In Memory of Mary Beth Ruskai

Mary Beth Ruskai passed away at her home in Vermont on September 27, 2023. Born in Cleveland, Ohio, in 1944, she earned degrees in chemistry, mathematics, and physical chemistry. She worked in mathematics and mathematical physics and was a Fellow of the American Mathematical Society, the Association for Women in Mathematics, the American Physical Society, and the American Association for the Advancement of Science. Beth spoke up whenever she saw injustice, and her work in this area included both analysis and advocacy. She was also an avid outdoors woman and chose her retirement home in Vermont for its access to hiking and cross-country skiing.

Remembrances⁠¹ from collaborators, colleagues, and friends will describe many of Beth’s research contributions. Beth gave her own summary (on a personal web page no longer available). She included her celebrated work with Lieb proving the strong subadditivity of quantum entropy (SSA) and her proof of what Barry Simon named the Ruskai-Sigal Theorem.⁠² She also described her move into quantum information theory, where the SSA theorem on quantum entropy plays an important role. Those of us who knew her can hear her voice in the conclusion of her summary:

¹

These follow in alphabetical order, except that Elisabeth Werner’s piece appears first to help orient non-specialists to some of Beth’s research.

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that an atom with a fixed nuclear charge can bind only finitely many electrons

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I worked with a group of young people at MIT who showed that quantum error correcting codes could be used to solve a question about $N$-representability that had been open for almost 40 years. In 2012, on the 40th anniversary of the proof of SSA, Isaac Kim (then a graduate student at Caltech) proved a stronger version of SSA, which I did not believe was possible….

As I near the end of my research career, I feel that in many ways I have come full circle. I enjoy meeting and working with young people who have new insights which enable them to move forward and prove stronger theorems.

By Elisabeth Werner

I first met Mary Beth when she was the Flora Stone Mather Visiting Professor at Case Western Reserve University in 1995. I was then an assistant professor at Case. Mary Beth taught a course on Wavelets and I took it. Mary Beth delivered her lectures, as was always the case with her, enthusiastically and with much gusto. I was thrilled.

My next meeting with Mary Beth was at the Georgia Institute of Technology. I was giving a talk on geometric inequalities, based on joint work with my colleague from Case Western, Stanislaw Szarek 19. Mary Beth was in the audience and she approached me after the talk, as she noticed that some of these inequalities, or refinements thereof, might be relevant for her work on one-dimensional analogs of the Coulomb potential for atoms in strong magnetic fields. We started to discuss and this led to many mutual visits in Cleveland and Boston and eventually to our earliest joint publications, one of which is also joint with R. Brummelhuis 216.

When I next heard from Mary Beth, she had started to work in quantum information theory. She later told me how much she enjoyed doing research in this particular area and that it had rekindled her joy in doing research. Relevant objects in quantum information theory are quantum channels. Important examples of quantum channels are completely positive trace preserving maps $\Phi$ on $\mathcal{M}_n$, the algebra of complex $n \times n$ matrices. These form convex sets. To get a better understanding of the quantum objects, it was important to characterize the boundary structure of these sets, and, for instance, identify the extreme points of these sets. If we know the extreme points, we can reconstruct the whole set. That was exactly the question Mary Beth asked Stanislaw Szarek and me to work on with her.

And then the fun started: the “battle” between Mary Beth and Stanislaw—and me in between! Stanislaw and I had a pretty good idea of how to approach this problem. But when we presented it to Mary Beth she vehemently objected, stating that this was not at all what was needed, but something completely different and we missed the point entirely. It was useless to convince her otherwise—which will not surprise anybody who knew Mary Beth. Discussions and email exchanges got rather heated—the more so as my colleague Stanislaw also enjoys a lively discussion, occasionally deliberately putting oil in the fire. Back and forth things went over weeks and months—but in the end, all was good and we had a very nice paper 15. We found a useful new “arithmetic” characterization of the set of all completely positive, trace-preserving maps $\Phi : \mathcal{M}_2 \to \mathcal{M}_2$ and described explicitly all extreme points of this set.

