minimal monotone)–for which no nontrivial crossover behavior is observed–instances. In the two [Formula: see text]-states cases, we derive analytical results; for [Formula: see text], we propose formulas that well-fit our numerical results; and for the other scenarios, rely presently upon large numerical analyses. The separability probability crossover regions found expand in length (lower [Formula: see text]) as [Formula: see text] increases. This report continues our efforts [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A 48 (2015) 035306.] from a univariate ([Formula: see text]) framework — in which they found separability probabilities to hold constant with [Formula: see text] — to a bivariate ([Formula: see text]) one. We also analyze the two-qutrit and qubit–qutrit counterparts reported in Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form [Formula: see text]. A physics.stack.exchange link to a contribution by Mark Fischler addressing, in considerable detail, the construction of suitable bivariate distributions is indicated at the end of the paper.
We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and (the 9-dimensional) two-rebits of 2964=2926≈0.453125\frac{29}{64} =\frac{29}{2^6} \approx 0.4531256429​=2629​≈0.453125. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of 271000=(310)3=3323⋅53=0.027\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027100027​=(103​)3=23⋅5333​=0.027 and a rebit-retrit one of 8606561=22⋅5⋅4338≈0.131078\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310786561860​=3822⋅5⋅43​≈0.131078. These four values for 2×m2 \times m2×m systems (m=2,3m=2,3m=2,3) suggest certain numerator/denominator sequences involving powers of mmm, which we further investigate for m>3m>3m>3. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit XXX-states all equal 163π2≈0.54038\frac{16}{3 π^2} \approx 0.540383π216​≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of XXX-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is 13ε2(4−ε2)\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)31​ε2(4−ε2), yielding 833\frac{8}{33}338​, its k=1k=1k=1 induced measure counterpart is 14ε2(3−ε2)2\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^241​ε2(3−ε2)2, yielding 61143=6111⋅13≈0.426573\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.42657314361​=11⋅1361​≈0.426573, where ε\varepsilonε is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".