Mary Beth and I wrote one more joint paper 17. Mutual visits continued, in Boston, Cleveland, and in Kiel, Germany. We both enjoyed having dinners at nice restaurants. Mary Beth liked outdoor activities, such as cross-country skiing and hiking. On one of our many walks together, in the Flats of Cleveland, she told me about growing up in the Hungarian community of Cleveland. She was interested in Science and Mathematics but in those days it was even more difficult for women to succeed there. Of course, Mary Beth prevailed.

Mary Beth was a force to be reckoned with. She was passionate about the issues she cared about. Among them was her engagement in promoting women in Science and Mathematics. She has made a difference and her legacy will stay.

Elisabeth Werner is a professor of mathematics at Case Western Reserve University. Her email address is emw2@case.edu.

By Michael Aizenman

Mary Beth Ruskai, or “Beth”—as she signed correspondence, was a notable mathematical physicist with interests in topics related to the quantum nature of physics. She was also a thoughtful colleague who did not hesitate to share her views on a range of subjects related to the way our profession has been evolving.

Mathematical physicists are a rare breed, straddling two fields which despite their obvious intertwining have different goals, tools, and criteria by which success is evaluated. Having a cross-disciplinary grasp is rare. Beth had it.

In research, she made her mark in two distinct areas of quantum physics. Particularly consequential was her work with Elliott Lieb on the strong subadditivity of quantum entropy (SSA) 6. Entropy is a subtle concept, originally introduced in thermodynamics as a quantifier of the irreversibility of realizable processes. Among its practical applications are bounds on the efficiency of possible energy transfers. The initially elusive intrinsic meaning of entropy was explained in a breakthrough contribution of Boltzmann along with his introduction of statistical mechanics. From that, in a creative leap, C. Shannon introduced an analogous concept of entropy as a measure of information, which allows quantification of communication channel capacity.

As Beth was starting her research career she was exposed to the discussions among mathematical physicists (D. Robinson, H. Araki, D. Ruelle, O. Lanford, E. Lieb) who were grappling with the fundamentals of quantum statistical mechanics. A simply stated extension of the Gibbs formula for the entropy of a quantum state was presented earlier by John von Neumann. It clearly shared some of the significant properties of the classical Gibbs state entropy function, but not all. There are many structural similarities between the theoretical formulations of classical and quantum physics, but famously there are also some striking differences between the two theories.

Still, quantum entropy shares its classical version’s sub-additivity $S(AB) \leq S(A)+S(B)$. Does it also share the classical entropy’s strong subadditivity $S(ABC) +S(B) \leq S(AB) + S(BC)$? The answer is Yes—but it took quite some effort to establish that. The proof was presented in the celebrated 1973 joint paper with Elliott Lieb. Further details and historical recollections on this work can be found in the authors’ contributions to the recent AMS memorial article on D. Robinson 1.

Their result is of lasting relevance. Initially it was appreciated for its significance for quantum statistical mechanics. With time it gained additional recognition within the emerging field of quantum information theory. Thirty years after its proof, SSA was referred to as the stepping-stone to a significant part of the theory of quantum entanglement 7. As we are now aware, von Neumann’s quantum entropy combines elements of Boltzmann/Gibbs statistical physics, with a quantum extension of Shannon’s notions of information theory. The differences from the classical theory can be traced to the phenomenon of quantum entanglement between $A$ and $B$.

In another direction, Beth’s early work included theorems regarding large atoms with Coulomb interaction, that placed upper bounds on the excess negative charge which large ions can support (see 14).

Over the following years, Beth continued to produce results which deepened our grasp of topics in quantum statistical mechanics and quantum information theory (some of which are mentioned in the contribution to this memoir by Elisabeth Werner). She also continued to express interest in streamlined and more transparent derivations of SSA, and in its different extensions (11).

During my service as its editor-in-chief, I invited Beth to join the editorial board of Communications in Mathematical Physics and serve as the editor of a new section on quantum information theory. In that role, which lasted through 2012, she was proactive, dedicated, and effective, and developed QIT’s presence in this flagship journal of its field.

Although most of the time we worked on different research topics, Beth and I sporadically had the chance to meet, update each other on our research, and enjoy friendly discussions. Beth tended to be opinionated, but she would also listen and take into account the feedback she got. We always found common ground, shared perspective, and reasons to laugh.

Beth’s career was launched at the time when being female could be a handicap in the world of mathematics and science. Seeking a suitable research environment, and not being one to yield to adversity, she complemented her regular university service with visiting appointments at a range of research institutes and universities, including Rockefeller, Courant, Bunting/Radcliffe Institutes, Vienna University, TU-Berlin, Dublin Institute of Technology, Georgia Tech, and Case Western Reserve.

Beth contributed to ongoing discussions concerning the underrepresentation of women in STEM departments. She called attention to the problem citing statistics, relevant research, and the occasional manifestations of harmful atmosphere 10:

In my experience, few scientists are overtly sexist and many have been extremely supportive toward women. But very few are willing to deal with sexism when they should, which gives a small number of “bad guys” disproportionate influence, particularly with junior women at critical phases in their careers.

She also contributed to the exchanges on ways to increase women’s presence in science. There, she was critical of presentations which take for granted the existence of gender-linked differences in skills in computer science, and more broadly mathematics and science, writing in the Newsletter of the Association for Women in Mathematics (1986, 5-6):

One recurrent idea in many articles of this type is that women are more intuitive than men, where intuition and logic are perceived of as opposites. In this context the notion that women are more intuitive seems suspiciously like a rewording of the old bigoted male accusation that women can’t think logically.

Addressing some of the circulating ideas on potential contrast between art and creative endeavors versus math and science she wrote there:

Many of my non-science [women] colleagues at Bunting were surprised to learn that scientists consider themselves creative and artistic; they were amazed that I used words like beautiful and elegant to describe theorems and proofs. I fear that such misunderstandings promote negative attitudes toward science which discourage young women from scientific careers.

Our correspondence was renewed in 2022, close to the 50th anniversary of the early SSA work. Beth’s continuing engagement with aspects of SSA has indirectly contributed motivation for my recent work with Giorgio Cipolloni, which presents an alternative path to the proof. The paper, dedicated to Lieb and Ruskai in celebration of their joint work, formed our submission to the special issue of Letters in Mathematical Physics dedicated to Beth 5.

The last time I heard from Beth was in June 2023, in a message sent in reply to the slides of my talk on the above work at Joel Lebowitz’s 124th Statistical Mechanics Conference (Rutgers, May 2023). She recalled attending some of the early meetings of this remarkable series in the mid-1970s, in its old venue—the Belfer Graduate School of Yeshiva University. That must have been where our paths crossed first.

Now, writing that she was switching to palliative care, Beth added, “For the past 5 years I was stable on a treatment with minimal side effects and could enjoy easy hiking, XC skiing and gardening. I was also able to finish 3 math papers which I feel good about.” These few lines, written at a sad moment, epitomize for me Beth’s character and courage.

Michael Aizenman is a professor of physics and mathematics at Princeton University. His email address is aizenman@princeton.edu.

By Ingrid Daubechies

Mary Beth Ruskai and I met for the first time on Monday June 2, 1980. The date is engraved so precisely in my mind because it was the start of a summer school on Rigorous Results for Scattering Theory in Quantum Mechanics at the Ettore Majorana Center in Erice, Sicily, to which I traveled just two days after the oral defense of my PhD thesis. Although it was over 40 years ago, the memories of that summer school are vivid.

Beth took me under her wing almost immediately. There were not many women in mathematical physics then, and as the new kid on the quantum mechanics block, I felt rather shy. She introduced herself to me, and then looked out for me all through the summer school. She made sure that I met all the other young students and postdocs who were there, and that I did not slink off for lunch in a corner because I didn’t know anyone yet. She introduced me to the more senior researchers, and in particular to Elliott Lieb, with whom she had worked on quantum mechanical entropy. It was as a direct consequence of this introduction, and Lieb’s subsequent interest in some coherent state estimates in my thesis, that I arrived in Princeton in the Spring of 1981 for postdoctoral work with Lieb. I was grateful then, and have been ever since—that summer school was a luminous experience for me, in great part due to Beth’s persistent efforts on my behalf.

I also learned about other aspects of Beth. As was not uncommon at the Ettore Majorana Center, two summer schools on entirely different topics were being held concurrently. The participants in the summer schools interacted sporadically, by having evening social events in adjacent areas, and were curious about each other. The directors of the two summer schools planned to each give a general lecture on successive evenings, to introduce the other community to what was going on in their own. The other summer school was centered on what was then called “genetic manipulation.” Recombination of DNA had started less than ten years before, and the first US FDA approval of a genetically modified organism wouldn’t happen until 1982.

Almost all of the mathematical physics participants in our summer school attended the biological evening lecture, which was riveting and gave rise to a lively discussion afterward, notably about the potential dangers of genetic engineering and the need for ethical guidelines. The next evening it was Arthur Wightman’s turn; he tried, without using many equations, to give a very precise description of some of the central issues in our summer school. There had been a lot of speculation that one of the several participating experts working on asymptotic completeness (not yet proved at the time) would announce a big result in one of his lectures. This did not happen. (Full proofs were published only in 1987 18 and 1994 4.) Maybe Wightman thought to captivate his audience by the drama of a fervently anticipated result that then didn’t materialize? His attempt to explain the rather technical problem of asymptotic completeness fell flat.

There were no questions from the biologists, during or after the lecture. After a few awkward moments of silence, the director of the biology summer school stood up and commented that the night before, there had been a lot of discussion of the possible dangers of their field of research, but it had also been clear that there was a lot of excitement. In contrast, he felt that the questions preoccupying us were less likely to lead to dangerous outcomes; he hoped we would not be offended if they seemed also less exciting, at least to him.

Many of the younger participants in the Quantum Mechanics summer school attended Wightman’s lecture, curious to hear how he would present mathematical physics to biologists. I was sitting next to Beth in the audience, and I remember her indignation when she realized that Wightman had chosen to construct his presentation with a narrow focus rather than standing up as the advocate for mathematical physics more generally. She seethed (rather quietly, by Beth’s standards, because she had a lot of respect for Wightman) for two days before joining the rest of us in laughing about the episode.

Figure 2.

Left to right: Richard Dudley, Mary Beth Ruskai, Alice Silverberg, Barbara Lee Keyfitz, AWM panel on Lawrence H. Summers at JMM 2006.

After I arrived in the US, a year later, Beth invited me several times to come visit her in Boston. She delighted in making me discover new places, new sights, new aspects of life in the US—I had never been out of Europe before. One memorable day in the Fall we drove up to New Hampshire, to climb Mount Washington. We had left very early and were already well on our way when the sun rose. In the red-tinted light of the dawn, one vividly colored tree on the dividing berm in the middle of the highway seemed to be on fire. I was only half convinced by Beth’s assurances that it was just a combination of the light and the spectacular Fall colors for which New England is renowned; on our way back to Boston that evening, Beth drew my attention to the same tree, which indeed had not burned down. The hike itself was truly memorable: as we walked up, we transitioned from a vale with trees just starting to turn, to the flamboyancy of a New England forest in peak fall color, to wintry trees that had lost all their leaves, and finally to the denuded top, above the treeline, where strong icy winds had turned freezing rain into virtually horizontal icicles, now glistening in the sun. Over time, Beth and I went on other hikes, in different parts of the world—Beth was a great enthusiast for the outdoors—but that first one stood out in both our memories.

In later years, our paths intersected many times, at conferences in which we both participated, or when one of us invited the other. We always reconnected easily, even if sometimes years had passed. Between meetings, Beth would sometimes draw my attention to issues that she knew would interest me and on which she was speaking out. For instance, at some point in the 90s, she alerted me to arguments from female social scientists who sought to explain the low percentage of women in mathematics by the nature of mathematical thinking itself, antithetical, according to their views, to the more intuitive, nurturing nature of women—I had missed that debate and joined Beth in expressing complete disagreement with that “take” on mathematics. She also introduced me to Rhonda Hughes who, with Sylvia Bozeman, started the EDGE (Enhancing Diversity in Graduate Education) program, which sought (and seeks!) to better prepare women students seeking a PhD in the mathematical sciences for grad school. I have been proud ever since to be affiliated with the EDGE program. Near the end of her life, Beth made a significant financial gift to EDGE. I am sorry that she missed the October 2023 25th anniversary of EDGE, a joyful mathematics and community celebration where her memory was honored. EDGE has established the Mary Beth Ruskai Research Fund for Women.

Ingrid Daubechies is James B. Duke Professor of Mathematics and Electrical and Computer Engineering at Duke University. Her email address is ingrid.daubechies@duke.edu.

By Mary Gray

Others write of Beth’s distinguished contributions in a variety of areas of the mathematical sciences, her warmth of friendship, her devotion to causes in which she believed. If I were to choose a single word to describe Beth, it would be persistence. Persistence, yes, in her broad field of research, consistently following up on notable results. But for me most memorable was her devotion to the use of documentation to promote the as-yet-unresolved long battle for the improvement of the position of women in the mathematical sciences.

Beth, like many of us, applauded the increase of women among PhD’s in mathematics from a low of less than 10% to consistent representation in the mid-twenties, while noting a regrettable lack not only of something more but of the uneven distribution of gains that had occurred. Beth would have been disappointed that a recent issue of the American Mathematical Society newsletter citing eight winners of major prizes in mathematics included only one woman. To those who rightly and enthusiastically cheered the recognition of a handful of women mathematicians at top universities and Fields and other medalists, Beth pointed out that data also showed the continued lack of progress at institutions, perhaps second tier, where many women were consistently overlooked or under valued, although perhaps not so much in hiring, as at the crucial steps of advancement to tenure, full professor, department chair, or holder of a named chair. Reporting on the latest overall results for women in math, Beth would have remarked that we still have far to go in achieving equality and that we should do something about it. How could we all give up while Beth was still reminding us. Her efforts to document the progress—or lack of it—of women researchers has provided a sound basis for the need for continued and increased effort on behalf of women in mathematics.

Indeed, many of us relied upon Beth’s regularly compiled statistics to urge more success, often unaware of her effort in compiling results of degrees, hiring, and recognition through appointments and awards. Looking at the numbers, we see the AMS’s long delayed recent election of three women as president, although it took 95 years from its founding to elect the first some 36 years ago, or the association’s more-than-a-quarter-century effort to adopt a blind refereeing policy as a waypoint in, not the culmination of, a campaign. These are just a few among the results of the persistent advocacy of Beth and others.

Another, perhaps less-recognized, manifestation of Beth’s persistence was her constant battle against governmental intrusion on our civil rights, in particular through the formulation and over-enthusiastic implementation of TSA airport searches. From an arrest—as a lawyer I was the recipient of “Do you know a good lawyer in Boston?”—to the courts, her persistence was part of the effort to make getting on a plane less traumatic.

Mary Gray is Distinguished Professor of Mathematics and Statistics at American University. Her email address is mgray@american.edu.

By Cathy Kessel

I first encountered Beth Ruskai through her articles in the Newsletter of the Association for Women in Mathematics,⁠³ beginning with her 1986 “Open Letter on Feminism in Science.” Her two years at the Bunting Institute (now the Radcliffe Institute of Advanced Study) had convinced her that there was cause for concern about the negative and inaccurate views of science and women scientists promulgated by a “vocal minority” of non-scientist women “in so-called feminist circles” as well as the popular press. (Here “science” and “scientist” refer to natural rather than social sciences.)

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All of Ruskai’s AWM Newsletter articles are indexed and available at the AWM website.

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The idea that science could be a creative and artistic endeavor surprised her Bunting colleagues. Many had heard that scientific fields used “numbers as their whole means of discovery,” “women are not interested in science—because it doesn’t deal with subtleties,” “women are more intuitive than men, where intuition and logic are perceived of as opposites,” “women are naturally more inclined to the biological sciences because of their ‘nurturing’ instincts,” and science “must change in fundamental ways in order to accommodate women.”

Beth wrote:

That non-scientists do regard the views of this vocal minority as orthodox was impressed upon me during my stay at the Bunting Institute. Most of the women I met at Bunting ordinarily had little or no contact with women scientists, whom they assume to be far rarer and more isolated than we actually are. (One seemed surprised to learn that I actually knew other women mathematicians.) Their attitudes toward science ranged from enthusiastic amateur to severe anxiety and avoidance. But most of them, regardless of attitude, received their information about women scientists from [social scientists], some of whom they regarded as scientists…. As a result, their views about science and women scientists were often quite distorted. Furthermore, because the social scientists in question are widely regarded as staunch feminists, dissenting views are sometimes regarded as non-feminist.

The Open Letter received many responses, both in the Newsletter of the AWM, and in the Gazette of the Committee on the Status of Women in Physics (CSWP),⁠⁴ where it had also been published. These were followed by talks and panels at meetings, including two at the American Association for the Advancement of Science annual meeting—one sponsored by AWM and one sponsored by CSWP—and a study conference on gender and mathematics education organized by the International Commission on Mathematical Instruction. These resulted in articles for a more general scientific audience in The Scientist 13 and Annals of the New York Academy of Sciences 12.

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See https://www.aps.org/programs/women/index.cfm.

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A second focus of Beth’s writings on gender, women, and science was mathematical ability. Looking back, it seems to me that Beth was one of very few in the natural sciences to write on this topic in a systematic, scholarly, and scientific way. Her 1991 article “Are There Innate Cognitive Gender Differences? Some Comments on the Evidence in Response to a Letter from M. Levin” was published in the American Journal of Physics, and reprinted in the CSWP Gazette and the Physics Teachers CD-ROM Toolkit distributed by the American Association of Physics Teachers.

In 2005, Harvard president Lawrence Summers’s conjectures about women in science, first mentioned in the Boston Globe, spread nationally, then internationally. A group of AWM members began discussing how to respond. Beth quickly produced an op-ed which she submitted to the Globe.⁠⁵ It began:

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It was rejected by the Globe and published with an addendum in the AWM Newsletter.

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I had hoped that I could resist the urge to comment on Harvard President Larry Summers’ remarks about women; however, none of the responses I’ve read adequately addressed one question…was it legitimate to call for research on the question of whether women have less innate mathematical ability?

As a scientist, I’ve learned that progress requires the acceptance of well-verified theories as well as the willingness to consider new hypotheses for unexplained phenomena. Engineers trying to design better cooling systems do not waste time with proposals that violate the second law of thermodynamics. In 1986, the British Royal Society (hardly a bastion of radical feminist theory) concluded that there was no convincing evidence for innate gender differences in mathematical ability. Does Summers have new evidence that would call for reopening this question?

Figure 3.

Mary Beth Ruskai giving an after-dinner talk in honor of Barry Simon.

Prominent among the studies discussed in connection with the Summers remarks were the widely publicized gender-gap findings of Camilla Benbow and her colleagues from the early 1980s. Beth pointed out in 1991 that Benbow had not mentioned subsequent findings showing smaller gaps during the intervening years, instead asserting that the gap had remained relatively constant. Although other researchers had noted changes since the 1980s, their articles received little notice. The 1980s findings continued to be cited in scholarly and popular books.

In 2006, Benbow was appointed as a member of the National Mathematics Advisory Panel, which was intended to “foster greater knowledge of and improved performance in mathematics among American students.” AWM petitioned for her removal. (I was AWM president-elect at the time.) As in the case of the Summers remarks, I (and AWM) benefited from Beth’s comments about what and how to communicate to other organizations, reporters, and a general audience.

One example of her wonderful humor and clear eye for what was important comes from an after-dinner talk in honor of the mathematical physicist Barry Simon.⁠⁶

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This is an edited excerpt of the talk that begins at minute 43 of http://www.math.caltech.edu/SimonFest/Photos/photos.html.

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I want to talk to you tonight about a side that some of you may not be so familiar with—Barry Simon, the radical feminist…[laughter, Beth smiles]

Barry is incredibly meticulous about references. His books have extensive historical notes and they have, as many people have said, influenced the subject. They have not only influenced the subject, but they’ve influenced people’s careers.

I particularly want to mention the work of Clasine van Winter whose work might have been completely forgotten had it not been for Barry, and also the book by Reed and Simon, naming the HVZ theorem, making sure everyone’s contributions to this theorem were recognized. I was shocked a year or two ago to meet a particle physicist who referred to the Weinberg equations of scattering theory when we all know that they’re the Weinberg–Van Winter equations. Periodically I have these conversations with people, they say, Well, why do people worry about whether there are women speakers on the program or whether there are women mathematicians or physicists on the faculty? Why can’t we just forget about affirmative action and do everything gender-blind, on the basis of merit? Of course, this is what we all want. But first we need more people like Barry Simon.

…and Mary Beth Ruskai.

Cathy Kessel is an independent scholar. Her email address is cbkessel@earthlink.net.

By Harriet Pollatsek

In 1998–1999, when I was first learning about error correction in quantum computing while on sabbatical in England, I was electronically introduced to Beth by our mutual friend Barbara Peskin. Even though I was effectively a novice (the intersection of my mathematical expertise and Beth’s was just barely non-empty), Beth warmly welcomed me as a colleague. When I returned to Massachusetts, we began meeting regularly. I couldn’t have had a more generous guide to this area of mathematics nor a more enthusiastic teacher. The main result of our collaboration was our joint paper 9, in which we obtained a number of new (non-additive) binary codes for quantum error correction and showed that the degeneracy arising from permutational symmetry facilitates the correction of certain two-bit errors.

Around the time Beth and I were first introduced, Roger Horn, then editor of the American Mathematical Monthly, invited Beth to write something for the Monthly about quantum computing and error correction. Beth didn’t really have time to do it herself, but she thought it would be possible if we did it jointly. I agreed, and our plan was that I would do first drafts of sections and she would comment. When I sent her my first attempt, the comments came back ALL IN CAPS. Second try, more CAPS, including exclamation marks. Third try, the same. So I wrote back to say that doing it jointly clearly wasn’t a good idea, and I should bow out. NO NO she replied; I should do it myself and she would help, and I could thank her fulsomely in my acknowledgments. And so it⁠⁷ came to pass.

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The article appeared in volume 108 of the Monthly in December 2001.

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Our mathematical collaboration eventually ended, but Beth and I remained in touch, visiting each other regularly—electronically after she moved to Vermont. Not only was Beth a superb mathematician and a stalwart defender of what she knew to be right, she was a generous colleague and a wonderful friend. I miss her.

Harriet Pollatsek is a professor emeritus of mathematics at Mount Holyoke College. Her email address is hpollats@mtholyoke.edu.

By Graeme Smith

I first met Mary Beth Ruskai while I was a graduate student at Caltech getting started in quantum information theory, probably around 2005. I was getting interested in questions about the additivity (and non-additivity) of entropy formulas that show up in information theory, and Beth had done some of the most interesting work in this area. In the following years, I had the good fortune to become her collaborator and we had many valuable discussions about quantum information. It was always a treat to see her at a conference and catch up on her thinking about (non) additivity, about which she invariably had interesting new ideas.

Of course Beth’s most famous work was proving strong subadditivity, but to me she was a guru of additivity. She made key contributions to the theory of additivity in quantum information, especially additivity of the Holevo information and related questions.

When Holevo information is additive, we can evaluate the capacity of a quantum channel for classical communication effectively. For many years it was conjectured that Holevo information was always additive. Beth played a key role in identifying this question as a central challenge for quantum information, proving additivity for special cases of channels, and formulating a simpler (a priori weaker) version of the question in the form of the minimum output entropy conjecture. She was also a kind of evangelist, looking to get others interested in the problem and patiently explaining the ins and outs of the question to graduate students and distinguished colleagues alike.

While it turns out that in general both the additivity of Holevo information and minimum output entropy have counterexamples, Beth’s proofs of additivity for specific channels as well as her reformulations of the question are essential to our understanding of the classical capacity of a quantum channel.

In addition to her technical contributions, Beth organized conferences, special sessions, and workshops regularly. These events helped many of us build new collaborations, meet new people, and learn about science outside our immediate areas of expertise. In particular, she was instrumental in forging connections among quantum information theorists, mathematical physicists, and operator theorists. This involved convincing quantum information folks that the mathematicians had really useful techniques and ideas that would help us solve problems we care about, and convincing the mathematicians that the information theorists had substantial questions worth tackling (and we were reasonably capable of rigorous mathematics). Over the years, Beth worked to bring quantum information theorists and mathematicians together. Both communities have benefited tremendously from her efforts.

We miss you Beth!

Figure 4.

Mary Beth Ruskai on left with Barry Simon and his coauthors, 2006; 60th birthday fest for Barry Simon.

Graeme Smith is an associate professor of applied mathematics at the University of Waterloo. His email address is graeme.smith@uwaterloo.ca.

By Bei Zeng

I had the honor of meeting Beth Ruskai during my PhD journey at MIT around 2008. She was a familiar presence at MIT, engaging in insightful discussions with luminaries like Peter Shor and others from the quantum information groups. Our initial conversations, centered around Hastings’s groundbreaking results on the superadditivity of communication capacity, marked the start of a deeply educational and inspiring journey with Beth.

Beth’s extensive knowledge led us into the intricate world of “$N$-representability,” a field intrinsically linked to her own doctoral research. This intellectual exploration was not only a profound learning experience but also led to our joint work, “Quantum Codes Give Counterexamples to the Unique Preimage Conjecture of the $N$-Representability Problem.” 8.

In 2009, my career transitioned to the Institute for Quantum Computing (IQC) at the University of Waterloo / University of Guelph. In a fortunate parallel, Beth became an associate member of IQC, which meant our paths continued to cross regularly. We delved further into $N$-representability, a topic that was attracting considerable attention in the quantum information community due to its relation to the “quantum marginal problem.” This collaboration led to another joint publication, “Comment on some results of Erdahl and the convex structure of reduced density matrices.” 3.

Figure 5.

Mary Beth Ruskai, ca. 2010.

As Beth prepared for her move from Boston to a retirement community in Vermont, she generously shared with me a treasure trove of her old materials, including her 1969 thesis and proceedings from the $N$-representability conferences held at Queen’s University around 1968–1969. These resources were not just a glimpse into the past but also a catalyst for future endeavors. Heeding Beth’s suggestion, we embarked on organizing workshops centered on the “quantum marginal problem.” In the summer of 2015, we realized this vision by hosting a workshop on “Quantum Marginals and Numerical Ranges” at the University of Guelph. This event was a collaborative effort, organized by a team of colleagues including David Kribs, Paul W. Ayers (a chemist from McMaster and a long-time friend of Beth’s), Isaac Kim, and myself. The success of this conference stood as a testament to Beth’s enduring influence and her unwavering commitment to advancing the frontiers of knowledge.

Working with Beth was an experience that went beyond intellectual enrichment; it was a source of personal inspiration. Her exceptional ability to inspire the younger generation made her a remarkable mentor and collaborator. Beth’s legacy in the quantum information field and her impact on all those she mentored and collaborated with will undoubtedly resonate for many years to come.

Bei Zeng is a professor of physics at the University of Texas at Dallas. Her email address is Bei.Zeng@UTDallas.edu.

Concluding remarks

From Beth’s sister Lois Melina:

At one point during her final days, Beth tried to explain quantum theory to her hospice nurse and social worker. Her eyes lit up. Her voice was strong. She was animated. Clearly, she was in her happy place.